# Positive eigenvectors and simple nonlinear maps

Marta Calanchi and Carlos Tomei

###### Abstract

For linear operators $\displaystyle L,T$ and nonlinear maps $\displaystyle P$,
we describe classes of simple maps $\displaystyle F=I-PT$, $\displaystyle
F=L-P$ between Banach and Hilbert spaces, for which no point has more than two
preimages. The classes encompass known examples (homeomorphisms, global folds)
and the weaker, geometric, hypotheses suggest new ones. The operator
$\displaystyle L$ may be the Laplacian with various boundary conditions, as in
the original Ambrosetti-Prodi theorem, or the operators associated with the
quantum harmonic oscillator, the hydrogen atom, a spectral fractional
Laplacian, elliptic operators in non-divergent form. The maps $\displaystyle
P$ include the Nemitskii map $\displaystyle P(u)=f(u)$ but may be non-local,
even non-variational. For self-adjoint operators $\displaystyle L$, we employ
familiar results on the nondegeneracy of the ground state. On Banach spaces,
we use a variation of the Krein-Rutman theorem.

Keywords: Ambrosetti-Prodi theorem, folds, Krein-Rutman theorem, positivity
preserving semigroups.

MSC-class: 35J65, 47H11, 47H30, 58K05.

##  1 Introduction and main results

The celebrated Ambrosetti-Prodi theorem provides a geometric description of a
class of differential maps. We state it in the context of Sobolev spaces ([1],
[2], [8]). For a smooth, bounded domain
$\displaystyle\Omega\subset{\mathbb{R}}^{n}$, denote by
$\displaystyle\lambda_{m}<\ \mu_{m}$ the two smallest eigenvalues of the
(Dirichlet) Laplacian $\displaystyle-\Delta_{D}:D=H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)\to L^{2}(\Omega)$ and by $\displaystyle\phi_{m}>0$ an
eigenvector associated with $\displaystyle\lambda_{m}$.

######  Theorem 1 (Ambrosetti-Prodi).

Let $\displaystyle{\bf F}:D\to L^{2}(\Omega),{\bf F}(u)=-\Delta_{D}u-{\bf
f}(u)$, where $\displaystyle{\bf f}:{\mathbb{R}}\to{\mathbb{R}}$ is a
$\displaystyle C^{2}$ strictly convex function for which

$\displaystyle{\bf a}=\inf{\bf f}^{\prime}(x)<\lambda_{m}<\lambda_{m}+{\bf
b}=\sup{\bf f}^{\prime}(x)<\mu_{m}\ .$

Then $\displaystyle{\bf F}$ folds downwards with respect to
$\displaystyle\phi_{m}$.

We define folds. Given real Banach spaces $\displaystyle X$ and $\displaystyle
Y$ and $\displaystyle\phi\in X\cap Y,\phi\neq 0$, let $\displaystyle
V=\langle\phi\rangle$ be the subspace generated by $\displaystyle\phi$. Denote
the vertical line through $\displaystyle y\in Y$ by $\displaystyle
L_{y}=\\{y+h\phi,h\in{\mathbb{R}}\\}\subset Y$. A continuous function
$\displaystyle F:X\to Y$ is simple if there is a subspace decomposition
$\displaystyle X=W_{X}\oplus V$ with the following property: for each
$\displaystyle y\in Y$, the inverse $\displaystyle F^{-1}(L_{y})$ is a curve (
a fiber)

$\displaystyle\Lambda_{y}=\\{u(y,t)=w(y,t)+t\phi,w(y,t)\in
W_{X},t\in{\mathbb{R}}\\}$

such that, for $\displaystyle y+h(y,t)\phi=F(u(y,t))$, the height
$\displaystyle h(y,.):{\mathbb{R}}\to{\mathbb{R}}$ is either strictly
monotonical or strictly unimodal for each $\displaystyle y\in Y$. Thus, no
point has three preimages under $\displaystyle F$. The definition clearly does
not depend on the choice of $\displaystyle W_{X}$.

The function $\displaystyle F$ folds downwards with respect to
$\displaystyle\phi$ if additionally, for all $\displaystyle y\in Y$,
$\displaystyle h(y,t)\to-\infty$ as $\displaystyle t\to\pm\infty$. Then, there
is a height $\displaystyle h(y)$ such that $\displaystyle F(u)=y+h\phi$ has
zero, one or two preimages, depending if $\displaystyle h>h(y),h=h(y)$ or
$\displaystyle h<h(y)$. Similarly, $\displaystyle F$ folds upwards if the
inequalities are reversed and $\displaystyle F$ folds folds vertically if
either case happens.

