# Lyapunov function and smooth periodic solutions to quasilinear 1D hyperbolic
systems

Irina Kmit Viktor Tkachenko Institute of Mathematics, Humboldt University of
Berlin. On leave from the Institute for Applied Problems of Mechanics and
Mathematics, Ukrainian National Academy of Sciences, Lviv. E-mail:
<EMAIL_ADDRESS>of Mathematics, National Academy of Sciences
of Ukraine, Kyiv. E-mail<EMAIL_ADDRESS>
###### Abstract

We apply a Lyapunov function to obtain conditions for the existence and
uniqueness of small classical time-periodic solutions to first order
quasilinear 1D hyperbolic systems with (nonlinear) nonlocal boundary
conditions in a strip. The boundary conditions cover different types of
reflections from the boundary as well as integral operators with delays. In
the first step we use a Lyapunov approach to derive sufficient conditions for
the robust exponential stability of the boundary value problems for a
linear(ized) homogeneous problem. Under those conditions and a number of non-
resonance conditions, in the second step we prove the existence and uniqueness
of smooth time-periodic solutions to the corresponding linear nonhomogeneous
problems. In the third step, we prove a perturbation theorem stating that the
periodic solutions survive under small perturbations of all coefficients of
the hyperbolic system. In the last step, we apply the linear results to
construct small and smooth time-periodic solutions to the quasilinear
problems.

Keywords: linear and quasilinear first order hyperbolic systems, (nonlinear)
nonlocal boundary conditions, robust exponential stability, Lyapunov function,
classical time-periodic solutions

##  1 Introduction

###  1.1 Problem setting and main result

In the domain $\Pi=\\{(x,t)\in{\mathbb{R}}^{2}\,:\,0\leq x\leq 1\\}$ we
consider first order quasilinear hyperbolic systems of the type

$\partial_{t}u+A(x,t,u)\partial_{x}u=F(x,t,u),\quad x\in(0,1),$ (1)

where $u=(u_{1},\ldots,u_{n})$ and $F=(F_{1},\dots,F_{n})$ are vectors of
real-valued functions, $A=\mathop{\rm diag}\nolimits(A_{1},\dots,A_{n})$ is a
matrix with real-valued entries, and $n\geq 2$ is an integer. Suppose that in
some neighbohrhood of $u=0$ in ${\mathbb{R}}^{n}$ we have

$\displaystyle
A_{1}(x,t,u)>\dots>A_{m}(x,t,u)>0>A_{m+1}(x,t,u)>\dots>A_{n}(x,t,u)$ (2)

for all $(x,t)\in\Pi$ and an integer $m$ in the range $0\leq m\leq n$.

We supplement the system (1) with the boundary conditions

$\displaystyle\begin{array}[]{ll}u_{j}(0,t)=R_{j}u(\cdot,t)+H_{j}\left(t,Q_{j}(t)u(\cdot,t)\right),\quad
1\leq j\leq m,\\\\[2.84526pt]
u_{j}(1,t)=R_{j}u(\cdot,t),+H_{j}\left(t,Q_{j}(t)u(\cdot,t)\right),\quad
m<j\leq n.\end{array}$ (5)

Here $R_{j}$ are linear bounded operators from $C([0,1];{\mathbb{R}}^{n})$ to
${\mathbb{R}}$ defined by

$\displaystyle\displaystyle
R_{j}v=\sum\limits_{k=m+1}^{n}r_{jk}v_{k}(0)+\sum\limits_{k=1}^{m}r_{jk}v_{k}(1),\quad
j\leq n,$ (6)

$r_{jk}$ are real constants, and $H_{j}(t,v)$ are real valued functions from
${\mathbb{R}}\times{\mathbb{R}}$ to ${\mathbb{R}}$. Moreover, for any
$t\in{\mathbb{R}}$, $Q_{j}(t)$ are linear bounded operators from
$C([0,1];{\mathbb{R}}^{n})$ to ${\mathbb{R}}$. Write

$\begin{array}[]{ll}R=(R_{1},\dots,R_{n}),\quad
Q(t)=\left(Q_{1}(t),\dots,Q_{n}(t)\right),\\\\[5.69054pt]
H(t,Q(t)v)=(H_{1}(t,Q_{1}(t)v),\dots,H_{n}(t,Q_{n}(t)v)).\end{array}$

The purpose of the paper is to establish conditions for the existence and
uniqueness of small classical time-periodic solutions to the problem (1), (5).
Since the nonlinearities $H_{j}$ in (5) will be treated by a perturbation
argument, using the concept of a Lyapunov function, we first establish
conditions for the robust exponential stability of the linearized homogeneous
problem without the nonlinearities in (5) (when $H_{j}\equiv 0$ for all $j\leq
n$). Next we prove the existence and uniqueness of classical time-periodic
solutions to the nonhomogeneous linear problem, under the conditions of the
robust exponential stability and a number of nonresonance conditions. Based on
this result and its proof, we then prove a perturbation theorem stating that
the time periodic solutions to the linearized problem survive under small
perturbations of the coefficients of the linear hyperbolic system. Finally, we
apply the obtained results for constructing the classical time-periodic
solutions to the quasilinear problem.

We will denote by $\langle\cdot,\cdot\rangle$ the scalar product in
${\mathbb{R}}^{n}$. We will use the same notation $\|\cdot\|$ for norms of
vectors and matrices of different sizes. Thus, if $A$ is an $k$-vector, then
$\|A\|=\max_{j\leq k}|A_{j}|$. If $A$ is an $k\times k$-matrix, then
$\|A\|=\max_{i,j\leq k}|A_{ij}|$. By ${\mathbb{M}}_{k}$ we denote the space of
$k\times k$-matrices with real entries.

For a given subdomain $\Omega$ of $\Pi$ or ${\mathbb{R}}$, let
$BC(\Omega;{\mathbb{R}}^{n})$ be the Banach space of all bounded and
continuous maps $u:\Omega\to{\mathbb{R}}^{n}$ with the $\sup$-norm

$\|u\|_{BC(\Omega;{\mathbb{R}}^{n})}=\sup\left\\{\|u(z)\|\,:\,z\in\Omega\right\\}.$

Analogously one introduces the spaces $BC^{k}_{t}(\Omega;{\mathbb{R}}^{n})$ of
functions $u\in BC(\Omega;{\mathbb{R}}^{n})$ such that
$\partial_{t}u,\dots,\partial_{t}^{k}u\in BC(\Omega;{\mathbb{R}}^{n}),$ with
norm

$\|u\|_{BC_{t}^{k}(\Omega;{\mathbb{R}}^{n})}=\sum_{j=0}^{k}\|\partial_{t}^{j}u\|_{BC(\Omega;{\mathbb{R}}^{n})}.$

We also use the spaces $BC^{k}(\Omega;{\mathbb{R}}^{n}),k=1,2$, of bounded and
$k$-times continuously differentiable functions.

Let $T>0$ be a fixed positive real. Denote by
$C_{per}(\Omega;{\mathbb{R}}^{n})$ the vector space of all continuous maps
$u:\Omega\to{\mathbb{R}}^{n}$ which are $T$-periodic in $t$, endowed with the
$BC$-norm. For $k=0,1,2$, we also introduce spaces

$\begin{array}[]{ll}C_{per}^{k}(\Omega;{\mathbb{R}}^{n})=BC^{k}(\Omega;{\mathbb{R}}^{n})\cap
C_{per}(\Omega;{\mathbb{R}}^{n})\end{array}$

and write $C_{per}(\Omega;{\mathbb{R}}^{n})$ for
$C_{per}^{0}(\Omega;{\mathbb{R}}^{n})$.

For the space of linear bounded operators from $X$ to $Y$ we use the usual
notation ${\cal L}(X,Y)$, and write ${\cal L}(X)$ for ${\cal L}(X,X)$. The
Sobolev space of all vector-functions from
$L^{2}\left((0,1),{\mathbb{R}}^{n}\right)$ whose distributional derivatives
belong to the same space $L^{2}\left((0,1),{\mathbb{R}}^{n}\right)$ is
denoted, as usual, by $H^{1}\left((0,1),{\mathbb{R}}^{n}\right)$.

Suppose that the data of the problem (1), (5) satisfy the following
conditions.

  * (A1)

There exists $\delta_{0}>0$ such that

    * –

the coefficients $A(x,t,v)$ and $F(x,t,v)$ are $T$-periodic in $t$ and have
bounded and continuous partial derivatives up to the second order in
$x\in[0,1]$, $t\in{\mathbb{R}}$, and $v\in{\mathbb{R}}^{n}$ with
$\|v\|\leq\delta_{0}$,

    * –

there exists $a_{0}>0$ such that for all $\|v\|\leq\delta_{0}$ it holds

$\begin{array}[]{lll}\inf\left\\{A_{j}(x,t,v)\,:\,(x,t)\in\Pi,\,1\leq j\leq
m\right\\}\geq a_{0},\\\\[2.84526pt]
\sup\left\\{A_{j}(x,t,v)\,:\,(x,t)\in\Pi,\,m+1\leq j\leq n\right\\}\leq-
a_{0},\\\\[2.84526pt]
\inf\left\\{|A_{j}(x,t,v)-A_{k}(x,t,v)|\,:\,(x,t)\in\Pi,\,1\leq j\neq k\leq
n\right\\}\geq a_{0}.\end{array}$ (7)

  * (A2)

$Q(t)$ is a a one-parameter family (where $t\in{\mathbb{R}}$ serves as the
parameter) of homogeneous bounded linear operators mapping
$C([0,1];{\mathbb{R}}^{n})$ to ${\mathbb{R}}^{n}$. Moreover, the family
$Q(t):C([0,1];{\mathbb{R}}^{n})\to{\mathbb{R}}^{n}$ is $T$-periodic and two-
times continuously differentiable in $t$. Furthermore, for all $v\in
BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})$ and $j\leq n$, it holds

$\begin{array}[]{rcl}\displaystyle\frac{d}{dt}[Q_{j}(t)v(\cdot,t)]&=&\displaystyle
Q_{j}^{\prime}(t)v(\cdot,t)+\widetilde{Q}_{j}(t)\partial_{t}v(\cdot,t),\\\\[5.69054pt]
\displaystyle\frac{d^{2}}{dt^{2}}[Q_{j}(t)v(\cdot,t)]&=&\displaystyle
Q_{0j}(t)v(\cdot,t)+Q_{1j}(t)\partial_{t}v(\cdot,t)+Q_{2j}(t)\partial_{t}^{2}v(\cdot,t),\end{array}$
(8)

where $Q_{j}^{\prime}(t),\widetilde{Q}_{j}(t),Q_{0j}(t),Q_{1j}(t),Q_{2j}(t)$
are certain one-parameter families of homogeneous bounded linear operators
mapping $C([0,1];{\mathbb{R}}^{n})$ to ${\mathbb{R}}$. Moreover, these
families are continuous and $T$-periodic in $t$.

  * (A3)

For all $j\leq n$, the function $H_{j}(t,v)$ is $T$-periodic in $t$ and have
continuous partial derivatives up to the second order in $t\in{\mathbb{R}}$
and $v\in{\mathbb{R}}$.

To formulate our main result, let us consider the problem (1), (5) with
$H_{j}\equiv 0$ for all $j\leq n$. The homogeneous part of a linearized
version of it at $u=0$ reads as follows:

$\partial_{t}u+a(x,t)\partial_{x}u+b(x,t)u=0,\quad x\in(0,1),$ (9)
$\begin{array}[]{ll}u_{j}(0,t)=R_{j}u(\cdot,t),\quad 1\leq j\leq
m,\\\\[2.84526pt] u_{j}(1,t)=R_{j}u(\cdot,t),\quad m<j\leq n,\end{array}$ (10)

where $a(x,t)=A(x,t,0)$, $b(x,t)=\partial_{3}F(x,t,0)$, and $\partial_{j}$
here and in what follows denotes the partial derivative with respect to the
$j$-th argument. Note that, due to (7), it holds

$\displaystyle\begin{array}[]{ll}\inf\left\\{a_{j}(x,t)\,:\,(x,t)\in\Pi,\,1\leq
j\leq m\right\\}\geq a_{0},\\\\[2.84526pt]
\sup\left\\{a_{j}(x,t)\,:\,(x,t)\in\Pi,\,m+1\leq j\leq n\right\\}\leq-
a_{0},\\\\[2.84526pt]
\inf\left\\{|a_{j}(x,t)-a_{k}(x,t)|\,:\,(x,t)\in\Pi,\,1\leq j\neq k\leq
n\right\\}\geq a_{0}.\end{array}$ (14)

For an arbitrary fixed initial time $s\in{\mathbb{R}}$, we supplement the
system (9)–(10) with the initial conditions

$\displaystyle u(x,s)=\varphi(x),\quad x\in[0,1].$ (15)

We start with formulating a result from [23] about the existence and
uniqueness of $L^{2}$-generalized solutions to the problem (9), (10), (15).
Due to [21, Theorem 2.1], for any $\varphi\in
C_{0}^{1}([0,1];{\mathbb{R}}^{n})$ and sufficiently smooth $a$ and $b$, the
problem (9), (10), (15) has a unique classical solution.

###### Definition 1.1

Let $\varphi\in L^{2}((0,1);{\mathbb{R}}^{n})$ and $a,b\in
BC^{1}(\Pi,{\mathbb{M}}_{n})$. A function $u$ that belongs to
$C\left([s,\infty);L^{2}\left((0,1);{\mathbb{R}}^{n}\right)\right)$ is called
an $L^{2}$-generalized solution to the problem (9), (10), (15) if, for any
sequence $\varphi^{l}\in C_{0}^{1}([0,1];{\mathbb{R}}^{n})$ with
$\varphi^{l}\to\varphi$ in $L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$, the
sequence $u^{l}\in C^{1}\left([0,1]\times[s,\infty);{\mathbb{R}}^{n}\right)$
of classical solutions to (9), (10), (15) with $\varphi$ replaced by
$\varphi^{l}$ fulfills the convergence

$\|u(\cdot,\theta)-u^{l}(\cdot,\theta)\|_{L^{2}\left((0,1);{\mathbb{R}}^{n}\right)}\to
0\mbox{ as }l\to\infty,$

uniformly in $\theta$ varying in the range $s\leq\theta\leq s+s_{1}$, for
every $s_{1}>0$.

###### Theorem 1.2

[23] Suppose that the coefficients $a$ and $b$ of the system (9) belong to
$BC^{1}(\Pi;{\mathbb{M}}_{n})$ and the conditions (14) are fulfilled. Then,
for given $s\in{\mathbb{R}}$ and $\varphi\in
L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$, there exists a unique
$L^{2}$-generalized solution $u:\Pi\to{\mathbb{R}}^{n}$ to the problem (9),
(10), (15).

To formulate our main result, some notations are in order. For given $j\leq
n$, $x\in[0,1]$, and $t\in{\mathbb{R}}$, the $j$-th characteristic of (9)
passing through the point $(x,t)\in\Pi$ is defined as the solution

$\xi\in[0,1]\mapsto\omega_{j}(\xi)=\omega_{j}(\xi,x,t)\in{\mathbb{R}}$

of the initial value problem

$\partial_{\xi}\omega_{j}(\xi,x,t)=\frac{1}{a_{j}(\xi,\omega_{j}(\xi,x,t))},\;\;\omega_{j}(x,x,t)=t.$
(16)

For $i=0,1,2$ and $j\leq n$, we introduce the notation

$c_{j}^{i}(\xi,x,t)=\exp\int_{x}^{\xi}\left[\frac{b_{jj}}{a_{j}}-i\frac{\partial_{t}a_{j}}{a_{j}^{2}}\right](\eta,\omega_{j}(\eta))\\!\;\mathrm{d}\eta,\
\ d_{j}^{i}(\xi,x,t)=\frac{c_{j}^{i}(\xi,x,t)}{a_{j}(\xi,\omega_{j}(\xi))}.$
(17)

We write $c_{j}(\xi,x,t)$ and $d_{j}(\xi,x,t)$ for $c_{j}^{0}(\xi,x,t)$ and
$d_{j}^{0}(\xi,x,t)$, respectively.

