# On transformation operators and
Riesz basis property of root vectors system
for $n\times n$ Dirac type operators.
Application to the Timoshenko beam model

Anton A. Lunyov<EMAIL_ADDRESS>Facebook, Inc., 1 Hacker Way, Menlo
Park, California, 94025, United States of America  Mark M. Malamud
<EMAIL_ADDRESS>Peoples Friendship University of Russia (RUDN University),
6 Miklukho-Maklaya St. Moscow, 117198, Russian Federation

###### Abstract

The paper is concerned with the following $n\times n$ Dirac type equation

$Ly=-iB(x)^{-1}\bigl{(}y^{\prime}+Q(x)y\bigr{)}=\lambda y,\quad
B(x)=B(x)^{*},\quad y=\operatorname{col}(y_{1},\ldots,y_{n}),\quad
x\in[0,\ell],$

on a finite interval $[0,\ell]$. Here $Q\in L^{1}([0,\ell];\mathbb{C}^{n\times
n})$ is a potential matrix and $B\in L^{\infty}([0,\ell];\mathbb{R}^{n\times
n})$ is an invertible self-adjoint diagonal “weight” matrix. If $n=2m$ and
$B(x)=\operatorname{diag}(-I_{m},I_{m})$ this equation is equivalent to Dirac
equation of order $n$.

We show the existence of triangular transformation operators for such equation
under additional uniform separation conditions on the entries of the matrix
function $B$. Here we apply this result to study direct spectral properties of
the boundary value problem (BVP) associated with the above equation subject to
the general boundary conditions $U(y)=Cy(0)+Dy(\ell)=0$,
$\operatorname{rank}(C\ D)=n$.

As a first application of this result, we show that the deviation of the
characteristic determinants of this BVP and the unperturbed BVP (with $Q=0$)
is a Fourier transform of some summable function explicitly expressed via
kernels of the transformation operators. In turn, this representation yields
asymptotic behavior of the spectrum in the case of regular boundary
conditions. Namely, $\lambda_{m}=\lambda_{m}^{0}+o(1)$ as $m\to\infty$, where
$\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$ and
$\\{\lambda_{m}^{0}\\}_{m\in\mathbb{Z}}$ are sequences of eigenvalues of
perturbed and unperturbed ($Q=0$) BVP, respectively.

Further, we prove that the system of root vectors of the above BVP constitutes
a Riesz basis in a certain weighted $L^{2}$-space, provided that the boundary
conditions are _strictly regular_. Along the way, we also establish
completeness, uniform minimality and asymptotic behavior of root vectors.

The main results are applied to establish asymptotic behavior of eigenvalues
and eigenvectors, and the Riesz basis property for the dynamic generator of
spatially non-homogenous damped Timoshenko beam model. We also found a new
case when eigenvalues have an explicit asymptotic, which to the best of our
knowledge is new even in the case of constant parameters of the model.

###### keywords:

Systems of ordinary differential equations , transformation operators ,
regular boundary conditions , eigenvalues asymptotic , Riesz basis property ,
Timoshenko beam model

###### MSC:

47E05 , 34L40 , 34L10 , 35L35

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Definition of the system of root vectors
    2. 2.2 Properties of Lipshitz functions
    3. 2.3 The Banach spaces $X_{1}$ and $X_{\infty}$
    4. 2.4 Properties of adjugate matrix
    5. 2.5 Determinants of matrix sums and products
  3. 3 Transformation operators
    1. 3.1 Similarity of operators $\mathcal{L}_{0}(Q)$ and $\mathcal{L}_{0}(0)$
    2. 3.2 Transformation operators
  4. 4 Fundamental matrix solution
    1. 4.1 Preliminaries
    2. 4.2 Key identities for fundamental matrix
    3. 4.3 Generalization of Liouville’s formula
  5. 5 Regular and strictly regular boundary conditions
    1. 5.1 General properties of BVP
    2. 5.2 Regular boundary conditions
    3. 5.3 Strictly regular boundary conditions
    4. 5.4 Examples of strictly regular boundary conditions
  6. 6 Asymptotic behavior of eigenvalues and eigenvectors
    1. 6.1 The key identity for characteristic determinant
    2. 6.2 Asymptotic behavior of eigenvalues
    3. 6.3 Asymptotic behavior of root vectors
    4. 6.4 The case of matrix function $Q$ with non-trivial block diagonal
  7. 7 Completeness property
  8. 8 Adjoint operator
    1. 8.1 General properties of the adjoint operator
    2. 8.2 Key identity for scalar product of eigenvectors
  9. 9 Uniform minimality and Riesz basis property
    1. 9.1 Uniform minimality
    2. 9.2 Riesz basis property
    3. 9.3 Riesz basis property with parentheses
  10. 10 Application to the Timoshenko beam model
    1. 10.1 Problem statement and reduction to Dirac type operator
    2. 10.2 Completeness and Riesz basis property with parentheses
    3. 10.3 Asymptotic behavior of eigenvalues and Riesz basis property

##  1 Introduction

In this paper we continue our investigation [32], [33] of the spectral
properties of non-self-adjoint boundary value problems (BVP) for the following
first order system of ordinary differential equations (ODE):

$\mathcal{L}(Q)y:=-iB(x)^{-1}\bigl{(}y^{\prime}+Q(x)y\bigr{)}=\lambda y,\qquad
y=\operatorname{col}(y_{1},\ldots,y_{n}),\qquad x\in[0,\ell],$ (1.1)

subject to the following boundary conditions with $n\times n$ matrices
$C,D\in\mathbb{C}^{n\times n}$:

$U(y):=Cy(0)+Dy(\ell)=0,\quad\text{and}\quad\operatorname{rank}(C\ D)=n.$
(1.2)

Here

$B=\operatorname{diag}(\beta_{1},\ldots,\beta_{n}),\qquad\beta_{k}\in
L^{1}([0,\ell];\mathbb{R}),\qquad k\in\\{1,\ldots,{n}\\},$ (1.3)

is a self-adjoint invertible diagonal summable matrix function, and

$Q=(Q_{jk})_{j,k=1}^{n},\qquad Q_{jk}\in
L^{1}[0,\ell]:=L^{1}([0,\ell];\mathbb{C}),\qquad j,k\in\\{1,\ldots,{n}\\},$
(1.4)

is a summable (generally non-self-adjoint) potential matrix.

Next we associate with the BVP (1.1)–(1.2) an operator $L_{U}(Q)$. To this end
we denote by $\mathfrak{H}_{k}:=L^{2}_{|\beta_{k}|}[0,\ell]$ the weighted
$L^{2}$-space with the weight $|\beta_{k}|$, $k\in\\{1,\ldots,{n}\\}$, and set
$\mathfrak{H}:=\mathfrak{H}_{1}\oplus\ldots\oplus\mathfrak{H}_{n}.$ Now the
operator $L_{U}(Q)$ in $\mathfrak{H}$ is defined as a restriction of the
maximal operator $L_{\max}(Q)$ generated in $\mathfrak{H}$ by the differential
expression $\mathcal{L}(Q)$ to the domain

$\operatorname{dom}(L_{U}(Q)):=\\{y\in\operatorname{AC}([0,\ell];\mathbb{C}^{n})\
:\ \mathcal{L}(Q)y\in\mathfrak{H},\ \ \ U(y)=Cy(0)+Dy(\ell)=0\\}.$ (1.5)

Systems (1.1) are of significant interest in some theoretical and practical
questions. For instance, if $n=2m$, $B(x)=\operatorname{diag}(-I_{m},I_{m})$,
$Q=\begin{pmatrix}0&Q_{12}\\\ Q_{21}&0\end{pmatrix}$, system (1.1) is
equivalent to the Dirac system (see [29], [44, Section 1.2]). Note also that
equation (1.1) with arbitrary constant not necessary self-adjoint matrix
$B(x)=\operatorname{diag}(b_{1},\ldots,b_{n})\in\mathbb{C}^{n\times n}$ is
used to integrate the $N$-waves problem arising in nonlinear optics [54,
Sec.III.4].

The spectral problem (1.1)–(1.2) (the operator $L_{U}(Q)$) has first been
investigated by G.D. Birkhoff and R.E. Langer [7]. Namely, they have extended
some previous results of Birkhoff and Tamarkin on non-self-adjoint boundary
value problem for ODE to the case of BVP (1.1)–(1.2). More precisely, they
introduced the concepts of _regular and strictly regular boundary conditions_
(1.2) and investigated the asymptotic behavior of eigenvalues and
eigenfunctions of the corresponding operator $L_{U}(Q)$. Moreover, they proved
_a pointwise convergence result_ on spectral decompositions of the operator
$L_{U}(Q)$ corresponding to the BVP (1.1)–(1.2) with regular boundary
conditions.

To the best of our knowledge, the problem of the completeness of the system of
root vectors _of general BVP_ (1.1)–(1.2) _with arbitrary constant not
necessary self-adjoint matrix
$B(x)=\operatorname{diag}(b_{1},\ldots,b_{n})\in\mathbb{C}^{n\times n}$_, was
first investigated in the recent papers [42, 43] by one of the authors and
L.L. Oridoroga. In these papers the concept of _weakly regular_ boundary
conditions for the system (1.1) was introduced and the completeness of root
vectors for this class of BVP was proved. Completeness property for general
BVP (1.1)–(1.2) with non-weakly regular and even with degenerate boundary
conditions was obtained in [32]. _Emphasize that in the case of non-weakly
regular boundary conditions the completeness property substantially depends on
the values $Q(0)$ and $Q(1)$._

Going over to the basis property note that during the last decade there
appeared numerous papers devoted mainly to the Riesz basis property for
$2\times 2$ Dirac system subject to the _regular or strictly regular_ boundary
conditions (see [69, 49, 50, 20, 6, 8, 11, 10, 12, 9, 13]). The most complete
result on the Riesz basis property for $2\times 2$ Dirac and Dirac-type
systems with $Q\in L^{1}$ and strictly regular boundary conditions was
obtained independently by different methods and at the same time by A.M.
Savchuk and A.A. Shkalikov [60] and by the authors [31, 33]. The case of
regular boundary conditions is treated in [60] for the first time. Other
proofs were obtained later in [57], [58] (see also their recent survey [59]
and references therein).

To the best of our knowledge the first result on the Riesz basis property for
BVP (1.1)–(1.2) generated by general $n\times n$ system (1.1) with
$B(x)=B=\operatorname{diag}(b_{1},\ldots,b_{n})\in\mathbb{C}^{n\times
n}\not=B^{*}$ and bounded $Q\in L^{\infty}([0,1];\mathbb{C}^{n\times n})$ was
obtained by the authors in [32]. Treated boundary conditions form rather broad
class that covers, in particular, periodic, antiperiodic, and regular
separated (not necessarily self-adjoint) boundary conditions. Note also that
BVP for $2m\times 2m$ Dirac equation ($B=\operatorname{diag}(-I_{m},I_{m})$)
was investigated in [51] (Bari-Markus property for Dirichlet BVP with $Q\in
L^{2}([0,1];\mathbb{C}^{2m\times 2m})$ and in [24, 25] (Bessel and Riesz basis
properties on abstract level).

Note also that _periodic and antiperiodic (necessarily non-strictly regular)
BVP_ for $2\times 2$ Dirac and Sturm-Liouville equations have also attracted
certain attention during the last decade. For instance, in [10, Theorem 13],
[9, Theorem 19] and [13], it is established a _criterion_ for the system of
root vectors to contain a Riesz basis for periodic (resp., antiperiodic)
$2\times 2$ Dirac operator in terms of the Fourier coefficients of $Q$ as well
as in terms of periodic (resp., antiperiodic) and Dirichlet spectra. It is
also worth mentioning that F. Gesztesy and V.A. Tkachenko [16, 17] for $q\in
L^{2}[0,\pi]$ and P. Djakov and B.S. Mityagin [9] for $q\in W^{-1,2}[0,\pi]$
established by different methods a _criterion_ for the system of root vectors
to contain a Riesz basis for Sturm-Liouville operator
$-\frac{d^{2}}{dx^{2}}+q(x)$ on $[0,\pi]$. See also recent survey [14] by P.
Djakov and B. Mityagin, surveys [34, 38] by A.S. Makin, and the references
therein.

Note in conclusion, that the Riesz basis property for abstract operators is
investigated in numerous papers. Let us mention [21, 46, 47, 1, 62, 4, 5], the
recent survey by A.A. Shkalikov [63], and the references therein.

Let us formulate our main results. To this end we need to impose certain
conditions on the entries of the matrix function $B(\cdot)$. We assume that
there exists $\theta\in(0,1)$ and $n_{-}\in\\{0,1,\ldots,n\\}$, such that

$\displaystyle-\infty<-\theta^{-1}<\beta_{1}(x)$
$\displaystyle\leqslant\ldots\leqslant\beta_{n_{-}}(x)<-\theta<0,\qquad
x\in[0,\ell],$ (1.6) $\displaystyle 0<\theta<\beta_{n_{-}+1}(x)$
$\displaystyle\leqslant\ldots\leqslant\beta_{n}(x)<\theta^{-1}<\infty,\qquad
x\in[0,\ell],$ (1.7)

and for each $k\in\\{1,\ldots,{n-1}\\}$

$\text{either}\quad\beta_{k}\equiv\beta_{k+1}\quad\text{or}\quad\beta_{k}(x)+\theta<\beta_{k+1}(x),\quad
x\in[0,\ell].$ (1.8)

Let us recall the definition of _regular boundary conditions_ (see [7, p.89])
confining ourselves to the case of $B(x)=B(x)^{*}$. Set

$S:=\operatorname{diag}(s_{1},\ldots,s_{n}),\qquad
s_{k}:=\operatorname{sign}(\beta_{k}(\cdot)),\quad k\in\\{1,\ldots,{n}\\}.$
(1.9)

Conditions (1.6)–(1.7) guarantee that, in fact,
$S=\operatorname{diag}(-I_{n_{-}},I_{n-n_{-}})$. Denote by $P_{+}$ and $P_{-}$
the spectral projections in $\mathbb{C}^{n}$ onto “positive” and “negative”
parts of the spectrum of $S=S^{*}$, respectively. Now the concept of
regularity of boundary conditions for the problem (1.1)–(1.2) reads as
follows:

$\det(CP_{+}+\ DP_{-})\neq 0\quad\text{and}\quad\det(CP_{-}+\ DP_{+})\neq 0.$
(1.10)

Additionally, boundary conditions (1.2) are called strictly regular if
eigenvalues of the unperturbed operator $L_{U}(0)$ are asymptotically
separated (see Definition 5.13 for details).

Our first main result establishes existence of triangular transformation
operators for equation (1.1) with $Q\in L^{1}$ and _non-constant_ self-adjoint
$n\times n$ matrix function $B(\cdot)$, satisfying conditions (1.6)–(1.8) (see
Theorem 3.7). Namely, assuming for simplicity that $\beta_{1}<\beta_{2}$ let
$A=\operatorname{col}(a_{1},\ldots,a_{n})$ have non-zero entries. Then we show
that the vector solution $y_{A}(\cdot,\lambda)$ of the equation (1.1) subject
to the initial condition $y_{A}(0,\lambda)=A$ admits a triangular
representation

$y_{A}(x,\lambda)=(I+\mathcal{K}_{A})e_{A}(x,\lambda)=e_{A}(x,\lambda)+\int^{x}_{0}K_{A}(x,t)B(t)e_{A}(t,\lambda)\,dt,$
(1.11)

where

$e_{A}(x,\lambda)=\operatorname{col}(a_{1}e^{i\lambda\rho_{1}(x)},\ldots,a_{n}e^{i\lambda\rho_{n}(x)}),\qquad\rho_{k}(x)=\int_{0}^{x}\beta_{k}(t)\,dt,$

is the solution to the unperturbed equation $\mathcal{L}(0)f=\lambda f$
subject to the same initial condition.

This result is substantial advancement comparing to the corresponding result
by one of the authors in [41], where this was established for $Q\in
L^{\infty}$ and _constant_ self-adjoint $n\times n$ matrix $B=B^{*}$, and the
corresponding result by the authors in [33], where this was established for
$Q\in L^{1}$ and _constant_ self-adjoint $2\times 2$ matrix
$B=\operatorname{diag}(b_{1},b_{2})=B^{*}$. This result has a wide area of
applications. Here we apply triangular transformation operators only to
investigation of the spectral properties of BVP (1.1)–(1.2) (operator
$L_{U}(Q)$).

As an immediate application of triangular transformation operators, we obtain
formulas relating the fundamental matrix solutions $\Phi(\cdot,\lambda)$ and
$\Phi_{0}(\cdot,\lambda)$ to equation (1.1) with $Q\not=0$ and $Q=0$,
respectively (see Proposition 4.2 and formula (4.29)). In fact, it leads to a
representation of the deviation $\Phi(\cdot,\lambda)-\Phi_{0}(\cdot,\lambda)$
as a Fourier transform of linear combinations of several transformation
operators kernels (see formula (4.45)).

In turn, starting with this Fourier representation of
$\Phi(\cdot,\lambda)-\Phi_{0}(\cdot,\lambda)$ we establish an important
identity relating characteristic determinants $\Delta_{Q}(\cdot)$ and
$\Delta_{0}(\cdot)$ of the operators $L_{U}(Q)$ and $L_{U}(0)$ (see
(5.11)–(5.13) for exact definitions). Namely, letting

$b_{-}:=b_{1}+\ldots+b_{n_{-}}\quad\text{and}\quad
b_{+}:=b_{n_{-}+1}+\ldots+b_{n},\quad\text{where}\quad
b_{k}:=\rho_{k}(\ell)=\int_{0}^{\ell}\beta_{k}(x)\,dx.$ (1.12)

_we show that the characteristic determinant $\Delta_{Q}(\cdot)$ admits the
following representation:_

$\Delta_{Q}(\lambda)=\Delta_{0}(\lambda)+\int_{b_{-}}^{b_{+}}g(u)e^{i\lambda
u}\,du\qquad\text{with}\qquad g\in
L^{1}[b_{-},b_{+}],\quad\lambda\in\mathbb{C}.$ (1.13)

It is worth mentioning that the second key ingredient in the proof of formula
(1.13) is an extension of the classical Liouville formula for the determinant
of a fundamental matrix $\Phi(x,\lambda)$ to the case of its $m$-th exterior
powers $\bigwedge^{m}\Phi(x,\lambda)$ obtained in Proposition 4.7.

For a special case of $2\times 2$ Dirac type operator (_constant_
$B\equiv\operatorname{diag}(b_{1},b_{2})=B^{*}\in\mathbb{C}^{2\times 2}$)
representation (1.13) was earlier established in [31], [33, Lemma 4.1]. In
recent papers by A. Makin [35, 36, 37] this representation was applied to
establish Riesz basis property of periodic BVP (regular but not strictly
regular) for $2\times 2$ Dirac equation under certain explicit algebraic
assumptions on a potential matrix.

Formula (1.13) gives a bridge between the spectral theory of the operator
$L_{U}(Q)$ and the theory of entire functions of exponential type due to the
simple fact: the spectrum
$\sigma(L_{U}(Q))=\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$ of $L_{U}(Q)$ coincides
with the set of zeros (counting multiplicity) of the entire function
$\Delta_{Q}(\cdot)$. Assuming _boundary conditions to be regular_ we easily
obtain from (1.13) that $\Delta_{Q}(\cdot)$ is an entire sine-type function of
the same types $b_{\pm}$ in $\mathbb{C}_{\mp}$ as the determinant
$\Delta_{0}(\cdot)$. Further, following the schema of the proof of [33,
Proposition 4.7], we extract the following asymptotic formula from
representation (1.13)

$\lambda_{m}=\lambda_{m}^{0}+o(1)\qquad\text{as}\qquad m\to\infty,$ (1.14)

relating the eigenvalues $\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$ of the operator
$L_{U}(Q)$ and eigenvalues $\\{\lambda_{m}^{0}\\}_{m\in\mathbb{Z}}$ of the
unperturbed operator $L_{U}(0)$. Note, that for $2\times 2$ Dirac equation
formula (1.14) was first established in [31] and [60] independently and by
different methods.

Moreover, _assuming boundary conditions to be strictly regular_ we complete
formula (1.14) by establishing similar formula for the normalized eigenvectors
$f_{m}(\cdot)$ and $f_{m}^{0}(\cdot)$ of the operators $L_{U}(Q)$ and
$L_{U}(0)$, respectively. Namely, using formula relating $\Phi(\cdot,\lambda)$
and $\Phi^{0}(\cdot,\lambda)$ as well as a simple abstract formula for simple
eigenvectors of the operator $L_{U}(Q)$, we establish the following formula
for their deviation which is valid _uniformly in $x\in[0,\ell]$_:

$f_{m}(x)=f_{m}^{0}(x)+o(1)\quad\text{as}\quad m\to\infty,\quad
m\in\mathbb{Z},$ (1.15)

In turn, this relationship and formula (1.14) are substantially involved in
the proof of the Riesz basis property of the operator $L_{U}(Q)$ provided that
boundary conditions are _strictly regular_. This proved to be challenging even
on the algebraic level (the case of $Q=0$) and required establishing a new
algebraic identity (see Proposition 8.5) for the inner product of the
eigenvectors of the unperturbed operator $L_{U}(0)$ and its adjoint
$L_{U}^{*}(0)$.

It is worth mentioning that in Section 5.4 we find necessary and sufficient
conditions for quasi-periodic boundary conditions

$y_{k}(\ell)=c_{k}y_{k}(0),\qquad c_{k}\neq 0,\qquad k\in\\{1,\ldots,{n}\\},$
(1.16)

to be strictly regular. In this case conditions (1.2) hold with invertible
$C=\operatorname{diag}(c_{1},\ldots,c_{n})$ and $D=-I_{n}$. In accordance with
(1.10), conditions (1.16) are always regular but not necessary strictly
regular. Morover, _antiperiodic_ boundary conditions $(c_{1}=\ldots=c_{n}=-1)$
are strictly regular if and only if for some $b_{0}>0$ the numbers
$b_{1},\ldots,b_{n}$ given by (1.12) can be ordered in such a way that the
following representation holds,

$b_{k}=2^{a_{k}}(2u_{k}+1)b_{0},\qquad a_{k},u_{k}\in\mathbb{Z},\quad
k\in\\{1,\ldots,{n}\\},\qquad 0\leqslant a_{1}<a_{2}<\ldots<a_{n}.$ (1.17)

In particular, antiperiodic boundary conditions are strictly regular if
$b_{k}=2^{k}$, $k\in\\{1,\ldots,{n}\\}$.

We also obtain completeness property in the case of regular boundary
conditions extending the corresponding result from [43] to the case of non-
constant matrix function $B(\cdot)\not=\operatorname{const}$. In Section 9 we
also establish the Riesz basis property with parentheses for the operator
$L_{U}(Q)$ provided that boundary conditions (1.2) are only regular (but not
strictly regular). For the proof we use the perturbation idea which goes back
to A.A. Shkalikov [61] and was applied later on to $2\times 2$ Dirac systems
in [60, 33].