Manes and Micheletti stated the theorem with the hypotheses above in the
context of Hölder spaces ([25]). With different techniques, Berger and Podolak
provided a different proof ([8], also [7]) for Hilbert spaces. The theorem was
a starting point of an active line of research, leading to characterizations
of folds and further examples. The interested reader may find a rich
collection of examples and techniques in the review papers by Church and
Timourian ([15], [16]), and Ruf ([30]), which cover extremely well the
material up to the mid nineties. Their approach is strongly influenced by the
original Ambrosetti-Prodi view of the problem: a combination of local theory
(the fold being the first, simplest singularity) with global information
related to the determination of the number of solutions in terms of some
parameter or forcing terms. Such results are frequently restricted by
technical requirements, such as smoothness: in this paper, we present
operational alternatives and provide generalizations to a wider class of
problems.

Recently, Sirakov et alii. ([35]) considered folds given by nonlinear
perturbations of second order operators $\displaystyle{\bf L}$ in non-
divergence form. For a bounded $\displaystyle C^{1,1}$ domain
$\displaystyle\Omega\subset{\mathbb{R}}^{n}$, $\displaystyle n\geq 2$,
$\displaystyle X=W^{2,n}(\Omega)\cap W^{1,n}_{0}(\Omega)$, $\displaystyle
Y=W^{0,n}(\Omega)=L^{n}(\Omega)$, set

$\displaystyle{\bf L}u=-A_{ij}\ \partial_{i}\partial_{j}u-bB_{i}\
\partial_{i}u-q\ u=-\operatorname{tr}(AD^{2}\ u)-B\ \nabla u-q\ u,$

where, for appropriate constants $\displaystyle\Lambda\geq\lambda>0$, the
matrix $\displaystyle A=(A_{ij})$ satisfies $\displaystyle A(x)\in
C(\overline{\Omega})$, $\displaystyle\sigma(A(x))\subset[\lambda,\Lambda]$ and
$\displaystyle|B|_{\infty},|q|_{\infty}\leq C$. Then $\displaystyle{\bf
L}\in{\cal B}(X,Y)$, the space of bounded linear operators from $\displaystyle
X$ to $\displaystyle Y$. Let $\displaystyle\lambda_{m}=\lambda_{m}({\bf L})$
be the eigenvalue of $\displaystyle{\bf L}$ of smallest real part and
$\displaystyle\phi_{m}>0$ an associated eigenvector.

######  Theorem 2 (Sirakov-Tomei-Zaccur).

Let $\displaystyle{\bf L}$ as above and $\displaystyle{\bf P}(u)={\bf f}(u)$
for a strictly convex function $\displaystyle{\bf
f}:{\mathbb{R}}\to{\mathbb{R}}$ such that

$\displaystyle{\bf a}\ =\ \inf_{r\neq s}\frac{f(r)-f(s)}{r-s}\ <\ \lambda_{m}\
<\ \lambda_{m}+{\bf b}\ =\ \sup_{r\neq s}\frac{f(r)-f(s)}{r-s}\ .$

Then, for some $\displaystyle{\bf b}>0$, $\displaystyle{\bf F}={\bf L}-{\bf
P}:X\to Y$ folds downwards with respect to $\displaystyle\phi_{m}$.

The proof in [35] combines elliptic estimates and geometric arguments. We
phrase an extension, Theorem 5, as a geometric statement with a geometric
proof.

Both results are inspirational, in the specification of hypotheses and in the
techniques of proof. Our results are more geometric and more general. We take
a different point of view: simple maps are obtained from nonlinear
perturbationsof special linear operators. As for the theorems above, the
positivity of an eigenvector of the Jacobian, or some appropriate
linearization, is an essential ingredient. We use the preservation of positive
eigenfunctions associated with the spectral radius (Krein-Rutman type
arguments, as in Section 2), or with the smallest eigenvalue of a self-adjoint
map (the perturbation theory of nondegenerate groundstates, Section 3). On a
global scale, the asymptotic linearity of the nonlinear terms yields a certain
uniformity of the linearizations, which belong to a set with privileged
compactness properties.

Let $\displaystyle Y$ be a real, separable, reflexive Banach space endowed
with a normal, generating cone $\displaystyle K\subset Y$ (Section 2.1
presents the terminology associated with cones). Denote by $\displaystyle
k\geq 0$ an element $\displaystyle k\in K$ and by $\displaystyle k>0$ a quasi-
interior point $\displaystyle k\in K$. For a bounded operator $\displaystyle
S:Y\to Y$, (i.e. $\displaystyle S\in{{\cal B}(Y)}$), denote its spectrum and
spectral radius by $\displaystyle\sigma(S)$ and $\displaystyle r(S)$. An
operator $\displaystyle S\in{{\cal B}(Y)}$ is r-special if it compact and
ergodic with respect to $\displaystyle K$ and $\displaystyle r(S)>0$.