Thanks to (14), the characteristic curve $\tau=\omega_{j}(\xi,x,t)$ passing
through the point $(x,t)\in\Pi$ reaches the boundary of $\Pi$ in two points
with distinct ordinates. Let $x_{j}$ denote the abscissa of that point whose
ordinate is smaller. Notice that the value of $x_{j}$ does not depend on $x$
and $t$ and, due to (14), it holds

$x_{j}=\left\\{\begin{array}[]{rl}0&\mbox{ if }\ 1\leq j\leq m\\\ 1&\mbox{ if
}\ m<j\leq n.\end{array}\right.$ (18)

Introduce operators $G_{0},G_{1},G_{2}\in{\cal
L}(BC({\mathbb{R}},{\mathbb{R}}^{n}))$ by

$[G_{i}\psi]_{j}(t)=c_{j}^{i}(x_{j},1-x_{j},t)\sum\limits_{k=1}^{n}r_{jk}\psi_{k}(\omega_{j}(x_{j},1-x_{j},t))\quad\mbox{for
all }j\leq n,\ \ i=0,1,2.$ (19)

###### Definition 1.3

Let $a,b\in BC^{1}(\Pi;{\mathbb{M}}_{n})$.

${\bf(i)}$ We say that the problem (9)–(10) is exponentially stable in $L^{2}$
if there exist constants $M\geq 1$ and $\alpha>0$ such that, for all
$s\in{\mathbb{R}}$ and $\varphi\in L^{2}((0,1);{\mathbb{R}}^{n})$, the
$L^{2}$-generalized solution $u$ to the problem (9), (10), (15) fulfills the
exponential decay estimate

$\|u(\cdot,t)\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\leq
Me^{-\alpha(t-s)}\|\varphi\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\quad\mbox{for all
}t\geq s.$ (20)

${\bf(ii)}$ We say that the problem (9)–(10) is robust exponentially stable in
$L^{2}$ if there exist constants $\gamma>0$, $M\geq 1$, and $\alpha>0$ such
that, for all $s\in{\mathbb{R}}$, $\varphi\in L^{2}((0,1);{\mathbb{R}}^{n})$,
and $\tilde{a},\tilde{b}\in BC^{1}(\Pi;{\mathbb{M}}_{n})$ with
$\|\tilde{a}-a\|_{BC^{1}(\Pi;{\mathbb{M}}^{n})}\leq\gamma$ and
$\|\tilde{b}-b\|_{BC^{1}(\Pi;{\mathbb{M}}^{n})}\leq\gamma$, the
$L^{2}$-generalized solution $\tilde{u}$ to the problem (9), (10), (15) with
$a$ and $b$ replaced by $\tilde{a}$ and $\tilde{b}$, respectively, fulfills
the (uniform) exponential decay estimate

$\|\tilde{u}(\cdot,t)\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\leq
Me^{-\alpha(t-s)}\|\varphi\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\quad\mbox{for all
}t\geq s.$ (21)

We are prepared to formulate the main result of the paper.

###### Theorem 1.4

Let the conditions $\bf(A1)$–$\bf(A3)$ be fulfilled. Assume also that the
conditions

$\displaystyle\|G_{i}\|_{{\cal L}(BC({\mathbb{R}},{\mathbb{R}}^{n}))}<1$ (22)

are satisfied for $i=0,1,2$ and the problem (9)–(10) is robust exponentially
stable in $L^{2}$. Then there exist $\varepsilon>0$ and $\delta>0$ such that,
if $\|F_{j}(\cdot,\cdot,0)\|_{BC_{t}^{2}(\Pi)}\leq\varepsilon$,
$\|H_{j}(\cdot,0)\|_{BC^{2}({\mathbb{R}})}\leq\varepsilon$, and
$\|\partial_{2}H_{j}(\cdot,0)\|_{BC^{1}({\mathbb{R}})}\leq\varepsilon$ for all
$j\leq n$, then the problem (1), (5) has a unique T-periodic classical
solution $u^{*}\in C^{2}_{per}(\Pi,{\mathbb{R}}^{n})$ such that
$\|u^{*}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}\leq\leavevmode\nobreak\ \delta$.

The paper is organized as follows. We complete the current introduction
section with Sections 1.2 and 1.3 including related work and some remarks on
our problem. In Section 2 we give a thorough analysis of linear problems.
Specifically, in Section 2.1 we prove that, if the homogeneous problem
(9)–(10) is exponentially stable and dissipative, then the nonhomogeneous
linear problem (23), (10) has a unique smooth time-periodic solution, for
every right hand side (Theorem 2.1). In Section 2.2 we use the Lyapunov
approach and derive sufficient conditions for the robust exponential stability
of (9)–(10) in $L^{2}$ (Theorem 2.2). We also give an example showing that the
conditions of Theorem 2.1 are consistent (Example 2.3). The time-periodic
solution survives under small perturbations of the coefficients of the linear
hyperbolic system and satisfies a uniform a priori estimate, what is the scope
of Theorem 2.4 in Section 2.3. Our main result stated in Theorem 1.4 about the
local existence and the local uniqueness of small time-periodic solutions to
the quasilinear problem is proved in Section 3.

###  1.2 Remarks

####  1.2.1 Condition (2) on the slope of the characteristic curves and
correctness of the problem (1), (5).

Accordingly to (2) and (5), the number of the boundary conditions posed on the
left hand side and on the right hand side of the boundary of $\Pi$ is strongly
related to the number of characteristic curves coming to the boundary as time
decreases. Remark that for the well-posedness of time-periodic hyperbolic
problems this relation is not essential and can easily be dropped. We
nevertheless keep this relation in our assumptions, because our approach to
solving quasilinear problems involves the corresponding linear(ized) initial
boundary value problems (in particular, the exponential stability issue for
those problems), where this relation is crucial for the well-posedness.

####  1.2.2 Dissipativity conditions (22) and small divisors.

The complication with small divisors naturally appears when addressing the
existence of time-periodic solutions to hyperbolic PDEs. The dissipativity or,
the same, the non-resonant conditions (22) on the coefficients of the
linearized problem are sufficient conditions served to prevent small divisors
(or resonances) from coming up. An interesting feature observed in our
previous work (see [25, Remark 1.4] and [26, Section 3.6]) is that the number
of those conditions in the nonautonomous setting essentially depends on the
regularity of the continuous solutions. Any higher order of the solution
regularity requires additional non-resonance conditions.

####  1.2.3 Exponential stability condition and existence of time-periodic
solutions.

The exponential stability condition not only ensures a "regular" behavior of
the linearized problem, but also plays a crucial role in proving the
solvability of the nonhomogeneous version of this problem. In particular, it
contributes to the uniqueness part of the proof and, in combination with the
Fredholm property, leads to the desired solvability.

####  1.2.4 Robust exponential stability condition and perturbation theorem.

The robustness part of the exponential stability assumption in our main
Theorem 1.4 plays an important role in the proof of the perturbation Theorem
2.4. Whether this part can be omitted here seems to be an interesting open
question. Remark that the conditions provided by Theorem 2.4 and derived by
the Lyapunov approach are sufficient conditions for the robust exponential
stability required in Theorem 1.4. In other words, Theorem 2.4 justifies the
robust exponential stability condition.

###  1.3 Related work

As mentioned above, one of the crucial assumptions in our main Theorem 1.4 is
the existence and robustness of the exponential stability of the linearized
problem. A number of (sufficient) conditions for these assumptions are
available in the literature. These include the spectral criterion [1, 31, 33,
36], the resolvent criterion [11, Theorem 1.11, p. 302], and the Lyapunov
approach (see, e.g. [1, 2, 3, 10, 13, 15, 16, 17]). How these criteria and
approaches can be used in concrete problems is illustrated by Example 2.3 of
the present paper, as well as by [29, Examples 1.7 and 1.8].

The robustness issue of exponential dichotomy, especially exponential
stability, has been extensively studied in the literature (see, e.g., [7, 18,
34]). These results are hardly applicable to hyperbolic PDEs in the case of
unbounded perturbations, mostly due to the well-known phenomenon of loss of
regularity. In [27, 28] we proved a robustness theorem for the system (9)–(10)
with the boundary conditions (10) of a smoothing type described in [22, 23].
In [20] the robustness issue is addressed in the case of the periodic boundary
conditions, which are, evidently, not of the smoothing type. There, the
authors give a number of sufficient conditions for robust exponential
stability. For 1D hyperbolic problems with more general boundary conditions,
the robustness issue remains a challenging open problem.

Note that the existence of small time-periodic solutions to quasilinear
hyperbolic problems has been investigated by other approaches in [30, 37, 38,
40, 41, 42]. The main difference lies in the linear analysis performed in
Section 2. More precisely, the main point is establishing conditions for the
unique classical solvability of the nonhomogeneous time-periodic linearized
problems. A number of sufficient conditions (different from ours) for this
solvability are obtained in [3, 9, 19, 23, 25, 29, 32, 35].

Speaking about potential applications of the models described by our system
(1), (5), we would like to mention the one-dimensional version of the
classical Saint-Venant system for shallow water in open channels and its
generalizations [5, 14], the Saint-Venant–Exner model [9, 35, 39], and the
one-dimensional Euler equations [12, 40, 41]. The system (1), (5) is also used
to describe rate-type materials in viscoelasticity [8, 32] and the
interactions between heterogeneous cancer cells [4].

##  2 Linear systems

###  2.1 Existence of time-periodic solutions

In this section we investigate a nonhomogeneous version of the problem
(9)–(10). Specifically, we consider the general linear nonhomogeneous
hyperbolic system

$\partial_{t}u+a(x,t)\partial_{x}u+b(x,t)u=f(x,t),\quad x\in(0,1),$ (23)

subjected to the linear nonhomogeneous boundary conditions

$\begin{array}[]{ll}u_{j}(0,t)=R_{j}u(\cdot,t)+h_{j}(t),\quad 1\leq j\leq
m,\\\\[2.84526pt] u_{j}(1,t)=R_{j}u(\cdot,t)+h_{j}(t),\quad m<j\leq
n.\end{array}$ (24)

The next theorem is a natural generalization of [29, Theorem 1.6] for the case
of the inhomogeneous boundary conditions.

###### Theorem 2.1

${\bf(i)}$ Assume that $a,b\in C^{1}_{per}(\Pi;{\mathbb{M}}_{n})$, $h\in
C^{1}_{per}({\mathbb{R}};{\mathbb{R}}^{n})$, and $f\in
C_{per}(\Pi;{\mathbb{R}}^{n})\cap BC^{1}_{t}(\Pi);{\mathbb{R}}^{n})$.
Moreover, suppose that the problem (9)–(10) is exponentially stable in
$L^{2}$. If the conditions (22) are fulfilled for $i=0$ and $i=1$, then the
system (23)–(24) has a unique classical time-periodic solution $u^{*}\in
C_{per}^{1}(\Pi;{\mathbb{R}}^{n})$. Moreover, the a priori estimate

$\|u^{*}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\leq
L_{1}\left(\|f\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})}\right)$
(25)

is fulfilled for a constant $L_{1}$ not depending on $f$ and $h$.

${\bf(ii)}$ Assume, additionally, that $a,b\in
BC_{t}^{2}(\Pi;{\mathbb{M}}_{n})$, $h\in
C^{2}_{per}({\mathbb{R}};{\mathbb{R}}^{n})$, and $f\in
BC^{1}(\Pi;{\mathbb{R}}^{n})\cap BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})$ and that
the condition (22) is true for $i=2$. Then $u^{*}\in
C^{2}_{per}(\Pi;{\mathbb{R}}^{n})$. Moreover, the a priori estimate

$\|u^{*}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}\leq
L_{2}\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC\left(\Pi;{\mathbb{R}}^{n}\right)}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right)$
(26)

is fulfilled for a constant $L_{2}$ not depending on $f$ and $h$.

Proof.  The proof essentially repeats the proof of [29, Theorem 1.6], where a
difference lies in the boundary conditions, which are inhomogeneous now. We
anyway provide this proof for the sake of completeness and for our further
purposes.

First write down the problem (23)–(24) in an equivalent operator form. To this
end, introduce operators $C,D,S\in{\cal L}(BC(\Pi;{\mathbb{R}}^{n}))$ and
$P\in{\cal L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))$ by

$\begin{array}[]{lll}&&[Cu]_{j}(x,t)=c_{j}(x_{j},x,t)R_{j}u(\cdot,\omega_{j}(x_{j},x,t))),\\\\[5.69054pt]
&&[Ph]_{j}(x,t)=c_{j}(x_{j},x,t)h_{j}(\omega_{j}(x_{j},x,t))),\\\\[5.69054pt]
&&[Du]_{j}(x,t)=\displaystyle-\int_{x_{j}}^{x}d_{j}(\xi,x,t)\sum_{k\neq
j}b_{jk}(\xi,\omega_{j}(\xi,x,t))u_{k}(\xi,\omega_{j}(\xi,x,t))\\!\;\mathrm{d}\xi,\\\\[5.69054pt]
&&[Sf]_{j}(x,t)=\displaystyle\int_{x_{j}}^{x}d_{j}(\xi,x,t)f_{j}(\xi,\omega_{j}(\xi,x,t))\\!\;\mathrm{d}\xi.\end{array}$
(27)

Then the problem (23)–(24) after the integration along the characteristic
curves can be written in the following operator form:

$\displaystyle u=Cu+Du+Ph+Sf.$ (28)

A function $u\in C_{per}(\Pi;{\mathbb{R}}^{n})$ satisfying the operator
equation (28) pointwise is called a continuous solution  to the problem
(23)–(24).

The proof of the theorem is divided into a number of claims. Specifically, the
proof of Part ${\bf(i)}$ goes through Claims 2.1–2.1, while the proof of Part
${\bf(ii)}$ is done in Claim 2.1–2.1.

In Claims 2.1–2.1 we, therefore, assume that the conditions of Part ${\bf(i)}$
of the theorem are fulfilled.

Claim 1. The operators $I-C:C_{per}(\Pi;{\mathbb{R}}^{n})\to
C_{per}(\Pi;{\mathbb{R}}^{n})$ and $I-C:C_{per}(\Pi;{\mathbb{R}}^{n})\cap
BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})\to C_{per}(\Pi;{\mathbb{R}}^{n})\cap
BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ defined by (27) are bijective.