Finally, we apply our main abstract results with
$B(x)=B(x)^{*}\in\mathbb{C}^{4\times 4}$ to the Timoshenko beam model
investigated under different restrictions in numerous papers (see [67, 23, 48,
65, 72, 71, 70, 66, 32, 33, 2] and the references therein). In our previous
papers [32, 33] we studied the Timoshenko beam model with relaxed smoothness
assumptions on the coefficients, when the beam is fixed at one end and with
the most general boundary condition at the other end. For this general model,
we established completeness and Riesz basis property with parentheses,
assuming certain identity for the coefficients of the model: the ratio of wave
speeds $\frac{K(\cdot)}{\rho(\cdot)}$ and $\frac{EI(\cdot)}{I_{\rho}(\cdot)}$
is constant. This assumption has to be added because in our previous papers
[32, 33] we treated BVP (1.1)–(1.2) with a _constant matrix_ $B(x)=B$.
However, the dynamic generator of the general Timoshenko beam model is similar
to the operator $L_{U}(Q)$ with
$B(x)=(-\beta_{1}(x),\beta_{1}(x),-\beta_{2}(x),\beta_{2}(x))$ and _functions
$\beta_{1}(x),\beta_{2}(x)$ with non-constant ratio_, and cannot be reduced to
Dirac-type operator with a constant matrix $B$. Since in this paper, we treat
more general BVP (1.1)–(1.2) with arbitrary non-constant matrix $B(\cdot)$,
this allows us to remove this algebraic assumption. Moreover, we establish
asymptotic behavior of the eigenvalues and eigenvectors of the dynamic
generator $L_{\operatorname{Tim}}$ of the Timoshenko beam model as well as the
Riesz basis property (without parentheses) of the root vectors system of the
operator $L_{\operatorname{Tim}}$, provided that the eigenvalues of
$L_{\operatorname{Tim}}$ are asymptotically separated. We also provide
comprehensive list of explicit conditions that guarantee this property.

When our preprint was almost ready we became aware of the short communication
[64] by A.A. Shkalikov, where results on Riesz basis property of regular BVP
(1.1)–(1.2) with $Q\in L^{1}$ were announced under the similar assumptions on
matrix function $B(\cdot)$.

The paper is organized as follows. Section 2 is devoted to some preliminaries.
In particular, we list some identities of determinants of sums and products of
matrices.

In Section 3 we prove our first main result, Theorem 3.7, establishing
existence of triangular transformation operators for equation (1.1). The case
of _non-constant_ matrix function $B(\cdot)$ poses significant difficulties
even for $Q\in C^{1}$.

In Section 4 we apply transformation operators to establish an important
identity for the fundamental matrix $\Phi(x,\lambda)$ of the equation (1.1).
Namely, in Proposition 4.2 we show that the deviation of the fundamental
matrices of equation (1.1) with $Q\neq 0$ and $Q=0$ admits a Fourier transform
representation involving the kernels of the transformation operators. In
Proposition 4.7 we generalize a classical Liouville’s formula and show that
$m$-th exterior power $\bigwedge^{m}\Phi(x,\lambda)$ of the fundamental matrix
$\Phi(x,\lambda)$ satisfies equation similar to (1.1). This result, in turn,
implies similar Fourier transform representation for the minors of
$\Phi(x,\lambda)$.

In Section 5 we establish some general properties of BVP (1.1)–(1.2) and
introduce concepts of regular and strictly regular boundary conditions. In
particular, we apply Jacobi’s formula to establish certain important uniform
estimates from below for eigenvectors of this BVP with $Q=0$, provided that
boundary conditions are strictly regular (see Proposition 5.16).

In Section 6 we establish key identity (1.13) and similar identity related to
eigenvectors of the operator $L_{U}(Q)$. These identities are used to
establish the asymptotic behavior of eigenvalues and eigenvectors in Theorems
6.6 and 6.9, respectively.

In Section 7, following [43], we establish completeness of the root vectors
system of the operator $L_{U}(Q)$ (see Subsection 2.1 for exact definition),
provided that $Q\in L^{1}$, matrix function $B(\cdot)$ meats conditions
(1.6)–(1.8), and boundary conditions (1.2) are regular.

In Section 8, we study adjoint operator $L_{U}(Q)^{*}$. In Proposition 8.5 we
establish an important identity for the inner product of eigenvectors of the
unperturbed operator $L_{U}(0)$ and its adjoint $L_{U}^{*}(0)$, which is
essential for proving uniform minimality property.

In Section 9, we prove our main results on uniform minimality and Riesz basis
property of the root vectors system of the operator $L_{U}(Q)$ with strictly
regular boundary conditions (see Theorems 9.4 and 9.11). Here we also
establish Riesz basis property with parentheses provided that boundary
conditions (1.2) are regular (see Theorem 9.20).

In Section 10 we apply our abstract results with
$B(x)=B(x)^{*}\in\mathbb{C}^{4\times 4}$ to the dynamic generator
$L_{\operatorname{Tim}}$ of the general model (10.1)–(10.5) of spatially non-
homogenous Timoshenko beam with both boundary and locally distributed damping.
By reducing this dynamic generator $L_{\operatorname{Tim}}$ to the special
$4\times 4$ Dirac type operator $L_{U}(Q)$, we show that the root vectors
system of $L_{\operatorname{Tim}}$ forms a Riesz basis in the suitable energy
space, when the corresponding operator $L_{U}(Q)$ is equipped with the
strictly regular boundary conditions (see Theorems (10.7) and (10.8)(ii)). We
also apply results of Section 6 to establish the asymptotic behavior of the
eigenvalues and the eigenvectors of the operator $L_{\operatorname{Tim}}$ (see
Theorems 10.6 and 10.8(i)). In particular, we found an interesting case when
eigenvalues of $L_{\operatorname{Tim}}$ have an explicit asymptotical formula,
which to the best of our knowledge is new even in the case of constant
parameters of the model (see Theorem 10.6(iii)).

##  2 Preliminaries

###  2.1 Definition of the system of root vectors

Let us also recall the notion of the system of root vectors of an operator
with compact resolvent. First, we recall a few basic facts regarding the
eigenvalues of a compact, linear operator
$T\in\mathcal{B}_{\infty}(\mathfrak{H})$ in a separable complex Hilbert space
$\mathfrak{H}$. The geometric multiplicity, $m_{g}(\mu_{0},T)$, of an
eigenvalue $\mu_{0}\in\sigma_{p}(T)$ of $T$ is given by
$m_{g}(\mu_{0},T):=\dim(\ker(T-\mu_{0})).$

The root subspace of $T$ corresponding to $\mu_{0}\in\sigma_{p}(T)$ is given
by

$\mathcal{R}_{\mu_{0}}(T)=\big{\\{}f\in\mathfrak{H}\,:\,(T-\mu_{0})^{k}f=0\ \
\text{for some}\ \ k\in\mathbb{N}\big{\\}}.$ (2.1)

Elements of $\mathcal{R}_{\mu_{0}}(T)$ are called root vectors. For
$\mu_{0}\in\sigma_{p}(T)\backslash\\{0\\}$, the set $\mathcal{R}_{\mu_{0}}(T)$
is a closed linear subspace of $\mathfrak{H}$ whose dimension equals to the
algebraic multiplicity, $m_{a}(\mu_{0},T)$, of $\mu_{0}$,
$m_{a}(\mu_{0},T):=\dim\big{(}\mathcal{R}_{\mu_{0}}(T)\big{)}<\infty.$

Denote by $\\{\mu_{j}\\}_{j=1}^{\infty}$ the sequence of non-zero eigenvalues
of $T$ ($\mu_{j}\neq\mu_{k}$) and let $m_{j}$ be the algebraic multiplicity of
$\mu_{j}$. By the system of root vectors of the operator $T$ we mean any
sequence of the form $\cup_{j=1}^{\infty}\\{e_{jk}\\}_{k=1}^{m_{j}},$ where
$\\{e_{jk}\\}_{k=1}^{m_{j}}$ is a basis in $\mathcal{R}_{\mu_{j}}(T)$,
$m_{j}=m_{a}(\mu_{j},T)<\infty$. The system or root vectors of the operator
$T$ is called _normalized_ if $\|e_{jk}\|_{\mathfrak{H}}=1$, $j\in\mathbb{N}$,
$k\in\\{1,\ldots,{m_{j}}\\}$.

We are particularly interested in the case where $A$ is a densely defined,
closed, linear operator in $\mathfrak{H}$ whose resolvent is compact, that is,
$R_{A}(\lambda):=(A-\lambda)^{-1}\in\mathcal{B}_{\infty}(\mathfrak{H}),\
\lambda\in\rho(A).$ Via the spectral mapping theorem all eigenvalues of $A$
correspond to eigenvalues of its resolvent $R_{A}(\lambda)$,
$\lambda\in\rho(A)$, and vice versa. Hence, we use the same notions of root
vectors, root subspaces, geometric and algebraic multiplicities associated
with the eigenvalues of $A$, and the system of root vectors of $A$.

###  2.2 Properties of Lipshitz functions

Recall that $\operatorname{Lip}_{1}(\mathcal{S})$ for $\mathcal{S}\subset X$
in any normed space $X$ is the class of functions $f$ acting from
$\mathcal{S}$ to $\mathbb{C}$ and satisfying the condition

$|f(u)-f(u^{\prime})|\leqslant\alpha\|u-u^{\prime}\|_{X},\quad
u,u^{\prime}\in\mathcal{S}\quad\text{for some}\quad\alpha=\alpha_{f}>0.$

It is well known that for any finite segment $[a,b]\in\mathbb{R}$,

$\operatorname{Lip}_{1}[a,b]=\\{f\in\operatorname{AC}[a,b]:f^{\prime}\in
L^{\infty}[a,b]\\}=W^{1,\infty}[a,b].$ (2.2)

We also denote by $L_{1,\operatorname{loc}}(\mathcal{S})$ a set of functions
$f$ that are Lipshitz on any compact subset of $\mathcal{S}$.

Our main target use case will be $X=\mathbb{R}^{2}$. To this end we denote by

$|u-u^{\prime}|:=\|u-u^{\prime}\|_{\mathbb{R}^{2}}:=|x-x^{\prime}|+|t-t^{\prime}|,\qquad
u=(x,t),\ u^{\prime}=(x^{\prime},t^{\prime})\in\mathbb{R}^{2},$

a Manhattan distance between points $u$ and $u^{\prime}$. Further, a simply
connected, closed bounded set $\mathcal{S}\subset\mathbb{R}^{2}$ is said to
have a Lipshitz boundary if its boundary $\partial S$ can be parametrized as

$\partial S=\\{(\gamma_{1}(t),\gamma_{2}(t)):t\in[a,b]\\},$ (2.3)

for some $-\infty<a<b<\infty$ where $\gamma_{j}\in\operatorname{Lip}_{1}[a,b]$
and $\gamma_{j}(a)=\gamma_{j}(b)$, $j\in\\{1,2\\}$. Throughout the paper we
will denote for $f:\mathbb{R}^{2}\to\mathbb{C}$,

$(D_{1}f)(x,t):=D_{1}f(x,t):=\frac{\partial}{\partial{x}}f(x,t),\qquad(D_{2}f)(x,t):=D_{2}f(x,t):=\frac{\partial}{\partial{t}}f(x,t),$
(2.4)

whenever corresponding partial derivatives exist.

In the sequel we will need the following simple properties of Lipshitz
functions.

######  Lemma 2.1.

(i) Let $\mathcal{S}\subset\mathbb{R}^{2}$ and let
$f_{m}\in\operatorname{Lip}_{1}(\mathcal{S})$, $m\in\mathbb{N}$, be such that

$\displaystyle f_{m}(x,t)\to f(x,t)\ \ \text{as}\ \ m\to\infty,\quad
u=(x,t)\in\mathcal{S},$ (2.5)
$\displaystyle|f_{m}(x,t)-f_{m}(x^{\prime},t^{\prime})|\leqslant\alpha(|x-x^{\prime}|+|t-t^{\prime}|),\quad(x,t),(x^{\prime},t^{\prime})\in\mathcal{S},\quad
m\in\mathbb{N},$ (2.6)

for some $\alpha>0$ and $f:\mathcal{S}\to\mathbb{C}$. Then

$f\in\operatorname{Lip}_{1}(\mathcal{S})\qquad\text{and}\qquad|f(x,t)-f(x^{\prime},t^{\prime})|\leqslant\alpha(|x-x^{\prime}|+|t-t^{\prime}|),\quad(x,t),(x^{\prime},t^{\prime})\in\mathcal{S}.$

Moreover, $\operatorname{Lip}_{1}(\mathcal{S})$ is a Banach space with the
norm

$\|f\|_{\operatorname{Lip}_{1}(\mathcal{S})}:=|f(x_{0},t_{0})|+\sup_{\genfrac{}{}{0.0pt}{2}{u,u^{\prime}\in\mathcal{S}}{u\neq
u^{\prime}}}\frac{|f(x,t)-f(x^{\prime},t^{\prime})|}{|u-u^{\prime}|},\qquad
u=(x,t),\ u^{\prime}=(x^{\prime},t^{\prime}),$

where $u_{0}=(x_{0},t_{0})\in\mathcal{S}$ is an arbitrary fixed point.

(ii) Let $\mathcal{S}\subset\mathbb{R}^{2}$ be a simply connected, closed
bounded set with a Lipshitz boundary and let
$f\in\operatorname{Lip}_{1}(\mathcal{S})$, be such that

$\mathcal{D}(f):=\max\\{\|D_{1}f\|_{L^{\infty}(\mathcal{S})},\|D_{2}f\|_{L^{\infty}(\mathcal{S})}\\}<\infty.$
(2.7)

Then

$|f(x,t)-f(x^{\prime},t^{\prime})|\leqslant\alpha(|x-x^{\prime}|+|t-t^{\prime}|),\quad(x,t),(x^{\prime},t^{\prime})\in\mathcal{S},$

where constant $\alpha\in(0,\infty)$ depends only on $\mathcal{D}(f)$ and the
set $\mathcal{S}$.

(iii) Let $\mathcal{S}\subset\mathbb{R}^{2}$ be a simply connected, closed
bounded set with a Lipshitz boundary and let
$f_{m}\in\operatorname{Lip}_{1}(\mathcal{S})$, $m\in\mathbb{N}$, be such that

$f_{m}(x,t)\to f(x,t)\ \ \text{as}\ \
m\to\infty,\quad(x,t)\in\mathcal{S},\qquad\|D_{1}f_{m}\|_{L^{\infty}(\mathcal{S})},\
\|D_{2}f_{m}\|_{L^{\infty}(\mathcal{S})}\leqslant\tau,\quad m\in\mathbb{N},$
(2.8)

for some $\tau>0$ and $f:\mathcal{S}\to\mathbb{C}$. Then
$f\in\operatorname{Lip}_{1}(\mathcal{S})$.

###### Proof.

Parts (i) and (ii) are well-known and are of folklore nature. Let us only
mention that the set $\mathcal{S}$ as a simply connected, closed and bounded
set with a Lipshitz boundary has the following important property: for any
interior points $u,u^{\prime}\in\operatorname{int}S$ there exists a
“Manhattan” curve $\Gamma_{u,u^{\prime}}$ (a finite sequence of alternating
vertical and horizontal segments) connecting $u$ and $u^{\prime}$ such the
length of $\Gamma_{u,u^{\prime}}$ is bounded by
$C_{\mathcal{S}}|u-u^{\prime}|$ for some $C_{\mathcal{S}}$ that only depend on
$\mathcal{S}$ (more precisely it only depends on Lipshitz constants of the
parametric curves $\gamma_{1}$, $\gamma_{2}$ of the boundary $\partial S$).
This property allows to utilize fundamental representation (2.2) for Lipshitz
space on a finite segment.

Let us also comment on part (iii). Part (ii) and condition (2.8) imply uniform
Lipshitz condition (2.5)–(2.6) with some $\alpha$ that only depends on $\tau$
and the set $\mathcal{S}$. Part (i) finishes the proof. ∎

###  2.3 The Banach spaces $X_{1}$ and $X_{\infty}$

Following [39, 33] denote by $X_{1}:=X_{1}(\Omega)$ and
$X_{\infty}:=X_{\infty}(\Omega)$ the linear spaces composed of (equivalent
classes of) measurable functions defined on

$\Omega:=\\{(x,t):0\leqslant t\leqslant x\leqslant\ell\\}$ (2.9)

satisfying

$\displaystyle\|f\|_{X_{1}}$
$\displaystyle:=\operatorname*{ess\,sup}_{t\in[0,\ell]}\int_{t}^{\ell}|f(x,t)|dx<\infty,$
(2.10) $\displaystyle\|f\|_{X_{\infty}}$
$\displaystyle:=\operatorname*{ess\,sup}_{x\in[0,\ell]}\int_{0}^{x}|f(x,t)|dt<\infty,$
(2.11)

respectively. It can easily be shown that the spaces $X_{1}$ and $X_{\infty}$
equipped with the norms (2.10) and (2.11) form Banach spaces that are not
separable. Denote by $X_{1,0}$ and $X_{\infty,0}$ the subspaces of $X_{1}$ and
$X_{\infty}$, respectively, obtained by taking the closure of continuous
functions $f\in C(\Omega)$. Clearly, the set $C^{1}(\Omega)$ of smooth
functions is also dense in both spaces $X_{1,0}$ and $X_{\infty,0}$.

The following simple property of the class $X_{\infty,0}(\Omega)$ established
in [33] will be essential in the sequel.

######  Lemma 2.2 (Lemma 2.2 in [33]).

For each $a\in[0,\ell]$ the trace mapping

$i_{a}:\ C(\Omega)\to C[0,a],\qquad(i_{a}R)(t):=R(a,t),\quad R\in C(\Omega),$
(2.12)

originally defined on $C(\Omega)$ admits a continuous extension (also denoted
by $i_{a}$) as a mapping from $X_{\infty,0}(\Omega)$ onto $L^{1}[0,a]$.

We will also need a property of intersections $X_{1}(\Omega)\cap
X_{\infty}(\Omega)$, $X_{1,0}(\Omega)\cap X_{\infty,0}(\Omega)$ established in
[33]. To this end, for any measurable on $\Omega$ kernel $R(\cdot,\cdot)$ we
define Volterra type operator $\mathcal{R}$ as follows,

$(\mathcal{R}f)(x)=\int^{x}_{0}R(x,t)f(t)dt.$ (2.13)

Denote by $\|\mathcal{R}\|_{p}:=\|\mathcal{R}\|_{L^{p}[0,\ell]\to
L^{p}[0,\ell]}$ the $L_{p}$-norm of the operator $\mathcal{R}$, provided that
it is bounded. Further, recall that a Volterra operator in a Banach space is a
compact operator with zero spectrum.

######  Lemma 2.3 (Lemma 2.1 in [33]).

Let $R\in X_{1}(\Omega)\cap X_{\infty}(\Omega)$ and $\mathcal{R}$ be a
Volterra type operator given by (2.13). Then:

(i) The operator $\mathcal{R}$ is bounded in $L^{p}[0,\ell]$ for each
$p\in[1,\infty]$ and

$\|\mathcal{R}\|_{p}\leqslant\|R\|_{X_{1}(\Omega)}^{1/p}\cdot\|R\|_{X_{\infty}(\Omega)}^{1-1/p}.$
(2.14)

Moreover,

$\|\mathcal{R}\|_{1}=\|R\|_{X_{1}(\Omega)},\qquad\|\mathcal{R}\|_{\infty}=\|R\|_{X_{\infty}}.$
(2.15)

(ii) If $R\in X_{1,0}(\Omega)\cap X_{\infty,0}(\Omega)$, then $\mathcal{R}$ is
a Volterra operator in $L^{p}[0,\ell]$ for each $p\in[1,\infty]$.

In what follows, we will also systematically use notations
$X_{1,0}\otimes\mathbb{C}^{n\times m}$ and
$X_{\infty,0}\otimes\mathbb{C}^{n\times m}$. In general, for any set $S$ of
complex-valued functions, notation $S\otimes\mathbb{C}^{n\times m}$ means a
set of all $n\times m$ matrices, where each entry of the matrix is an element
of the set $S$.

###  2.4 Properties of adjugate matrix

Denote by $\operatorname{GL}(n):=\operatorname{GL}(n,\mathbb{C})$ the set of
invertible $n\times n$ matrices with complex entries. For a matrix
$\mathcal{A}\in\mathbb{C}^{n\times n}$ denote by $\mathcal{A}^{a}$ its
adjugate matrix, i.e.

$\mathcal{A}\mathcal{A}^{a}=\mathcal{A}^{a}\mathcal{A}=\det(\mathcal{A})I_{n}.$
(2.16)

Let us recall some properties of the adjugate matrix,

$\displaystyle\mathcal{A}^{a}=\det(\mathcal{A})\cdot\mathcal{A}^{-1},\qquad\mathcal{A}\in\operatorname{GL}(n),$
(2.17)
$\displaystyle[\mathcal{A}^{a}]^{*}=[\mathcal{A}^{*}]^{a},\qquad\mathcal{A}\in\mathbb{C}^{n\times
n},$ (2.18)
$\displaystyle[\mathcal{A}\mathcal{B}]^{a}=\mathcal{B}^{a}\mathcal{A}^{a},\qquad\mathcal{A},\mathcal{B}\in\mathbb{C}^{n\times
n},$ (2.19)
$\displaystyle[\mathcal{A}_{1}\mathcal{A}\mathcal{A}_{2}]^{a}=\mathcal{A}_{2}^{a}\mathcal{A}^{a}\mathcal{A}_{1}^{a}=\det(\mathcal{A}_{1}\mathcal{A}_{2})\cdot\mathcal{A}_{2}^{-1}\mathcal{A}^{a}\mathcal{A}_{1}^{-1},\qquad\mathcal{A}_{1},\mathcal{A}_{2}\in\operatorname{GL}(n,\mathbb{C}).$
(2.20)

Let $\mathcal{A}(\cdot)=(\mathfrak{a}_{jk}(\cdot))_{j,k=1}^{n}$ be an $n\times
n$ matrix function differentiable at a point $\lambda\in\mathbb{C}$ and let
$\mathcal{A}^{a}(\cdot)=:(\mathcal{A}_{jk}(\cdot))_{j,k=1}^{n}$ be its
adjugate matrix function. Then in accordance with the Jacobi identity,

$\frac{d}{d\lambda}\det(\mathcal{A}(\lambda))=\operatorname{tr}\left(\mathcal{A}^{a}(\lambda)\mathcal{A}^{\prime}(\lambda)\right)=\sum_{j,k=1}^{n}\mathcal{A}_{jk}(\lambda)\mathfrak{a}_{kj}^{\prime}(\lambda).$
(2.21)

###  2.5 Determinants of matrix sums and products

If what follows we will need a few classical formulas for determinant of the
sum and the product of matrices. Further, assuming $n\in\mathbb{N}$ is fixed
throughout entire paper, we introduce the following set:

$\mathfrak{P}_{m}:=\\{\mathfrak{p}:=(p_{1},\ldots,p_{m}):1\leqslant
p_{1}<\ldots<p_{m}\leqslant n\\},\qquad m\in\\{1,\ldots,{n}\\},$ (2.22)

i.e. $\mathfrak{P}_{m}$ is the set of all increasing sequences with exactly
$m$ elements from $1$ to $n$. Additionally, we define
$\mathfrak{P}_{0}:=\\{\mathfrak{E}_{0}\\}$, where $\mathfrak{E}_{0}:=()$ is an
empty sequence. In what follows, we also denote
$\sigma(\mathfrak{p}):=p_{1}+\ldots+p_{m}$. Clearly
$\sigma(\mathfrak{E}_{0})=0$.