Consider a r-special operator $\displaystyle T\in{{\cal B}(Y)}$ with
$\displaystyle r(T)=1$. For $\displaystyle b,R>0$ and an r-special operator
$\displaystyle S\in{{\cal B}(Y)}$, define the set of perturbations

$\displaystyle{\mathscr{P}}^{b}\ =\ {\mathscr{P}}^{R,b,S}_{T}\ =\\{H=A+B,\
\hbox{for}\ \ A,B\in{{\cal B}(Y)},\ \|A\|\leq R,\ \|B\|\leq b,$
$\displaystyle\ S\leq AT\leq T,\ B\geq 0\ \\}\ .$

The map $\displaystyle P:Y\to Y$ admits linearizations if for $\displaystyle
y,z\in Y$, there exists $\displaystyle G(y,z)=G(z,y)\in{{\cal B}(Y)}$ for
which $\displaystyle P(y)-P(z)=G(y,z)(y-z)$.

The proof of Theorem 1 in [2] uses linearizations. Linearizations
$\displaystyle G(y,z)\in{{\cal B}(Y)}$ are not necessarily continuous in
$\displaystyle y,z\in Y$.

We enumerate possible properties of $\displaystyle P:Y\to Y$. Let
$\displaystyle F:Y\to Y,F=I-PT$.

  1. (r-H)

$\displaystyle P:Y\to Y$ is a Lipschitz map admitting linearizations
$\displaystyle G(y,z)\in{\mathscr{P}}^{b}$.

  2. (r-Conv)

Let $\displaystyle y_{1},y_{2},z_{1},z_{2}\in Y$ such that $\displaystyle
y_{1}>z_{1}$, $\displaystyle y_{2}>z_{2}$, $\displaystyle y_{1}\geq y_{2}$,
$\displaystyle z_{1}\geq z_{2}$ and either $\displaystyle y_{1}>y_{2}$ or
$\displaystyle z_{1}>z_{2}$. Then $\displaystyle
G(y_{1},z_{1})-G(y_{2},z_{2})\in{{\cal B}(Y)}$ is strictly positive.

  3. (r-Hs)

$\displaystyle P:Y\to Y$ is a $\displaystyle C^{1}$ map whose Jacobians
$\displaystyle J(y)=DP(y)$ belong to $\displaystyle{\mathscr{P}}^{b}$.

  4. (r-Convs)

Let $\displaystyle y,z\in Y$, $\displaystyle y>z$. Then $\displaystyle
DP(y)-DP(z)\in{{\cal B}(Y)}$ is strictly positive.

  5. (r-Crit)

$\displaystyle P:Y\to Y$ is $\displaystyle C^{1}$ and some $\displaystyle y\in
Y$ is critical: $\displaystyle 0\in\sigma(DF(y))$.

######  Theorem 3.

Suppose $\displaystyle Y$ and $\displaystyle K$ as above, $\displaystyle
T\in{{\cal B}(Y)}$ an r-special operator for which $\displaystyle
r(T)=1,T\phi=\phi>0$ and $\displaystyle P:Y\to Y$ a Lipschitz map. Define the
function $\displaystyle F:Y\to Y,y\mapsto y-P(Ty)$. For small $\displaystyle
b>0$,

(1) suppose that (r-H) holds and, for $\displaystyle y,z\in Y$, $\displaystyle
r(G(Ty,Tz)T)\neq 1$ . Then $\displaystyle F$ is a homeomorphism onto its
image.

(2) suppose that (r-H), (r-Conv) hold. Then $\displaystyle F$ is a simple map.

(3) if (r-H), (r-Conv) hold and $\displaystyle F:Y\to Y$ is proper, then it is
a homeomorphism or folds vertically.

(4) suppose that (r-Hs), (r-Convs) and (r-Crit) hold. Then $\displaystyle F$
folds downwards.

Item (1) is a non-symmetric version of the Dolph-Hammerstein theorem ([18],
[23]). In items (2) and (3), we circumvent the arguments in [2] and [8] which,
under smoothness hypotheses, identify the critical points of $\displaystyle F$
as being folds, a fact which follows from the theorem. For Nemitskii maps
$\displaystyle P(u)=f(u)$, we obtain finer results (Theorem 5), from which
Theorem 2 follows. Theorem 3 admits variations, but we are interested in
displaying a set of techniques, more than specific results. Theorem 3 (4), for
example, is extended in Section 2.8.