Proof of Claim. First note that $C$ maps both $C_{per}(\Pi;{\mathbb{R}}^{n})$
into itself and $C_{per}(\Pi;{\mathbb{R}}^{n})\cap
BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ into itself. Hence, to prove that $I-C$ is
bijective from $C_{per}(\Pi;{\mathbb{R}}^{n})$ to itself, it sufffices to show
that the system

$\begin{array}[]{rcl}u_{j}(x,t)&=&[Cu]_{j}(x,t)+g_{j}(x,t)\\\\[5.69054pt]
&=&c_{j}(x_{j},x,t)R_{j}u(\cdot,\omega_{j}(x_{j},x,t))+g_{j}(x,t),\quad j\leq
n,\end{array}$ (29)

is uniquely solvable in $BC(\Pi;{\mathbb{R}}^{n})$ for any $g\in
C_{per}(\Pi;{\mathbb{R}}^{n})$. Since $Cu$ is a certain vector-function of
$u_{1},\dots,u_{n}$ evaluated at the boundary of $\Pi$, we first consider (29)
at $x=0$ for $m<j\leq n$ and at $x=1$ for $1\leq j\leq m$. On the account of
(19), we get the following system:

$\begin{array}[]{rcl}z_{j}(t)&=&\displaystyle
c_{j}(x_{j},1-x_{j},t)\sum\limits_{k=1}^{n}r_{jk}z_{k}(\omega_{j}(x_{j},1-x_{j},t))+g_{j}(1-x_{j},t)\\\\[11.38109pt]
&=&[G_{0}z]_{j}(t)+g_{j}(1-x_{j},t),\quad j\leq n,\end{array}$ (30)

with respect to the new unknown

$z(t)=(z_{1}(t),\dots,z_{n}(t))=(u_{1}(1,t),\dots,u_{m}(1,t),u_{m+1}(0,t),\dots,u_{n}(0,t)).$

Evidently, the system (29) is uniquely solvable in $BC(\Pi;{\mathbb{R}}^{n})$
if and only if

$I-G_{0}\mbox{ is bijective from }BC^{l}({\mathbb{R}};{\mathbb{R}}^{n})\mbox{
to }BC^{l}({\mathbb{R}};{\mathbb{R}}^{n})$ (31)

for $l=0$. This statement straightforwardly follows from the condition (22)
for $i=0$, and this is the scope of the first part of the claim.

As it follows from the above, to prove that the operator $I-C$ is bijective
from $C_{per}(\Pi;{\mathbb{R}}^{n})\cap BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ to
itself, we have to prove the condition (31) for $l=1$. To this end, we use one
of the equivalent norms in the space $BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})$,
namely the norm

$\|v\|_{\sigma}=\|v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\|\partial_{t}v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})},$
(32)

where a positive constant $\sigma$ will be specified later. We are done if we
prove that there exist constants $\sigma<1$ and $\nu<1$ such that

$\|G_{0}v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\left\|\frac{d}{dt}[G_{0}v]\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\leq\nu\left(\|v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\|v^{\prime}\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\right)$
(33)

for all $v\in BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})$. For given $v\in
BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})$ and $j\leq n$, it holds

$\frac{d}{dt}[G_{0}v]_{j}(t)=[Wv]_{j}(t)+[G_{1}v^{\prime}]_{j}(t),$ (34)

where the operator $G_{1}$ is given by (19) and the operator
$W\in\mathcal{L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))$ is defined by

$[Wv]_{j}(t)=\partial_{t}c_{j}(x_{j},1-x_{j},t)\sum\limits_{k=1}^{n}r_{jk}v_{k}(\omega_{j}(x_{j},1-x_{j},t)),\quad
j\leq n.$

Because of the assumption (22), we have the bounds
$\left\|G_{0}\right\|_{{\cal L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}<1$ and
$\left\|G_{1}\right\|_{{\cal L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}<1$. Hence,
we can fix $\sigma<1$ such that $\left\|G_{0}\right\|_{{\cal
L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}+\sigma\left\|W\right\|_{{\cal
L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}<1$. Then the number

$\nu=\max\left\\{\left\|G_{0}\right\|_{{\cal
L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}+\sigma\left\|W\right\|_{{\cal
L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))},\left\|G_{1}\right\|_{{\cal
L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}\right\\}<1$

makes the estimate (33) correct. Indeed,

$\begin{array}[]{rcl}\|G_{0}v\|_{\sigma}&\leq&\displaystyle\left\|G_{0}v\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\|Wv\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\left\|G_{1}v^{\prime}\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\\\\[5.69054pt]
&\leq&\displaystyle\nu\left(\|v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\left\|v^{\prime}\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\right)=\nu\|v\|_{\sigma},\end{array}$

as desired. It follows that

$\begin{array}[]{rcl}\|(I-G_{0})^{-1}v\|_{BC_{t}^{1}({\mathbb{R}};{\mathbb{R}}^{n})}&\leq&\displaystyle\frac{1}{\sigma}\|(I-G_{0})^{-1}v\|_{\sigma}\leq\frac{1}{\sigma(1-\nu)}\|v\|_{\sigma}\\\
&\leq&\displaystyle\frac{1}{\sigma(1-\nu)}\|v\|_{BC_{t}^{1}({\mathbb{R}};{\mathbb{R}}^{n})}.\end{array}$
(35)

This implies that the system (30) can be written in the form

$\displaystyle z=(I-G_{0})^{-1}\tilde{g},$ (36)

where
$\tilde{g}(t)=\left(g_{1}(1,t),\dots,g_{m}(1,t),g_{m+1}(0,t),\dots,g_{n}(0,t)\right)$.
Substituting (36) into (29), we obtain

$\begin{array}[]{rcl}u_{j}(x,t)&=&\displaystyle
c_{j}(x_{j},x,t)\sum\limits_{k=1}^{n}r_{jk}\left[(I-G_{0})^{-1}\tilde{g}\right]_{k}(\omega_{j}(x_{j},x,t))+g_{j}(x,t)\\\\[14.22636pt]
&=&\left[(I-C)^{-1}g\right]_{j}(x,t)\end{array}$ (37)

for all $j\leq n$. Combining (37) with (35) gives the estimate

$\|(I-C)^{-1}\|_{{\cal L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}\leq
1+\frac{1}{\sigma(1-\nu)}\|C\|_{{\cal L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}$
(38)

that we will use later on. The proof of the second part of the claim is
therewith complete. $\;\;\;\Box$

Claim 2. The operator $I-C-D$ is Fredholm operator of index zero from
$C_{per}(\Pi;{\mathbb{R}}^{n})$ to itself.

Proof of Claim. By Claim 2.1, the operator $I-C$ is bijective from
$C_{per}(\Pi;{\mathbb{R}}^{n})$ to itself. Hence,
$I-C-D:C_{per}(\Pi;{\mathbb{R}}^{n})\to C_{per}(\Pi;{\mathbb{R}}^{n})$ is
Fredholm opeeator of index zero if and only if
$I-(I-C)^{-1}D:C_{per}(\Pi;{\mathbb{R}}^{n})\to C_{per}(\Pi;{\mathbb{R}}^{n})$
is Fredholm operator of index zero. Accordingly to the Fredholmness criterion
stated in [43, Proposition 1, pp. 298–299], the last statement is true if
there exist linear continuous operators
$P_{l},P_{r}:C_{per}(\Pi;{\mathbb{R}}^{n})\to C_{per}(\Pi;{\mathbb{R}}^{n})$
and compact operators $Q_{l},Q_{r}:C_{per}(\Pi;{\mathbb{R}}^{n})\to
C_{per}(\Pi;{\mathbb{R}}^{n})$ such that

$P_{l}\left[I-(I-C)^{-1}D\right]=I+Q_{l}\quad\mbox{and}\quad\left[I-(I-C)^{-1}D\right]P_{r}=I+Q_{r}.$

Put $P_{l}=P_{r}=I+(I-C)^{-1}D.$ Then we have

$\begin{array}[]{rcl}P_{l}\left[I-(I-C)^{-1}D\right]=\left[I-(I-C)^{-1}D\right]P_{r}=I-(I-C)^{-1}D(I-C)^{-1}D.\end{array}$

Accordingly to [43, Proposition 1, pp. 298–299], the proof will, therefore, be
done if we prove the compactness of the operator $(I-C)^{-1}D(I-C)^{-1}D$ or,
on the account of the boundedness of $(I-C)^{-1}$, the compactness of the
operator $D(I-C)^{-1}D:C_{per}(\Pi;{\mathbb{R}}^{n})\to
C_{per}(\Pi;{\mathbb{R}}^{n})$. With this aim, we rewrite this operator as
follows:

$D(I-C)^{-1}D=D\left[I+C(I-C)^{-1}\right]D=D^{2}+DC(I-C)^{-1}D.$

Because of the boundedness of $(I-C)^{-1}D$, we are done if we prove the
compactness of $D^{2}$ and $DC$. This follows from [25, Equation (4.2)], where
it is shown that the operators $D^{2}$ and $DC$ map continuously
$C_{per}(\Pi;{\mathbb{R}}^{n})$ into $C_{per}^{1}(\Pi;{\mathbb{R}}^{n})$. The
main idea of the Fredholmness proof above lies in the fact that we regularize
the operator $I-(I-C)^{-1}D$ by means of the operators
$P_{l}=P_{r}=I+(I-C)^{-1}D$. The main point of this regularisation is that the
Fredholmness of the regularized and original operators are equivalent.
Specifically, the operator $D$, being a partial integral operator, is not
compact. At the same time, the ‘‘regularized’’ operators $D^{2}$ and $DC$ are
already integral operators and, hence, are compact. This completes the proof
of the claim. $\;\;\;\Box$

Claim 3. The operator $I-C-D:C_{per}(\Pi;{\mathbb{R}}^{n})\to
C_{per}(\Pi;{\mathbb{R}}^{n})$ is injective.

Proof of Claim. Assume, contrary to this claim, that, for given $f$ and $h$,
the equation (28) has two continuous periodic solutions, say $u^{1}$ and
$u^{2}$. Then the difference $u^{1}-u^{2}$ satisfies the equation (28) with
$f=0$ and $h=0$ and, consequently, the following iterated equation:

$\displaystyle u^{1}-u^{2}=C(u^{1}-u^{2})+[DC+D^{2}](u^{1}-u^{2}).$ (39)

As mentioned above, the operators $D^{2}$ and $DC$ map continuously
$C_{per}(\Pi;{\mathbb{R}}^{n})$ into $C_{per}^{1}(\Pi;{\mathbb{R}}^{n})$.
Moreover, accordingly to Claim 2.1, the operator $(I-C)^{-1}$ maps
$C_{per}(\Pi;{\mathbb{R}}^{n})\cap BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})$ into
itself. Hence, the equation (39) implies that $u^{1}-u^{2}\in
BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})$. Using additionally that $u_{1}-u_{2}$ is a
distributional solution to (23), we get that $u^{1}-u^{2}\in
C_{per}^{1}(\Pi;{\mathbb{R}}^{n})$ and, therefore, that $u^{1}-u^{2}$ is a
classical time-periodic solution to (23)–(24).

Let $s\in{\mathbb{R}}$ be arbitrary fixed. Then the function
$u^{1}(\cdot,s)-u^{2}(\cdot,s)$ belongs to $C([0,1];{\mathbb{R}}^{n})$ and,
hence, to $L^{2}((0,1);{\mathbb{R}}^{n})$. By Definition 1.1, the function
$u^{1}-u^{2}$ is also the $L^{2}$-generalized solution to the problem (23),
(24), (15). Due to the exponential decay estimate (20), $u^{1}-u^{2}$ decays
to zero as $t\to\infty$. As $u^{1}-u^{2}$ is $T$-periodic, this yields that
$u^{1}(x,t)=u^{2}(x,t)=0$ for all $x\in[0,1]$ and $t\in{\mathbb{R}}$. The
injectivity is therewith proved. $\;\;\;\Box$

Claim 4. The operator $I-C-D:C_{per}(\Pi;{\mathbb{R}}^{n})\to
C_{per}(\Pi;{\mathbb{R}}^{n})$ is bijective and, hence, the problem (23)–(24)
has a unique continuous time-periodic solution, say $u^{*}$. Moreover, $u^{*}$
satisfies the following a priori estimate:

$\|u^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
K\left(\|f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\right)$
(40)

with a constant $K$ not depending on $f$ and $h$.

Proof of Claim. By Claim 2.1, the problem (28) satisfies the Fredholm
Alternative in $C_{per}(\Pi;{\mathbb{R}}^{n})$. Combining this with Claim 2.1
gives the desired bijectivity of the operator
$I-C-D:C_{per}(\Pi;{\mathbb{R}}^{n})\to C_{per}(\Pi;{\mathbb{R}}^{n})$. In
other words, the problem (23)–(24) has a unique continuous time-periodic
solution $u^{*}$ given by the formula

$u^{*}=[I-C-D]^{-1}(Ph+Sf).$

It follows also that the map $(f,h)\in C_{per}(\Pi;{\mathbb{R}}^{n})\times
C_{per}({\mathbb{R}};{\mathbb{R}}^{n})\to u\in C_{per}(\Pi;{\mathbb{R}}^{n})$
is continuous. The a priori estimate (40) now easily follows.

$\;\;\;\Box$

Claim 5. The continuous time-periodic solution $u^{*}$ to the problem
(23)–(24) belongs to $C^{1}(\Pi;{\mathbb{R}}^{n})$ and satisfies the a priori
estimate (25).

Proof of Claim. We iterate the equation (28) by substituting (28) into the
second summand of this equation, and afterwards apply the operator
$(I-C)^{-1}$ to both sides of the obtained equation. The resulting equation
for $u=u^{*}$ then reads

$\displaystyle
u^{*}=(I-C)^{-1}\left[DC+D^{2}\right]u^{*}+(I-C)^{-1}\left[I+D\right]\left[Ph+Sf\right].$
(41)

We again use the smoothing property of the operators $DC$ and $D^{2}$, that
map $BC$ into $BC^{1}_{t}$ and satisfy the estimate

$\displaystyle\left\|\partial_{t}\left[(DC+D^{2})u\right]\right\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
K_{1}\|u\|_{BC(\Pi;{\mathbb{R}}^{n})}$ (42)

for a constant $K_{1}$ which is uniform for all $u\in
BC(\Pi;{\mathbb{R}}^{n})$ (in particular, for $u=u^{*}$). Since $u^{*}$ is a
distributinal solution to (23), this entails that $u^{*}$ is $BC^{1}$-smooth
and is, therefore, the classical solution to the problem (23)–(24). The a
priori estimate (25) then easily follows from the operator equation (41), from
the estimates (38), (40), (42), and from the differential equation (23). This
finishes the proof of the claim. $\;\;\;\Box$

The proof of Part ${\bf(i)}$ of the theorem is, therefore, complete. Now we
assume that the conditions of Part ${\bf(ii)}$ of the theorem are fulfilled.

Claim 6. The operator $I-C:C_{per}(\Pi;{\mathbb{R}}^{n})\cap
BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})\to C_{per}(\Pi;{\mathbb{R}}^{n})\cap
BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})$ is bijective.

Proof of Claim. The proof follows the same line as the proof of Claim 2.1, but
now we have to show that the system (29) is uniquely solvable in
$BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})$ or, the same, that the system (30) is
uniquely solvable in $BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})$. To this end, we
additionally use the condition (31) for $l=2$. Now we use one of the
equivalent norms in the space $BC_{t}^{2}({\mathbb{R}};{\mathbb{R}}^{n})$,
namely the norm

$\|v\|_{\sigma_{1},\sigma_{2}}=\|v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma_{1}\|\partial_{t}v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma_{2}\|\partial_{t}^{2}v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})},$
(43)

where the constants $\sigma_{1}<1$ and $\sigma_{2}<1$ have to be chosen to
satisfy the inequality

$\begin{array}[]{rr}\displaystyle\|G_{0}v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma_{1}\left\|\frac{d}{dt}[G_{0}v]\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma_{2}\left\|\frac{d^{2}}{dt^{2}}[G_{0}v]\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\\\\[14.22636pt]
\leq\nu\left(\|v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma_{1}\|v^{\prime}\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma_{2}\|v^{\prime\prime}\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\right)\end{array}$

for a constant $\nu<1$ and all $v\in BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})$.
To show the existence of such constants, we differentiate the function (34)
and get

$\begin{array}[]{rcl}\displaystyle\frac{d^{2}}{dt^{2}}[G_{0}v]_{j}(t)&=&\displaystyle\partial_{t}^{2}c_{j}(x_{j},1-x_{j},t)\sum\limits_{k=1}^{n}r_{jk}v_{k}(\omega_{j}(x_{j},1-x_{j},t))+[G_{2}v^{\prime\prime}]_{j}(t)\\\\[11.38109pt]
&&+\left(\partial_{t}c_{j}(x_{j},1-x_{j},t)\partial_{t}\omega_{j}(x_{j};1-x_{j},t)+\partial_{t}c_{j}^{1}(x_{j},1-x_{j},t)\right)\\\\[8.53581pt]
&&\times\displaystyle\sum\limits_{k=1}^{n}r_{jk}v_{k}^{\prime}(\omega_{j}(x_{j},1-x_{j},t)).\end{array}$

Further we proceed similarly to the proof of Claim 2.1, but now taking
additionally into account the condition (22) for $i=2$. Moreover, similarly to
(38), we derive the estimate

$\|(I-C)^{-1}\|_{{\cal L}(BC^{2}_{t}(\Pi;{\mathbb{R}}^{n}))}\leq
1+\frac{1}{\max\\{\sigma_{1},\sigma_{2}\\}(1-\nu)}\|C\|_{{\cal
L}(BC^{2}_{t}(\Pi;{\mathbb{R}}^{n}))}.$ (44)

$\;\;\;\Box$

Claim 7. The classical time-periodic solution $u^{*}$ to the problem (23)–(24)
belongs to $C^{2}_{per}(\Pi;{\mathbb{R}}^{n})$ and satisfies the a priori
estimate (26).