Further, for any $n\times n$ matrix $\mathcal{A}=(a_{jk})_{j,k=1}^{n}$ and
elements $\mathfrak{p}=(p_{1},\ldots,p_{m})$ and
$\mathfrak{q}=(q_{1},\ldots,q_{m})$ of $\mathfrak{P}_{m}$,
$m\in\\{0,1,\ldots,n\\}$, we set

$\mathcal{A}[\mathfrak{q},\mathfrak{p}]:=\det(a_{q_{j}p_{k}})_{j,k=1}^{m}=\det\begin{pmatrix}a_{q_{1}p_{1}}&\ldots&a_{q_{1}p_{m}}\\\
\vdots&\ddots&\vdots\\\ a_{q_{m}p_{1}}&\ldots&a_{q_{m}p_{m}}\\\
\end{pmatrix},\qquad\mathfrak{q},\mathfrak{p}\in\mathfrak{P}_{m},$ (2.23)

i.e. $\mathcal{A}[\mathfrak{q},\mathfrak{p}]$ is a minor of the matrix
$\mathcal{A}$ generated by the rows with indexes $q_{1}<\ldots<q_{m}$ and
columns with indexes $p_{1}<\ldots<p_{m}$. If $m=0$ then
$\mathcal{A}[\mathfrak{q},\mathfrak{p}]:=\det(I_{0}):=1$, where $I_{0}$ is an
empty matrix, where $\mathfrak{q}=\mathfrak{p}=\mathfrak{E}_{0}$ are empty
sequences.

Further, for $\mathfrak{p}\in\mathfrak{P}_{m}$ we denote by
$\widehat{\mathfrak{p}}\in\mathfrak{P}_{n-m}$ the complement of $\mathfrak{p}$
in the set $\\{1,\ldots,{n}\\}$. Namely, let
$\mathfrak{p}=(p_{1},\ldots,p_{m})\in\mathfrak{P}_{m}$, i.e. $1\leqslant
p_{1}<\ldots<p_{m}\leqslant n$, and let

$\\{1,\ldots,{n}\\}\setminus\\{p_{1},\ldots,p_{m}\\}=:\\{r_{1},\ldots,r_{n-m}\\},$

where $1\leqslant r_{1}<\ldots<r_{n-m}\leqslant n$. Then by definition
$\widehat{\mathfrak{p}}:=(r_{1},\ldots,r_{n-m})$.

Now we are ready to formulate a classical “folklore” formula for the
determinant of the sum of matrices (see e.g. [45]).

######  Lemma 2.4.

Let $\mathcal{A},\mathcal{B}\in\mathbb{C}^{n\times n}$. Then

$\det(\mathcal{A}+\mathcal{B})=\det(\mathcal{A})+\sum_{m=1}^{n}\sum_{\mathfrak{q},\mathfrak{p}\in\mathfrak{P}_{m}}(-1)^{\sigma(\mathfrak{p})+\sigma(\mathfrak{q})}\mathcal{A}[\widehat{\mathfrak{q}},\widehat{\mathfrak{p}}]\cdot\mathcal{B}[\mathfrak{q},\mathfrak{p}].$
(2.24)

Here we utilized all of the above notations, including a complement notation
$\widehat{\mathfrak{p}}$. It is clear, that the summand in r.h.s of (2.24) for
$m=n$ equals to $\det(\mathcal{B})$. Indeed, if $m=n$ then
$\mathfrak{P}_{m}=\\{\mathfrak{p}_{0}\\}$, where
$\mathfrak{p}_{0}:=(1,\ldots,n)$, and inner sum degenerates to
$\mathcal{A}[\widehat{\mathfrak{p}_{0}},\widehat{\mathfrak{p}_{0}}]\cdot\mathcal{B}[\mathfrak{p}_{0},\mathfrak{p}_{0}]$.
Complement $\widehat{\mathfrak{p}_{0}}=\mathfrak{E}_{0}$ is an empty sequence.
Hence
$\mathcal{A}[\widehat{\mathfrak{p}_{0}},\widehat{\mathfrak{p}_{0}}]=\det(I_{0})=1$.
It is also clear that
$\mathcal{B}[\mathfrak{p}_{0},\mathfrak{p}_{0}]=\det(\mathcal{B})$.

Next, we formulate a straightforward extension of the classical Cauchy–Binet
formula.

######  Lemma 2.5 (Subsection 1.2.6 in [15]).

Let $\mathcal{B},\mathcal{C}\in\mathbb{C}^{n\times n}$,
$m\in\\{1,\ldots,{n}\\}$ and $\mathfrak{p},\mathfrak{q}\in\mathfrak{P}_{m}$.
Then

$(\mathcal{B}\mathcal{C})[\mathfrak{q},\mathfrak{p}]=\sum_{\mathfrak{r}\in\mathfrak{P}_{m}}\mathcal{B}[\mathfrak{q},\mathfrak{r}]\cdot\mathcal{C}[\mathfrak{r},\mathfrak{p}].$
(2.25)

Some remarks. If $m=1$, this formula is nothing more than a definition of the
matrix product. If $m=n$, then this formula turns into
$\det(\mathcal{B}\mathcal{C})=\det(\mathcal{B})\cdot\det(\mathcal{C})$.

Finally, combining Lemmas 2.4 and 2.5 we arrive at the following formula that
will be useful for studying characteristic determinant of the BVP (1.1)–(1.2).

######  Lemma 2.6.

Let $\mathcal{A},\mathcal{B},\mathcal{C}\in\mathbb{C}^{n\times n}$. Then

$\det(\mathcal{A}+\mathcal{B}\mathcal{C})=\det(\mathcal{A})+\sum_{m=1}^{n}\sum_{\mathfrak{q},\mathfrak{p},\mathfrak{r}\in\mathfrak{P}_{m}}(-1)^{\sigma(\mathfrak{p})+\sigma(\mathfrak{q})}\mathcal{A}[\widehat{\mathfrak{q}},\widehat{\mathfrak{p}}]\cdot\mathcal{B}[\mathfrak{q},\mathfrak{r}]\cdot\mathcal{C}[\mathfrak{r},\mathfrak{p}].$
(2.26)

To estimate root vectors of the operator $L_{U}(Q)$ we will also need version
of Lemma 2.6 for cofactors of $\mathcal{A}+\mathcal{B}\mathcal{C}$. Let us
recall the corresponding definition. To this end, let
$\mathcal{A}=(\mathfrak{a}_{jk})_{j,k=1}^{n}\in\mathbb{C}^{n\times n}$. Then
by definition, cofactor $\mathcal{A}\\{j,k\\}$ of the element
$\mathfrak{a}_{jk}$ of the matrix $\mathcal{A}$ is the element at the $j$-th
row and $k$-th column of the matrix $\mathcal{A}^{a}$ adjugate to
$\mathcal{A}$ (introduced in Subsection 2.4), i.e.
$\mathcal{A}^{a}=:(\mathcal{A}\\{j,k\\})_{j,k=1}^{n}$. Let us express it via
our notation $\mathcal{A}[\mathfrak{q},\mathfrak{p}]$. It is easily seen that

$\mathcal{A}^{a}=\left(\mathcal{A}\\{j,k\\}\right)_{j,k=1}^{n}=\left((-1)^{j+k}\mathcal{A}[\mathfrak{p}_{k},\mathfrak{p}_{j}]\right)_{j,k=1}^{n},$
(2.27)

where

$\mathfrak{p}_{k}:=(1,\ldots,k-1,k+1,\ldots
n)=\widehat{(k)}\in\mathfrak{P}_{n-1}.$ (2.28)

With this observation we can easily derive the following versions of Lemmas
2.4 and 2.6 for cofactors.

######  Lemma 2.7.

Let $\mathcal{A},\mathcal{B}\in\mathbb{C}^{n\times n}$ and let
$j,k\in\\{1,\ldots,{n}\\}$. Then

$\displaystyle(\mathcal{A}+\mathcal{B})\\{j,k\\}$
$\displaystyle=\mathcal{A}\\{j,k\\}+(-1)^{j+k}\sum_{m=1}^{n-1}\sum_{\genfrac{}{}{0.0pt}{2}{\mathfrak{q},\mathfrak{p}\in\mathfrak{P}_{m}}{k\not\in\mathfrak{q},\,j\not\in\mathfrak{p}}}(-1)^{\sigma(\mathfrak{p})+\sigma(\mathfrak{q})}\mathcal{A}[\widehat{\mathfrak{q}},\widehat{\mathfrak{p}}]\cdot\mathcal{B}[\mathfrak{q},\mathfrak{p}],$
(2.29) $\displaystyle(\mathcal{A}+\mathcal{B}\mathcal{C})\\{j,k\\}$
$\displaystyle=\mathcal{A}\\{j,k\\}+(-1)^{j+k}\sum_{m=1}^{n-1}\sum_{\genfrac{}{}{0.0pt}{2}{\mathfrak{q},\mathfrak{p},\mathfrak{r}\in\mathfrak{P}_{m}}{k\not\in\mathfrak{q},\,j\not\in\mathfrak{p}}}(-1)^{\sigma(\mathfrak{p})+\sigma(\mathfrak{q})}\mathcal{A}[\widehat{\mathfrak{q}},\widehat{\mathfrak{p}}]\cdot\mathcal{B}[\mathfrak{q},\mathfrak{r}]\cdot\mathcal{C}[\mathfrak{r},\mathfrak{p}].$
(2.30)

##  3 Transformation operators

In this section we prove the existence of triangular transformation operators
for the system

$\mathcal{L}(Q)y:=-iB(x)^{-1}(y^{\prime}+Q(x)y)=\lambda y,\qquad
y=\operatorname{col}(y_{1},\ldots,y_{n}),\quad x\in[0,\ell],\\\ $ (3.1)

expressing solution to the certain Cauchy problem for equation (3.1) via the
solution to the same Cauchy problem for the simplest equation

$\mathcal{L}(0)y:=-iB(x)^{-1}y^{\prime}=\lambda y,\qquad
y=\operatorname{col}(y_{1},\ldots,y_{n}),\quad x\in[0,\ell].$ (3.2)

This result extends Theorem 1.2 from [41] and Theorem 2.5 from [33] to the
case of non-constant matrix $B(\cdot)$. Following the scheme of reasonings of
Theorem 1.2 from [41] we first establish the similarity of certain
restrictions $\mathcal{L}_{0}(Q)$ and $\mathcal{L}_{0}(0)$ of the operators
$\mathcal{L}(Q)$ and $\mathcal{L}(0)$, respectively.

###  3.1 Similarity of operators $\mathcal{L}_{0}(Q)$ and $\mathcal{L}_{0}(0)$

First we introduce the main objects, the operators $\mathcal{L}_{0}(Q)$ and
$\mathcal{L}_{0}(0)$. To this end we need to change notation comparing to
(1.3)–(1.4) and work with a block-matrix decomposition for matrix functions
$B(\cdot)$ and $Q(\cdot)$. Namely, let

$\displaystyle B(x)$
$\displaystyle=\operatorname{diag}(B_{1}(x),\ldots,B_{r}(x))=B(x)^{*},\qquad
x\in[0,\ell],$ (3.3) $\displaystyle B_{k}(x)$
$\displaystyle=\beta_{k}(x)I_{n_{k}},\quad x\in[0,\ell],\qquad\beta_{k}\in
L^{1}([0,\ell];\mathbb{R}),\qquad k\in\\{1,\ldots,{r}\\},$ (3.4)

be a self-adjoint invertible diagonal summable matrix function, where
$n_{1}+\ldots+n_{r}=n$, and

$Q=:(Q_{jk})_{j,k=1}^{r},\qquad Q_{jk}\in
L^{1}([0,\ell];\mathbb{C}^{n_{j}\times n_{k}}),\qquad Q_{jj}\equiv 0,\qquad
j,k\in\\{1,\ldots,{r}\\},$ (3.5)

be a summable (generally non-self-adjoint) potential matrix with zero “block
diagonal” with respect to the decomposition
$\mathbb{C}^{n}=\mathbb{C}^{n_{1}}\oplus\ldots\oplus\mathbb{C}^{n_{r}}$. It
will be shown in Lemma 6.11 that the case of arbitrary $Q$ can be reduced to
it. We deliberately reused existing notation for $\beta_{k}$ and $Q_{jk}$ to
avoid introducing new notation. But notation (3.3)–(3.5) will be used solely
in this section, which should avoid any confusion.

Let us rewrite conditions (1.6)–(1.8) on matrix function $B(\cdot)$ with a new
notation (3.3)–(3.4) in mind. Namely, in this section we assume that for some
$\theta>0$ and $r_{-}\in\\{0,1,\ldots,r\\}$ the following relations hold:

$\displaystyle\beta_{k},\ 1/\beta_{k}\in
L^{\infty}[0,\ell],\qquad\operatorname{sign}(\beta_{k}(\cdot))\equiv\operatorname{const}\neq
0,\qquad k\in\\{1,\ldots,{r}\\},$ (3.6)
$\displaystyle\beta_{1}(x)<\ldots<\beta_{r_{-}}(x)<-\theta<0<\theta<\beta_{r_{-}+1}(x)<\ldots<\beta_{r}(x),\qquad
x\in[0,\ell],$ (3.7) $\displaystyle\beta_{k}(x)+\theta<\beta_{k+1}(x),\quad
x\in[0,\ell],\qquad k\in\\{1,\ldots,{r-1}\\}.$ (3.8)

Let us also set

$b_{j}:=\rho_{j}(\ell),\qquad\rho_{j}(x):=\int_{0}^{x}\beta_{j}(t)dt,\qquad
j\in\\{1,\ldots,{r}\\}.$ (3.9)

Next we denote by $\mathcal{L}_{0}(Q)$ the restriction of the maximal operator
$\mathcal{L}_{\max}(Q)$ in $\mathcal{H}:=L^{2}([0,\ell];\mathbb{C}^{n})$
generated by the expression $\mathcal{L}(Q)$ on the domain

$\operatorname{dom}\mathcal{L}_{0}(Q)=\\{f\in\operatorname{dom}\mathcal{L}_{\max}(Q):\
f(0)=0\\}\subset\widetilde{W}^{1,1}_{0}([0,\ell];\mathbb{C}^{n}).$ (3.10)

Here

$\widetilde{W}^{1,p}_{0}([0,\ell];\mathbb{C}^{n}):=\\{f\in
W^{1,p}([0,\ell];\mathbb{C}^{n}):f(0)=0\\}.$

Moreover, if $Q\in L^{2}\bigl{(}[0,\ell];\mathbb{C}^{n\times n}\bigr{)}$, then

$\operatorname{dom}\mathcal{L}_{\max}(Q)=W^{1,2}([0,\ell];\mathbb{C}^{n})\qquad\text{and}\qquad\operatorname{dom}\mathcal{L}_{0}(Q)=\widetilde{W}^{1,2}_{0}([0,\ell];\mathbb{C}^{n}).$
(3.11)

In particular, one has

$\mathcal{L}_{0}(0)=B(x)^{-1}\otimes
D_{0},\quad\operatorname{dom}\mathcal{L}_{0}(0)=\widetilde{W}^{1,2}_{0}([0,\ell];\mathbb{C}^{n}),\quad\text{where}\quad
D_{0}:=-i\frac{d}{dx}\upharpoonright\widetilde{W}^{1,2}_{0}[0,\ell].$

Note also that the operator $\mathcal{L}_{0}(0)$ is invertible and
$\mathcal{L}_{0}(0)^{-1}=B(x)\otimes(iJ)$ where $J$ is the Volterra
integration operator, $J:\ f\to\int^{x}_{0}f(t)\,dt$.

To state the main result of this subsection let us recall the following
definition.

######  Definition 3.1.

Let $L_{1}$ and $L_{2}$ be closed densely defined operators in a Banach space
$X$ with domains $\operatorname{dom}L_{1}$ and $\operatorname{dom}L_{2}$,
respectively. It is said that a bounded operator $T$ intertwines the operators
$L_{1}$ and $L_{2}$ if:

a) $T$ maps $\operatorname{dom}L_{1}$ onto $\operatorname{dom}L_{2}$;

b) $L_{2}Tf=TL_{1}f,\quad f\in\operatorname{dom}L_{1}$.

If in addition, $0\in\rho(T)$, i.e. $T$ has a bounded inverse, then the
operators $L_{1}$ and $L_{2}$ are called similar.

To establish similarity of the operators $\mathcal{L}_{0}(Q)$ and
$\mathcal{L}_{0}(0)$ for $Q\in L^{1}([0,\ell];\mathbb{C}^{n\times n})$ we
first establish this under additional smoothness assumptions of the potential
$Q$ related to Lipshitz properties (see Subsection 2.2).

######  Proposition 3.2.

Let matrix functions $B(\cdot)$ and $Q(\cdot)$ satisfy conditions (3.3)–(3.8).
In particular, we assume that $Q_{jj}=0$ for $j\in\\{1,\ldots,{r}\\}$ and
$|\beta_{j}(x)-\beta_{k}(x)|>\theta$ for a.e. $x\in[0,\ell]$ and $j\neq k$.
Let also

$\widetilde{Q}_{jk}:=\frac{Q_{jk}}{\beta_{j}-\beta_{k}}\in\operatorname{Lip}_{1}[0,\ell],\quad
j\neq k,\qquad Q\in L^{\infty}([0,\ell];\mathbb{C}^{n\times n}).$ (3.12)

Then the operators $\mathcal{L}_{0}(Q)$ and
$\mathcal{L}_{0}(0)=B(x)^{-1}\otimes D_{0}$ are similar in
$L^{p}([0,\ell];\mathbb{C}^{n\times n})$, $p\in[1,\infty]$. Moreover, there
exists an $n\times n$ matrix kernel

$R\in\operatorname{Lip}_{1}(\Omega)\otimes\mathbb{C}^{n\times n},$ (3.13)

where domain $\Omega$ is given by (2.9), such that triangular Volterra type
operator $I+\mathcal{R}$,

$(I+\mathcal{R})f:=f(x)+\int^{x}_{0}R(x,t)B(t)f(t)\,dt,\qquad f\in
L^{p}([0,\ell];\mathbb{C}^{n}),$ (3.14)

is bounded on $L^{p}([0,\ell];\mathbb{C}^{n})$, has a bounded inverse, and
intertwines the operators $\mathcal{L}_{0}(Q)$ and $\mathcal{L}_{0}(0)$, i.e.

$\mathcal{L}_{0}(Q)(I+\mathcal{R})f=(I+\mathcal{R})\mathcal{L}_{0}(0)f,\qquad
f\in\operatorname{dom}\mathcal{L}_{0}(0)=\widetilde{W}^{1,2}_{0}([0,\ell];\mathbb{C}^{n}).$
(3.15)

###### Proof.

The proof will be divided into multiple steps.

(i) At this step we show that the intertwining property (3.15) is equivalent
to a certain boundary value problem for the kernel $R(\cdot,\cdot)$ in the
triangle $\Omega=\\{0\leqslant t\leqslant x\leqslant\ell\\}$. It is easily
seen that

$\bigl{(}-iB\mathcal{L}_{0}(Q)(I+\mathcal{R})f\bigr{)}(x)=\left[\frac{d}{dx}+Q\right]\bigl{(}f(x)+\int^{x}_{0}R(x,t)B(t)f(t)\,dt\bigr{)}\\\
=f^{\prime}(x)+Q(x)f(x)+\frac{d}{dx}\int^{x}_{0}R(x,t)B(t)f(t)\,dt+Q(x)\int^{x}_{0}R(x,t)B(t)f(t)\,dt\\\
=f^{\prime}(x)+Q(x)f(x)+R(x,x)B(x)f(x)+\int^{x}_{0}\frac{\partial}{\partial{x}}R(x,t)B(t)f(t)\,dt+Q(x)\int^{x}_{0}R(x,t)B(t)f(t)\,dt.$
(3.16)

On the other hand, integrating by parts one derives

$\bigl{(}-iB(I+\mathcal{R})\mathcal{L}_{0}(0)f\bigr{)}(x)=\bigl{(}B(I+\mathcal{R})(B^{-1}f^{\prime})\bigr{)}(x)\\\
=f^{\prime}(x)+B(x)\int^{x}_{0}R(x,t)B(t)\cdot
B^{-1}(t)f^{\prime}(t)\,dt=f^{\prime}(x)+B(x)\int^{x}_{0}R(x,t)f^{\prime}(t)\,dt\\\
=f^{\prime}(x)+B(x)R(x,x)f(x)-B(x)R(x,0)f(0)-B(x)\int^{x}_{0}\left(\frac{\partial}{\partial{t}}R(x,t)\right)f(t)\,dt.$
(3.17)

Equating right hand sides of both equations and noting that
$f\in\widetilde{W}^{1,2}_{0}([0,\ell];\mathbb{C}^{n})$ is arbitrary satisfying
$f(0)=0$, leads to the following boundary value problem for the matrix kernel
$R(x,t):$

$\displaystyle\frac{\partial}{\partial{x}}R(x,t)B(t)+B(x)\frac{\partial}{\partial{t}}R(x,t)+Q(x)R(x,t)B(t)$
$\displaystyle=0,$ (3.18) $\displaystyle R(x,x)B(x)-B(x)R(x,x)+Q(x)$
$\displaystyle=0,$ (3.19)

for a.e. $(x,t)\in\Omega$. Let us write the matrix kernel $R(\cdot,\cdot)$ in
the block-matrix form $R(x,t)=\left(R_{jk}(x,t)\right)_{j,k=1}^{r}$ with
respect to the decomposition
$\mathbb{C}^{n}=\mathbb{C}^{n_{1}}\oplus\ldots\oplus\mathbb{C}^{n_{r}}$. Since
$B(\cdot)$ is block-diagonal it follows that the problem (3.18)–(3.19) splits
into $r$ independent problems on columns $R_{k}(x,t):=(R_{jk}(x,t))_{j=1}^{r}$
of the matrix kernel $R(\cdot,\cdot)$. Fixing $k\in\\{1,\ldots,{r}\\}$, using
the block-matrix representation of matrix functions $B(\cdot)$ and $Q(\cdot)$
and taking into account condition $Q_{jj}\equiv 0$, $j\in\\{1,\ldots,{r}\\}$,
corresponding problem for the $k$-th column of the matrix kernel
$R(\cdot,\cdot)$ takes the following form for a.e. $(x,t)\in\Omega$,

$\displaystyle\frac{\partial}{\partial{x}}R_{jk}(x,t)+\frac{\beta_{j}(x)}{\beta_{k}(t)}\frac{\partial}{\partial{t}}R_{jk}(x,t)$
$\displaystyle=-\sum^{r}_{p=1}Q_{jp}(x)R_{pk}(x,t),\quad
j\in\\{1,\ldots,{r}\\},$ (3.20) $\displaystyle
R_{jk}(x,x)=\frac{Q_{jk}(x)}{\beta_{j}(x)-\beta_{k}(x)}$
$\displaystyle=\widetilde{Q}_{jk}(x),\qquad j\neq k,\quad
j\in\\{1,\ldots,{r}\\}.$ (3.21)

Emphasize that $\widetilde{Q}_{jk}$, $j\neq k$, is well-defined and summable
on $[0,\ell]$ due to conditions (3.5)–(3.8). It is clear now that to finish
the proof it is sufficient to show that for each $k\in\\{1,\ldots,{r}\\}$
there exists a vector kernel $R_{k}(\cdot,\cdot)$ that satisfies the
incomplete Cauchy problem (3.20)–(3.21).