Here is a sketch of proof. From (r-H) or (r-Hs), linearizations (or Jacobians)
belong to $\displaystyle{\mathscr{P}}^{b}$. We begin with a perturbation
result: for $\displaystyle H\in{\mathscr{P}}^{b}$ and a special operator
$\displaystyle T\in{{\cal B}(Y)}$, $\displaystyle HT$ is also r-special. We
use an extension of the Krein-Rutman theorem for cones with possibly empty
interior ([17]; for an alternative approach, [9]). It is a diluted version of
Theorem 10 in the Appendix, a condensation of results by Marek, Bonsall,
Schaefer and Sawashima ([26]).

The convex set $\displaystyle{\mathscr{P}}^{b}$ has good compactness
properties, implying a uniform coercive bound for operators $\displaystyle
I-HT$. An argument with covering spaces, the Banach-Mazur theorem, then
converts $\displaystyle F$ into adapted coordinates (Proposition 4)
$\displaystyle F^{a}:Y\to Y$ for $\displaystyle F$, a nonlinear rank one
perturbation of the identity, already present in the original proof of Theorem
1 by Berger and Podolak ([8]) and in the proof of Theorem 2 in [35]. At this
point, the proof of Theorem 3(2) is at hand.

In combination with the convexity-like hypothesis (r-Conv), no point in the
counterdomain has more than two preimages under $\displaystyle F$ (Proposition
6), so that $\displaystyle F^{a}$ and $\displaystyle F:Y\to Y$ are simple maps
(Theorem 3(2)). In Theorem 1, the convexity of $\displaystyle f$ is
essentially necessary for $\displaystyle F(u)=-\Delta u-f(u)$ to be a fold
([14]): the search for further examples led us to the material in this text.

From the adapted coordinates, the inverse under $\displaystyle F$ of a
vertical line $\displaystyle L$ is a fiber $\displaystyle\Lambda_{y}$: our
geometric description of $\displaystyle F$ is complete once we compute the
asymptotic behavior of $\displaystyle F$ at endpoints of the fiber,
informally, how $\displaystyle F(\Lambda_{y})$ goes up and down along
$\displaystyle L$. Here the arguments bifurcate. For Nemitskii maps, a
standard argument (Theorem 5) provides our proof to Theorem 2. For more
general $\displaystyle C^{1}$, the existence of one critical point of
$\displaystyle F$ (hypothesis (m-Crit)) yields a full fledged critical set
(Corollary 2), from which the properness of $\displaystyle F$ follows
(Proposition 10), and the asymptotic behavior is restricted. Finally, the fact
that the images of fibers fold downwards is a consequence of degree theory
(Proposition 13).

Inspired by the fact that Theorem 1 holds for Sobolev and Hölder spaces, we
present a set up in Section 2.8 where folds transfer between scales of spaces.

In Section 3, we consider additive, nonlinear perturbations $\displaystyle
F(u)=Lu-P(u)$ of a self-adjoint operator $\displaystyle L:D\subset H\to H$ on
Hilbert spaces $\displaystyle H$ of functions, possibly defined on unbounded
domains. We concentrate on the smallest eigenvalue $\displaystyle\lambda_{m}$
of $\displaystyle L$, which we require to be basic, an isolated point in
$\displaystyle\sigma(L)$, whose invariant subspace is generated by a positive
eigenfunction $\displaystyle\phi>0$. As in Theorem 1, the corresponding
Theorem 8 does not restrict the analogous parameter $\displaystyle b$.

Again, the appropriate perturbation theory, from Faris and Simon ([21], [28]),
delimits a set containing linearizations and Jacobians. The estimates are
tighter, being of a spectral nature, and allow for a different technique to
derive properness of $\displaystyle F$, which we apply to spaces of functions
on unbounded sets. The compactness of resolvents $\displaystyle(L-\gamma
I)^{-1}$ is not needed.

We complete the text with some examples in Section 3.4.

Acknowledgements: Calanchi is partially supported by INdAM-GNAMPA Project
2018. Tomei is supported by CAPES, CNPq and FAPERJ. He thanks Boyan Sirakov
for a number of illuminating conversations.