Proof of Claim. One can easily show that, under the regularity assumptions on
the coefficients of (23), the operators $DC$ and $D^{2}$ have the smoothing
property of the type that they map $BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ into
$BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})$. Moreover, the following smoothing estimate
is fulfilled:

$\displaystyle\left\|\partial_{t}^{2}\left[(DC+D^{2})u\right]\right\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
K_{2}\|u\|_{BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})}$ (45)

for some $K_{2}>0$ and all $u\in BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ (in
particular, for $u=u^{*}$). Since $u^{*}$ satisfies (41), the claim now
follows from Claim 2.1, the estimates (25), (44), (45), and the differential
system (23) and its differentiations in $t$ and $x$. $\;\;\;\Box$

The proof of the theorem is complete.

$\;\;\;\Box$

###  2.2 Lyapunov function and sufficient conditions for the robust
exponential stability in $L^{2}$

In this section we consider general linear initial-boundary value problem of
the type (9), (10), (15). Write $u=(u^{1},u^{2}),$ where
$u^{1}=(u_{1},\dots,u_{m})$ and $u^{2}=(u_{m+1},\dots,u_{n}).$ We will also
use the following notation:

$\displaystyle z(t)=\left[\begin{array}[]{c}u^{1}(1,t)\\\ u^{2}(0,t)\\\
\end{array}\right],\ J_{0}=\left[\begin{array}[]{cc}R_{11}&R_{12}\\\ 0&I\\\
\end{array}\right],\ J_{1}=\left[\begin{array}[]{cc}I&0\\\ R_{21}&R_{22}\\\
\end{array}\right],$ (52)

where $R_{11}=(r_{jk})_{j,k=1}^{m},\ R_{12}=(r_{jk})_{j=1,k=m+1}^{m,n},\
R_{21}=(r_{jk})_{j=m+1,k=1}^{n,m},\ R_{22}=(r_{jk})_{j,k=m+1}^{n},$ and
$r_{jk}$ are the boundary coefficients from (6). Then the boundary conditions
(10) can be written in the form

$\displaystyle u(0,t)=J_{0}z(t),\quad u(1,t)=J_{1}z(t).$ (53)

###### Theorem 2.2

Let $a,b\in BC^{1}(\Pi;{\mathbb{M}}_{n})$. Assume that there exist positive
constants $\beta_{i}$, $i\leq 4$, and a diagonal matrix $V(x,t)=\mathop{\rm
diag}\nolimits(V_{1}(x,t),\dots,V_{n}(x,t))$ with continuously differentiable
entries in $\Pi$ such that the following conditions are satisfied:
${\bf(i)}$ $\beta_{1}\|v\|^{2}\leq\langle Vv,v\rangle\leq\beta_{2}\|v\|^{2}$
for all $(x,t)\in\Pi$ and $v\in{\mathbb{R}}^{n}$,
${\bf(ii)}$
$\left\langle\left(\partial_{t}V+\partial_{x}(Va)-Vb-b^{T}V\right)v,v\right\rangle<-\beta_{3}\|v\|^{2}$
for all $(x,t)\in\Pi$ and $v\in{\mathbb{R}}^{n}$,
${\bf(iii)}$
$\left\langle\left(J_{0}^{T}V(0,t)a(0,t)J_{0}-J_{1}^{T}V(1,t)a(1,t)J_{1}\right)v,v\right\rangle<-\beta_{4}\|v\|^{2}$
for all $t\in{\mathbb{R}}$ and $v\in{\mathbb{R}}^{n}$.
Then the problem (9)–(10) is robust exponentially stable in
$L^{2}((0,1),{\mathbb{R}}^{n})$.

Proof.  Due to the continuous embedding
$C_{0}^{1}([0,1];{\mathbb{R}}^{n})\hookrightarrow
L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$ and Definition 1.1 of the
$L^{2}$-generalized solution, it suffices to prove the estimate (21) for any
$\varphi\in C_{0}^{1}([0,1];{\mathbb{R}}^{n})$. Fix an arbitrary $\varphi\in
C_{0}^{1}([0,1];{\mathbb{R}}^{n})$ and denote by $u$ the classical solution to
the problem (9), (10), (15) (see [21, Theorem 2.1]).

Set ${\cal V}(t)=\int_{0}^{1}\langle
V(x,t)u(x,t),u(x,t)\rangle\\!\;\mathrm{d}x$. The derivative of ${\cal V}(t)$
along the trajectories of the system (9)–(10) is calculated directly, as
follows:

$\begin{array}[]{rcl}\displaystyle\frac{d{\cal
V}}{dt}&=&\displaystyle\int_{0}^{1}\left[\left\langle\partial_{t}Vu,u\right\rangle+\left\langle
V\partial_{t}u,u\right\rangle+\langle
Vu,\partial_{t}u\rangle\right]\\!\;\mathrm{d}x=\\\\[8.53581pt]
&=&\displaystyle\int_{0}^{1}\left[\left\langle\partial_{t}Vu,u\right\rangle-\left\langle
Va\partial_{x}u,u\right\rangle-\left\langle Vbu,u\right\rangle-\left\langle
Vu,a\partial_{x}u\right\rangle-\left\langle
Vu,bu\right\rangle\right]\\!\;\mathrm{d}x.\end{array}$

Since

$\displaystyle\displaystyle\int_{0}^{1}\partial_{x}\langle
Vau,u\rangle\\!\;\mathrm{d}x=\langle Vau,u\rangle\Bigl{|}_{0}^{1}$
$\displaystyle=\int_{0}^{1}\langle\partial_{x}(Va)u,u\rangle\\!\;\mathrm{d}x+\int_{0}^{1}\langle
Va\partial_{x}u,u\rangle dx+\int_{0}^{1}\langle
Vau,\partial_{x}u\rangle\\!\;\mathrm{d}x,$

the following equality holds:

$\displaystyle\langle
Vau,u\rangle\Bigl{|}_{0}^{1}-\int_{0}^{1}\langle\partial_{x}(Va)u,u\rangle\\!\;\mathrm{d}x=\int_{0}^{1}\langle
Va\partial_{x}u,u\rangle\\!\;\mathrm{d}x+\int_{0}^{1}\langle
Vu,a\partial_{x}u\rangle\\!\;\mathrm{d}x.$

It follows that

$\begin{array}[]{rcl}\displaystyle\frac{d{\cal V}}{dt}&=&\displaystyle-\langle
Vau,u\rangle\Bigl{|}_{0}^{1}+\int_{0}^{1}\langle\left[\partial_{t}V+\partial_{x}(Va)-Vb-b^{T}V\right]u,u\rangle\\!\;\mathrm{d}x.\end{array}$

Taking into account (53), the boundary term in the last equation is computed
as follows:

$\begin{array}[]{rcl}-\langle Vau,u\rangle\Bigl{|}_{0}^{1}&=&\left\langle
V(0,t)a(0,t)u(0,t),u(0,t)\right\rangle-\langle
V(1,t)a(1,t)u(1,t),u(1,t)\rangle\\\\[8.53581pt] &=&\langle
V(0,t)a(0,t)J_{0}z(t),J_{0}z(t)\rangle-\langle
V(1,t)a(1,t)J_{1}z(t),J_{1}z(t)\rangle\\\\[8.53581pt]
&=&\left\langle\left[J_{0}^{T}V(0,t)a(0,t)J_{0}-J_{1}^{T}V(1,t)a(1,t)J_{1}\right]z(t),z(t)\right\rangle.\end{array}$

By Assumption ${\bf(iii)}$, $-\langle Vau,u\rangle\Bigl{|}_{0}^{1}\leq 0$.
Then, thanks to the assumptions ${\bf(i)}$ and ${\bf(ii)}$, it holds

$\frac{d{\cal
V}(t)}{dt}\leq-\beta_{3}\|u(\cdot,t)\|_{L^{2}((0,1),{\mathbb{R}}^{n})}^{2}\leq-\frac{\beta_{3}}{\beta_{2}}{\cal
V}(t)\quad\mbox{for all }t\in{\mathbb{R}}.$

Consequently, ${\cal V}(t)\leq{\cal
V}(s)e^{-\frac{\beta_{3}}{\beta_{2}}(t-s)}$ for all $t\geq s$. Now, using the
assumption ${\bf(i)}$, we derive the inequality

$\|u(t)\|_{L^{2}((0,1),{\mathbb{R}}^{n})}^{2}\leq\frac{\beta_{2}}{\beta_{1}}e^{-\frac{\beta_{3}}{\beta_{2}}(t-s)}\|u(s)\|_{L^{2}(0,1),{\mathbb{R}}^{n})}^{2}\quad\mbox{for
all }t\geq s.$ (54)

Since the conditions ${\bf(ii)}$ and ${\bf(iii)}$ are stable with respect to
small perturbations of $a$ and $b$, the estimate (54) is fulfilled for
sufficiently small perturbations of $a$ and $b$. This comletes the proof.
$\;\;\;\Box$

###### Example 2.3

Here we show that the conditions of Theorem 2.2 are consistent. More
precisely, our aim is to show that the robust exponential stability assumption
does not contradict to the dissipativity conditions (22).

Consider the $2\times 2$-hyperbolic system

$\displaystyle\partial_{t}u_{1}+(2-x)\,\partial_{x}u_{1}+2\sin t\,u_{2}=0$
$\displaystyle\partial_{t}u_{2}-(2+\sin t)\,\partial_{x}u_{2}-\sin
t\,u_{1}+2u_{2}=0$

with the boundary conditions

$u_{1}(0,t)=\frac{1}{e^{3}}u_{1}(1,t)+\frac{1}{2e^{3}}u_{2}(0,t),\quad
u_{2}(1,t)=\frac{1}{e^{3}}u_{1}(1,t)+\frac{1}{e^{3}}u_{2}(0,t).$

Let us first show that the Lyapunov function $V(x,t)$ as in Theorem 2.2 can be
chosen to be the unit diagonal matrix $V(x,t)=\mathop{\rm
diag}\nolimits(1,1)$. In other words, we intend to show that the matrix $V$
satisfies all conditions of Theorem 2.2.

The condition ${\bf(i)}$ is evident. To check ${\bf(ii)}$, we calculate the
matrix

$\displaystyle\partial_{t}V+\partial_{x}(Va)-Vb-b^{T}V$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}-1&0\\\ 0&0\\\
\end{array}\right]-\left[\begin{array}[]{cc}0&2\sin t\\\ -\sin t&2\\\
\end{array}\right]-\left[\begin{array}[]{cc}0&-\sin t\\\ 2\sin t&2\\\
\end{array}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}-1&-\sin t\\\ -\sin t&-4\\\
\end{array}\right].$

Hence, the matrix $\left(\partial_{t}V+\partial_{x}(Va)-Vb-b^{T}V\right)$ is
uniformly negative definite, which means that the condition ${\bf(ii)}$ of
Theorem 2.2 is satisfied.

To verify the condition ${\bf(iii)}$, we calculate the matrix

$\displaystyle
J_{0}^{T}V(0,t)a(0,t)J_{0}-J_{1}^{T}V(1,t)a(1,t)J_{1}=\left[\begin{array}[]{cc}r_{11}&0\\\
r_{12}&1\\\ \end{array}\right]\times\left[\begin{array}[]{cc}2&0\\\ 0&\
-2-\sin t\\\ \end{array}\right]\times\left[\begin{array}[]{cc}r_{11}&r_{12}\\\
0&1\\\ \end{array}\right]$ $\displaystyle-\left[\begin{array}[]{cc}1&r_{21}\\\
0&r_{22}\\\ \end{array}\right]\times\left[\begin{array}[]{cc}1&0\\\ 0&\
-2-\sin t\\\ \end{array}\right]\times\left[\begin{array}[]{cc}1&0\\\
r_{21}&r_{22}\\\ \end{array}\right]$
$\displaystyle=\left[\begin{array}[]{cc}2r_{11}^{2}+(2+\sin t)r_{21}^{2}-1&\
2r_{11}r_{12}+(2+\sin t)r_{21}r_{22}\\\ 2r_{11}r_{12}+(2+\sin t)r_{21}r_{22}&\
2r_{12}^{2}+(2+\sin t)r_{22}^{2}-2\end{array}\right].$

Since $r_{11}=r_{21}=r_{22}=1/e^{3}$ and $r_{12}=1/2e^{3}$, the matrix
$J_{0}^{T}V(0,t)a(0,t)J_{0}-J_{1}^{T}V(1,t)a(1,t)J_{1}$ is strictly uniformly
negative definite, which is the scope of the condition ${\bf(iii)}$.

It remains to check that the dissipativity conditions (22) are fulfilled for
$i=0,1,2$. Indeed, for any $i=0,1,2$ and $\psi\in
BC({\mathbb{R}};{\mathbb{R}}^{n})$ with
$\|\psi\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}=1$, it holds

$\begin{array}[]{rcl}\left|[G_{i}\psi]_{2}(t)\right|&\leq&\displaystyle\exp\left\\{\int_{0}^{1}\left[\frac{2}{2+\sin(\omega_{j}(\eta))}+i\frac{\cos(\omega_{j}(\eta))}{(2+\sin(\omega_{j}(\eta))^{2}}\right]\\!\;\mathrm{d}\eta\right\\}\\\\[14.22636pt]
&&\times\left\|R\right\|_{\mathcal{L}(BC([0,1];{\mathbb{R}}^{n});{\mathbb{R}}^{n})}\leq
e^{2}\left\|R\right\|_{\mathcal{L}(BC([0,1];{\mathbb{R}}^{n});{\mathbb{R}}^{n})}<1,\end{array}$

as desired. Analogously one checks the dissipativity condition (22) for
$[G_{i}\psi]_{1}(t)$.

###  2.3 Perturbation theorem

In this section we prove that the time-periodic solution to the problem
(23)–(24) survives under sufficiently small perturbations of the coefficients
$a$ and $b$ and that the inverse operator is bounded uniformly in those
perturbations. With this aim, we consider the perturbed system

$\partial_{t}u+\tilde{a}(x,t)\partial_{x}u+\tilde{b}(x,t)u=f(x,t),\quad
x\in(0,1),$ (60)

where $\tilde{a}=\mathop{\rm
diag}\nolimits(\tilde{a}_{1},\dots,\tilde{a}_{n})$ and
$\tilde{b}=(\tilde{b}_{jk})_{j,k=1}^{n}$ are matrices of $T$-periodic real-
valued functions.