(ii) To prepare for the next step, we need to extend functions
$\beta_{j}(\cdot)$ and $\rho_{j}(\cdot)$ to be defined on $\mathbb{R}$ and
satisfy conditions (3.6)–(3.8) (for each $x\in\mathbb{R}$). By definition,
$\beta_{j}\in L^{\infty}$ is a class of functions equivalent to a certain base
function and conditions (3.6)–(3.8) are valid only for a.e. $x\in[0,\ell]$. It
is clear that for each $j\in\\{1,\ldots,{r}\\}$ we can select an appropriate
representative from this class of equivalence (and call it $\beta_{j}$ for
simplicity) such that the selected functions $\beta_{j}(\cdot)$ are defined
for each $x\in[0,\ell]$ and satisfy conditions (3.6)–(3.8) for each
$x\in[0,\ell]$ (and not just for a.e. $x\in[0,\ell]$). With this remark in
mind, it is clear that the following extension satisfies conditions
(3.6)–(3.8) for each $x\in\mathbb{R}$,

$\beta_{j}(x):=\begin{cases}\beta_{j}(0),&x<0,\\\
\beta_{j}(x),&x\in[0,\ell],\\\
\beta_{j}(\ell),&x>\ell,\end{cases},\qquad\rho_{j}(x):=\int_{0}^{x}\beta_{j}(t)dt,\quad
x\in\mathbb{R},\qquad j\in\\{1,\ldots,{r}\\}.$ (3.22)

It is clear from (3.6) that $\rho_{j}(\mathbb{R})=\mathbb{R}$ and
$\rho_{j}(\cdot)$ is a strictly monotonous function, $j\in\\{1,\ldots,{r}\\}$.
Hence there exists strictly monotonous inverse $\rho_{j}^{-1}$ that also maps
$\mathbb{R}$ onto $\mathbb{R}$, $j\in\\{1,\ldots,{r}\\}$. Summarizing this we
have,

$\rho_{j}(\mathbb{R})=\rho_{j}^{-1}(\mathbb{R})=\mathbb{R},\qquad\text{and}\qquad\rho_{j}\
\ \text{and}\ \ \rho_{j}^{-1}\quad\text{are strictly monotonous},\qquad
j\in\\{1,\ldots,{r}\\}.$ (3.23)

More importantly, condition (3.8) implies the same property for differences
$\rho_{j}-\rho_{k}$,

$(\rho_{j}-\rho_{k})(\mathbb{R})=(\rho_{j}-\rho_{k})^{-1}(\mathbb{R})=\mathbb{R},\quad\text{and}\quad\rho_{j}-\rho_{k},\
(\rho_{j}-\rho_{k})^{-1}\ \ \text{are strictly monotonous},\quad j\neq k.$
(3.24)

Note also that conditions (3.6)–(3.8) implie the following important Lipshitz
property,

$\rho_{j},\ \rho_{j}^{-1},\ \ \rho_{j}-\rho_{k},\
(\rho_{j}-\rho_{k})^{-1}\in\operatorname{Lip}_{1}(\mathbb{R}),\qquad j\neq
k,\quad j,k\in\\{1,\ldots,{r}\\}.$ (3.25)

Local Lipshitz property is implied directly by conditions (3.6)–(3.8). Global
Lipshitz property holds because functions $\rho_{j}$ are linear outside of
$[0,\ell]$.

We also need to extend matrix function $Q(\cdot)$ to be defined on
$\mathbb{R}$. For simplicity we will use the same notation for it and the same
notation for $\widetilde{Q}_{jk}$, $j\neq k$, given by (3.21). In the future
we will need to work with a certain system of integral equations without the
assumption $\widetilde{Q}_{jk}\in\operatorname{Lip}_{1}[0,\ell]$. Hence, we
start by extending $\widetilde{Q}_{jk}$ (that outside of this Proposition
might be only summable) in any way such that the following properties hold,

$\displaystyle\widetilde{Q}_{jk}(x)=0,\qquad
x\notin(-\delta,\ell+\delta),\qquad j\neq k,$ (3.26)
$\displaystyle\widetilde{Q}_{jk}\in
L^{p}(\mathbb{R})\quad\text{whenever}\quad\widetilde{Q}_{jk}\in
L^{p}[0,\ell],\quad p\in[1,\infty],\qquad j\neq k,$ (3.27)
$\displaystyle\widetilde{Q}_{jk}\in\operatorname{Lip}_{1}(\mathbb{R})\quad\text{whenever}\quad\widetilde{Q}_{jk}\in\operatorname{Lip}_{1}[0,\ell],\qquad
j\neq k,$ (3.28)

with some $\delta>0$. Then we naturally define $Q_{jk}(x)$ for
$x\in\mathbb{R}$ by formula (3.21) and also extend block-diagonal entries of
the matrix function $Q(\cdot)$ to be zero,

$\displaystyle Q_{jk}(x)$
$\displaystyle:=(\beta_{j}(x)-\beta_{k}(x))\cdot\widetilde{Q}_{jk}(x),\qquad
x\in\mathbb{R},\qquad j\neq k,$ (3.29) $\displaystyle Q_{jj}(x)$
$\displaystyle:=\widetilde{Q}_{jj}(x):=0,\qquad x\in\mathbb{R},\qquad
j\in\\{1,\ldots,{r}\\}.$ (3.30)

It is clear from the construction and conditions (3.5)–(3.8) that

$Q(x)=0,\quad x\notin(-\delta,\ell+\delta),\qquad\text{and}\qquad Q\in
L^{p}(\mathbb{R})\quad\text{whenever}\quad Q\in L^{p}[0,\ell],\quad
p\in[1,\infty].$ (3.31)

(iii) At this step assuming $k\in\\{1,\ldots,{r}\\}$ to be fixed, we consider
the incomplete Cauchy problem (3.20)–(3.21) and complete it up to a special
Goursat problem in a certain extended domain $\Omega_{k}\supset\Omega$
assuming functions $\beta_{j}$, $\rho_{j}$ and $Q_{jp}$ to be extended on
$\mathbb{R}$ as constructed in step (ii). Along the way, we establish
equivalent system of integral equations.

The corresponding characteristic system is
${\beta_{j}(x)}\,dx={\beta_{k}(t)}\,dt$, $j\in\\{1,\ldots,{r}\\}$. Since
$\rho_{j}^{\prime}(x)=\beta_{j}(x)$ for a.e. $x\in[0,\ell]$, this
characteristic system defines the system of characteristic curves of the
equation (3.20):

$\rho_{j}(x)-\rho_{k}(t)=c=\operatorname{const},\qquad
j\in\\{1,\ldots,{r}\\}.$ (3.32)

Condition (3.23) implies that we can present the characteristic at “level”
$c\in\mathbb{R}$ as

$\Gamma_{jk}^{c}:=\\{(x,\gamma_{jk}^{c}(x)):x\in\mathbb{R}\\},\qquad\gamma_{jk}^{c}(x):=\rho_{k}^{-1}(\rho_{j}(x)-c),\quad
x\in\mathbb{R},\qquad j\in\\{1,\ldots,{r}\\}.$ (3.33)

It is clear that $\gamma_{jk}^{c}(\mathbb{R})=\mathbb{R}$, and
$\gamma_{jk}^{c}$ is strictly monotonous, linear outside of a certain interval
(that depends on $j$, $k$ and $c$) and globally Lipshitz just like functions
$\rho_{k}^{-1}$ and $\rho_{j}$.

It can be easily seen that the incomplete Cauchy problem (3.20)–(3.21) is not
characteristic. To integrate it we need to look for solution of this problem
in the extended domain. To this end, let the column
$R_{k}\in\operatorname{Lip}_{1,\operatorname{loc}}(\mathbb{R}^{2};\mathbb{C}^{n})$
satisfy (3.20)–(3.21) for all $x,t\in\mathbb{R}$. Let us obtain some important
properties of $R_{k}$. It follows from the formula for derivative of the
inverse function of the absolutely continuous function that

$\rho_{j}^{\prime}(x)=\beta_{j}(x),\qquad(\rho_{j}^{-1})^{\prime}(x)=\frac{1}{\beta_{j}(\rho_{j}^{-1}(x))},\qquad\text{for
a.e.}\ \ x\in\mathbb{R},\qquad j\in\\{1,\ldots,{r}\\}.$ (3.34)

Combining (3.33) and (3.34) we arrive at

$(\gamma_{jk}^{c})^{\prime}(u)=\frac{\beta_{j}(u)}{\beta_{k}(\gamma_{jk}^{c}(u))},\qquad\text{for
a.e.}\ \ u\in\mathbb{R},\qquad j\in\\{1,\ldots,{r}\\}.$ (3.35)

Recall that as per (2.4),

$D_{1}R_{jk}(x,t)=\frac{\partial}{\partial{x}}R_{jk}(x,t),\qquad
D_{2}R_{jk}(x,t)=\frac{\partial}{\partial{t}}R_{jk}(x,t),\qquad
x,t\in\mathbb{R},\quad j\in\\{1,\ldots,{r}\\}.$ (3.36)

With this notation in mind we can rewrite (3.20) as follows

$D_{1}R_{jk}(u,v)+\frac{\beta_{j}(u)}{\beta_{k}(v)}D_{2}R_{jk}(u,v)=-\sum^{r}_{p=1}Q_{jp}(u)R_{pk}(u,v),\qquad\text{for
a.e.}\ \ u,v\in\mathbb{R},\quad j\in\\{1,\ldots,{r}\\}.$ (3.37)

Using standard rules for differentiating functions of two variables, and
combining (3.35) and (3.37), we obtain for $j\in\\{1,\ldots,{r}\\}$,

$\displaystyle\frac{d}{du}\bigl{[}R_{jk}(u,\gamma_{jk}^{c}(u))\bigr{]}$
$\displaystyle=D_{1}R_{jk}(u,\gamma_{jk}^{c}(u))+(\gamma_{jk}^{c})^{\prime}(u)D_{2}R_{jk}(u,\gamma_{jk}^{c}(u))$
$\displaystyle=D_{1}R_{jk}(u,\gamma_{jk}^{c}(u))+\frac{\beta_{j}(u)}{\beta_{k}(\gamma_{jk}^{c}(u))}D_{2}R_{jk}(u,\gamma_{jk}^{c}(u))$
$\displaystyle=-\sum^{r}_{p=1}Q_{jp}(u)R_{pk}(u,\gamma_{jk}^{c}(u)),\qquad\text{for}\
\ c\in\mathbb{R}\quad\text{and for a.e.}\ \ u\in\mathbb{R}.$ (3.38)

It is clear that the function $R_{jk}(\cdot,\gamma_{jk}^{c}(\cdot))$ is
locally Lipshitz. Hence, integrating formula (3.38) from $a$ to $x$, we arrive
at

$R_{jk}(x,\gamma_{jk}^{c}(x))-R_{jk}(a,\gamma_{jk}^{c}(a))=-\int_{a}^{x}\sum^{r}_{p=1}Q_{jp}(u)R_{pk}(u,\gamma_{jk}^{c}(u))du,\quad
x,a,c\in\mathbb{R},\quad j\in\\{1,\ldots,{r}\\}.$ (3.39)

Let us fix $j\in\\{1,\ldots,{r}\\}$ and $-\infty<t\leqslant x<\infty$.
Consider the characteristic curve $\Gamma_{jk}^{c}$ passing through the point
$(x,t)$. It is clear from definition (3.33) of $\gamma_{jk}^{c}(\cdot)$ that

$\gamma_{jk}^{c}(x)=t\qquad\text{and}\qquad c=\rho_{j}(x)-\rho_{k}(t).$ (3.40)

Let us also set for brevity for $j\in\\{1,\ldots,{r}\\}$ and
$x,t,u\in\mathbb{R}$

$\gamma_{jk}^{x,t}(u):=\gamma_{jk}^{c}(u)=\rho_{k}^{-1}(\rho_{j}(u)-\rho_{j}(x)+\rho_{k}(t)),\qquad
x,t,u\in\mathbb{R},\qquad\gamma_{jk}^{\cdot,\cdot}(\cdot)\in\operatorname{Lip}_{1}(\mathbb{R}^{3}),$
(3.41)

where the last inclusion follows from the property (3.25).

First let $j\neq k$. Then there exists $a=a_{jk}(x,t)\in\mathbb{R}$ such that
$\gamma_{jk}^{x,t}(a)=a$. Indeed, this equation is equivalent to
$\rho_{j}(a)-\rho_{j}(x)+\rho_{k}(t)=\rho_{k}(a)$ and hence condition (3.24)
implies that

$a_{jk}(x,t):=(\rho_{j}-\rho_{k})^{-1}(\rho_{j}(x)-\rho_{k}(t))=\Bigl{.}(\rho_{j}-\rho_{k})^{-1}(u)\Bigr{|}_{u=\rho_{j}(x)-\rho_{k}(t)},\qquad
a_{jk}\in\operatorname{Lip}_{1}(\mathbb{R}^{2}).$ (3.42)

Setting $a=a_{jk}(x,t)$ in (3.39) and taking into account (3.40), (3.42) and
(3.21), we arrive at

$-\int_{a}^{x}\sum^{r}_{p=1}Q_{jp}(u)R_{pk}(u,\gamma_{jk}^{x,t}(u))du=R_{jk}(x,\gamma_{jk}^{x,t}(x))-R_{jk}(a,\gamma_{jk}^{x,t}(a))\\\
=R_{jk}(x,t)-R_{jk}(a,a)=R_{jk}(x,t)-\widetilde{Q}_{jk}(a),\quad
a=a_{jk}(x,t).$ (3.43)

or

$R_{jk}(x,t)=\widetilde{Q}_{jk}(a_{jk}(x,t))-\int_{a_{jk}(x,t)}^{x}\sum^{r}_{p=1}Q_{jp}(u)R_{pk}(u,\gamma_{jk}^{x,t}(u))du,\qquad
t\leqslant x,\quad j\neq k.$ (3.44)

Now we are ready to define the domain $\Omega_{k}$ on which we will set the
Goursat problem. For convenience, $[u,v]$ and $[v,u]$ will denote the same
segment of the real line,

$[u,v]:=[v,u]=\\{t\in\mathbb{R}:v\leqslant t\leqslant u\\},\qquad v\leqslant
u.$ (3.45)

Let us also define the part of the characteristic curve $\Gamma_{jk}^{c}$ that
starts at the point $(x,t)$ of the triangle $\Omega$ and ends on the diagonal
$\mathbf{D}:=\\{(u,v)\in\mathbb{R}^{2}:u=v\\}$ of $\mathbb{R}^{2}$,

$\Gamma_{jk}^{x,t}:=\left\\{\left(u,\gamma_{jk}^{x,t}(u)\right):\
u\in[x,a_{jk}(x,t)]\right\\},\qquad(x,t)\in\Omega,\qquad j\neq k.$ (3.46)

With this notation in mind, we formally define

$\Omega_{k}:=\bigcup_{j\neq
k}\Omega_{jk},\qquad\Omega_{jk}:=\bigcup_{(x,t)\in\Omega}\Gamma_{jk}^{x,t},\qquad
j\neq k.$ (3.47)

Let us fix $j\neq k$. It is clear from the continuity and monotonicity of
$\gamma_{jk}^{x,t}(\cdot)$ and $(\rho_{j}-\rho_{k})^{-1}$, and general
geometrical reasoning that we can only have three cases:

  * 1.

$\Omega_{jk}=\Omega$ (when the characteristic curve segments
$\Gamma_{jk}^{x,t}$ do not “go outside” of the triangle $\Omega$);

  * 2.

$\Omega_{jk}=\\{(u,v):\ u\in[a_{jk}(\ell,0),\ell],\
v\in[\gamma_{jk}^{\ell,0}(u),u]\\}$ (when $\Gamma_{jk}^{x,t}$ that “go
outside” of the triangle $\Omega$ intersect the diagonal $\mathbf{D}$ at
points with negative coordinate $u$);

  * 3.

$\Omega_{jk}=\Omega\cup\\{(u,v):\ u\in(\ell,a_{jk}(\ell,0)],\
v\in[\gamma_{jk}^{\ell,0}(u),u]\\}$ (when $\Gamma_{jk}^{x,t}$ that “go
outside” of $\Omega$ intersect the diagonal $\mathbf{D}$ at points with
positive coordinate $u$).

It is clear that we can unify all three cases with a single representation of
the following form,

$\Omega_{jk}=\\{(u,v):\ u\in[a_{jk}^{-},a_{jk}^{+}],\ \
v\in[\gamma_{jk}^{-}(u),u]\\},$ (3.48)

where numbers $a_{jk}^{-}\leqslant 0<\ell\leqslant a_{jk}^{+}$ and function
$\gamma_{jk}^{-}(\cdot)\leqslant 0$ are defined as follows in each of the
three cases:

$\displaystyle\quad\begin{array}[]{llll}\bullet&a_{jk}^{-}=0,&a_{jk}^{+}=\ell,&\qquad\gamma_{jk}^{-}(u)=0,\quad
u\in[0,\ell],\\\
\bullet&a_{jk}^{-}=a_{jk}(\ell,0),&a_{jk}^{+}=\ell,&\qquad\gamma_{jk}^{-}(u)=\gamma_{jk}^{\ell,0}(u),\quad
u\in[0,\ell],\\\
\bullet&a_{jk}^{-}=0,&a_{jk}^{+}=a_{jk}(\ell,0),&\qquad\gamma_{jk}^{-}(u)=\begin{cases}0,\quad&u\in[0,\ell],\\\
\gamma_{jk}^{\ell,0}(u),\quad&u\in(\ell,a_{jk}(\ell,0)],\\\ \end{cases}\\\
\end{array}$ (3.52)

(see Remark 3.3 for more details). It now follows from (3.48)–(3.52) that

$\displaystyle\Omega_{k}=\\{(u,v):\ u\in[a_{k}^{-},a_{k}^{+}],\ \
v\in[\gamma_{k}^{-}(u),u]\\},\quad\text{where}$ (3.53) $\displaystyle
a_{k}^{-}:=\min\\{a_{jk}^{-}:j\neq k\\}\leqslant 0,\qquad
a_{k}^{+}:=\max\\{a_{jk}^{+}:j\neq k\\}\geqslant\ell,$ (3.54)
$\displaystyle\gamma_{k}^{-}(u):=\min\\{\gamma_{jk}^{-}(u):\ j\neq k\ \
\text{such that}\ \ u\in[a_{jk}^{-},a_{jk}^{+}]\\},\qquad
u\in[a_{k}^{-},a_{k}^{+}],$ (3.55)
$\displaystyle\Gamma_{k}^{-}:=\\{(u,\gamma_{k}^{-}(u)):\
u\in[a_{k}^{-},a_{k}^{+}]\\}\quad\text{is the ``lower boundary'' of the
domain}\ \ \Omega_{k}.$ (3.56)

Properties of $\gamma_{jk}^{c}(\cdot)$ outlined after formula (3.33) and the
fact, that the minimum of finite number of Lipshitz non-decreasing functions
preserves these properties, imply that

$\gamma_{k}^{-}\in\operatorname{Lip}_{1}[a_{k}^{-},a_{k}^{+}],\qquad\gamma_{jk}^{-}\in\operatorname{Lip}_{1}[a_{jk}^{-},a_{jk}^{+}]\qquad\text{and}\qquad\gamma_{k}^{-},\gamma_{jk}^{-}\quad\text{are
non-decreasing},\quad j\neq k.$ (3.57)

This in turn imply that $\Omega_{k}$ has a Lipshitz boundary. It is also
simply connected, closed and bounded.

In order to formulate Goursat problem, observe that the definition of
$\Omega_{jk}$ implies that the characteristic curve segment
$\Gamma_{jk}^{x,t}$, defined in (3.46) and considered for points
$(x,t)\in\Omega_{jk}$, does not go outside of $\Omega_{jk}$. The same is valid
for the union $\Omega_{k}$ (though this needs a bit more considerations
involving “ordering” of characteristic curves implied by (3.7)). Hence we can
restrict kernels $R_{jk}(x,t)$ to be only defined on $\Omega_{k}$ and the
system of equations (3.44) will still be valid,

$R_{jk}(x,t)=\widetilde{Q}_{jk}(a_{jk}(x,t))-\int_{a_{jk}(x,t)}^{x}\sum^{r}_{p=1}Q_{jp}(u)R_{pk}(u,\gamma_{jk}^{x,t}(u))du,\qquad(x,t)\in\Omega_{k},\quad
j\neq k.$ (3.58)

Let us now go back to our raw equation (3.39) and handle the diagonal entry
$R_{kk}(\cdot,\cdot)$ (recall that $k\in\\{1,\ldots,{r}\\}$ is fixed). We
don’t have any initial data for it yet and are free to impose any appropriate
initial condition. To this end note that the “characteristic function”
$\gamma_{kk}^{x,t}$ is strictly increasing for any given point $(x,t)$,

$\gamma_{kk}^{x,t}(u)=\rho_{k}^{-1}(\rho_{k}(u)-\rho_{k}(x)+\rho_{k}(t)),\quad
u\in\mathbb{R},\qquad\gamma_{kk}^{x,t}\quad\text{is increasing},\quad
x,t\in\mathbb{R}.$ (3.59)

Note also, that the characteristic curve $\Gamma_{kk}^{c}$ never intersects
the diagonal $\mathbf{D}$ of $\mathbb{R}^{2}$ if $c\neq 0$, while
$\Gamma_{kk}^{0}=\mathbf{D}$. With account of these observations and noting
that $\Gamma_{k}^{-}$ (the “lower boundary” of $\Omega_{k}$) intersects the
diagonal $\mathbf{D}$ of $\mathbb{R}^{2}$ at the point $a_{k}^{-}$, we see
that the characteristic curve $\Gamma_{kk}^{x,t}$, that passes through any
point $(x,t)$ of the domain $\Omega_{k}$, intersects with $\Gamma_{k}^{-}$.
Namely, there exists $a_{kk}=a_{kk}(x,t)\in[a_{k}^{-},a_{k}^{+}]$ such that

$\gamma_{kk}^{x,t}(a_{kk}(x,t))=\gamma_{k}^{-}(a_{kk}(x,t)),\qquad(x,t)\in\Omega_{k}.$
(3.60)

Let us show that $a_{kk}\in\operatorname{Lip}(\Omega_{k})$. Let
$(x,t)\in\Omega_{k}$ and $j\neq k$ be fixed and let $a_{kk,j}(x,t)$ be the
intersection of $\Gamma_{kk}^{x,t}$ with the “lower boundary” of
$\Omega_{jk}$, i.e.