##  2 Proof of Theorem 3

###  2.1 Cones and the spectral radius of r-special operators

We recall some basic facts about cones ([17], [26], [22]). Let $\displaystyle
Y$ be a real Banach space. We denote its norm by $\displaystyle|\cdot|$. A
cone $\displaystyle K\subset Y$ is a closed set for which

$\displaystyle 0\in K\ ,\quad K+K\subset K\ ,\quad t\geq 0\ \Rightarrow\
tK\subset K,\quad K\cap(-K)=\\{0\\}\ .$

It induces a partial order in $\displaystyle Y$: $\displaystyle y\leq
z\Leftrightarrow z-y\in K$. A cone $\displaystyle K$ is reproducing if
$\displaystyle K-K=Y$. A cone is normal if, for some
$\displaystyle\alpha\in{\mathbb{R}}$, $\displaystyle 0\leq y\leq
z\Longrightarrow|y|\leq\alpha|z|$. For a reproducing cone $\displaystyle K$,
the dual cone is $\displaystyle K^{\ast}=\\{k^{\ast}\in Y^{\ast}\ |\ \langle
k^{\ast},k\rangle\geq 0,\forall k\in K\\}$ (brackets denote the coupling
between $\displaystyle Y$ and $\displaystyle Y^{\ast}$). According to a result
of Krein ([22], [26]), $\displaystyle K$ is a normal, reproducing cone if and
only if $\displaystyle K^{*}$ also is.

An element $\displaystyle k\in K$ is quasi-interior if $\displaystyle\langle
k^{\ast},k\rangle\neq 0$, for any $\displaystyle k^{\ast}\in
K^{\ast}\setminus\\{0\\}$. Similarly, a linear functional $\displaystyle
k^{\ast}\in K^{\ast}$ is strictly positive if $\displaystyle\langle
k^{\ast},k\rangle>0$ for all $\displaystyle k\in K\setminus\\{0\\}$. To
simplify notation, we denote a quasi-interior point $\displaystyle k$ by
$\displaystyle k>0$ and a strictly positive functional $\displaystyle
k^{\ast}$ by $\displaystyle k^{\ast}>0$.

A cone $\displaystyle K$ is unflattened if, for some $\displaystyle M>0$ and
any $\displaystyle y\in Y$, there are $\displaystyle k_{1},k_{2}\in K$ such
that $\displaystyle y=k_{1}-k_{2},|k_{1}|,|k_{2}|\leq M|y|$. A reproducing
cone is unflattened (a consequence of the Krein-Smulian theorem [39]).

Denote by $\displaystyle{\cal B}(Y)$ is the space of bounded linear operators
from $\displaystyle Y$ to $\displaystyle Y$ equipped with the sup norm. As
usual, the spectral properties of an operator $\displaystyle S\in{\cal B}(Y)$
refer to its complexification. Let $\displaystyle r(S)$ be the spectral radius
of $\displaystyle S\in{\cal B}(Y)$.

An eigenvalue $\displaystyle\lambda\in\sigma(S)$ of $\displaystyle S\in{\cal
B}(Y)$ is basic if the following properties hold.

  1. (b-1)

There are associated eigenvectors $\displaystyle\phi>0$ of $\displaystyle S$,
$\displaystyle\phi^{\ast}>0$ of $\displaystyle S^{\ast}$.

  2. (b-2)

Eigenvectors of $\displaystyle S$ (resp. $\displaystyle S^{\ast}$) associated
with $\displaystyle\lambda$ are multiples of $\displaystyle\phi$ (resp.
$\displaystyle\phi^{\ast}$).

  3. (b-3)

No $\displaystyle y\in Y$ satisfies $\displaystyle(S-\lambda)y=\phi$.

An eigenvalue $\displaystyle\lambda\in\sigma(S)$ satisfying properties (b-2)
and (b-3) above is simple. A simple eigenvalue of $\displaystyle S$ is also of
$\displaystyle S^{\ast}$.

An operator $\displaystyle S\in{\cal B}(Y)$ is positive ($\displaystyle S\geq
0$) with respect to a cone $\displaystyle K$ if $\displaystyle SK\subset K$;
$\displaystyle S$ is strictly positive ($\displaystyle S>0$) if $\displaystyle
S(K\setminus\\{0\\})\subset K\setminus\\{0\\}$, and $\displaystyle S$ is
ergodic  if for any $\displaystyle k\in K\setminus\\{0\\}$, $\displaystyle Sk$
is a quasi-interior element of $\displaystyle K$, $\displaystyle Sk>0$. An
operator $\displaystyle S\in{\cal B}(Y)$ is r-special if it is compact and
ergodic with $\displaystyle r(S)>0$. If $\displaystyle S$ is r-special,
$\displaystyle S^{\ast}\in{\cal B}(Y^{\ast})$ also is (recall that
$\displaystyle K^{\ast}$ is also a normal, reproducing cone).