###### Theorem 2.4

Assume that there exists $\gamma>0$ such that the problem (9)–(10) is robust
exponentially stable in $L^{2}$ in the sense of Definition 1.3 $({\bf ii})$.
Moreover, assume that the condition (14) is fulfilled with $\tilde{a}$ in
place of $a$ for all $\tilde{a}$ with
$\|\tilde{a}-a\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma$. Then the following
is true.

$(\bf i)$ Let the conditions (22) be fulfilled for $i=0,1$. Then there exists
$\gamma_{1}\leq\gamma$ such that, for given $f\in
C_{per}(\Pi;{\mathbb{R}}^{n})\cap BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$, $h\in
C^{1}_{per}({\mathbb{R}};{\mathbb{R}}^{n})$, and $\tilde{a},\tilde{b}\in
C_{per}^{2}(\Pi,{\mathbb{M}}_{n})$ with
$\|\tilde{a}-a\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$ and
$\|\tilde{b}-b\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1},$ the problem
(60), (24) has a unique classical $T$-periodic solution, say $\tilde{u}$.
Moreover, $\tilde{u}$ satisfies the uniform a priori estimate

$\|\tilde{u}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\leq
L_{3}\left(\|f\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})}\right)$
(61)

with a constant $L_{3}$ not depending on $\tilde{a},\tilde{b}$, $f$, and $h$.

$(\bf ii)$ Let the conditions (22) be fulfilled for $i=0,1,2$. Then there
exists $\gamma_{1}\leq\gamma$ such that, for given $f\in
C_{per}^{1}(\Pi;{\mathbb{R}}^{n})\cap BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})$, $h\in
C_{per}^{2}({\mathbb{R}},{\mathbb{R}}^{n})$, and $\tilde{a},\tilde{b}\in
C_{per}^{2}(\Pi;{\mathbb{M}}_{n})$ with
$\|\tilde{a}-a\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$ and
$\|\tilde{b}-b\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$, the classical
solution $\tilde{u}$ to the problem (60), (24) belongs to
$C_{per}^{2}(\Pi,{\mathbb{R}}^{n})$ (see Theorem 2.1 $({\bf ii})$) and
satisfies the uniform a priori estimate

$\|\tilde{u}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}\leq
L_{4}\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right)$
(62)

with a constant $L_{4}$ not depending on $\tilde{a},\tilde{b}$, $f$, and $h$.

Proof.  Let the conditions of Part $(\bf i)$ of the theorem be fulfilled.
Denote by $\widetilde{\omega}_{j}(\xi,x,t)$ the continuous solution to the
problem (16) with $a_{j}$ replaced by $\tilde{a}_{j}$. We will write
$\widetilde{G}_{i}$, $\widetilde{C}$, $\widetilde{D}$, $\widetilde{P}$, and
$\widetilde{S}$ for the operators $G_{i}$, $C$, $D$, $P$, and $S$ defined by
(19) and (27) where $a$, $b$, and $\omega$ are replaced by $\tilde{a}$ and
$\tilde{b}$, and $\widetilde{\omega}$, respectively. Fix
$\gamma_{1}\leq\gamma$ so small that the inequalities
$\|\widetilde{G}_{0}\|_{{\cal L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}<c_{0}$
and $\|\widetilde{G}_{1}\|_{{\cal
L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}<c_{0}$ are fulfilled for some $c_{0}<1$
and all $\tilde{a},\tilde{b}\in C_{per}^{2}(\Pi,{\mathbb{M}}_{n})$ varying in
the range $\|\tilde{a}-a\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$ and
$\|\tilde{b}-b\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$. Let $f\in
C_{per}(\Pi;{\mathbb{R}}^{n})\cap BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$, $h\in
C^{1}_{per}({\mathbb{R}};{\mathbb{R}}^{n})$, and $\tilde{a},\tilde{b}\in
C_{per}^{2}(\Pi;{\mathbb{M}}_{n})$ in the range
$\|\tilde{a}-a\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$ and
$\|\tilde{b}-b\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$ be arbitrary
fixed. Then, by Part $({\bf i})$ of Theorem 2.1, the problem (60), (24) has a
unique classical $T$-periodic solution, that will be denoted by $\tilde{u}$.
Moreover, $\tilde{u}$ satisfies the a priori estimate (61) with a constant
$L_{3}$ not depending on $f$ and $h$ (but possibly depending on $\tilde{a}$
and $\tilde{b}$). To show that $L_{3}$ can be chosen to be the same for all
$\tilde{a}$ and $\tilde{b}$, we first prove for $\tilde{u}$ the uniform upper
bound as in (40). In other words, we have to prove that the following
inequality is true for all $g\in C_{per}(\Pi,{\mathbb{R}}^{n})$:

$\left\|[I-\widetilde{C}-\widetilde{D}]^{-1}g\right\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
K_{3}\|g\|_{BC(\Pi;{\mathbb{R}}^{n})},$ (63)

where the constant $K_{3}$ can be chosen to be the same for all $\tilde{a}$
and $\tilde{b}$ in the $\gamma_{1}$-ball of $a$ and $b$.

To prove (63), assume, on the contrary, that there exist sequences
$(\tilde{a}^{l})$ and $(\tilde{b}^{l})$ with
$C_{per}^{2}(\Pi;{\mathbb{M}}_{n})$-elements, $(g^{l})$ with
$C_{per}^{2}(\Pi;{\mathbb{R}}^{n})$-elements, and the corresponding them
sequences $(\widetilde{\omega}^{l}(\xi))$ of functions defining characteristic
curves of the system (60), such that

$\|v^{l}\|_{BC(\Pi;{\mathbb{R}}^{n})}\geq
l\|g^{l}\|_{BC(\Pi;{\mathbb{R}}^{n})}.$ (64)

Here $v^{l}=(I-\widetilde{C}^{l}-\widetilde{D}^{l})^{-1}g^{l}$,
$\|v^{l}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}=1$ for all $l\in{\mathbb{N}}$, and
$(\widetilde{C}^{l})$ and $(\widetilde{D}^{l})$ are the corresponding
sequences of perturbed operators. It follows that

$\|[I-\widetilde{C}^{l}-\widetilde{D}^{l}]v^{l}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq\frac{1}{l}\|v^{l}\|_{BC(\Pi;{\mathbb{R}}^{n})}.$
(65)

Since the sequences $(\tilde{a}^{l})$ and $(\tilde{b}^{l})$ are uniformly
bounded in $C^{2}_{per}(\Pi;{\mathbb{M}}_{n})$, by the Arzelà–Ascoli theorem,
there exist subsequences of $(\tilde{a}^{l})$ and $(\tilde{b}^{l})$ (here and
below we will keep the same notation for the subsequences) converging in
$C_{per}^{1}(\Pi;{\mathbb{M}}_{n})$ to functions, say $a^{*}$ and $b^{*}$.
Hence, the corresponding subsequence of ($\widetilde{\omega}^{l}(\xi)$)
converges in $BC^{1}([0,1]\times\Pi;{\mathbb{R}}^{n})$ to, say
$\omega^{*}(\xi)$. Note that for all $l\geq 1$ it holds

$\widetilde{\omega}^{l}(\xi;x,t+T)=\widetilde{\omega}^{l}(\xi;x,t)+T$

and, hence

$\omega^{*}(\xi;x,t+T)=\omega^{*}(\xi;x,t)+T.$

This, in its turn, implies that the corresponding subsequence of $(v^{l})$
converges in $C^{1}_{per}(\Pi;{\mathbb{R}}^{n})$ to a function $v^{*}\in
C_{per}^{1}(\Pi;{\mathbb{R}}^{n})$. Note that $v^{*}\not\equiv 0$, since
$\|v^{l}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}=1.$ By $C^{*}$ and $D^{*}\in{\cal
L}(C_{per}(\Pi;{\mathbb{R}}^{n}))$ we denote the limit operators of
$(\widetilde{C}^{l})$ and $(\widetilde{D}^{l})$, respectively. Passing to the
limit as $l\to\infty$ results in the equation

$[I-C^{*}-D^{*}]v^{*}=0.$

This means that $v^{*}$ is a nontrivial continuous time-periodic solution to
the homogeneous problem (60), (24) (where $f\equiv 0$ and $h\equiv 0$) with
$a^{*}$ and $b^{*}$ in place of $\tilde{a}$ and $\tilde{b}$, respectively.
This, in its turn, contradicts to Claim 2.1 in the proof Theorem 2.1 stating
that the operator $I-C^{*}-D^{*}:C_{per}(\Pi;{\mathbb{R}}^{n})\to
C_{per}(\Pi;{\mathbb{R}}^{n})$ is injective. We, therefore, conclude that the
a priori estimate (63) is fulfilled with a constant $K_{3}$ not depending on
$f$, $h$, $\tilde{a}$, $\tilde{b}$. This means, in particular, that

$\begin{array}[]{rcl}\|u^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}&=&\left\|[I-\widetilde{C}-\widetilde{D}]^{-1}(\widetilde{P}h+\widetilde{S}f)\right\|_{BC(\Pi;{\mathbb{R}}^{n})}\\\\[8.53581pt]
&\leq&K_{4}\left(\|f\|_{BC(\Pi,{\mathbb{R}}^{n})}+\|h\|_{BC({\mathbb{R}},{\mathbb{R}}^{n})}\right).\end{array}$
(66)

for a constant $K_{4}$ not depending on $f$, $h$, $\tilde{a}$, $\tilde{b}$.

Next we prove the uniform a priori estimate (61). The operators
$\widetilde{C}$ and $\widetilde{D}$ are bounded in the norm of ${\cal
L}(BC^{1}_{t})$ uniformly in $\tilde{a}$ and $\tilde{b}$ varying in the range
$\|\tilde{a}-a\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$ and
$\|\tilde{b}-b\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$. On the account
of (38) and (42), the following estimates:

$\|(I-\widetilde{C})^{-1}\|_{{\cal L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}\leq
1+\frac{1}{\sigma(1-\nu)}\|\widetilde{C}\|_{{\cal
L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}$ (67)

and

$\displaystyle\left\|\partial_{t}\left[(\widetilde{D}\widetilde{C}+\widetilde{D}^{2})u\right]\right\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
K_{5}\|u\|_{BC(\Pi;{\mathbb{R}}^{n})}$ (68)

are fulfilled for some $\sigma<1$, $\nu<1$, and $K_{5}>0$ uniformly in all
$\tilde{a}$, $\tilde{b}$, and $u$ under consideration. In particular, (68) is
true for $u=u^{*}$. Further, following the proof of Claim 2.1 in Section 2.1
with $\tilde{a}$ and $\tilde{b}$ in place of $a$ and $b$ and taking into
account the estimates (67) and (68), we derive the desired estimate (61). Part
$(\bf i)$ of the theorem is therewith proved.

To prove Part $(\bf ii)$, we fix $\gamma_{1}\leq\gamma$ so small that there
exists $c_{0}<1$ such that $\|\widetilde{G}_{i}\|_{{\cal
L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}<c_{0}$ for $i=0,1,2$ and all
$\tilde{a},\tilde{b}\in C_{per}^{2}(\Pi,{\mathbb{M}}_{n})$ varying in the
range $\|\tilde{a}-a\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$ and
$\|\tilde{b}-b\|_{BC^{2}(\Pi;{\mathbb{M}}_{n})}\leq\gamma_{1}$. Then, by Claim
$(\bf ii)$ of Theorem 2.1, the classical solution $\tilde{u}$ to the problem
(60), (24) belongs to $C_{per}^{2}(\Pi,{\mathbb{R}}^{n})$. To prove the
estimate (62), we take additionally into account that the estimate (45) with
$C$ and $D$ replaced by $\tilde{C}$ and $\tilde{D}$ is uniform in $\tilde{a}$
and $\tilde{b}$ varying in the $\gamma_{1}$-balls of $a$ and $b$,
respectively, with appopriately chosen $K_{2}$. Then, repeating the proof of
Claim 2.1 in Section 2.1 with $\tilde{a}$ and $\tilde{b}$ in place of $a$ and
$b$, respectively, results in the desired estimate (62).

The proof of Theorem 2.4 is complete. $\;\;\;\Box$

##  3 Quasilinear problems: proof of Theorem 1.4.

We first rewrite the system (1) in a suitable form, namely

$\partial_{t}u+A(x,t,u)\partial_{x}u+B(x,t,u)u=f(x,t),$

where $f(x,t)=F(x,t,0)$ and $B(x,t,u)u=F(x,t,0)-F(x,t,u)$. By our assumptions,
the functions $f$ and $B$ are $T$-periodic in $t$.

Let $a_{0}$ be a constant fulfilling the condition (7). Let $\gamma_{1}$ and
$L_{4}$ be constants as in Theorem 2.4 $(\bf ii)$.

Put $u^{0}(x,t)=0.$ Determine a sequence of iterations
$(u^{k}(x,t))_{k=1}^{\infty}$ such that any subsequent iteration $u^{k+1}$ is
the unique classical $T$-periodic solution to the linear system

$\displaystyle\partial_{t}u^{k+1}+A^{k}\partial_{x}u^{k+1}+B^{k}u^{k+1}=f(x,t),\
\ k=0,1,2,\dots$ (69)

subjected to the boundary conditions

$\begin{array}[]{ll}u^{k+1}_{j}(0,t)=R_{j}u^{k+1}(\cdot,t)+H_{j}^{k}(t),\quad
1\leq j\leq m,\\\\[2.84526pt]
u^{k+1}_{j}(1,t)=R_{j}u^{k+1}(\cdot,t)+H_{j}^{k}(t),\quad m<j\leq
n,\end{array}$ (70)

where $H^{k}_{j}(t)=H_{j}(t,Q_{j}(t)u^{k}(\cdot,t))$. Here and below we will
simply write $A^{k}$ and $B^{k}$ for $A(x,t,u^{k}(x,t))$ and
$B(x,t,u^{k}(x,t))$, respectively.

Let us separate the linear part of the boundary operator $H_{j}(t,Q_{j}(t)u)$
near $u=0$, by writing $H_{j}$ as follows:

$\begin{array}[]{rcl}H_{j}(t,Q_{j}(t)u(\cdot,t))&=&h_{j}(t)+\partial_{2}H_{j}(t,0)Q_{j}(t)u(\cdot,t)\\\\[5.69054pt]
&&\hskip-22.76219pt\displaystyle+\left[Q_{j}(t)u(\cdot,t)\right]^{2}\int_{0}^{1}\partial_{2}^{2}H_{j}(t,\sigma
Q_{j}(t)u(\cdot,t))\\!\;\mathrm{d}\sigma,\quad j\leq n,\end{array}$ (71)

where $h(t)=H(t,0)$. Write

$\|Q\|=\sum_{k=0}^{2}\sup\left\\{\left\|\frac{d^{k}}{dt^{k}}Q(t)\right\|_{{\cal
L}\left(BC([0,1];{\mathbb{R}}^{n}),{\mathbb{R}}^{n}\right)}\,:\,t\in{\mathbb{R}}\right\\}$

and set

$\begin{array}[]{ll}\varrho_{1}=\displaystyle\max\left\\{\left\|\partial_{2}H_{j}(\cdot,0)\right\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\,:\,j\leq
n\right\\},\\\\[8.53581pt]
\displaystyle\varrho_{2}=\sup\left\\{\left\|\partial_{2}^{2}H_{j}(\cdot,v(\cdot))\right\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\,:\,\|v\|_{BC^{2}({\mathbb{R}})}\leq\delta_{0}\|Q\|,\,j\leq
n\right\\},\end{array}$ (72)

where the constant $\delta_{0}$ is as in Assumption ${\bf(A1)}$. We will
suppose that the number $\varrho_{1}$ is small enough to fulfill the
inequality $\varrho_{1}\|Q\|<1/L_{4}$ for some $L_{4}$ satisfying the a priori
estimate (62). Set $a^{\psi}(x,t)=A(x,t,\psi(x,t))-A(x,t,0)$ and
$b^{\psi}(x,t)=B(x,t,\psi(x,t))-B(x,t,0).$ Since the functions $A$ and $B$ are
$C^{2}$-smooth, then there exists a positive real

$\delta_{1}\leq\min\left\\{\delta_{0},\frac{1}{\varrho_{2}\|Q\|}\left(\frac{1}{L_{4}\|Q\|}-\varrho_{1}\right)\right\\}$
(73)

such that, for all $\psi\in C^{2}_{per}(\Pi;{\mathbb{R}}^{n})$ with
$\|\psi\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}\leq\delta_{1}$, we have

$\|a^{\psi}\|_{BC^{2}(\Pi;{\mathbb{M}}^{n})}\leq\gamma_{1},\quad\|b^{\psi}\|_{BC^{2}(\Pi;{\mathbb{M}}^{n})}\leq\gamma_{1}.$
(74)

Hence, by Theorem 2.4 ${\bf(ii)}$, for given $\psi\in
C^{2}_{per}(\Pi;{\mathbb{R}}^{n})$, the system

$\partial_{t}u+A(x,t,\psi)\partial_{x}u+B(x,t,\psi)u=f(x,t)$

with the boundary conditions (24) has a unique classical solution $u^{\psi}$,
which belongs to $C^{2}_{per}(\Pi,{\mathbb{R}}^{n})$ and satisfies the a
priori estimate (62).