$\gamma_{kk}^{x,t}(a_{kk,j}(x,t))=\gamma_{jk}^{-}(a_{kk,j}(x,t)),\qquad(x,t)\in\Omega_{k},\quad
j\neq k.$ (3.61)

In each of the three cases in (3.52) we can find explicit form of
$a_{kk,j}(x,t)$. For the first two cases we have,

$\displaystyle\quad\begin{array}[]{llll}\bullet&\gamma_{jk}^{-}(u)=0,\quad
u\in[0,\ell],&a_{kk,j}(x,t)=\rho_{k}^{-1}(\rho_{k}(x)-\rho_{k}(t)),\\\
\bullet&\gamma_{jk}^{-}(u)=\gamma_{jk}^{\ell,0}(u),\quad
u\in[0,\ell],&a_{kk,j}(x,t)=(\rho_{j}-\rho_{k})^{-1}(\rho_{j}(\ell)-\rho_{k}(x)+\rho_{k}(t)),\\\
\end{array}$ (3.64)

while in the third $a_{kk,j}(x,t)$ is in a way “a union” of two cases. Here is
the detailed proof for the second case for posterity,

$\displaystyle\gamma_{kk}^{x,t}(a)$
$\displaystyle=\gamma_{jk}^{\ell,0}(a)\quad\Leftrightarrow$
$\displaystyle\rho_{k}^{-1}(\rho_{k}(a)-\rho_{k}(x)+\rho_{k}(t))$
$\displaystyle=\rho_{k}^{-1}(\rho_{j}(a)-\rho_{j}(\ell))\quad\Leftrightarrow$
$\displaystyle\rho_{j}(\ell)-\rho_{k}(x)+\rho_{k}(t)$
$\displaystyle=\rho_{j}(a)-\rho_{k}(a)\quad\Leftrightarrow$ $\displaystyle
a_{kk,j}(x,t)$
$\displaystyle=(\rho_{j}-\rho_{k})^{-1}(\rho_{j}(\ell)-\rho_{k}(x)+\rho_{k}(t)).$

Note that for some points $(x,t)\in\Omega_{k}$ the found point $a_{kk,j}(x,t)$
intersects the curve $\Gamma_{jk}^{-}:=\\{(u,\gamma_{jk}^{-}(u)):\
u\in\mathbb{R}\\}$ outside of $\Omega_{jk}$. Despite that, it can be shown
that

$a_{kk}(x,t)=\min\\{a_{kk,j}(x,t):j\neq k\\},\qquad(x,t)\in\Omega_{k}.$ (3.65)

Formulas (3.64), (3.65) and property (3.25) now easily imply that

$a_{kk},a_{kk,j}\in\operatorname{Lip}(\Omega_{k}),\qquad j\neq k.$ (3.66)

With this preparation in mind, we can impose the following initial condition
on $R_{kk}(\cdot,\cdot)$,

$R_{kk}(x,\gamma_{k}^{-}(x))=0,\qquad x\in[a_{k}^{-},a_{k}^{+}].$ (3.67)

Setting $a=a_{kk}(x,t)$ in (3.39) and taking into account (3.67) we get
similar to (3.43),

$R_{kk}(x,t)=-\int_{a_{kk}(x,t)}^{x}\sum^{r}_{p=1}Q_{kp}(u)R_{pk}(u,\gamma_{kk}^{x,t}(u))du,\qquad(x,t)\in\Omega_{k}.$
(3.68)

With account of convention $\widetilde{Q}_{kk}=0$ (see (3.30)), we can combine
(3.58) and (3.68) into a single formula for all $r$ equations,

$R_{jk}(x,t)=\widetilde{Q}_{jk}(a_{jk}(x,t))-\int_{a_{jk}(x,t)}^{x}\sum^{r}_{p=1}Q_{jp}(u)R_{pk}(u,\gamma_{jk}^{x,t}(u))du,\qquad(x,t)\in\Omega_{k},\quad
j\in\\{1,\ldots,{r}\\},$ (3.69)

where, as before, $k\in\\{1,\ldots,{r}\\}$ is fixed. It is clear from
considerations in formulas (3.38)–(3.39) that the system of integral equations
(3.69) is equivalent to the following Goursat problem whenever
$R_{k}\in\operatorname{Lip}_{1}(\Omega_{k})$,

$\displaystyle\frac{\partial}{\partial{x}}R_{jk}(x,t)+\frac{\beta_{j}(x)}{\beta_{k}(t)}\frac{\partial}{\partial{t}}R_{jk}(x,t)$
$\displaystyle=-\sum^{r}_{p=1}Q_{jp}(x)R_{pk}(x,t),\quad(x,t)\in\Omega_{k},\quad
j\in\\{1,\ldots,{r}\\}.$ (3.70) $\displaystyle R_{jk}(x,x)$
$\displaystyle=\widetilde{Q}_{jk}(x),\qquad x\in[a_{k}^{-},a_{k}^{+}],\qquad
j\neq k,\quad j\in\\{1,\ldots,{r}\\}.$ (3.71) $\displaystyle
R_{kk}(x,\gamma_{k}^{-}(x))$ $\displaystyle=0,\qquad
x\in[a_{k}^{-},a_{k}^{+}].$ (3.72)

Moreover, it is obvious that any vector kernel solution $R_{k}(\cdot,\cdot)$
of this Goursat problem satisfies the desired incomplete Cauchy problem
(3.20)–(3.21). Therefore, to finish the proof it is sufficient to show
solvability of the system of integral equations (3.69) in
$\operatorname{Lip}_{1}(\Omega_{k})$.

(iv) At this step, again assuming $k\in\\{1,\ldots,{r}\\}$ to be fixed, we
apply the method of successive approximation to prove the existence of
solution to the system (3.69) in $C(\Omega_{k})$. First, note that if the
$k$-th block column $Q_{k}=(Q_{jk})_{j=1}^{r}$ of the matrix $Q$ is zero, then
$R_{k}\equiv 0$ is a valid solution of the system (3.69). Going forward we
assume that $Q_{k}\neq 0$. To this end we set

$R_{jk}^{(0)}(x,t):=0,\qquad(x,t)\in\Omega_{k},\qquad j\in\\{1,\ldots,{r}\\}.$
(3.73)

Assuming functions $R_{jk}^{(m-1)}$ to be defined for a given
$m\in\mathbb{N}$, we set for $(x,t)\in\Omega_{k}$,

$\displaystyle R_{jk}^{(m)}(x,t)$
$\displaystyle:=\widetilde{Q}_{jk}(a_{jk}(x,t))-\int_{a_{jk}(x,t)}^{x}\sum^{r}_{p=1}Q_{jp}(u)R_{pk}^{(m-1)}(u,\gamma_{jk}^{x,t}(u))\,du,\quad
j\neq k,\quad j\in\\{1,\ldots,{r}\\}.$ (3.74) $\displaystyle
R_{kk}^{(m)}(x,t)$
$\displaystyle:=-\int_{a_{kk}(x,t)}^{x}\sum^{r}_{p=1}Q_{kp}(u)R_{pk}^{(m)}(u,\gamma_{kk}^{x,t}(u))\,du,$
(3.75)

To explain the correctness of this definition let us mention that since
$Q_{kk}(\cdot)=0$ by the assumption, the entries $R_{kk}^{(m)}$ of the vector
function $\bigl{(}R_{jk}^{(m)}\bigr{)}_{j=1}^{r}$ are expressed by means of
equalities (3.75) via the entries $R_{jk}^{(m)}$, $j\neq k$, defined on the
previous step by equalities (3.74). Note in particular, that

$R_{jk}^{(1)}(x,t)=\widetilde{Q}_{jk}(a_{jk}(x,t)),\qquad(x,t)\in\Omega_{k},\qquad
j\neq k,\quad j\in\\{1,\ldots,{r}\\},$ (3.76)

Note that since

$\widetilde{Q}_{jk}\in\operatorname{Lip}_{1}(\mathbb{R}),\quad
a_{jk}\in\operatorname{Lip}_{1}(\mathbb{R}^{2}),\quad Q_{jp}\in
L^{\infty}(\mathbb{R}),\quad\gamma_{jk}^{\cdot,\cdot}(\cdot)\in\operatorname{Lip}_{1}(\mathbb{R}^{3}),\qquad
j,p\in\\{1,\ldots,{r}\\},$ (3.77)

then

$R_{jk}^{(m)}\in\operatorname{Lip}_{1}(\Omega_{k}),\qquad
m\in\mathbb{N}_{0},\quad j\in\\{1,\ldots,{r}\\}.$ (3.78)

First we show that for any $j\in\\{1,\ldots,{r}\\}$ the (uniform) limit
$\lim_{m\to\infty}R_{jk}^{(m)}=:R_{jk}$ exists in $C(\Omega_{k})$ and defines
a solution to the system (3.69). Clearly, this convergence is equivalent to
the convergence in $C(\Omega_{k})$ of the series

$R_{jk}^{(0)}+\sum_{m=1}^{\infty}\left[R_{jk}^{(m)}-R_{jk}^{(m-1)}\right]$
(3.79)

Let us prove by induction on $N$ that

$\displaystyle\left|{R^{(N+1)}_{jk}(x,t)-R^{(N)}_{jk}(x,t)}\right|\leqslant
C_{jk}\tau_{k}^{N}\frac{|\rho_{k}(x)-\rho_{k}(t)|^{N}}{N!},\quad(x,t)\in\Omega_{k},\quad
j\in\\{1,\ldots,{r}\\},\quad N\in\mathbb{N}_{0},$ (3.80)

with constants $C_{1k},\ldots,C_{rk}\geqslant 0$ and $\tau_{k}>0$ given by

$\displaystyle C_{jk}$
$\displaystyle:=\widetilde{q}_{k}:=\max\\{\|\widetilde{Q}_{pk}\|_{\infty}:p\neq
k\\}>0,\qquad j\neq k,\quad j\in\\{1,\ldots,{r}\\},$ (3.81) $\displaystyle
C_{kk}$ $\displaystyle:=\widetilde{q}_{k}\cdot
q_{k}\cdot(a_{k}^{+}-a_{k}^{-}),\qquad
q_{k}:=\sum^{r}_{p=1}\|Q_{kp}\|_{\infty}\geqslant 0,$ (3.82)
$\displaystyle\tau_{k}$
$\displaystyle:=\theta^{-1}\max\Bigl{\\{}q_{k}\|Q_{jk}\|_{\infty}+\sum^{r}_{p=1}\|Q_{jp}\|_{\infty}:j\neq
k\Bigr{\\}}>0.$ (3.83)

where $[a_{k}^{-},a_{k}^{+}]$ is the projection of $\Omega_{k}$ on
$\mathbb{R}$ and is given by (3.54) and parameter $\theta$ is from the
condition (3.8). We also set for brevity,
$\|f\|_{\infty}:=\|f\|_{L^{\infty}(\Omega_{k})}$ for any
$f:\Omega_{k}\to\mathbb{C}$. Note that $\widetilde{q}_{k}>0$ and $\tau_{k}>0$
because of the current assumption that the $k$-th block column
$Q_{k}=(Q_{jk})_{j=1}^{r}$ of the matrix $Q$ is non-zero. In addition,
$C_{kk}=0=q_{k}$ whenever the block row $(Q_{kp})_{p=1}^{r}$ of $Q$ is zero.
In this case r.h.s. of equation (3.75) vanishes and $R_{kk}^{(m)}$ is
necessarily zero for all $m\in\mathbb{N}_{0}$.

Let $N=0$ and $j\neq k$. It follows from (3.73), (3.76) and (3.81) that

$\left|{R_{jk}^{(1)}(x,t)-R_{jk}^{(0)}(x,t)}\right|=\left|{\widetilde{Q}_{jk}(a_{jk}(x,t))}\right|\leqslant\|\widetilde{Q}_{jk}\|_{\infty}\leqslant\widetilde{q}_{k}=C_{jk},\qquad(x,t)\in\Omega_{k},\qquad
j\neq k,$ (3.84)

for $j\in\\{1,\ldots,{r}\\}$. Hence (3.80) is valid for $N=0$ and $j\neq k$.

Assume that the estimate (3.80) is valid for $N=m\in\mathbb{N}_{0}$ and $j\neq
k$ and let us prove it for the same $N=m$ and $j=k$. First observe that
relation (3.41) implies that

$\rho_{k}(u)-\rho_{k}(\gamma_{kk}^{x,t}(u))=\rho_{k}(x)-\rho_{k}(t),\qquad
x,t,u\in\mathbb{R}.$ (3.85)

Taking into account relations (3.85), (3.81)–(3.82) and the fact that
$Q_{kk}\equiv 0$, we subtract two equations (3.75) with $m$ and $m+1$,
respectively, and insert the estimate (3.80), valid by induction hypothesis
for $N=m$ and $p\neq k$, into this difference,

$\displaystyle|R^{(m+1)}_{kk}(x,t)-R^{(m)}_{kk}(x,t)|$
$\displaystyle\leqslant\sum^{r}_{p=1}\left|{\int_{a_{kk}(x,t)}^{x}|Q_{kp}(u)|\cdot\left|{R_{pk}^{(m+1)}(u,\gamma_{kk}^{x,t}(u))-R_{pk}^{(m)}(u,\gamma_{kk}^{x,t}(u))}\right|\,du}\right|$
$\displaystyle\leqslant\tau_{k}^{m}\widetilde{q}_{k}\sum^{r}_{p=1}\|Q_{kp}\|_{\infty}\left|\int_{a_{kk}(x,t)}^{x}\frac{|\rho_{k}(u)-\rho_{k}(\gamma_{kk}^{x,t}(u))|^{m}}{m!}\,du\right|$
$\displaystyle\leqslant\widetilde{q}_{k}q_{k}\tau_{k}^{m}\left|\int_{a_{kk}(x,t)}^{x}\frac{|\rho_{k}(x)-\rho_{k}(t)|^{m}}{m!}\,du\right|$
$\displaystyle\leqslant\widetilde{q}_{k}q_{k}\tau_{k}^{m}(a_{k}^{+}-a_{k}^{-})\frac{|\rho_{k}(x)-\rho_{k}(t)|^{m}}{m!}=C_{kk}\tau_{k}^{m}\frac{|\rho_{k}(x)-\rho_{k}(t)|^{m}}{m!},$
(3.86)

which yields the desired relation (3.80) for $N=m$ and $j=k$.

Let $m\in\mathbb{N}_{0}$ and assume that the estimate (3.80) is valid for
$N=m$ and $j\in\\{1,\ldots,{r}\\}$. Let us prove it for $N=m+1$ and $j\neq k$.
Observe that relation (3.41) implies that

$\rho_{k}(\gamma_{jk}^{x,t}(u))=\rho_{j}(u)-\rho_{j}(x)+\rho_{k}(t),\qquad
x,t,u\in\mathbb{R}.$ (3.87)

Let $j\neq k$ and $(x,t)\in\Omega_{k}$ be fixed. With account of identity
(3.87) and taking difference of (3.74) for $m+1$ and $m$, the induction
hypothesis implies,

$\displaystyle\left|{R^{(m+1)}_{jk}(x,t)-R^{(m)}_{jk}(x,t)}\right|$
$\displaystyle\leqslant\sum^{r}_{p=1}\left|{\int_{a_{jk}(x,t)}^{x}|Q_{jp}(u)|\cdot\left|{R_{pk}^{(m)}(u,\gamma_{jk}^{x,t}(u))-R_{pk}^{(m-1)}(u,\gamma_{jk}^{x,t}(u))}\right|\,du}\right|$
$\displaystyle\leqslant\tau_{k}^{m}\sum^{r}_{p=1}C_{pk}\|Q_{jp}\|_{\infty}\left|{\int_{a_{jk}(x,t)}^{x}\frac{\left|{\rho_{k}(u)-\rho_{k}(\gamma_{jk}^{x,t}(u))}\right|^{m}}{m!}\,du}\right|$
$\displaystyle=C_{0jk}\tau_{k}^{m}\left|{\int_{a_{jk}(x,t)}^{x}\frac{\left|{\rho_{k}(u)-\rho_{j}(u)+\rho_{j}(x)-\rho_{k}(t)}\right|^{m}}{m!}\,du}\right|,$
(3.88)

where definitions (3.81)–(3.83) imply

$C_{0jk}:=\sum^{r}_{p=1}C_{pk}\|Q_{jp}\|_{\infty}=\widetilde{q}_{k}\cdot\Bigl{(}q_{k}\|Q_{jk}\|_{\infty}+\sum^{r}_{\genfrac{}{}{0.0pt}{2}{p=1}{p\neq
k}}\|Q_{jp}\|_{\infty}\Bigr{)},\qquad C_{0jk}\theta^{-1}\leqslant
C_{jk}\tau_{k}.$ (3.89)

Since $j\neq k$, then relations (3.24)–(3.25) imply that the function

$f(\cdot):=\rho_{k}(\cdot)-\rho_{j}(\cdot)+\rho_{j}(x)-\rho_{k}(t)$

is strictly monotonous and Lipshitz on $\mathbb{R}$ (recall, that $j,k,x,t$
are fixed, hence we didn’t add them into notation of $f$). Moreover, from
definition (3.42) of $a_{jk}(x,t)$ if follows that $f(a_{jk}(x,t))=0$, while
clearly $f(x)=\rho_{k}(x)-\rho_{k}(t)$. Hence, monotonicity of $f(\cdot)$
implies that

$\text{either}\quad f(u)\geqslant 0,\quad
u\in[a_{jk}(x,t),x]\qquad\text{or}\qquad f(u)\leqslant 0,\quad
u\in[a_{jk}(x,t),x].$ (3.90)

Let $g:=f^{-1}$ be the function inverse to $f$, which exists due to the above
observations. It follows from the standard formula for derivate of the inverse
function that

$g^{\prime}(v)=\frac{1}{\beta_{k}(f(v))-\beta_{j}(f(v))},\qquad|g^{\prime}(v)|\leqslant\theta^{-1},\qquad
v\in\mathbb{R},$ (3.91)

where inequality is implied by conditions (3.7)–(3.8) and construction (3.22).
Therefore, making a change of variable $u=g(v)$ in the integral (3.88) we
obtain

$\left|{\int_{a_{jk}(x,t)}^{x}\frac{\left|{\rho_{k}(u)-\rho_{j}(u)+\rho_{j}(x)-\rho_{k}(t)}\right|^{m}}{m!}\,du}\right|\\\
=\left|{\int_{a_{jk}(x,t)}^{x}\frac{f(u)^{m}}{m!}\,du}\right|=\left|{\int_{0}^{\rho_{k}(x)-\rho_{k}(t)}\frac{v^{m}}{m!}\cdot
g^{\prime}(v)\,dv}\right|\leqslant\theta^{-1}\frac{|\rho_{k}(x)-\rho_{k}(t)|^{m+1}}{(m+1)!}.$
(3.92)

Inserting (3.92) into (3.88) and taking into account estimate (3.89) we arrive
at

$\left|{R^{(m+1)}_{jk}(x,t)-R^{(m)}_{jk}(x,t)}\right|\leqslant
C_{0jk}\tau_{k}^{m}\theta^{-1}\frac{|\rho_{k}(x)-\rho_{k}(t)|^{m+1}}{(m+1)!}\leqslant
C_{jk}\tau_{k}^{m+1}\frac{|\rho_{k}(x)-\rho_{k}(t)|^{m+1}}{(m+1)!},$ (3.93)

which proves (3.80) for $N=m+1$ and $j\neq k$.

It is clear now, that the crucial estimate (3.80) is proved. In turn, this
estimate implies the absolute and uniform convergence of the series (3.79) in
$\Omega_{k}$ which ensures the existence in $\Omega_{k}$ of the continuous
solution $R_{jk}=\lim_{m\to\infty}R_{jk}^{(m)}$ to the integral system (3.69).
Moreover, inserting estimate (3.80) in (3.79) leads to the following estimate
for the vector solution $(R_{jk})_{j=1}^{r}$:

$|R_{jk}(x,t)|\leqslant
C_{jk}\sum_{m=0}^{\infty}\tau_{k}^{m}\frac{|\rho_{k}(x)-\rho_{k}(t)|^{m}}{m!}\leqslant\|\widetilde{Q}\|_{\infty}\widetilde{C}_{jk}\cdot
e^{\tau_{k}\cdot|\rho_{k}(x)-\rho_{k}(t)|},\qquad(x,t)\in\Omega_{k},$ (3.94)

where $\widetilde{C}_{jk}=1$ for $j\neq k$,
$\widetilde{C}_{kk}=q_{k}\cdot(a_{k}^{+}-a_{k}^{-})$ and

$\|\widetilde{Q}\|_{\infty}=\max\\{\|\widetilde{Q}_{jp}\|_{\infty}:j,p\in\\{1,\ldots,{r}\\}\\}=\max\\{\widetilde{q}_{p}:p\in\\{1,\ldots,{r}\\}\\}.$

(v) At this final step, assuming $k\in\\{1,\ldots,{r}\\}$ to be fixed, we show
that $R_{jk}\in\operatorname{Lip}_{1}(\Omega)$, $j\in\\{1,\ldots,{r}\\}$.
According to Lemma 2.1(iii) with account of inclusion (3.78) and the fact that
$\Omega_{k}$ is a simply connected, closed and bounded set with a Lipshitz
boundary, to show that $R_{jk}\in\operatorname{Lip}_{1}(\Omega_{k})$ it is
sufficient to show uniform boundedness of the derivatives
$\frac{\partial}{\partial{x}}R_{jk}^{(m)}(x,t)$ and
$\frac{\partial}{\partial{t}}R_{jk}^{(m)}(x,t)$ in $\Omega_{k}$.