The result below is a special case of Theorem 10 in the Appendix.

######  Theorem 4.

Let $\displaystyle K$ be a normal reproducing cone of a real Banach space
$\displaystyle Y$. Then an r-special operator $\displaystyle S\in{\cal B}(Y)$
has a basic eigenvalue $\displaystyle r(S)=r(S^{\ast})$. Eigenvectors of
$\displaystyle S$ in $\displaystyle K$ are multiples of $\displaystyle\phi$.
For $\displaystyle\mu>r(S)$, $\displaystyle(\mu I-S)^{-1}K\subset K$. If
$\displaystyle\mu\in\sigma(S)\setminus\\{r(S)\\}$, $\displaystyle|\mu|<r(S)$.

We list simple properties of r-special operators frequently used in the text.

######  Corollary 1.

Let $\displaystyle S\in{{\cal B}(Y)}$ be an r-special operator with basic
eigenvalue $\displaystyle r(S)$ and associated eigenvectors
$\displaystyle\phi>0$ and $\displaystyle\phi^{\ast}>0$ of $\displaystyle S$
and $\displaystyle S^{\ast}$. Then the operator $\displaystyle S-r(S)\ I:Y\to
Y$ is a Fredholm operator of index $\displaystyle 0$, the subspaces
$\displaystyle V=\langle\phi\rangle$ and $\displaystyle
W=\operatorname{Ker}\phi^{\ast}=\operatorname{Ran}(S-r(S)I)$ are invariant
under $\displaystyle S$ and $\displaystyle Y=W\oplus V$.

Proof:  A simple consequence of $\displaystyle S$ being r-special.

The basic eigenvalue $\displaystyle r(S)=r(S^{\ast})$ of $\displaystyle S$ and
$\displaystyle S^{\ast}$ and related positive, normalized eigenvectors
$\displaystyle\phi$, $\displaystyle\phi^{\ast}$, vary smoothly with
$\displaystyle S$. We collect some known formulas (Proposition 79.15 in [42],
Theorem 4.3 in [26]).

######  Lemma 1.

Let $\displaystyle S\in{{\cal B}(Y)}$ be an r-special operator. There is an
open neighborhood $\displaystyle{\cal U}$ of $\displaystyle S$ such that, for
$\displaystyle\hat{S}\in{\cal U}$ the spectral radius $\displaystyle
r(\hat{S})$ is a simple eigenvalue. The map $\displaystyle\hat{S}\in{\cal
U}\mapsto r(\hat{S})$ is real analytic. For appropriate local normalizations,
the eigenfunctions $\displaystyle\phi(\hat{S})$ and
$\displaystyle\phi^{\ast}(\hat{S}^{\ast})$ are real analytic maps. For a
normalization such that
$\displaystyle\langle\phi^{\ast}(\hat{S}),\phi(\hat{S})\rangle=1$,

$\displaystyle
Dr(\hat{S})\cdot{\overline{S}}=\langle\phi^{\ast}(\hat{S}),{\overline{S}}\
\phi(\hat{S})\rangle\ .$

Let $\displaystyle S_{\star}\in{{\cal B}(Y)}$ be r-special. If $\displaystyle
S_{\star}\geq S$ , then $\displaystyle r(S_{\star})\geq r(S)$. Similarly, if
$\displaystyle S_{\star}>S$, then $\displaystyle r(S_{\star})>r(S)$.

###  2.2 Spectral theory in $\displaystyle{\mathscr{P}}^{b}$

Let $\displaystyle T\in{{\cal B}(Y)}$ be an r-special operator with
$\displaystyle r(T)=r(T^{\ast})=1$ and associated normal eigenvectors
$\displaystyle\phi=\phi(T)>0$ and $\displaystyle\phi^{\ast}=\phi(T^{\ast})>0$
as in Corollary 1.

######  Proposition 1.

For $\displaystyle H\in{\mathscr{P}}^{b}$, the operator $\displaystyle
HT\in{{\cal B}(Y)}$ is r-special, with a basic eigenvalue $\displaystyle
r(HT)=r((HT)^{\ast})$ bounded away from zero uniformly in
$\displaystyle{\mathscr{P}}^{b}$.

Proof:  Since $\displaystyle S$ is r-special and $\displaystyle HT\geq S$ from
(r-H), $\displaystyle HT$ is r-special and, by Lemma 1, $\displaystyle
r(HT)\geq r(S)>0$ uniformly in $\displaystyle{\mathscr{P}}^{b}$. The
statements about basic eigenvalues follow from Theorem 4 and its corollary.