We divide the proof into a number of claims and proceed similarly to the proof
of [26, Theorem 1].

Claim 1. If $\varrho_{1}\|Q\|<1/L_{4}$ for some $L_{4}$ satisfying the a
priori estimate (62) and

$\begin{array}[]{c}\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\\\\[5.69054pt]
\displaystyle\leq\frac{\delta_{1}}{L_{4}}\left[1-\left(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|\right)\|Q\|L_{4}\right],\end{array}$
(75)

then the problem (69)–(70) generates a sequence $(u^{k})$ of classical time-
periodic solutions $u^{k}\in C^{2}_{per}(\Pi;{\mathbb{R}}^{n})$ such that

$\|u^{k}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}\leq\delta_{1}\quad\mbox{for all
}k\geq 1.$ (76)

Proof of Claim. Note that the first iteration $u^{1}$ satisfies the system
(23)–(24), where $a(x,t)=A(x,t,0)$, $b(x,t)=\partial_{3}F(x,t,0)$, and
$h(t)=H(t,0)$. Due to Theorem 2.1 (ii), the last system has a unique time-
periodic classical solution $u^{1}\in C^{2}_{per}(\Pi;{\mathbb{R}}^{n})$.
Moreover, this solution satisfies the a priori estimate (26) and, obviously,
the weaker estimate (62). Hence, if $f$ and $h$ satisfy the estimate (75),
then, on the account of (73), $u^{1}$ satisfies the estimate (76) with $k=1$.
Furthermore, due to (74), it holds that

$\|A^{1}-A^{0}\|_{BC^{2}(\Pi;{\mathbb{M}}^{n})}\leq\gamma_{1},\quad\|B^{1}-B^{0}\|_{BC^{2}(\Pi;{\mathbb{M}}^{n})}\leq\gamma_{1}.$

Now we use Theorem 2.4 (ii) to state that there exists a unique classical
time-periodic solution $u^{2}\in C^{2}_{per}(\Pi,{\mathbb{R}}^{n})$ to the
system (69)–(70) with $k=1$. Moreover, due to (62), (72), (75), and (76), the
following desired a priori estimate is fulfilled:

$\displaystyle\|u^{2}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}$ $\displaystyle\leq$
$\displaystyle
L_{4}\Bigl{(}\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|H^{1}\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\Bigl{)}$
$\displaystyle\leq$
$\displaystyle\delta_{1}\left[1-\left(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|\right)\|Q\|L_{4}\right]+\delta_{1}\left(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|\right)\|Q\|L_{4}=\delta_{1}.$

We again use (74) to immediately conclude that

$\|A^{2}-A^{0}\|_{BC^{2}(\Pi;{\mathbb{M}}^{n})}\leq\gamma_{1},\quad\|B^{2}-B^{0}\|_{BC^{2}(\Pi;{\mathbb{M}}^{n})}\leq\gamma_{1}.$

Proceeding by induction, assume that the problem (69)–(70) has a unique
classical time-periodic solution $u^{k}$ belonging to
$C^{2}_{per}(\Pi,{\mathbb{R}}^{n})$ and satisfying the bound (76). This
implies, in particular, that

$\|A^{k}-A^{0}\|_{BC^{2}(\Pi;{\mathbb{M}}^{n})}\leq\gamma_{1},\quad\|B^{k}-B^{0}\|_{BC^{2}(\Pi;{\mathbb{M}}^{n})}\leq\gamma_{1}.$

By Theorem 2.4 (ii), the problem (69)–(70) has a unique classical time-
periodic solution $u^{k+1}\in C^{2}_{per}(\Pi,{\mathbb{R}}^{n})$. That this
solution fulfills the bound (76), with $k+1$ in place of $k$, again follows
from (62), (72), (75), and (76). $\;\;\;\Box$

Claim 2. There exists

$\varepsilon_{1}\leq\frac{\delta_{1}}{L_{4}}\left[1-\left(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|\right)\|Q\|L_{4}\right]$

such that, if
$\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}},{\mathbb{R}}^{n})}<\varepsilon_{1},$
then the sequence $(u^{k})$ converges in $C_{per}^{1}(\Pi;{\mathbb{R}}^{n})$
to a classical $T$-periodic solution to the problem (1), (5).

Proof of Claim. Set $z^{k+1}=u^{k+1}-u^{k}.$ The function $z^{k+1}$ belongs to
$C_{per}^{2}(\Pi;{\mathbb{R}}^{n})$ and satisfies the system

$\displaystyle\partial_{t}z^{k+1}+A^{k}\partial_{x}z^{k+1}+B^{k}z^{k+1}=f^{k}(x,t)$
(77)

and the boundary conditions

$\begin{array}[]{ll}z^{k+1}_{j}(0,t)=R_{j}z^{k+1}(\cdot,t)+H_{j}^{k}(t)-H_{j}^{k-1}(t),\quad
1\leq j\leq m,\\\\[2.84526pt]
z^{k+1}_{j}(1,t)=R_{j}z^{k+1}(\cdot,t)+H_{j}^{k}(t)-H_{j}^{k-1}(t),\quad
m<j\leq n,\end{array}$ (78)

for all $\ t\in{\mathbb{R}}$, where

$\displaystyle f^{k}$ $\displaystyle=$
$\displaystyle-\left(B^{k}-B^{k-1}\right)u^{k}-\left(A^{k}-A^{k-1}\right)\partial_{x}u^{k}$
$\displaystyle=$
$\displaystyle-\left[\left\langle\int_{0}^{1}\,\nabla_{u}B_{ij}\left(x,t,\sigma
u^{k}+(1-\sigma)u^{k-1}\right)\\!\;\mathrm{d}\sigma,z^{k}\right\rangle\right]_{i,j=1}^{n}u^{k}$
$\displaystyle-\left[\left\langle\int_{0}^{1}\nabla_{u}A_{ij}\left(x,t,\sigma
u^{k}+(1-\sigma)u^{k-1}\right)\\!\;\mathrm{d}\sigma,z^{k}\right\rangle\right]_{i,j=1}^{n}\partial_{x}u^{k}\,$

and

$\displaystyle
H^{k}_{j}(t)-H_{j}^{k-1}(t)=\partial_{2}H_{j}(t,0)Q_{j}(t)z^{k}(\cdot,t)$
$\displaystyle\qquad\quad+Q_{j}(t)z_{j}^{k}(\cdot,t)\int_{0}^{1}\biggl{\\{}\left[\sigma
Q_{j}(t)u^{k}(\cdot,t)+(1-\sigma)Q_{j}(t)u^{k-1}(\cdot,t)\right]$
$\displaystyle\qquad\quad\times\int_{0}^{1}\partial_{2}^{2}H_{j}\left(t,\sigma_{1}\left[\sigma
Q_{j}(t)u^{k}(\cdot,t)+(1-\sigma)Q_{j}(t)u^{k-1}(\cdot,t)\right]\right)\\!\;\mathrm{d}\sigma_{1}\biggl{\\}}\\!\;\mathrm{d}\sigma.$

Our aim is now to show that the problem (77)–(78) for $z^{k+1}$ fulfills the
conditions of the perturbaion Theorem 2.4 $(\bf i)$. For that, we first derive
an upper bound for $u^{k}$ in terms of $f$ and $h$. Accordingly to the proof
of Claim 3, the first iteration $u^{1}$ satisfies the inequality

$\|u^{1}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}\leq
L_{4}\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right),$

while the second iteration $u^{2}$ satisfies the following inequality (see
(72) and (73)):

$\displaystyle\|u^{2}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}$ $\displaystyle\leq$
$\displaystyle
L_{4}(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|H^{1}\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})})$
$\displaystyle\leq$ $\displaystyle
L_{4}\left(1+(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|)\|Q\|L_{4}\right)$
$\displaystyle\times\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right).$

Proceeding by induction, assume that the problem (69)–(70) has a unique
classical time-periodic solution $u^{k-1}$ belonging to
$C^{2}_{per}(\Pi,{\mathbb{R}}^{n})$ and satisfying the bound

$\begin{array}[]{rcl}\|u^{k-1}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}&\leq&L_{4}\Bigl{(}1+(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|)\|Q\|L_{4}+\dots\\\\[5.69054pt]
&&\hskip-14.22636pt+\left[(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|)\|Q\|L_{4}\right]^{k-2}\Bigr{)}\\\\[5.69054pt]
&&\hskip-14.22636pt\times\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right).\end{array}$
(79)

and, hence, the bound (76). Using (72), (73), (76), and (79), we derive the
following estimate for $u^{k}$:

$\begin{array}[]{ll}\|u^{k}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}\leq
L_{4}\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|H^{k-1}\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right)\\\\[5.69054pt]
\qquad\leq
L_{4}\left(1+(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|)\|Q\|L_{4}+\dots+\left[(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|)\|Q\|L_{4}\right]^{k-1}\right)\\\\[8.53581pt]
\qquad\quad\times\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right)\\\\[5.69054pt]
\qquad\leq\displaystyle\frac{L_{4}}{1-(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|)\|Q\|L_{4}}\\\\[11.38109pt]
\qquad\quad\times\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right).\end{array}$
(80)

It follows that

$\begin{array}[]{ll}\|f^{k}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\\\\[8.53581pt]
\qquad\leq
N_{1}\left(\|u^{k}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{x}u^{k}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\right)\|z^{k}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\\\\[8.53581pt]
\qquad\leq\displaystyle\frac{N_{1}L_{4}}{1-(\varrho_{1}+\varrho_{2}\delta_{1})\|Q\|L_{4}}\|z^{k}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\\\\[11.38109pt]
\qquad\quad\times\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right),\\\\[11.38109pt]
\|H^{k}-H^{k-1}\|_{BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})}\leq\displaystyle\|z^{k}\|_{BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})}\displaystyle\biggl{(}\varrho_{1}+\frac{N_{2}L_{4}}{1-(\varrho_{1}+\varrho_{2}\delta_{1})\|Q\|L_{4}}\\\\[11.38109pt]
\qquad\quad\times\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right)\biggr{)},\end{array}$
(81)

where the constants $N_{1}$ and $N_{2}$ depend on $A$, $B$, $H$, and
$\delta_{1}$ but not on $z^{k}$.

Applying now the estimate (61) to $z^{k+1}$ and using (81), we derive the
inequality

$\displaystyle\begin{array}[]{ll}\|z^{k+1}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\leq
L_{3}\left(\|f^{k}\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}+\|H^{k}-H^{k-1}\|_{BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})}\right)\\\\[8.53581pt]
\qquad\leq\displaystyle\|z^{k}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\biggl{(}\varrho_{1}+\frac{L_{3}L_{4}(N_{1}+N_{2})}{1-(\varrho_{1}+\varrho_{2}\delta_{1})\|Q\|L_{4}}\\\\[11.38109pt]
\qquad\times\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right)\biggr{)}.\end{array}$
(85)

Set

$\varepsilon_{1}=\min\left\\{\frac{\delta_{1}}{L_{4}}\left[1-\left(\varrho_{1}+\delta_{1}\varrho_{2}\|Q\|\right)\|Q\|L_{4}\right],\frac{(1-\varrho_{1})\left[1-(\varrho_{1}+\varrho_{2}\delta_{1})\|Q\|L_{4}\right]}{L_{3}L_{4}(N_{1}+N_{2})}\right\\}.$
(86)

If
$\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}}<\varepsilon_{1},$
then, due to (85), the sequence $(z^{k})$ is contracting and, hence, tends to
zero in $C^{1}_{per}(\Pi;{\mathbb{R}}^{n}).$

Consequently, the sequence $(u^{k})$ converges to a function $u^{*}$ in
$C^{1}_{per}(\Pi;{\mathbb{R}}^{n})$. It follows that $u^{*}$ is a classical
solution to the problem (1), (5). The proof of the claim is complete.
$\;\;\;\Box$

Claim 3. There exist positive reals $\varepsilon\leq\varepsilon_{1}$ and
$\delta\leq\delta_{1}$ such that, if $\varrho_{1}<\varepsilon$ and

$\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\leq\varepsilon,$

then the solution $u^{*}$ belongs to $C^{2}_{per}(\Pi;{\mathbb{R}}^{n})$ and
satisfies the estimate

$\displaystyle\|u^{*}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}\leq\delta.$ (87)

Proof of Claim. It is sufficient to show that the sequence $(u^{k})$ converges
in $C^{2}_{per}(\Pi;{\mathbb{R}}^{n})$. First show that the sequence
$(\partial_{t}u^{k+1})$ converges in $BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})\cap
C_{per}(\Pi;{\mathbb{R}}^{n})$. With this aim, we differentiate the system
(69)–(70) with respect to $t$ and write down the resulting system with respect
to $v^{k+1}=\partial_{t}u^{k+1}$, namely

$\partial_{t}v^{k+1}+A^{k}\partial_{x}v^{k+1}+B^{1}(x,t,u^{k})v^{k+1}=g^{1k}(x,t,v^{k})v^{k+1}+g^{2k}(x,t,v^{k}),$
(88)
$\begin{array}[]{rcl}v_{j}^{k+1}(0,t)&=&R_{j}v^{k+1}(\cdot,t)+h^{\prime}_{j}(t)\\\
&&+\displaystyle\frac{d}{dt}\left[\partial_{2}H_{j}(t,0)Q_{j}(t)u^{k}(\cdot,t)+p_{j}^{k}(t)\right],\
1\leq j\leq m,\\\\[8.53581pt]
v_{j}^{k+1}(1,t)&=&R_{j}v^{k+1}(\cdot,t)+h^{\prime}_{j}(t)\\\
&&\displaystyle+\frac{d}{dt}\left[\partial_{2}H(t,0)Q(t)u^{k}(\cdot,t)+p_{j}^{k}(t)\right],\
m<j\leq n,\end{array}$ (89)

where

$\begin{array}[]{rcl}B^{1}(x,t,u^{k})&=&B^{k}-\partial_{2}A^{k}(A^{k})^{-1},\\\\[5.69054pt]
g^{1k}(x,t,v^{k})&=&\left(\partial_{3}A^{k}v^{k}\right)(A^{k})^{-1},\\\\[5.69054pt]
g^{2k}(x,t,v^{k})&=&-\partial_{2}B^{k}u^{k+1}+\partial_{2}A^{k}\,(A^{k})^{-1}B^{k}u^{k+1}-\left(\partial_{3}B^{k}v^{k}\right)u^{k+1}\\\\[5.69054pt]
&&+\partial_{t}f-\partial_{2}A^{k}\,(A^{k})^{-1}f+\left(\partial_{3}A^{k}\,v^{k}\right)(A^{k})^{-1}(B^{k}u^{k+1}-f),\\\\[5.69054pt]
p_{j}^{k}(t)&=&\displaystyle\left[Q_{j}(t)u^{k}(\cdot,t)\right]^{2}\int_{0}^{1}\partial_{2}^{2}H_{j}(t,\sigma
Q_{j}(t)u^{k}(\cdot,t))\\!\;\mathrm{d}\sigma,\\\\[8.53581pt]
\partial_{3}A^{k}v^{k}&=&\mathop{\rm
diag}\nolimits\left(\langle\nabla_{u}A_{1}(x,t,u^{k}),v^{k}\rangle,\dots,\langle\nabla_{u}A_{n}(x,t,u^{k}),v^{k}\rangle\right),\\\\[8.53581pt]
\partial_{3}B^{k}v^{k}&=&\left[\langle\nabla_{u}B_{ij}(x,t,u^{k}),v^{k}\rangle\right]_{i,j=1}^{n}.\end{array}$