To this end, let us obtain formulas for
$D_{2}R_{jk}^{(m)}(x,t):=\frac{\partial}{\partial{t}}R_{jk}^{(m)}(x,t)$ by
differentiating formulas (3.74)–(3.75). Inclusions (3.77) and (3.78) allow us
to apply standard rules of differentiation (for a.e. $(x,t)\in\Omega_{k}$).
Applying the operator $\frac{\partial}{\partial{t}}$ to equations
(3.74)–(3.75) yields for $j\in\\{1,\ldots,{r}\\}$,

$D_{2}R_{jk}^{(m)}(x,t)=\frac{\partial}{\partial{t}}\widetilde{Q}_{jk}(a_{jk}(x,t))-\sum_{p=1}^{r}\int_{a_{jk}(x,t)}^{x}Q_{jp}(u)\left(D_{2}R_{pk}^{(m-1)}(u,\gamma_{jk}^{x,t}(u))\right)\frac{\partial}{\partial{t}}\gamma_{jk}^{x,t}(u)\,du\\\
+\left(\frac{\partial}{\partial{t}}a_{jk}(x,t)\right)\sum^{r}_{p=1}Q_{jp}(a_{jk}(x,t))\cdot
R_{pk}^{(m-1)}\left(a_{jk}(x,t),\gamma_{jk}^{x,t}(a_{jk}(x,t))\right),\quad
j\neq k,$ (3.95)
$D_{2}R_{kk}^{(m)}(x,t)=-\sum_{p=1}^{r}\int_{a_{kk}(x,t)}^{x}Q_{kp}(u)\left(D_{2}R_{pk}^{(m)}(u,\gamma_{kk}^{x,t}(u))\right)\frac{\partial}{\partial{t}}\gamma_{kk}^{x,t}(u)\,du\\\
+\left(\frac{\partial}{\partial{t}}a_{kk}(x,t)\right)\sum^{r}_{p=1}Q_{kp}(a_{kk}(x,t))\cdot
R_{pk}^{(m)}\left(a_{kk}(x,t),\gamma_{kk}^{x,t}(a_{kk}(x,t))\right),$ (3.96)

for a.e. $(x,t)\in\Omega_{k}$. Equations (3.95)–(3.96) have exact same form as
(3.74)–(3.75) with only three notable differences that do not prevent the
application of successive approximation procedure used in the previous step:

  * 1.

The “initial data” (out-of-integral term in (3.95)) now belongs to
$L^{\infty}(\Omega_{k})$, which is implied by inclusions
$a_{jk},\widetilde{Q}_{jk}\circ a_{jk}\in\operatorname{Lip}_{1}(\Omega_{k})$
and $Q_{jp}\in L^{\infty}(\mathbb{R})$. This only changes the smoothness of
approximations $D_{2}R_{jk}^{(m)}$ to $L^{\infty}(\Omega_{k})$, but does not
affect the proof in any way. The only difference is that the uniform limit of
$D_{2}R_{jk}^{(m)}$ will be also in $L^{\infty}(\Omega_{k})$ instead of
$C(\Omega_{k})$;

  * 2.

The “initial data” now includes the term $R_{pk}^{(N)}$. When we estimate the
difference $D_{2}R_{jk}^{(m+1)}-D_{2}R_{jk}^{(m)}$ this term will generate the
difference $R_{jk}^{(N+1)}-R_{jk}^{(N)}$ (either with $N=m-1$ or $N=m$), for
which we already have the key estimate (3.80) and hence this can be handled
properly during estimation;

  * 3.

The coefficient $Q_{kp}(u)$ of $R_{pk}(\ldots)$ in (3.69) is replaced with
$Q_{jp}(u)\frac{\partial}{\partial{t}}\gamma_{jk}^{x,t}(u)$. It now depends on
$x,t,u$, but inclusions $Q_{jp}\in L^{\infty}(\mathbb{R})$ and
$\gamma_{jk}^{\cdot,\cdot}(\cdot)\in\operatorname{Lip}_{1}(\mathbb{R}^{3})$
imply that it belongs to $L^{\infty}(\mathbb{R}^{3})$. Hence in the estimates
for the difference $D_{2}R_{jk}^{(m+1)}-D_{2}R_{jk}^{(m)}$ similar to (3.86),
(3.88) we can still estimate this coefficient from above as before.

There remarks prove the absolute and uniform convergence of the series

$\sum_{m=1}^{\infty}D_{2}\left[R_{jk}^{(m)}-R_{jk}^{(m-1)}\right]=\sum_{m=1}^{\infty}\frac{\partial}{\partial{t}}\left[R_{jk}^{(m)}-R_{jk}^{(m-1)}\right]$
(3.97)

in $\Omega_{k}$, which implies in particular uniform boundedness of
$\frac{\partial}{\partial{t}}R_{jk}^{(m)}(x,t)$ in $\Omega_{k}$ over
$m\in\mathbb{N}_{0}$.

Applying the operator $\frac{\partial}{\partial{x}}$ to equations
(3.74)–(3.75) yields similar formula for
$\frac{\partial}{\partial{x}}R_{jk}^{(m)}(x,t)$ except in the integral we
still have $D_{2}R_{pk}^{(N)}(u,\gamma_{jk}^{x,t}(u))$ term (either with
$N=m-1$ or $N=m$). Hence uniform boundedness of
$\frac{\partial}{\partial{x}}R_{jk}^{(m)}(x,t)$, $m\in\mathbb{N}_{0}$, is
implied by uniform boundedness of
$\frac{\partial}{\partial{t}}R_{jk}^{(m)}(x,t)$, $m\in\mathbb{N}_{0}$, and
Lipshitz and boundedness properties of the involved functions
$\widetilde{Q}_{jk}$, $Q_{jp}$, $a_{jk}$, $\gamma_{jk}^{\cdot,\cdot}(\cdot)$.
Lemma 2.1(iii) now finishes the proof. The proof of uniqueness is proved by
applying the Grönwall’s lemma and is omitted. ∎

######  Remark 3.3.

Let us prove properties (3.48)–(3.52) more formally. First note that

$\gamma_{jk}^{x,t}(u)\leqslant u,\quad u\in[x,a_{jk}(x,t)],\quad 0\leqslant
t\leqslant x\leqslant\ell.$ (3.98)

This follows from the definition (3.41) of $\gamma_{jk}^{x,t}(\cdot)$,
relations $\gamma_{jk}^{x,t}(x)=t\leqslant x$ and $\gamma_{jk}^{x,t}(a)=a$ for
$a=a_{jk}(x,t)$, and monotonicity of functions $\rho_{k}$ and $\rho_{k}^{-1}$.
Further, since $\rho_{j}$ and $\rho_{k}$ are (absolutely) continuous and
monotonous it follows that $\left\\{\rho_{j}(x)-\rho_{k}(t):\ \ 0\leqslant
t\leqslant x\leqslant\ell\right\\}$ is a smallest segment of the real line
$\mathbb{R}$ containing points $0$, $\rho_{j}(\ell)$ and
$\rho_{j}(\ell)-\rho_{k}(\ell)$. I.e.

$\left\\{\rho_{j}(x)-\rho_{k}(t):\ 0\leqslant t\leqslant
x\leqslant\ell\right\\}=:[\rho_{jk}^{-},\rho_{jk}^{+}],$ (3.99)

where

$\rho_{jk}^{-}:=\min\\{0,\ \rho_{j}(\ell),\
\rho_{j}(\ell)-\rho_{k}(\ell)\\}\leqslant 0,\qquad\rho_{jk}^{+}:=\max\\{0,\
\rho_{j}(\ell),\ \rho_{j}(\ell)-\rho_{k}(\ell)\\}\geqslant 0.$ (3.100)

Since $(\rho_{j}-\rho_{k})^{-1}$ is continuous and monotonous, it follows that
the set of all values $a_{jk}(x,t)$ when $(x,t)$ runs through $\Omega$ is also
a finite segment of $\mathbb{R}$,

$\\{a_{jk}(x,t):\ 0\leqslant t\leqslant
x\leqslant\ell\\}=(\rho_{j}-\rho_{k})^{-1}([\rho_{jk}^{-},\rho_{jk}^{+}])=:[a_{jk}^{-},a_{jk}^{+}]\supset[0,\ell],$
(3.101)

where the last inclusion follows from formulas (3.100) for $\rho_{jk}^{\pm}$.
Namely, relations

$a_{jk}(0,0)=(\rho_{j}-\rho_{k})^{-1}(0)=0,\qquad
a_{jk}(\ell,0)=(\rho_{j}-\rho_{k})^{-1}(\rho_{j}(\ell)),\qquad
a_{jk}(\ell,\ell)=\ell,$

imply inclusion $[a_{jk}^{-},a_{jk}^{+}]\supset[0,\ell]$. These formulas also
imply one of the formula (3.52) for $a_{jk}^{\pm}$. Which of the cases we will
have depends on the relation of $a_{jk}(\ell,0)$ to the numbers $0$ and
$\ell$.

Observations (3.98) and (3.101) already imply boundedness of $\Omega_{jk}$
from “three sides”,

$\Omega_{jk}\subset\\{(u,v):u\in[a_{jk}^{-},a_{jk}^{+}],v\leqslant u\\}.$
(3.102)

Let us fix $u\in[a_{jk}^{-},a_{jk}^{+}]$ and find the intersection of
$\Omega_{k}$ with the vertical line
$\mathbf{L}_{u}:=\\{(u,v):v\in\mathbb{R}\\}$. Since
$u\in[a_{jk}^{-},a_{jk}^{+}]$, then by definition of $a_{jk}^{-}$ and
$a_{jk}^{+}$ there exists $(x,t)\in\Omega$ such that $a_{jk}(x,t)=u$. Hence
$\gamma_{jk}^{x,t}(u)=u$ by definition of $a_{jk}(x,t)$, which implies that
$(u,u)\in\Omega_{jk}\cup\mathbf{L}_{u}$. From general continuity and
monotonicity reasoning it is clear that $\Omega_{jk}\cup\mathbf{L}_{u}$ is a
finite segment of the form $[\gamma_{jk}^{-}(u),u]$, where
$\gamma_{jk}^{-}(\cdot)$ satisfy one of the cases in (3.52).

Now we are ready to state the main result of this subsection which, in
particular, states the similarity of the operators $\mathcal{L}_{0}(Q)$ and
$\mathcal{L}_{0}(0)$.

######  Theorem 3.4.

Let matrix functions $B(\cdot)$ and $Q(\cdot)$ satisfy conditions (3.3)–(3.8).
In particular, we assume that $Q_{kk}=0$ for $j\in\\{1,\ldots,{r}\\}$. Then
the operators $\mathcal{L}_{0}(Q)$ and $\mathcal{L}_{0}(0)=B(x)^{-1}\otimes
D_{0}$ are similar in $L^{p}([0,\ell];\mathbb{C}^{n\times n})$,
$p\in[1,\infty]$. Moreover, there exists a bounded on
$L^{p}([0,\ell];\mathbb{C}^{n})$ triangular Volterra type operator
$I+\mathcal{R}$,

$(I+\mathcal{R})f=f(x)+\int^{x}_{0}R(x,t)B(t)f(t)\,dt,\qquad f\in
L^{p}([0,\ell];\mathbb{C}^{n}),$ (3.103)

with a bounded inverse that intertwines the operators $\mathcal{L}_{0}(Q)$ and
$\mathcal{L}_{0}(0)$, i.e.

$\mathcal{L}_{0}(Q)(I+\mathcal{R})f=(I+\mathcal{R})\mathcal{L}_{0}(0)f,\qquad
f\in\operatorname{dom}\mathcal{L}_{0}(0)=W^{1,2}_{0}([0,\ell];\mathbb{C}^{n}).$
(3.104)

Here $R(x,t)=(R_{jk}(x,t))_{j,k=1}^{r}$ is the block-matrix kernel of the
operator $\mathcal{R}$ that meets the condition

$R\in\left(X_{1,0}(\Omega)\cap
X_{\infty,0}(\Omega)\right)\otimes\mathbb{C}^{n\times n},$ (3.105)

where the function spaces $X_{1,0}(\Omega)$, $X_{\infty,0}(\Omega)$ are
defined in Subsection 2.3 above.

###### Proof.

First, let us extend matrix functions $B(\cdot)$ and $Q(\cdot)$ to be defined
on $\mathbb{R}$ as it was done in the step (ii) of the proof of Proposition
3.2. Let us show that under the assumption $Q\in
L^{1}(\mathbb{R};\mathbb{C}^{n\times n})$ (see (3.5) and (3.31)) the integral
system (3.69) has a solution $R$ satisfying inclusion (3.105). It can be
treated as a generalized solution to the problem (3.70)–(3.72). For simplicity
we restrict ourselves to the case $Q\in
L^{\infty}(\mathbb{R};\mathbb{C}^{n\times n})$ and $\widetilde{Q}_{jk}\in
C(\mathbb{R})$, $j\neq k$ (see (3.5) and (3.12)) General case is treated
similarly to that of our treatment of Theorem 2.5 from [33].

To this end, we choose a sequence of smooth finite matrix functions
$\widetilde{Q}_{m}=(\widetilde{Q}_{jk,m})^{r}_{j,k=1}\in\operatorname{Lip}_{1}(\mathbb{R};\mathbb{C}^{n\times
n})$, $m\in\mathbb{N}$, where $Q_{jj,m}\equiv 0$, $j\in\in\\{1,\ldots,{r}\\}$,
that approximate finite matrix function
$\widetilde{Q}=(\widetilde{Q}_{jk})^{r}_{j,k=1}\in
C(\mathbb{R};\mathbb{C}^{n\times n_{k}})$. Following (3.29)–(3.30) we set

$\displaystyle Q_{jk,m}(x)$
$\displaystyle:=(\beta_{j}(x)-\beta_{k}(x))\cdot\widetilde{Q}_{jk,m}(x),\qquad
x\in\mathbb{R},\qquad j\neq k,$ (3.106) $\displaystyle Q_{jj,m}(x)$
$\displaystyle:=\widetilde{Q}_{jj,m}(x):=0,\qquad x\in\mathbb{R},\qquad
j\in\\{1,\ldots,{r}\\}.$ (3.107)

It follows that there exists positive constants $C_{q}$ and
$C_{\widetilde{q}}$ such that

$\|\widetilde{Q}_{m}\|_{C(\mathbb{R})}\leqslant
C_{\widetilde{q}}\qquad\Longrightarrow\qquad\|Q_{m}\|_{L^{\infty}(\mathbb{R})}\leqslant
C_{q},\qquad m\in\mathbb{N}.$ (3.108)

Moreover

$\|Q_{jk,m}-Q_{jk}\|_{\infty}\to 0,\quad\text{as}\quad m\to\infty,\qquad
j,k\in\\{1,\ldots,{r}\\}.$ (3.109)

Let us fix $k\in\\{1,\ldots,{r}\\}$. In accordance with Proposition 3.2, the
system of equations (3.69) with $Q_{m}$ in place of $Q$, has
$\operatorname{Lip}_{1}$-smooth matrix block solution $(R_{jk,m})_{j=1}^{r}$,
i.e. for $j\in\\{1,\ldots,{r}\\}$ and $(x,t)\in\Omega_{k}$ we have,

$R_{jk,m}(x,t)=\widetilde{Q}_{jk,m}\bigl{(}a_{jk}(x,t)\bigr{)}-\int_{a_{jk}(x,t)}^{x}\sum^{r}_{p=1}Q_{jp,m}(u)R_{pk,m}\left(u,\gamma_{jk}^{x,t}(u)\right)\,du.$
(3.110)

To evaluate the difference $R_{jk,m}-R_{jk,s}$ we set

$\displaystyle\widehat{R}_{jk,m,s}(x,t):=R_{jk,m}(x,t)-R_{jk,s}(x,t),\qquad\widehat{Q}_{jk,m,s}:=Q_{jk,m}-Q_{jk,s},$
$\displaystyle\widehat{Q}_{m,s}:=Q_{m}-Q_{s},\quad\text{and}\qquad\widetilde{Q}_{jk,m,s}:=\widetilde{Q}_{jk,m}-\widetilde{Q}_{jk,s},\qquad
m,s\in\mathbb{N}.$ (3.111)

Taking the difference of two equations (3.110) with $m=m$ and $m=s$ and using
the notations (3.1) one easily rewrites it in the form

$\displaystyle\widehat{R}_{jk,m,s}(x,t)=\widetilde{Q}_{jk,m,s}\bigl{(}a_{jk}(x,t)\bigr{)}-\sum_{p=1}^{r}\int^{x}_{a_{jk}}\widehat{Q}_{jp,m,s}(u)R_{pk,s}(u,\gamma_{jk}^{x,t}(u)\bigr{)}du$
$\displaystyle-\sum_{p=1}^{r}\int^{x}_{a_{jk}}Q_{jp,m}(u)\widehat{R}_{pk,m,s}(u,\gamma_{jk}^{x,t}(u)\bigr{)}du,$
(3.112)

where for brevity we set $a_{jk}:=a_{jk}(x,t)$. Further, first we show that
the family $\\{R_{jk,m}\\}_{m\in\mathbb{N}}$ is uniformly bounded in
$L^{\infty}$-norm. Indeed, one gets from (3.94) with account of definitions
(3.108) that

$\|R_{jk,m}\|_{L^{\infty}(\Omega_{k})}\leqslant
C_{jk}\cdot\exp\left(\tau_{k}\cdot|\rho_{k}(a_{k}^{+})-\rho_{k}(a_{k}^{-})|\right)=M_{jk},\quad
m\in\mathbb{N}.$ (3.113)

It follows from (3.1) with account of estimate (3.113) and notations (3.108)
that

$\displaystyle|\widehat{R}_{jk,m,s}(x,t)|=\|\widetilde{Q}_{jk,m}-\widetilde{Q}_{jk,s}\|_{L^{\infty}}+M_{jk}\sum_{p=1}^{r}\int^{x}_{a_{jk}}|Q_{jp,m}(u)-Q_{jp,s}(u)|\,du$
$\displaystyle+\sum_{p=1}^{r}\int^{x}_{a_{jk}}|Q_{jp,m}(u)|\cdot|\widehat{R}_{pk,m,s}(u,\gamma_{jk}^{x,t}(u)|\,du$
$\displaystyle\leqslant\|\widetilde{Q}_{jk,m}-\widetilde{Q}_{jk,s}\|_{L^{\infty}}+M_{jk}r\|Q_{m}-Q_{s}\|_{L^{\infty}}+C_{q}\sum_{p=1}^{r}\int^{x}_{a_{jk}}|\widehat{R}_{pk,m,s}(u,\gamma_{jk}^{x,t}(u)|\,du.$
(3.114)

Emphasize that all the constants in (3.1) do not depend on $m,s\in\mathbb{N}$.

Applying the method of successive approximation to system of equations (3.1)
and repeating the reasoning from the step (iv) of the proof of Proposition 3.2
we arrive at the estimate similar to (3.94):

$\displaystyle|\widehat{R}_{jk,m,s}(x,t)|\leqslant\|\widetilde{Q}_{m}-\widetilde{Q}_{s}\|_{L^{\infty}}C_{1}\exp(C_{2}|\rho_{k}(x)-\rho_{k}(t)|),$
(3.115)

with some $C_{1},C_{2}>0$ that do not depend on $m,s,x,t,j,k$. (see (3.113)
and recall the definition of $C_{jk}$). It follows that the sequence of
solutions $R_{jk,m}$ to equations (3.110) is a Cauchy sequence in
$C(\Omega_{k})\otimes\mathbb{C}^{n\times n}.$ Therefore for any pair
$j,k\in\\{1,\ldots,{r}\\}$ there exists a uniform limit
$R_{jk}(x,t):=\lim_{s\to\infty}R_{jk,s}(x,t)$ that meets the following uniform
in $(x,t)\in\Omega_{k}$ estimate

$\displaystyle|R_{jk}(x,t)-R_{jk,m}(x,t)|\leqslant\|\widetilde{Q}_{m}-\widetilde{Q}\|_{L^{\infty}}C_{1}\exp(C_{2}|\rho_{k}(x)-\rho_{k}(t)|).$
(3.116)

Moreover, due to this estimate and since
$\|\widetilde{Q}_{m}-\widetilde{Q}\|_{\infty}\to 0$ and
$\|Q_{m}-Q\|_{\infty}\to 0$ as $m\to\infty$, we can pass to the limit as
$m\to\infty$ in equations (3.110) to show that
$R(x,t)=\\{R_{jk}(x,t)\\}_{j,k=1}^{r}$ is a matrix solution to system of
integral equations (3.69) and define the Volterra operator $\mathcal{R}$ with
the matrix kernel $R(x,t)$.

On the other hand, by Proposition 3.2, since
$Q_{m}\in\operatorname{Lip}_{1}([a_{k}^{-},a_{k}^{+}];\mathbb{C}^{n\times
n})$, the operator $I+\mathcal{R}_{m}$ intertwines the operators
$\mathcal{L}_{0}(Q_{m})$ and $\mathcal{L}_{0}(0)$, i.e. equation (3.104) holds
with $Q_{m}$ and $\mathcal{R}_{m}$ in place of $Q$ and $\mathcal{R}$,
respectively,

$\mathcal{L}_{0}(Q_{m})(I+R_{m})f=(I+R_{m})\mathcal{L}_{0}(0)f,\qquad
f\in\operatorname{dom}\mathcal{L}_{0}(0)=\widetilde{W}^{1,2}_{0}([0,\ell];\mathbb{C}^{n}).$
(3.117)

Taking inverses we rewrite these equations in the form

$(I+R_{m})^{-1}(\mathcal{L}_{0}(Q_{m}))^{-1}=(\mathcal{L}_{0}(0))^{-1}(I+R_{m})^{-1}.\qquad$
(3.118)

Noting that $\mathcal{R}$ is a Volterra operator, we can pass here to the
limit as $m\to\infty$ to arrive to the equation

$(I+\mathcal{R})^{-1}(\mathcal{L}_{0}(Q))^{-1}=(\mathcal{L}_{0}(0))^{-1}(I+\mathcal{R})^{-1}.\qquad$
(3.119)

It follows that $I+\mathcal{R}$ maps $\operatorname{dom}\mathcal{L}_{0}(0)$
onto $\operatorname{dom}\mathcal{L}_{0}(Q)$. Therefore taking the inverses in
(3.119) we arrive at identity (3.104). Since $0\in\rho(I+\mathcal{R})$, this
proves the similarity and completes the proof.

∎

###  3.2 Transformation operators

In this subsection we prove the existence of triangular transformation
operators for equation (3.1). As in [41] our proof is substantially relies on
the similarity result, Theorem 3.4.

It is well known that the commutant $\\{J\\}^{\prime}$ of the Volterra
integration operator $J(J:\ f\to\int^{x}_{0}f(t)dt)$ on $L^{2}[0,\ell]$
consists of convolutions with distributions (see [53] and [40]). In
particular, a convolution operator $K:f\to k*f=\int_{0}^{x}k(x-t)f(t)dt$ with
$k\in L^{1}[0,\ell]$ belongs to $\\{J\\}^{\prime}$.

We complete this subsection by a simple lemma on commutant of the operator
$J\mathcal{B}$ on $L^{2}[0,\ell]$ with $\mathcal{B}:\ f\to b(t)f(t)$ being a
multiplication operator. This result is a substantial ingredient in the proof
of the existence of transformation operators in the next subsection (see
Theorem 3.7).

Surprisingly, that partial differential equations technique is highly involved
in a description of the commutant which is not so explicit as in the case of
$b_{0}\equiv\operatorname{const}$.