Endow $\displaystyle{{\cal B}(Y)}$ with the weak operator topology: for
$\displaystyle H_{k},H_{\infty}\in{{\cal B}(Y)}$,

$\displaystyle H_{k}{\stackrel{{\scriptstyle{\scriptstyle
w}}}{{\longrightarrow}}\ }H_{\infty}\quad\Leftrightarrow\ \ \forall y\in Y,\
y^{\ast}\in Y^{\ast},\quad\langle y^{\ast},H_{k}\ y\rangle\to\langle
y^{\ast},H_{\infty}\ y\rangle\ .$

Since $\displaystyle K$ and $\displaystyle K^{\ast}$ are reproducing cones, to
show $\displaystyle H_{k}{\stackrel{{\scriptstyle{\scriptstyle
w}}}{{\longrightarrow}}\ }H_{\infty}$, it suffices to prove

$\displaystyle\lim\ \langle y^{\ast},H_{k}\ y\rangle=\langle
y^{\ast},H_{\infty}\ y\rangle\quad\hbox{for}\quad\ y\in K,\ y^{\ast}\in
K^{\ast}\ .$

For a separable space $\displaystyle Y$, the weak topology metrizable.

######  Lemma 2.

The sets $\displaystyle{\mathscr{P}}^{b}$ and $\displaystyle{\mathscr{P}}^{0}$
are convex, compact and sequentially compact in the weak operator topology.

Proof:  We obtain compactness in the weak operator topology. Closed balls
$\displaystyle\\{||H||\leq C\\}\subset{\cal B}(Y)$ are compact and
sequentially compact in the weak operator topology for a reflexive, separable
Banach space $\displaystyle Y$ ([20], [40]). It suffices to show that
$\displaystyle{\mathscr{P}}^{b}$ is closed in the strong operator topology:
for $\displaystyle S\leq H_{k}T\leq T,\ \|H_{k}\|\leq R$, if $\displaystyle
H_{k}{\stackrel{{\scriptstyle{\scriptstyle s}}}{{\longrightarrow}}\ }H$, the
same inequalities hold for $\displaystyle H$.

Spectral objects behave well under weak operator convergence.

######  Lemma 3.

Let $\displaystyle H_{k}\in{\mathscr{P}}^{b}$ such that $\displaystyle
H_{k}{\stackrel{{\scriptstyle{\scriptstyle w}}}{{\longrightarrow}}\
}H_{\infty}\in{\mathscr{P}}^{b}$. Consider sequences $\displaystyle\\{y_{k}\in
Y,|y_{k}|=1\\}$, $\displaystyle\\{g_{k}\in Y,g_{k}\to g_{\infty}\\}$,
$\displaystyle\\{\lambda_{k}\in{\mathbb{R}},\ 0<c<\lambda_{k}<C\\}$ for which

$\displaystyle H_{k}Ty_{k}=\lambda_{k}y_{k}+g_{k}\ .$

Then, for some (relabeled) subsequences, $\displaystyle y_{k}\to
y_{\infty}\neq 0$, $\displaystyle\lambda_{k}\to\lambda_{\infty}>0$ and
$\displaystyle H_{\infty}Ty_{\infty}=\lambda_{\infty}y_{\infty}+g_{\infty}$.

Proof:  Since $\displaystyle Y$ is separable and reflexive, from the
sequential Banach-Alaoglu theorem, there is a subsequence $\displaystyle
y_{k}\rightharpoonup y_{\infty}\in Y$ and, from the bounds on
$\displaystyle\\{\lambda_{k}\\}$,
$\displaystyle\lambda_{k}\to\lambda_{\infty}\neq 0$. As $\displaystyle T$ is
compact, for yet another subsequence $\displaystyle y_{k}\rightharpoonup
y_{\infty}\in Y$, $\displaystyle Ty_{k}\to Ty_{\infty}$. Take
$\displaystyle\ell\in Y^{\ast}$ and use $\displaystyle
H_{k}{\stackrel{{\scriptstyle{\scriptstyle w}}}{{\longrightarrow}}\
}H_{\infty}$:

$\displaystyle\langle\ell,H_{\infty}Ty_{\infty}\rangle=\lim\langle\ell,H_{k}Ty_{\infty}\rangle=\lim\
\big{(}\langle\ell,H_{k}T(y_{\infty}-y_{k})\rangle+\
\langle\ell,H_{k}Ty_{k}\rangle\big{)}\ .$

As $\displaystyle T(y_{k}-y_{\infty})\to 0$ and the operators $\displaystyle
H_{k}$ are uniformly bounded by (r-H), the first term goes to zero and then