Similarly to $B^{k}$, we will write also $B^{1k}$ for $B^{1}(x,t,u^{k})$. It
is evident that the sequence $(v^{k+1})$ of $T$-periodic solutions to the
problem (88)–(89) converges in $BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ if and only
if the sequence $(w^{k+1})$ with $w^{k+1}=\partial_{t}v^{k+1}$ converges in
$C_{per}(\Pi;{\mathbb{R}}^{n})$. We, therefore, will prove the convergence of
$w^{k+1}$. To this end, by differentiating the system (88) in a distributional
sense in $t$ and the boundary conditions (89) pointwise in $t$ we get the
following problem with respect to $w^{k+1}$:

$\displaystyle\partial_{t}w^{k+1}+A^{k}\partial_{x}w^{k+1}+B^{2}(x,t,u^{k})w^{k+1}={\cal
G}_{1}(k)w^{k+1}+{\cal G}_{2}(k)w^{k}+g^{3k},$ (90)
$\begin{array}[]{rcl}w_{j}^{k+1}(0,t)&=&R_{j}w^{k+1}(\cdot,t)+h^{\prime\prime}_{j}(t)\\\
&&+\displaystyle\frac{d^{2}}{dt^{2}}\left[\partial_{2}H_{j}(t,0)Q_{j}(t)u^{k}(\cdot,t)+p_{j}^{k}(t)\right],\
1\leq j\leq m,\\\\[8.53581pt]
w_{j}^{k+1}(1,t)&=&R_{j}w^{k+1}(\cdot,t)+h^{\prime\prime}_{j}(t)\\\
&&+\displaystyle\frac{d^{2}}{dt^{2}}\left[\partial_{2}H_{j}(t,0)Q_{j}(t)u^{k}(\cdot,t)+p_{j}^{k}(t)\right],\
m<j\leq n,\end{array}$ (91)

where

$\displaystyle B^{2}(x,t,u^{k})$ $\displaystyle=$ $\displaystyle
B^{1k}-\partial_{2}A^{k}\,(A^{k})^{-1}=B^{k}-2\partial_{2}A^{k}\,(A^{k})^{-1},$
$\displaystyle g^{3k}(x,t,v^{k})$ $\displaystyle=$
$\displaystyle\Bigl{(}\partial_{2}g^{1k}-\partial_{3}B^{1k}v^{k}+\left[\left(\partial_{3}A^{k}v^{k}\right)(A^{k})^{-1}+\partial_{2}A^{k}(A^{k})^{-1}\right](B^{1k}-g^{1k}),$
$\displaystyle-\partial_{2}B^{1}\Bigl{)}v^{k+1}+\partial_{2}g^{2k}-\left(\partial_{3}A^{k}v^{k}\right)(A^{k})^{-1}g^{2k}-\partial_{2}A^{k}(A^{k})^{-1}g^{2k},$
$\displaystyle\partial_{3}B^{1}v^{k}$ $\displaystyle=$
$\displaystyle\left[\left\langle\nabla_{u}B_{ij}^{1}(x,t,u^{k}),v^{k}\right\rangle\right]_{i,j=1}^{n},$

and the operators ${\cal G}_{1}(k),{\cal G}_{2}(k)\in{\cal
L}(C_{per}(\Pi;{\mathbb{R}}^{n}))$ are defined by

$\begin{array}[]{rcl}\left[{\cal
G}_{1}(k)w^{k+1}\right](x,t)&=&\left[g^{1k}+\left(\partial_{3}A^{k}v^{k}\right)(A^{k})^{-1}\right]w^{k+1},\\\\[5.69054pt]
\left[{\cal
G}_{2}(k)w^{k}\right](x,t)&=&\left(\partial_{3}A^{k}w^{k}\right)(A^{k})^{-1}v^{k+1}+\left(\partial_{3}A^{k}\,w^{k}\right)(A^{k})^{-1}(B^{k}u^{k+1}-f).\end{array}$
(92)

On the account of (3) and (92), the following estimate is true:

$\displaystyle\|{\cal G}_{2}(k)w^{k}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
L_{5}\delta_{1}\|w^{k}\|_{BC(\Pi;{\mathbb{R}}^{n})},$ (93)

where constant $L_{5}$ does not depend on $w^{k}.$ Because of (8), we have

$\displaystyle\frac{d^{2}}{dt^{2}}Q_{j}(t)u^{k}(\cdot,t)=\displaystyle
Q_{0j}(t)u^{k}(\cdot,t)+Q_{1j}(t)v^{k}(\cdot,t)+Q_{2j}(t)w^{k}(\cdot,t).$ (94)

For every $j\leq n$, decompose

$\displaystyle\frac{d^{2}}{dt^{2}}\left[\partial_{2}H_{j}(t,0)Q_{j}(t)u^{k}(\cdot,t)+p_{j}^{k}(t)\right]={\cal
H}_{1j}(k)+{\cal H}_{2j}(k)w^{k}$

into two summands, where the second part ${\cal H}_{2j}(k)w^{k}$ consists of
all summands with $w^{k}$. Taking into account (94), the summand ${\cal
H}_{2j}(k)w$ allows the representation

${\cal
H}_{2j}(k)w^{k}=\left[\partial_{2}H_{j}(t,0)+2Q_{j}(t)u^{k}(\cdot,t)\int_{0}^{1}\partial_{2}^{2}H_{j}(t,\sigma
Q_{j}(t)u^{k}(\cdot,t))\\!\;\mathrm{d}\sigma\right]Q_{2j}(t)w^{k}$

and, therefore, satisfies the estimate

$\displaystyle\|{\cal H}_{2}(k)w^{k}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
L_{6}(\varrho_{1}+2\delta_{1}\varrho_{2}\|Q\|)\|w^{k}\|_{BC(\Pi;{\mathbb{R}}^{n})}$
(95)

with a constant $L_{6}$ independent of $w^{k}.$

Evidently, the function $w^{k+1}$ satisfies (90) in a distributional sense and
(91) pointwise if and only if it satisfies the following operator equation
obtained after the integration of (90) along characteristic curves:

$\begin{array}[]{rcl}w^{k+1}&=&C(k)w^{k+1}+D(k)w^{k+1}+P(k)\left(h^{\prime\prime}+{\cal
H}_{1}(k)+{\cal H}_{2}(k)w^{k}\right)\\\\[5.69054pt] &&+S(k)\left({\cal
G}_{1}(k)w^{k+1}+{\cal G}_{2}(k)w^{k}+g^{3k}\right),\end{array}$ (96)

where the operators $C(k),D(k),S(k),P(k)\in{\cal
L}(C_{per}(\Pi;{\mathbb{R}}^{n}))$ are defined by

$\displaystyle\begin{array}[]{rcl}\displaystyle[C(k)u]_{j}(x,t)&=&c_{jk}^{2}(x_{j},x,t)R_{j}u(\cdot,\omega_{j}^{k}(x_{j},x,t))),\\\\[5.69054pt]
[P(k)h]_{j}(x,t)&=&c_{jk}^{2}(x_{j},x,t)h_{j}(\omega_{j}(x_{j},x,t))),\\\\[5.69054pt]
\displaystyle[D(k)u]_{j}(x,t)&=&\displaystyle-\int_{x_{j}}^{x}d_{jk}^{2}(\xi,x,t)\sum_{i\neq
j}b_{ji}^{2k}(\xi,\omega_{j}^{k}(\xi,x,t))u_{i}(\xi,\omega_{j}^{k}(\xi,x,t))\\!\;\mathrm{d}\xi,\\\\[5.69054pt]
\displaystyle[S(k)f]_{j}(x,t)&=&\displaystyle\int_{x_{j}}^{x}d_{jk}^{2}(\xi,x,t)f_{j}(\xi,\omega_{j}^{k}(\xi,x,t))\\!\;\mathrm{d}\xi.\end{array}$

Here the functions $b_{ji}^{2k}$ are elements of the matrix $B^{2k}$, while
the functions $\omega_{j}^{k},c_{jk}^{l},d_{jk}^{l}$ are defined by (16) and
(17) with $a_{j}$ and $b_{jj}$ replaced by $A_{j}^{k}$ and $b_{jj}^{k}$,
respectively. Note that
$b_{jj}^{2k}=b_{jj}^{1k}-\partial_{2}A_{j}^{k}(A_{j}^{k})^{-1}=b_{jj}^{k}-2\partial_{2}A_{j}^{k}(A_{j}^{k})^{-1}.$
Analogously, we write $G_{0}(k),G_{1}(k),$ and $G_{2}(k)$ for the operators
$G_{0},G_{1},$ and $G_{2}$ defined by (22) where $\omega_{j}$ is replaced by
$\omega_{j}^{k}$.

After an iteration of (96), we get the equation

$\displaystyle\begin{array}[]{rcl}w^{k+1}&=&C(k)w^{k+1}+\left(D(k)C(k)+D^{2}(k)\right)w^{k+1}\\\\[5.69054pt]
&&+(I+D(k))S(k)\left({\cal G}_{1}(k)w^{k+1}+{\cal
G}_{2}(k)w^{k}+g^{3k}\right)\\\\[5.69054pt]
&&+(I+D(k))P(k)\left(h^{\prime\prime}+{\cal H}_{1}(k)+{\cal
H}_{2}(k)w^{k}\right),\end{array}$ (101)

Now we intend to show that for all sufficiently small $\delta\leq\delta_{1}$
(and, hence for all sufficiently small $u^{k}$, see (76), the last system can
uniquelly be solved with respect to $w^{k+1}$. In other words, the system
(101) can be written in the following form:

$w^{k+1}={\cal A}(k)w^{k}+X^{k},$ (102)

where, for each $k$, we have ${\cal A}(k)\in{\cal
L}(C_{per}(\Pi;{\mathbb{R}}^{n}))$, $X^{k}\in C_{per}(\Pi;{\mathbb{R}}^{n})$,
and

$\begin{array}[]{rcl}{\cal A}(k)w&=&\left[I-C(k)-(I+D(k))S(k){\cal
G}_{1}(k)\right]^{-1}\\\\[5.69054pt] &&\times(I+D(k))S(k){\cal
G}_{2}(k)w+(I+D(k))P(k){\cal H}_{2}(k)w,\\\\[5.69054pt]
X^{k}&=&\left[I-C(k)-(I+D(k))S(k){\cal
G}_{1}(k)\right]^{-1}\Bigl{(}\left(D(k)C(k)+D^{2}(k)\right)w^{k+1}\\\\[5.69054pt]
&&+(I+D(k))S(k)g^{3k}+(I+D(k))P(k)\left(h^{\prime\prime}+{\cal
H}_{1}(k)\right)\Bigl{)}.\end{array}$ (103)

To prove the equivalence of (101) and (102), it suffices to show that, for any
$k\geq 1$, the operator $I-C(k)-(I+D(k))S(k){\cal G}_{1}(k)$ is invertable
and, moreover, the inverse operator is bounded uniformly in
$k\in{\mathbb{Z}}$. We first prove the invertibility of the operators $I-C(k)$
in $C_{per}(\Pi,{\mathbb{R}}^{n})$. To this end, we prove the unique
solvability of the system

$v_{j}(x,t)=c_{jk}^{2}(x_{j},x,t)R_{j}v(\cdot,\omega_{j}^{k}(x_{j},x,t))+g_{j}(x,t),\quad
j\leq n,$ (104)

in $C_{per}(\Pi,{\mathbb{R}}^{n})$ for any $g\in
C_{per}(\Pi,{\mathbb{R}}^{n})$. Putting $x=0$ for $m<j\leq n$ and $x=1$ for
$1\leq j\leq m$ in (104), we get the following system of $n$ equations with
respect to $z(t)=(v_{1}(1,t),\dots,v_{m}(1,t),v_{m+1}(0,t),\dots,v_{n}(0,t))$:

$\begin{array}[]{rcl}z_{j}(t)&=&\displaystyle
c_{jk}^{2}(x_{j},1-x_{j},t)\sum_{i=1}^{n}r_{ji}z_{i}(\omega_{j}^{k}(x_{j},1-x_{j},t))+g_{j}(1-x_{j},t)\\\
&=&[G_{2}(k)z]_{j}(t)+g_{j}(1-x_{j},t),\quad j\leq n.\end{array}$ (105)

Note that the system (104) is uniquely solvable with respect to $v\in
C_{per}(\Pi;{\mathbb{R}}^{n})$ if and only if the system (105) is uniquely
solvable with respect to $z\in C_{per}({\mathbb{R}};{\mathbb{R}}^{n})$.

From the estimate (76) follows the bound $\|G_{2}(k)\|_{{\cal
L}(BC({\mathbb{R}},{\mathbb{R}}^{n}))}<c_{0}<1$, which uniform in
$k\in{\mathbb{N}}$ accordingly to the proof of Theorem 2.4. Hence, the
operator $I-C(k)$ is invertible and, moreover, fulfills the estimate

$\|(I-C(k))^{-1}\|_{{\cal L}(BC(\Pi;{\mathbb{R}}^{n}))}\leq
1+(1-c_{0})^{-1}\|C(k)\|_{{\cal L}(BC(\Pi;{\mathbb{R}}^{n}))}.$

Since the set of all invertible operators having bounded inverse is open, it
suffices to show that the operator $(I+D(k))S(k){\cal G}_{1}(k)$ is
sufficiently small whenever $\delta_{1}$ is sufficiently small. As Claim 3 is
true with $\delta_{2}$ in place of $\delta_{1}$ for any
$\delta_{2}\leq\delta_{1}$, for any $\sigma>0$ there is $\delta_{2}$ such that
for all $u^{k}$ fulfilling (76) with $\delta_{2}$ in place of $\delta_{1}$, we
have $\|{\cal G}_{1}(k)\|_{{\cal L}(BC(\Pi;{\mathbb{R}}^{n}))}\leq\sigma$ for
all $k$. Moreover, the operators $D(k),P(k)$ and $S(k)$ are bounded uniformly
in $k$. Consequently, for any $f$ and $h$ satisfying (75), the operator
$I-C(k)-(I+D(k))S(k){\cal G}_{1}(k)$ is invertible and the inverse is bounded
uniformly in $k$. The desired conclusion about the equivalence of (101) and
(102) follows.

To finish the proof of this claim, it suffices to show that the sequence
$w^{k}$ converges in $C_{per}(\Pi;{\mathbb{R}}^{2})$ as $k\to\infty$. To this
end, we apply to the equation (102) a linear version of the fiber contraction
principle (see [6, Section 1.11.3] ) or, more precisely, a version of it given
by [24, Lemma A.1] and written in terms of our problem as follows.