######  Lemma 3.5.

Let $b\in L^{\infty}[0,\ell]$ and let $b(\cdot)$ be either positive or
negative for a.e. $x\in[0,\ell]$ and let

$\mathcal{P}:\ f\to\int^{x}_{0}P(x,t)b(t)f(t)\,dt$ (3.120)

be a Volterra operator with $P\in C(\Omega)$. Then the operator $\mathcal{P}$
commutes with the operator

$J\mathcal{B}:\ f\to\int^{x}_{0}b(t)f(t)dt$ (3.121)

if and only if the kernel $P(\cdot,\cdot)$ is given by

$P(x,t)=P(\xi(x,t),0).$ (3.122)

Here $\xi(x,t)$ is the implicit solution to the equation

$\quad\rho(\xi(x,t))-\rho(x)+\rho(t)=0,\qquad\rho(x):=\int^{x}_{0}b(s)\,ds.$
(3.123)

###### Proof.

Changing the order of integrals one easily gets

$PJBf=\int^{x}P(x,s)b(s)\,ds\int^{s}b(t)f(t)\,ds=\int^{x}\left(\int^{x}_{t}P(x,s)b(s)\,ds\right)b(t)f(t)\,dt,$
(3.124)
$JBPf=\int^{x}b(s)\,ds\int^{s}P(s,t)b(t)f(t)\,dt=\int^{x}\left(\int^{x}_{t}b(s)P(s,t)\,ds\right)b(t)f(t)\,dt.$
(3.125)

Equating these relations one concludes that the commutation relation
$[P,JB]=0$ is equivalent to

$\int^{x}_{t}P(x,s)b(s)\,ds=\int^{x}_{t}b(s)P(s,t)\,ds.$ (3.126)

Assume first that $P\in C^{1}(\Omega)$. Then differentiating this equation
with respect to $x$ yields

$P(x,x)b(x)+\int^{x}_{t}\frac{\partial}{\partial{x}}P(x,s)b(s)\,ds=b(x)P(x,t).$
(3.127)

In turn, applying the operator $\frac{\partial}{\partial{t}}$ to this equation
leads to the first order partial differential equation

$\frac{\partial}{\partial{x}}P(x,t)b(t)+b(x)D_{t}P(x,t)=0.$ (3.128)

Conversely, if the kernel $P\in C^{1}(\Omega)$ and satisfies equation (3.128)
we obtain by reversing the reasonings that the operator $\mathcal{P}$ of the
form (3.120) commutes with $J\mathcal{B}$, i.e.
$P\in\\{J\mathcal{B}\\}^{\prime}$.

Next we extend the function $b(\cdot)$ to the whole line $\mathbb{R}$
preserving its $L^{\infty}$-norm and the sign. In what follows we keep the
notation $b(\cdot)$ for this extension.

It is easily seen that in the coordinates $\\{\xi,\eta\\}$ the characteristic
of equation (3.128) passing through the point $(x,t)$, is given by

$\Gamma(\xi,\eta):\quad\rho(\eta)=\rho(\xi)-\rho(x)+\rho(t),\qquad\rho(\xi)=\int^{\xi}_{0}b(s)\,ds,$
(3.129)

where $\rho$ is also defined on the line. The “explicit” form
$\eta=\gamma(\xi;x,t)$ of the characteristic is defined to be the unique
solution to the equation

$\Gamma(\xi,\gamma(\xi;x,t)=\rho(\gamma(\xi;x,t)-\rho(\xi)+\rho(x)-\rho(t)=0\quad\text{and}\quad\gamma(x;x,t)=t.$
(3.130)

Since $\rho$ is monotone on $\mathbb{R}$, each characteristic is well defined
on $\mathbb{R}$.

$\eta(\xi)=\gamma(\xi;x,t):=\rho^{-1}\circ(\rho(\xi)-\rho(x)+\rho(t)),\qquad\xi\in\mathbb{R}.$

Next we denote by $M(\xi(x,t),0)$ the point of interaction of the
characteristic $\eta=\gamma(\xi;x,t)$ with the axis $\eta=0$. Then in
accordance with (3.130) $\xi(x,t)$ satisfies the equation

$0=\rho(0)=\rho(\xi(x,t))-\rho(x)+\rho(t),$ (3.131)

It follows that

$\frac{\partial\xi(x,t)}{\partial
x}=\frac{\rho^{\prime}(x)}{\rho^{\prime}\bigl{(}\xi(x,t)\bigr{)}}=\frac{\beta(x)}{\beta\bigl{(}\xi(x,t)\bigr{)}}\qquad\text{and}\qquad\frac{\partial\xi(x,t)}{\partial
t}=-\frac{\beta(t)}{\beta\bigl{(}\xi(x,t)\bigr{)}}.$ (3.132)

In turn, using these relations imply that alongside $P(x,t)$ the function

$\widetilde{P}(x,t):=P(\xi(x,t),0)$ (3.133)

satisfies the equation (3.128). Besides, the identity $\xi(x,0)=x$ yields

$\widetilde{P}(x,0)=P(\xi(x,0),0)=P(x,0)=:g(x).$ (3.134)

So, we have two solutions $\widetilde{P}(\cdot,\cdot)$ and $P(\cdot,\cdot)$ to
the Cauchy problem (3.128), (3.134) in $\Omega$. However, since the slope of
the characteristic $\gamma(\xi;x,t)$ is positive,

$\kappa(\xi_{0},\eta_{0})=\frac{d\gamma(\xi;x,t)}{d\xi}\big{|}_{\xi=\xi_{0}}=\frac{\beta(\xi_{0})}{\beta(\eta_{0})}=\frac{\beta(\xi_{0})}{\beta(\gamma_{jk}(\xi_{0};x,t))}>0,$
(3.135)

the Cauchy problem (3.128), (3.134) is not characteristic, and hence has the
unique solution in $\Omega$, i.e. $\widetilde{P}(x,t)=P(x,t)$. ∎

######  Remark 3.6.

Note that as a byproduct we proved that if $P\in\\{J\mathcal{B}\\}^{\prime}$
and the kernel of $P$ is smooth, $P\in C^{1}(\Omega),$ then it is a solution
to equation (3.128). If $P\in C(\Omega),$ then it is a generalized solution to
equation (3.128). A complete description of the commutant
$\\{J\mathcal{B}\\}^{\prime}$ will be published elsewhere.

Now we are ready to establish our main result on transformation operators for
the equation (3.1).

######  Theorem 3.7.

Let matrix functions $B(\cdot)$ and $Q(\cdot)$ satisfy conditions (3.3)–(3.8).
In particular, we assume that $Q_{jj}=0$ for $j\in\\{1,\ldots,{r}\\}$.
Further, let

$\displaystyle A=\begin{pmatrix}A_{1}\\\ \cdots\\\
A_{r}\end{pmatrix}\in\mathbb{C}^{n\times n_{\min}},\qquad
A_{j}\in\mathbb{C}^{n_{j}\times n_{\min}},\qquad j\in\\{1,\ldots,{r}\\},$
(3.136)

where $n_{\min}:=\min\\{n_{1},\ldots,n_{r}\\}$, and let all matrices $A_{j}$
be of maximal rank, i.e. $\operatorname{rank}(A_{j})=n_{\min}$.
$j\in\\{1,\ldots,{r}\\}$. Further, let

$Y_{A}(x,\lambda)=\begin{pmatrix}Y_{1}(x,\lambda)\\\ \cdots\\\
Y_{r}(x,\lambda)\end{pmatrix}\qquad\text{and}\qquad
e_{A}(x,\lambda):=\begin{pmatrix}e^{i\lambda\rho_{1}(x)}A_{1}\\\ \cdots\\\
e^{i\lambda\rho_{r}(x)}A_{r}\end{pmatrix}$ (3.137)

are the $n\times n_{\min}$ block-matrix solutions to equations (3.1) and
(3.2), respectively, satisfying the initial conditions

$Y_{A}(0,\lambda)=e_{A}(0,\lambda)=A.$ (3.138)

Then solution $Y_{A}(x,\lambda)$ admits a triangular representation

$Y_{A}(x,\lambda)=(I+\mathcal{K}_{A})e_{A}(x,\lambda)=e_{A}(x,\lambda)+\int^{x}_{0}K_{A}(x,t)B(t)e_{A}(t,\lambda)\,dt,$
(3.139)

where the block-matrix kernel $K_{A}=(K_{jk})_{j,k=1}^{r}$ in the operator
$\mathcal{K}_{A}$ satisfies

$K_{A}\in\left(X_{1,0}(\Omega)\cap
X_{\infty,0}(\Omega)\right)\otimes\mathbb{C}^{n\times n_{1}}.$ (3.140)

Here domain $\Omega$ and function spaces $X_{1,0}(\Omega)$,
$X_{\infty,0}(\Omega)$ are defined in Subsection 2.3 above.

######  Remark 3.8.

Note that Lemma 2.2 and inclusion $K_{A}\in
X_{\infty,0}(\Omega)\otimes\mathbb{C}^{n\times n_{1}}$ ensure that the traces
$K(x,\cdot)$ are well-defined and summable for each $x\in[0,\ell]$. In
particular, end trace $K(\ell,\cdot)$ is well defined. Hence formula (3.139)
is well-defined and valid for each $x\in[0,\ell]$. A more relaxed inclusion
$K_{A}\in X_{\infty}(\Omega)\otimes\mathbb{C}^{n\times n}$ would only yield
this formula for a.e $x\in[0,\ell]$.

###### Sketch of the proof.

Assume for definiteness that $n_{\min}=n_{1}$.

(i) At this step assuming the validity of representation (3.139) with
$K_{A}\in C^{1}(\Omega)$ we indicate the boundary value problem for the kernel
$K_{A}(\cdot,\cdot)$. Inserting representation (3.139) into equation (3.1) we
obtain

$Y^{\prime}(x,\lambda)+Q(x)Y(x,\lambda)=e^{\prime}_{A}(x,\lambda)+K_{A}(x,x)B(x)e_{A}(x,\lambda)\\\
+\int^{x}_{0}\frac{\partial}{\partial{x}}K_{A}(x,t)B(t)e_{A}(t,\lambda)\,dt+Q(x)e_{A}(x,\lambda)+Q(x)\cdot\int^{x}_{0}K_{A}(x,t)B(t)e_{A}(t,\lambda)\,dt\\\
=e^{\prime}_{A}(x,\lambda)+\bigl{(}K_{A}(x,x)B(x)+Q(x)\bigr{)}e_{A}(x,\lambda)\\\
+\int_{0}^{x}\bigl{(}\frac{\partial}{\partial{x}}K_{A}(x,t)+Q(x)K_{A}(x,t)\bigr{)}B(t)e_{A}(t,\lambda)\,dt$
(3.141)

On the other hand, it follows from (3.139) after integrating by parts that

$\displaystyle i\lambda B(x)Y(x,\lambda)$ $\displaystyle=i\lambda
B(x)e_{A}(x,\lambda)+i\lambda
B(x)\int^{x}_{0}K_{A}(x,t)B(t)e_{A}(t,\lambda)\,dt$
$\displaystyle=e^{\prime}_{A}(x,\lambda)+B(x)\int^{x}_{0}K_{A}(x,t)e_{A}^{\prime}(t,\lambda)\,dt$
$\displaystyle=e^{\prime}_{A}(x,\lambda)+B(x)K_{A}(x,x)e_{A}(x,\lambda)$
$\displaystyle\qquad\qquad-B(x)K_{A}(x,0)e_{A}(0,\lambda)-B(x)\int^{x}_{0}D_{t}\bigl{(}K_{A}(x,t)\bigr{)}e_{A}(t,\lambda)\,dt.$
(3.142)

Equating both sides of this identities we arrive at the following boundary
value problem for the $n\times n$-matrix kernel $K_{A}(\cdot,\cdot)$:

$D_{x}K_{A}(x,t)B(t)+B(x)D_{t}K_{A}(x,t)+Q(x)K_{A}(x,t)B(t)=0,$ (3.143)
$B(x)K_{A}(x,x)-K_{A}(x,x)B(x)=Q(x),$ (3.144) $K_{A}(x,0)A=0.$ (3.145)

Writing the kernel $K_{A}$ in the block-matrix form
$K_{A}=\left(K_{jk}\right)_{j,k=1}^{r}$ with respect to the decomposition
$\mathbb{C}^{n}=\mathbb{C}^{n_{1}}\oplus\ldots\oplus\mathbb{C}^{n_{r}}$ and
using the block-matrix form of $Q$ we rewrite the problem (3.143)–(3.145) in
the following form

$\displaystyle\beta_{k}(t)D_{x}K_{jk}(x,t)+\beta_{j}(x)D_{t}K_{jk}(x,t)$
$\displaystyle=-\sum^{r}_{p=1}\beta_{k}(t)Q_{jp}(x)K_{pk}(x,t),\quad
j,k\in\\{1,\ldots,{r}\\},$ (3.146) $\displaystyle K_{jk}(x,x)$
$\displaystyle=\frac{Q_{jk}(x)}{\beta_{j}(x)-\beta_{k}(x)},\qquad j\neq
k,\quad j,k\in\\{1,\ldots,{r}\\},$ (3.147)
$\displaystyle\sum^{r}_{k=1}K_{jk}(x,0)A_{k}$ $\displaystyle=0,\qquad
j\in\\{1,\ldots,{r}\\}.$ (3.148)

Emphasise that formula (3.147) has sense due to the conditions (1.6)–(1.7).

Conversely, reversing the reasonings one proves that any $C^{1}$-solution to
the problem (3.146)–(3.148) generates representation (3.139). So, to prove the
result it suffices to show the solvability of the problem (3.146)–(3.148).

(ii) At this step we prove the solvability. To construct a solution $K_{A}$ to
the problem (3.146)–(3.148) we use a solution $R(x,t)$ to the problem
(3.146)–(3.147), constructed in Theorem 3.4. Besides, we introduce a Volterra
operator

$P=\bigoplus_{1}^{r}P_{j},\quad\text{where}\quad P_{j}:\
f_{j}\to\int^{x}_{0}P_{j}(x,t)B_{j}(t)f_{j}(t)\,dt,\qquad f_{j}\in
L^{2}\bigl{(}[0,\ell];\mathbb{C}^{n_{j}}\bigr{)}.$ (3.149)

Moreover, we assume that $P_{j}$ has a smooth kernel $P_{j}\in
C^{1}((\Omega);\mathbb{C}^{n_{j}\times n_{j}})$ and satisfies

$[P_{j},J\otimes B_{j}]=[P_{j},\left(J\beta_{j}(t)\right)\otimes
I_{n_{j}}]=0,\qquad j\in\\{1,\ldots,{r}\\},$

i.e. $P_{j}\in\\{J\otimes B_{j}\\}^{\prime}$.

Starting with the operator $I+\mathcal{R}$ constructed in Theorem 3.4, we
define the operator $I+\mathcal{K}_{A}$ as the product of two operators:

$I+\mathcal{K}_{A}:=(I+\mathcal{R})(I+\mathcal{P}).$ (3.150)

In terms of the kernels of integral operators $\mathcal{K}_{A}$,
$\mathcal{R}$, and $\mathcal{P}$, equality (3.150) can be rewritten as

$K_{A}(x,t)=R(x,t)+P(x,t)+\int^{x}_{t}R(x,s)B(s)P(s,t)\,ds.$ (3.151)

In fact, identity (3.150) is equivalent to (3.151) after multiplying the last
equality by the factor $B(t)$ from the right. This factor is canceled in
(3.151) because the matrix $B(t)$ is non-singular for every $t\in[0,\ell]$.

Since the operators $I+\mathcal{P}$ and $\mathcal{L}_{0}(0)$ commutes, the
operator $I+\mathcal{K}_{A}$ intertwines the operators $\mathcal{L}_{0}(Q)$
and $\mathcal{L}_{0}(0)$ alongside the operator $I+\mathcal{R}$, i.e. identity
(3.104) holds with $I+\mathcal{K}_{A}$ in place of $I+\mathcal{R}$. Therefore
Theorem 3.4 (sufficiency) applies and ensures that the kernel
$K_{A}(\cdot,\cdot)$ is also a solution to the problem (3.146)–(3.147). To
complete the proof it suffices to find a kernel $P(\cdot,\cdot)$ in such a way
that $K_{A}(\cdot,\cdot)$ meets the condition (3.148). To this end we insert
the right hand side of equality (3.151) in (3.148) and obtain

$P_{j}(x,0)A_{j}+\sum^{r}_{k=1}\left[R_{jk}(x,0)+\int^{x}_{0}R_{jk}(x,s)B_{k}(s)P_{k}(s,0)\,ds\right]A_{k}=0,\qquad
j\in\\{1,\ldots,{r}\\}.$ (3.152)

Rewriting this equality as

$(I+\mathcal{R})\begin{pmatrix}P_{1}(x,0)A_{1}\\\ \ldots\ldots\\\
P_{r}(x,0)A_{r}\end{pmatrix}=\begin{pmatrix}\widetilde{R}_{1}(x,0)\\\
\ldots\ldots\\\
\widetilde{R}_{r}(x,0)\end{pmatrix}\,,\qquad\widetilde{R}_{j}(x,0):=-\sum^{r}_{k=1}R_{jk}(x,0)A_{k}$
(3.153)

we find the unique solution

$\begin{pmatrix}P_{1}(x,0)A_{1}\\\ \ldots\ldots\\\
P_{r}(x,0)A_{r}\end{pmatrix}=(I+\mathcal{R})^{-1}\begin{pmatrix}\widetilde{R}_{1}(x,0)\\\
\ldots\ldots\\\
\widetilde{R}_{r}(x,0)\end{pmatrix}\,=:\begin{pmatrix}\widetilde{g}_{1}(x)\\\
\ldots\ldots\\\ \widetilde{g}_{r}(x)\end{pmatrix}\,.$ (3.154)

In turn, due to the condition $\operatorname{rank}(A_{j})=n_{\min}=n_{1}$,
meaning that each matrix $A_{j}$ is of the maximal rank, there exists (in
general, non-unique) matrix solution
$\\{g_{j}(x)\\}_{j=1}^{r}:=\\{P_{j}(x,0)\\}_{j=1}^{r}$ to the system (3.154),
i.e. $g_{j}A_{j}=\widetilde{g}_{j}$, $j\in\\{1,\ldots,{r}\\}$. A solution
$\\{P_{j}(x,0)\\}_{j=1}^{r}$ is definitely unique whenever
$n_{1}=n_{2}=\ldots=n_{r}$, and hence $\det A_{j}\neq 0$.

Further, to find a system of matrix functions $\\{P_{j}(x,t)\\}_{j=1}^{r}$ we
apply Lemma (3.5). In accordance with this lemma each $P_{j}(x,t)$ is a
solution to the following Cauchy problem

$\displaystyle D_{x}P_{j}(x,t)B_{j}(t)+B_{j}(x)D_{t}P_{j}(x,t)=0,$ (3.155)
$\displaystyle P_{j}(x,0)=g_{j}(x),\qquad j\in\\{1,\ldots,{r}\\}.$ (3.156)

and is given by (3.122) with $P_{j}(x,t)$ in place of $P(x,t)$, i.e.

$P_{j}(x,t)=P_{j}(\xi_{j}(x,t),0),\qquad P_{j}(\xi_{j}(x,0),0)=P_{j}(x,0).$

Here $\xi_{j}(x,t)$ is a solution to the equation

$\quad\rho_{j}(\xi_{j}(x,t))-\rho_{j}(x)+\rho_{j}(t)=0,\qquad\rho_{j}(x):=\int^{x}_{0}b_{j}(s)\,ds.$
(3.157)

This completes the proof. ∎

######  Remark 3.9.

(i) For Dirac $2\times 2$ system $(B=\operatorname{diag}(-1,1))$ with
continuous $Q$ the triangular transformation operators have been constructed
in [29, Ch.10.3] and [44, Ch.1.2]. For $Q\in(L^{1}[0,1];\mathbb{C}^{2\times
2})$ it is proved in [3] by an appropriate generalization of Marchenko’s
method.

(ii) Let $J:f\to\int_{0}^{x}f(t)dt$ denote the Volterra integration operator
on $L^{p}[0,1]$. Note that the similarity of integral Volterra operators given
by (2.13) to the simplest Volterra operator of the form $B\otimes J$ acting in
the spaces $L^{p}([0,1];\mathbb{C}^{2})$ has been investigated in [41, 56].
The technique of investigation of integral equations for the kernels of
transformation operators in the spaces $X_{\infty,1}(\Omega)$ and
$X_{1,1}(\Omega)$ goes back to the paper [39].

##  4 Fundamental matrix solution

In this section we apply results of the previous section to obtain an
important representation for the fundamental matrix solution of equation (1.1)
and its minors, involving Fourier transform of kernels from the transformation
operators.

###  4.1 Preliminaries

For reader’s convenience, recall the main equation (1.1) and its version with
zero potential:

$\displaystyle\mathcal{L}(Q)y:=-iB(x)^{-1}(y^{\prime}+Q(x)y)=\lambda y,\qquad
y=\operatorname{col}(y_{1},\ldots,y_{n}),\quad x\in[0,\ell],$ (4.1)
$\displaystyle\mathcal{L}_{0}y:=\mathcal{L}(0)y:=-iB(x)^{-1}y^{\prime}=\lambda
y,\qquad y=\operatorname{col}(y_{1},\ldots,y_{n}),\quad x\in[0,\ell].$ (4.2)

Let us recall definitions (1.3)–(1.4) of matrix functions $B(\cdot)$ and
$Q(\cdot)$

$B=\operatorname{diag}(\beta_{1},\ldots,\beta_{n}),\qquad\beta_{k}\in
L^{1}([0,\ell];\mathbb{R}\setminus\\{0\\}),\qquad k\in\\{1,\ldots,{n}\\},$
(4.3)

is a self-adjoint invertible diagonal summable matrix function, and

$Q=(Q_{jk})_{j,k=1}^{n},\qquad Q_{jk}\in
L^{1}[0,\ell]:=L^{1}([0,\ell];\mathbb{C}),\qquad j,k\in\\{1,\ldots,{n}\\},$
(4.4)

is a summable (generally non-self-adjoint) potential matrix.

Throughout this section and many results in sections below we will assume
conditions (1.6)–(1.8) on entries of the matrix function $B(\cdot)$. Namely,
we assume that

$\displaystyle\beta_{k},1/\beta_{k}\in L^{\infty}[0,\ell],\qquad
s_{k}:=\operatorname{sign}(\beta_{k}(\cdot))\equiv\operatorname{const}\neq
0,\qquad k\in\\{1,\ldots,{n}\\},$ (4.5)
$\displaystyle\beta_{1}(x)\leqslant\ldots\leqslant\beta_{n_{-}}(x)<0<\beta_{n_{-}+1}(x)\leqslant\ldots\leqslant\beta_{n}(x),\qquad
x\in[0,\ell],$ (4.6)

and there exists $\theta>0$ such that for each $k\in\\{1,\ldots,{n-1}\\}$

$\text{either}\quad\beta_{k}\equiv\beta_{k+1}\quad\text{or}\quad\beta_{k}(x)+\theta<\beta_{k+1}(x),\quad
x\in[0,\ell].$ (4.7)

Here $n_{-}\in\\{0,1,\ldots,n\\}$ is the number of negative functions among
$\beta_{1},\ldots,\beta_{n}$. Let us also set $n_{+}:=n-n_{-}$. See Remark 4.1
for some discussion about these conditions.

Throughout this section we will also assume the following “zero block
diagonality” condition on entries of the matrix function $Q(\cdot)$,

$Q_{jk}\equiv 0\quad\text{whenever}\quad\beta_{j}\equiv\beta_{k},\qquad
j,k\in\\{1,\ldots,{n}\\}.$ (4.8)

In particular $Q_{jj}\equiv 0$, $j\in\\{1,\ldots,{n}\\}$.

######  Remark 4.1.

(i) Note, that as opposed to the previous section, we work with notation (1.3)
for $B(x)$. To avoid confusion, we rewrite block-matrix decomposition
(3.3)–(3.4) as

$B=:\operatorname{diag}(\widetilde{\beta}_{1}I_{n_{1}},\ldots,\widetilde{\beta}_{r}I_{n_{r}}),\qquad
n_{1}+\ldots+n_{r}=n.$ (4.9)

It is clear that

$\displaystyle\beta_{1}\equiv\ldots\equiv\beta_{n_{1}}\equiv\widetilde{\beta}_{1},$
(4.10)
$\displaystyle\beta_{n_{1}+1}\equiv\ldots\equiv\beta_{n_{1}+n_{2}}\equiv\widetilde{\beta}_{2},$
(4.11) $\displaystyle\qquad\qquad\ldots$
$\displaystyle\beta_{n-n_{r}+1}\equiv\ldots\equiv\beta_{n}\equiv\widetilde{\beta}_{r}.$
(4.12)

These relations imply that conditions (4.5)–(4.7) on functions
$\beta_{1},\ldots,\beta_{n}$ from representation (4.3) are equivalent to
conditions (3.6)–(3.8) on functions $\beta_{1},\ldots,\beta_{r}$ from
representation (3.3)–(3.4).

Note also that relations (4.10)–(4.12) and condition (4.7) imply that
condition (4.8) on $Q$ means that $Q$ has zero block diagonal with respect to
decomposition
$\mathbb{C}^{n}=\mathbb{C}^{n_{1}}\oplus\ldots\oplus\mathbb{C}^{n_{r}}$.

(ii) Sometimes it is useful to work with the matrix function $B(\cdot)$
without ordering its entries (see e.g. (10.12)). Let us reformulate conditions
(4.6)–(4.7) for such general “unordered” case. Namely, it is easy to verify
that equivalent form of these conditions is the following,

$\displaystyle\text{there exists}\ \ \theta>0\ \ \text{such that for
each}\quad j,k\in\\{1,\ldots,{n}\\}\ \ \text{the following condition holds},$
(4.13) $\displaystyle\text{either}\quad\nu_{jk}:=\beta_{j}-\beta_{k}\equiv
0,\quad\text{or}\quad\nu_{jk}(x)>\theta,\quad
x\in[0,\ell],\quad\text{or}\quad\nu_{jk}(x)<-\theta,\quad x\in[0,\ell].$
(4.14)

Further, we set

$\rho_{k}(x):=\int_{0}^{x}\beta_{k}(t)dt\quad\text{and}\quad
b_{k}:=\rho_{k}(\ell)\in\mathbb{R}\setminus\\{0\\},\qquad
k\in\\{1,\ldots,{n}\\}.$ (4.15)

Going forward for $u\leqslant v$, notations $[u,v]$ and $[v,u]$ will mean the
same segment of real line and will be used interchangeably. It follows from
(4.5) that

$\displaystyle\rho_{1}(x)\leqslant\ldots\leqslant\rho_{n_{-}}(x)<0<\rho_{n_{-}+1}(x)\leqslant\ldots\leqslant\rho_{n}(x),\qquad
x\in[0,\ell],$ (4.16) $\displaystyle b_{1}\leqslant\ldots\leqslant
b_{n_{-}}<0<b_{n_{-}+1}\leqslant\ldots\leqslant b_{n},$ (4.17)
$\displaystyle\rho_{k}\in\operatorname{Lip}[0,\ell],\quad\rho_{k}^{-1}\in\operatorname{Lip}[0,b_{k}],\quad\rho_{k},\rho_{k}^{-1}\
\ \text{are strictly monotonous},\quad k\in\\{1,\ldots,{n}\\},$ (4.18)

where $\rho_{k}^{-1}$ denotes the function inverse to $\rho_{k}$ on the
segment $[0,\ell]$, $k\in\\{1,\ldots,{n}\\}$. Here
$\operatorname{Lip}[0,\alpha]$ is the class of functions
$f:[0,\alpha]\to\mathbb{C}$ satisfying $|f(x)-f(y)|\leqslant C|x-y|$,
$x,y\in[0,\alpha]$, for some $C>0$. As per remark above,
$[0,\alpha]=[\alpha,0]$ if $\alpha<0$.

Next, we introduce the fundamental matrices $\Phi(\cdot,\lambda)$ and
$\Phi_{0}(\cdot,\lambda)$ as the solutions to the equations
$\mathcal{L}(Q)\Phi=\lambda\Phi$ and $\mathcal{L}(0)\Phi_{0}=\lambda\Phi_{0}$,
respectively, satisfying the initial condition
$\Phi(0,\lambda)=\Phi_{0}(0,\lambda)=I_{n}$. Clearly,

$\displaystyle\Phi(x,\lambda)$
$\displaystyle=\begin{pmatrix}\Phi_{1}(x,\lambda)&\ldots&\Phi_{n}(x,\lambda)\end{pmatrix},\qquad\Phi_{p}(x,\lambda)=\operatorname{col}(\varphi_{jp}(x,\lambda),\ldots,\varphi_{np}(x,\lambda)),$
(4.19) $\displaystyle\Phi_{0}(x,\lambda)$
$\displaystyle=\begin{pmatrix}\Phi^{0}_{1}(x,\lambda)&\ldots&\Phi^{0}_{n}(x,\lambda)\end{pmatrix},\qquad\Phi^{0}_{k}(x,\lambda)=e^{i\lambda\rho_{k}(x)}\operatorname{col}\left(\delta_{1k},\ldots,\delta_{nk}\right).$
(4.20)

Note in this connection, that the matrix equation
$\mathcal{L}(Q)\Phi=\lambda\Phi$ is equivalent to

$\Phi^{\prime}(x,\lambda)=(i\lambda B(x)-Q(x))\Phi(x,\lambda),\qquad
x\in[0,\ell],\quad\lambda\in\mathbb{C},$ (4.21)

where $i\lambda B(\cdot)-Q(\cdot)$ is a summable function on $[0,\ell]$ for
each $\lambda\in\mathbb{C}$. General theory of ODE implies the existence of
global solution $\Phi(x,\lambda)$ on segment $[0,\ell]$, such that
$\Phi(\cdot,\lambda)\in\operatorname{AC}([0,\ell];\mathbb{C}^{n\times n})$ for
each $\lambda\in\mathbb{C}$ and $\Phi(x,\cdot)$ is an entire function for each
$x\in[0,\ell]$. Moreover, Liouville’s formula (see (4.49) below) implies that
$\Phi(\cdot,\lambda)^{-1}\in\operatorname{AC}([0,\ell];\mathbb{C}^{n\times
n})$ for each $\lambda\in\mathbb{C}$.

###  4.2 Key identities for fundamental matrix

In the following proposition we relate the columns of fundamental matrices
$\Phi(\cdot,\lambda)$ and $\Phi_{0}(\cdot,\lambda)$.

######  Proposition 4.2.

Let matrix functions $B(\cdot)$ and $Q(\cdot)$ satisfy conditions (4.3)–(4.8)
and let $p\in\\{1,\ldots,{n}\\}$. Then, there exist vector kernels

$R_{q}^{[p]}\in\left(X_{1,0}(\Omega)\cap
X_{\infty,0}(\Omega)\right)\otimes\mathbb{C}^{n},\qquad
q\in\\{1,\ldots,{n}\\},$ (4.22)

such that the following representation holds

$\Phi_{p}(x,\lambda)=\Phi_{p}^{0}(x,\lambda)+\sum_{q=1}^{n}\int_{0}^{x}R_{q}^{[p]}(x,t)e^{i\lambda\rho_{q}(t)}\beta_{q}(t)\,dt,\qquad
x\in[0,\ell],\quad\lambda\in\mathbb{C}.$ (4.23)

###### Proof.

It is clear that matrix functions $B(\cdot)$ and $Q(\cdot)$ satisfy
assumptions of Theorem 3.7. For simplicity let’s assume that $r=n$ and
$n_{1}=\ldots=n_{r}=1$ in block-matrix decomposition (3.3)–(3.4). Then
notations (3.3)–(3.4) and (4.3) coincide. Let $\widetilde{A}$ be some
invertible matrix with non-zero entries:

$\widetilde{A}=(a_{jk})_{j,k=1}^{n},\qquad\det(\widetilde{A})\neq 0,\qquad
a_{jk}\neq 0,\quad j,k\in\\{1,\ldots,{n}\\}.$ (4.24)

E.g. one can set $a_{jk}:=j^{k}$ to obtain invertible Vandermonde matrix with
non-zero entries. Denote by $A^{[k]}$, the $k$-th column of $\widetilde{A}$:

$\widetilde{A}=\begin{pmatrix}A^{[1]}&\ldots&A^{[n]}\end{pmatrix},\qquad
A^{[k]}=\operatorname{col}(a_{1k},\ldots,a_{nk})\quad k\in\\{1,\ldots,{n}\\}.$
(4.25)

It is clear, that for a given $k\in\\{1,\ldots,{n}\\}$, $n\times 1$ matrix
$A=A^{[k]}$ satisfy conditions of Theorem 3.7. Hence triangular representation
(3.139) takes place with some

$K_{A^{[k]}}=:K^{[k]}=:\bigl{(}K_{jp}^{[k]}\bigr{)}_{j,p=1}^{n}\in\left(X_{1,0}(\Omega)\cap
X_{\infty,0}(\Omega)\right)\otimes\mathbb{C}^{n\times n},\qquad
k\in\\{1,\ldots,{n}\\}.$ (4.26)

Further, note that due to Cauchy uniqueness theorem,

$Y_{A^{[k]}}(x,\lambda)=\Phi(x,\lambda)A^{[k]},\qquad
e_{A^{[k]}}(x,\lambda)=\Phi_{0}(x,\lambda)A^{[k]},\qquad
k\in\\{1,\ldots,{n}\\},$ (4.27)

where $Y_{A}(x,\lambda)$ and $e_{A}(x,\lambda)$ for $A=A^{[k]}$ are defined in
(3.137)–(3.138). Inserting (4.27) into (3.139) we arrive at

$\Phi(x,\lambda)A^{[k]}=\Phi_{0}(x,\lambda)A^{[k]}+\int^{x}_{0}K^{[k]}(x,t)B(t)\Phi_{0}(t,\lambda)A^{[k]}\,dt,\qquad
k\in\\{1,\ldots,{n}\\}.$ (4.28)

Formulas (4.28) and (4.25) now imply

$\Phi(x,\lambda)\widetilde{A}=\Phi_{0}(x,\lambda)\widetilde{A}+\int^{x}_{0}\left(K^{[k]}(x,t)B(t)\Phi_{0}(t,\lambda)A^{[k]}\right)_{k=1}^{n}\,dt.$
(4.29)

Let $\widetilde{A}^{-1}=(\alpha_{kp})_{k,p=1}^{n}$. Then with account of
notations (4.26), (4.3), (4.20) and (4.25) we have for $0\leqslant t\leqslant
x\leqslant 1$:

$\left(K^{[k]}(x,t)B(t)\Phi_{0}(t,\lambda)A^{[k]}\right)_{k=1}^{n}\widetilde{A}^{-1}=\left(\sum_{q=1}^{n}K_{jq}^{[k]}(x,t)\beta_{q}(t)e^{i\lambda\rho_{q}(t)}a_{qk}\right)_{j,k=1}^{n}\cdot(\alpha_{kp})_{k,p=1}^{n}\\\
=\left(\sum_{k,q=1}^{n}K_{jq}^{[k]}(x,t)\beta_{q}(t)e^{i\lambda\rho_{q}(t)}a_{qk}\alpha_{kp}\right)_{j,p=1}^{n}=\left(\sum_{q=1}^{n}R_{jq}^{[p]}(x,t)e^{i\lambda\rho_{q}(t)}\beta_{q}(t)\right)_{j,p=1}^{n},$
(4.30)

where we set

$R_{jq}^{[p]}(x,t):=\sum_{k=1}^{n}K_{jq}^{[k]}(x,t)a_{qk}\alpha_{kp},\qquad
p,j,q\in\\{1,\ldots,{n}\\},\quad 0\leqslant t\leqslant x\leqslant 1.$ (4.31)

Multiplying (4.29) by $\widetilde{A}^{-1}$ from the right with account of
(4.30) and (4.31) and taking $p$-th column in the resulting matrix equation,
we arrive at the desired formula (4.23) by setting

$R_{q}^{[p]}:=\operatorname{col}\bigl{(}R_{1q}^{[p]},\ldots,R_{nq}^{[p]}\bigr{)}_{j=1}^{n}.$
(4.32)

Desired inclusion (4.22) follows from inclusion (4.26) and formula (4.31).

The general case can be treated as follows. Start with the matrix
$\widetilde{A}$ as above. For each column $A^{[k]}$ of $\widetilde{A}$ extend
it be an $n\times n_{\min}$ matrix $A$ with blocks of maximal rank. After
applying Theorem 3.7, pick only the first column of the solutions
$Y_{A}(x,\lambda)$ and $e_{A}(x,\lambda)$ as $Y_{A^{[k]}}(x,\lambda)$ and
$e_{A^{[k]}}(x,\lambda)$, respectively. ∎

To study the integrals appearing in formula (4.23), we need the following
generalization of Riemann-Lebesgue Lemma for space $X_{\infty,0}(\Omega)$.

######  Lemma 4.3.

Let $\beta\in L^{\infty}([0,\ell];\mathbb{R})$ and $\beta$ do not change sign
on $[0,\ell]$. Set $\rho(x):=\int_{0}^{x}\beta(t)dt$. Let $R\in
X_{\infty,0}(\Omega)$. Then for any $\delta>0$ there exists $R_{\delta}>0$
such that

$\left|{\int_{0}^{x}R(x,t)e^{i\lambda\rho(t)}\beta(t)dt}\right|<\delta\cdot\left(\left|{e^{i\lambda\rho(x)}}\right|+1\right),\qquad|\lambda|>R_{\delta},\quad
x\in[0,\ell].$ (4.33)

###### Proof.

Let $\varepsilon>0$. By the definition of the space $X_{\infty,0}(\Omega)$,
the inclusion $R\in X_{\infty,0}(\Omega)$ ensures that there exists
$R_{\varepsilon}\in C^{1}(\Omega)$ such that

$\|R-R_{\varepsilon}\|_{X_{\infty}}=\operatorname*{ess\,sup}_{x\in[0,\ell]}\int_{0}^{x}\left|{R(x,t)-R_{\varepsilon}(x,t)}\right|dt<\varepsilon.$
(4.34)

In particular, we get the following uniform estimate

$\left|{\int_{0}^{x}\left(R(x,t)-R_{\varepsilon}(x,t)\right)e^{i\lambda\rho(t)}\beta(t)dt}\right|\leqslant\varepsilon\cdot\|\beta\|_{\infty}\cdot\max_{t\in[0,x]}\left|{e^{i\lambda\rho(t)}}\right|,\quad
x\in[0,\ell],\quad\lambda\in\mathbb{C}.$ (4.35)

Since $R_{\varepsilon}\in C^{1}(\Omega)$, $\rho\in\operatorname{AC}[0,\ell]$
and $\rho^{\prime}=\beta$, integrating by parts we get for $x\in[0,\ell]$ and
$\lambda\neq 0$,

$\left|{\int_{0}^{x}R_{\varepsilon}(x,t)e^{i\lambda\rho(t)}\beta(t)\,dt}\right|=\left|{\int_{0}^{x}\frac{R_{\varepsilon}(x,t)}{i\lambda}\,d\left(e^{i\lambda\rho(t)}\right)}\right|=\frac{1}{|\lambda|}\left|{\int_{0}^{x}e^{i\lambda\rho(t)}\frac{\partial}{\partial
t}R_{\varepsilon}(x,t)\,dt}\right|\\\
\leqslant\frac{1}{|\lambda|}\cdot\max_{t\in[0,x]}\left|{e^{i\lambda\rho(t)}}\right|\cdot\max_{t\in[0,x]}\left|{\frac{\partial}{\partial
t}R_{\varepsilon}(x,t)}\right|\leqslant\frac{\|R_{\varepsilon}\|_{C^{1}(\Omega)}}{|\lambda|}\cdot\max_{t\in[0,x]}\left|{e^{i\lambda\rho(t)}}\right|.$
(4.36)

Since function $\rho(\cdot)$ is real-valued and strictly monotonous on
$[0,\ell]$, and $\rho(0)=0$, it follows that

$\max_{t\in[0,x]}\left|{e^{i\lambda\rho(t)}}\right|=\max\\{\left|{e^{i\lambda\rho(x)}}\right|,1\\}<\left|{e^{i\lambda\rho(x)}}\right|+1.$
(4.37)

Setting

$\varepsilon=\frac{\delta}{2\|\beta\|_{\infty}},\qquad
R_{\delta}=\frac{2}{\delta}\|R_{\varepsilon}\|_{C^{1}(\Omega)},$ (4.38)

and combining estimates (4.35)–(4.37) we arrive at the desired estimate
(4.33). ∎

Going forward, for $h\geqslant 0$ we will denote by $\Pi_{h}$ a horizontal
strip of semi-width $h$ symmetrical with respect to the real line:

$\Pi_{h}:=\\{\lambda\in\mathbb{C}:|\operatorname{Im}\lambda|\leqslant
h\\},\qquad\Pi_{0}:=\mathbb{R}.$ (4.39)

######  Corollary 4.4.

Let matrix functions $B(\cdot)$ and $Q(\cdot)$ satisfy conditions (4.3)–(4.8).
Let $h\geqslant 0$ and $p\in\\{1,\ldots,{n}\\}$. Then the following uniform
asymptotic formula holds:

$\Phi_{p}(x,\lambda)=\Phi_{p}^{0}(x,\lambda)+o(1),\quad
x\in[0,\ell],\quad\text{as}\quad\lambda\to\infty,\quad\lambda\in\Pi_{h}.$
(4.40)

###### Proof.

Proposition 4.2 implies representation (4.23). Due to condition (4.22), Lemma
4.3 implies the following uniform at $x\in[0,\ell]$ asymptotic behavior for
integrals in (4.23):

$\int_{0}^{x}R_{q}^{[p]}(x,t)e^{i\lambda\rho_{p}(t)}\beta_{p}(t)\,dt=o\left(\left|{e^{i\lambda\rho_{q}(x)}}\right|+1\right),\quad\text{as}\quad\lambda\to\infty,\quad\lambda\in\mathbb{C},\qquad
q\in\\{1,\ldots,{n}\\}.$ (4.41)

Since functions $\rho_{q}(\cdot)$ are real-valued and bounded, these estimates
imply that all integrals in (4.23) are $o(1)$ uniformly at $x\in[0,\ell]$ as
$\lambda\to\infty$ and $\lambda\in\Pi_{h}$, which finishes the proof. ∎

The following representation will be useful for studying characteristic
determinant of the BVP (5.1)–(5.2). Let us set

$\rho_{1}^{-}(x)=\min\\{\rho_{1}(x),0\\},\qquad\rho_{n}^{+}(x)=\max\\{\rho_{n}(x),0\\},\qquad
x\in[0,\ell].$ (4.42)

It follow from (4.16) that

$\rho_{1}^{-}(x)\leqslant
0\leqslant\rho_{n}^{+}(x),\qquad[0,\rho_{q}(x)]\subset[\rho_{1}^{-}(x),\rho_{n}^{+}(x)],\quad
x\in[0,\ell],\quad q\in\\{1,\ldots,{n}\\}.$ (4.43)

Here $[0,\rho_{q}(x)]$ means $[\rho_{q}(x),0]$ if $\rho_{q}(x)<0$.

######  Corollary 4.5.

Let matrix functions $B(\cdot)$ and $Q(\cdot)$ satisfy conditions (4.3)–(4.8).
Let $p\in\\{1,\ldots,{n}\\}$. Then there exists a measurable vector kernel
$\widetilde{R}_{p}$ defined on
$\widetilde{\Omega}:=\\{(x,u):x\in[0,\ell],u\in[\rho_{1}^{-}(x),\rho_{n}^{+}(x)]\\}$
such that for each $x\in[0,\ell]$, a trace function
$\widetilde{R}_{p}(x,\cdot)$ is well-defined, summable,

$\sup_{x\in[0,\ell]}\int_{\rho_{1}^{-}(x)}^{\rho_{n}^{+}(x)}\left\|{\widetilde{R}_{p}(x,u)}\right\|_{\mathbb{C}^{n}}\,du<\infty,$
(4.44)

and the following representation holds

$\Phi_{p}(x,\lambda)=\Phi_{p}^{0}(x,\lambda)+\int_{\rho_{1}^{-}(x)}^{\rho_{n}^{+}(x)}\widetilde{R}_{p}(x,u)e^{i\lambda
u}\,du,\qquad x\in[0,\ell],\quad\lambda\in\mathbb{C}.$ (4.45)

###### Proof.

Proposition 4.2 implies representation (4.23). Let $q\in\\{1,\ldots,{n}\\}$ be
fixed and consider the integral in (4.23) that contains $R_{q}^{[p]}(x,t)$.
Condition (4.18) allows us to make a change of variable $u=\rho_{q}(t)$ in
this integral, which with account of (4.43) yields

$\int_{0}^{x}R_{q}^{[p]}(x,t)e^{i\lambda\rho_{q}(t)}\beta_{q}(t)\,dt=\int_{0}^{\rho_{q}(x)}R_{q}^{[p]}(x,\rho_{q}^{-1}(u))e^{i\lambda
u}\,du=\int_{\rho_{1}^{-}(x)}^{\rho_{n}^{+}(x)}\widetilde{R}_{q}^{[p]}(x,u)e^{i\lambda
u}\,du,$ (4.46)

where

$\widetilde{R}_{q}^{[p]}(x,u)=\begin{cases}R_{q}^{[p]}(x,\rho_{q}^{-1}(u)),&u\in[0,\rho_{q}(x)],\\\
0,&u\in[\rho_{1}^{-}(x),\rho_{n}^{+}(x)]\setminus[0,\rho_{q}(x)].\end{cases}$
(4.47)