###### Lemma 3.1

Let the following conditions be fulfilled:

$(i)$ ${\cal A}(k)u$ converges in $C_{per}(\Pi;{\mathbb{R}}^{n})$ as
$k\to\infty$ for all $u\in C_{per}(\Pi;{\mathbb{R}}^{2})$,

$(ii)$ there exists $c<1$ such that $\|{\cal A}(k)\|_{{\cal
L}(BC(\Pi;{\mathbb{R}}^{2}))}\leq c,$ for all $k\in{\mathbb{N}}$,

$(iii)$ $X^{k}\mbox{ converges in }C_{per}(\Pi;{\mathbb{R}}^{n})\mbox{ as
}k\to\infty.$
Then the sequence $w^{k}$ given by (102) converges in
$C_{per}(\Pi;{\mathbb{R}}^{n})$ as $k\to\infty$.

To varify the conditions of this lemma, we follow the same line as in [26, p.
4204–4205]. Thus, since all the operators in the right-hand side of first
formula of $(\ref{oper6*})$ do not depend on $w^{k}$ for all $k$, the
condition $(i)$ of the lemma easily follows from Claim 3.

By (93) and (95), for any $\sigma>0$ there exist $\delta\leq\delta_{2}$ and
$\varrho>0$ such that for all $f$ and $h$ satisfying the bound (75) with
$\delta$ in place of $\delta_{1}$ it holds $\|{\cal G}_{2}(k)\|_{{\cal
L}(BC(\Pi;{\mathbb{R}}^{n}))}+\|{\cal H}_{2}(k)\|_{{\cal
L}(BC(\Pi;{\mathbb{R}}^{n}))}\leq\sigma$. Hence, taking into the account that
all other operators in the right hand side of the first equality in (103) are
bounded uniformly in $k$ whenever $\delta$ is sufficiently small, we state
that, for sufficiently small $\delta$ and $\varrho_{1}$, the condition $(ii)$
is fulfilled for all $f$ and $h$ satisfying the bound (75) with $\delta$ in
place of $\delta_{1}.$

For the remaining property $(iii)$, note that the operators $D(k)$ and $C(k)$
depend neither on $w^{k}$ nor on $v^{k}$ for all $k$. Moreover, the operators
$D(k)C(k)$ and $D(k)^{2}$ are smoothing mapping $BC(\Pi;{\mathbb{R}}^{n}))$
into $BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))$. This entails that the expressions
$D(k)C(k)w^{k+1}$ and $D(k)^{2}w^{k+1}$, actually, do not depend on $w^{k+1}$
but only on $v^{k+1}$. Moreover, analogously to (42), the following estimate
is true:

$\displaystyle\left\|\partial_{t}\left(D(k)C(k)+D(k)^{2}\right)w^{k+1}\right\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
L_{7}\|v^{k+1}\|_{BC(\Pi;{\mathbb{R}}^{n})}$

for some $L_{7}$ not depending on $k$. It follows that the right hand side of
the second formula in (103) does not depend on $w^{k}$ for all $k$ and,
therefore, the convergence of $X^{k}$ immediately follows from Claim 3.

Set

$\varepsilon=\min\left\\{\frac{\delta}{L_{4}}\left[1-\left(\varrho_{1}+\delta\varrho_{2}\|Q\|\right)\|Q\|L_{4}\right],\frac{(1-\varrho_{1})\left[1-(\varrho_{1}+\varrho_{2}\delta)\|Q\|L_{4}\right]}{L_{3}L_{4}(N_{1}+N_{2})}\right\\}.$

Therefore, by assumptions of Claim 3 the sequence $w^{k}$ converges in
$C_{per}(\Pi;{\mathbb{R}}^{n})$ as $k\to\infty$.

Now, by Claim 3, we conclude that the second derivative in $t$ of $u^{*}$
exists and that the sequence $\partial^{2}_{t}u^{k}$ converges to
$\partial^{2}_{t}u^{*}$ in $C_{per}(\Pi;{\mathbb{R}}^{n})$ as $k\to\infty$.
Differentiating (69) first in $t$ and then in $x$, we conclude that $u^{k}$
converges to $u^{*}$ in $C_{per}^{2}(\Pi;{\mathbb{R}}^{n})$ as $k\to\infty$.
It follows also that the estimate (87) is fulfilled. $\;\;\;\Box$

Claim 4. Let the conditions of Claim 3 are fulfilled. Then the bounded
classical solution $u^{*}$ to the problem (1), (5) is unique.

Proof of Claim. Assume, contrary to the claim, that $\tilde{u}$ is a classical
solution to the problem (1), (5) which satisfies the estimate (87) and differs
from $u^{*}$. Then functions $\widetilde{A}(x,t)=A(x,t,\tilde{u}(x,t))$ and
$\widetilde{B}(x,t)=B(x,t,\tilde{u}(x,t))$ satisfy inequalities

$\|\widetilde{A}-A^{0}\|_{BC^{2}(\Pi,{\mathbb{R}}^{n})}\leq\gamma_{1},\quad\|\widetilde{B}-B^{0}\|_{BC^{2}(\Pi,{\mathbb{R}}^{n})}\leq\gamma_{1}.$

The difference $\tilde{y}^{k+1}=u^{k+1}-\tilde{u}$ belongs to
$C^{2}_{per}(\Pi;{\mathbb{R}}^{n})$ and satisfies the system

$\partial_{t}y^{k+1}+A(x,t,\tilde{u})\partial_{x}y^{k+1}+B(x,t,\tilde{u})y^{k+1}=\tilde{f}^{k}(x,t)$

and the boundary conditions

$\displaystyle
y^{k+1}_{j}(0,t)=R_{j}y^{k+1}(\cdot,t)+H_{j}(t,Q_{j}u^{k+1}(\cdot,t))-H_{j}(t,Q_{j}\tilde{u}(\cdot,t)),\
1\leq j\leq m,$ $\displaystyle
y^{k+1}_{j}(1,t)=R_{j}y^{k+1}(\cdot,t)+H_{j}(t,Q_{j}u^{k+1}(\cdot,t))-H_{j}(t,Q_{j}\tilde{u}(\cdot,t)),\
m<j\leq n,$

for all $\ t\in{\mathbb{R}}$, where

$\tilde{f}^{k}(x,t)=\left(\widetilde{B}-B^{k}\right)u^{k+1}+\left(\widetilde{A}-A^{k}\right)\partial_{x}u^{k+1}.$

Analogously to (85), we derive the inequality

$\displaystyle\begin{array}[]{ll}\|y^{k+1}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\leq
L_{3}\left(\|\tilde{f}^{k}\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}+\|H^{k+1}-\widetilde{H}\|_{BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})}\right)\\\\[5.69054pt]
\quad\qquad\leq\displaystyle\displaystyle\|y^{k}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\biggl{(}\varrho_{1}+\frac{L_{3}L_{4}(N_{1}+N_{2})}{1-(\varrho_{1}+\varrho_{2}\delta_{1})\|Q\|L_{4}}\\\\[11.38109pt]
\quad\qquad\times\left(\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}\right)\biggr{)},\end{array}$
(109)

where $\widetilde{H}_{j}(t)=H_{j}(t,Q_{j}\tilde{u}(\cdot,t))$ for all $j\leq
n$. If
$\|f\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|h\|_{BC^{2}({\mathbb{R}};{\mathbb{R}}^{n})}<\varepsilon_{1},$
where $\varepsilon_{1}$ is defined by (86), then, by the inequality (109), the
sequence $y^{k}$ is contracting and, hence tends to zero in
$BC^{1}(\Pi;{\mathbb{R}}^{n}).$ This means that $\tilde{u}=u^{*},$ and this is
the desired contradiction. $\;\;\;\Box$

The proof of Theorem 1.4 is complete.

## References

  * [1] Bastin, G., Coron, J.-M.: Dissipative boundary conditions for one-dimensional quasi-linear hyperbolic systems: Lyapunov stability for the $C^{1}$-norm. SIAM J. Control Optim. 53, 1464–1483 (2015).
  * [2] Bastin, G., Coron, J.-M.: Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Progress in Nonlinear Differential Equations and Their Applications 88, Birkhäuser, 2016.
  * [3] Bastin, G., Coron, J. M., d’Andréa-Novel, B.: On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Networks Heterog. Media 4.2, 177–187 (2009).
  * [4] Bitsouni, V., Eftimie, R.: Non-local parabolic and hyperbolic models for cell polarisation in heterogeneous cancer cell populations. Bull. Math. Biol. 80(10), 2600–2632 (2018).
  * [5] Bouchut, F., Mangeney-Castelnau, A., Perthame, B., Vilotte, J. P.: A new model of Saint-Venant and Savage-Hutter type for gravity driven shallow water flows. Comptes rendus mathematique, 336(6), 531-536 (2003).
  * [6] C. Chicone, Ordinary Differential Equations with Applications, Second Edition. Texts in Appl. Math. 34, Springer, 2006
  * [7] Chow, S.N., Leiva, H.; Unbounded perturbation of the exponential dichotomy for evolution equations. J. Differential Equations 129(2), 509–531 (1996).
  * [8] Cristescu, N.: Dynamic plasticity. Elsevier (2012).
  * [9] Diagne, A., Diagne, M., Tang, S., Krstic, M.: Backstepping stabilization of the linearized Saint-Venant–Exner model. Automatica 76, 345–354 (2017).
  * [10] Dick, M., Gugat, M., Leugering, G.: A strict $H^{1}$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control and Optimization, 1(2), 225-244 (2011).
  * [11] Engel K. J., Nagel R.: One-parameter semigroups for linear evolution equations. Vol. 194, New York: Springer, 2000.
  * [12] Gugat, M., Dick, M., Leugering, G.: Gas flow in fan-shaped networks: Classical solutions and feedback stabilization. SIAM Journal on Control and Optimization, 49(5), 2101–2117 (2011).
  * [13] Gugat, M., Leugering, G., Tamasoiu, S., Wang, K.: $H^{2}$-stabilization of the isothermal Euler equations: a Lyapunov function approach. Chinese Annals of Mathematics, Series B 33(4), 479–500 (2012).
  * [14] de Halleux, J., Prieur, C., Coron, J. M., d’Andréa-Novel, B., Bastin, G.: Boundary feedback control in networks of open channels. Automatica 39(8), 1365–1376 (2003).
  * [15] Hayat, A.: Boundary stability of 1-D nonlinear inhomogeneous hyperbolic systems for the $C^{1}$ norm. SIAM Journal on Control and Optimization 57(6), 3603–3638 (2019).
  * [16] Hayat, A., Shang, P.: A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope. Automatica 100, 52–60 (2019).
  * [17] Hayat, A., Shang, P.: Exponential stability of density-velocity systems with boundary conditions and source term for the $H^{2}$ norm. Journal de mathématiques pures et appliquées 153, 187–212 (2021).
  * [18] Henry, D: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin-New York (1981).
  * [19] Kirilich, V. M., Myshkis, A. D.: A boundary value problem without initial conditions for a linear one-dimensional system of hyperbolic-type equations. Differentsial’nye Uravneniya 28(3), 393–399 (1992).
  * [20] Klyuchnyk, R., Kmit, I., Recke, L.: Exponential Dichotomy for Hyperbolic Systems with Periodic Boundary Conditions. J. Differential Equations 262, 2493–2520 (2017).
  * [21] Kmit I.: Classical solvability of nonlinear initial-boundary problems for first-order hyperbolic systems. Int. J. Dyn. Syst. Differ. Equ. 1(3), 191–195 (2008).
  * [22] Kmit I.: Smoothing solutions to initial-boundary problems for first-order hyperbolic systems. Appl. Anal. 90(11), 1609–1634 (2011).
  * [23] Kmit, I., Lyul’ko, N.: Perturbations of superstable linear hyperbolic systems. J. Math. Anal. Appl. 460(2), 838–862 (2018).
  * [24] Kmit, I., Recke, L.: Hopf bifurcation for semilinear dissipative hyperbolic systems. J. Differential Equations 257, 264–309 (2014).
  * [25] Kmit, I., Recke, L.: Fredholm alternative and solution regularity for time-periodic hyperbolic systems. Differ. Integral Equ. 29(11/12), 1049–1070 (2016).
  * [26] Kmit, I., Recke, L., Tkachenko, V.: Bounded and almost periodic solvability of nonautonomous quasilinear hyperbolic systems. J. Evol. Equ. 21, 4171–4212 (2021).
  * [27] Kmit, I., Recke, L., Tkachenko, V.: Classical bounded and almost periodic solutions to quasilinear first-order hyperbolic systems in a strip. J. Differ. Equ. 269, 2532–2579 (2020).
  * [28] Kmit, I., Recke, L., Tkachenko, V.: Robustness of exponential dichotomies of boundary-value problems for general first-order hyperbolic systems. Ukrainian Math. J. 65(2), 260–276 (2013).
  * [29] Kmit, I., Tkachenko, V.: On the bounded smooth solutions to exponentially stable linear nonautonomous hyperbolic systems, accepted in Comm. Pure Appl. Anal. (2024).
  * [30] Li, Ta Tsien: Global classical solutions for quasilinear hyperbolic systems. RAM Res. Appl. Math., 32. Masson, Paris; John Wiley & Sons, Ltd., Chichester (1994).
  * [31] Lichtner, M.: Spectral mapping theorem for linear hyperbolic systems. Proc. Amer. Math. Soc. 136 (6), 2091–2101 (2008).
  * [32] Manganaro, N., Valenti, G.: Group analysis and linearization procedure for a nonautonomous model describing rate-type materials. Journal of mathematical physics 34(4), 1360–1369 (1993).
  * [33] Neves A. F.,Ribeiro H. de Souza , Lopes O.: On the spectrum of evolution operators generated by hyperbolic systems. J. Funct. Anal. 67 (3), 320–344.(1986).
  * [34] Pliss, V.A., Sell, G.R.: Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dynam. Different. Equat.11, 471–513 (1999).
  * [35] Prieur, C., Winkin, J. J.: Boundary feedback control of linear hyperbolic systems: Application to the Saint-Venant–Exner equations. Automatica 89, 44-51 (2018).
  * [36] Renardy M.: On the type of certain $C^{0}$-semigroups. Commun. Partial Differ. Equ. 18.7-8, 1299–1307 (1993).
  * [37] Qu, P.: Time-periodic solutions to quasilinear hyperbolic systems with time-periodic boundary conditions. J. Math. Pures Appl. 139, 356–382 (2020).
  * [38] Qu, P., Yu, H., Zhang, X.: Subsonic time-periodic solution to compressible Euler equations with damping in a bounded domain. J. Differ. Equ. 352, 122–152 (2023).
  * [39] Siviglia, A., Nobile, G., Colombini, M.: Quasi-conservative formulation of the one-dimensional Saint-Venant–Exner model. Journal of Hydraulic Engineering 134(10), 1521–1526 (2008).
  * [40] Temple, B., Young, R.: Time periodic linearized solutions of the compressible Euler equations and a problem of small divisors. SIAM J. Math. Anal. 43(1), 1–49 (2011).
  * [41] Yuan, H.: Time-periodic isentropic supersonic Euler flows in one-dimensional ducts driving by periodic boundary conditions. Acta Mathematica Scientia 39, 403–412 (2019).
  * [42] Zhang, X., Sun, J., Yu, H.: Subsonic time-periodic solution to the compressible Euler equations triggered by boundary conditions. Nonlinear Anal. Real World Appl. 74, 103954 (2023).
  * [43] Zeidler, E.: Applied Functional Analysis - Applications to Mathematical Physics. Springer, New York (1985).