$\displaystyle\langle\ell,H_{\infty}Ty_{\infty}\rangle=\lim\
\langle\ell,H_{k}Ty_{k}\rangle=\lim\
\langle\ell,\lambda_{k}y_{k}+g_{k}\rangle=\langle\ell,\lambda_{\infty}y_{\infty}+g_{\infty}\rangle\
,$

and thus $\displaystyle
H_{\infty}Ty_{\infty}=\lambda_{\infty}y_{\infty}+g_{\infty}$. If
$\displaystyle y_{\infty}=0$, then $\displaystyle Ty_{k}\to Ty_{\infty}=0$ and
$\displaystyle g_{\infty}=0$. so that, from $\displaystyle
H_{k}Ty_{k}=\lambda_{k}y_{k}+g_{k}$, since $\displaystyle|\lambda_{k}y_{k}|$
is bounded away from zero and $\displaystyle\\{H_{k}\\}$ is bounded, we must
have $\displaystyle y_{\infty}\neq 0$, a contradiction. Thus $\displaystyle
y_{\infty}\neq 0$. Finally, $\displaystyle
y_{k}=(H_{k}Ty_{k}-g_{k})/\lambda_{k}$ and the right hand side converges:
$\displaystyle y_{k}\to y_{\infty}$.

Set $\displaystyle V=\langle\phi(T)\rangle$, $\displaystyle
W=\operatorname{Ker}\phi^{\ast}(T)$ as in Corollary 1, so that $\displaystyle
Y=W\oplus V$. Define the associated projection $\displaystyle\Pi_{W}:Y\to W$.

######  Proposition 2.

For $\displaystyle a\in(0,1]$, there exists $\displaystyle{\bf b}>0$ such
that, for $\displaystyle H\in{\mathscr{P}}^{{\bf b}}$, the following
properties hold.

  1. (i)

If $\displaystyle 1\in\sigma(HT)$ is an eigenvalue, then $\displaystyle
1=r(HT)$.

  2. (ii)

If $\displaystyle(I-HT)w=c\ \phi(T)$ for $\displaystyle w\in W$,
$\displaystyle c\in{\mathbb{R}}$, then $\displaystyle w=0$.

  3. (iii)

$\displaystyle HT$ does not have more than one eigenvalue larger than 1.

  4. (iv)

Assume $\displaystyle{\bf b}$ for which items (i)-(iii) hold. Then the
operators $\displaystyle I-\Pi_{W}HT:W\to W$ are uniformly coercive:

$\displaystyle\exists\ C>0,\ \forall\ v\in W,\ \forall\
t\in{\mathbb{R}},\quad|(I-\Pi_{W}HT)v|\ \geq\ C\ |v|\ ,$

Proof:  All items will be proved by contradiction.

(i) Suppose $\displaystyle H_{k}\in{\mathscr{P}}^{b_{k}}$, $\displaystyle
b_{k}<1/k$ such that $\displaystyle 1\in\sigma(H_{k}T)$ and $\displaystyle
r(H_{k}T)>1$. As $\displaystyle T\in{{\cal B}(Y)}$ is compact, so is
$\displaystyle H_{k}T$, and thus $\displaystyle\sigma(H_{k}T)\setminus\\{0\\}$
contains only eigenvalues: there are eigenvectors (normalized in
$\displaystyle Y$) for which $\displaystyle H_{k}T\psi_{k}=\psi_{k}$ and
$\displaystyle H_{k}T\phi_{k}=r(H_{k}T)\phi_{k}$, such that $\displaystyle
r(H_{k}T)>1$ is a basic eigenvalue, $\displaystyle\phi_{k}>0$. From Lemma 2,
up to subsequences, $\displaystyle H_{k}{\stackrel{{\scriptstyle{\scriptstyle
w}}}{{\longrightarrow}}\ }H_{\infty}\in{\mathscr{P}}^{0}$. From Lemma 3,
setting $\displaystyle\lambda_{k}=1$ and $\displaystyle g_{k}=0$, there exists
$\displaystyle\psi_{\infty}\in Y$ such that (up to subsequences)
$\displaystyle\psi_{k}\to\psi_{\infty}\neq 0$, with $\displaystyle
H_{\infty}T\psi_{\infty}=\psi_{\infty}$.

From the monotonicity of the spectral radius (Lemma 1), $\displaystyle 1\leq
r(H_{k}T)\leq(1+b_{k})r(T)\to 1$ and, again by Lemma 3 with
$\displaystyle\lambda_{k}=r(H_{k}T)$, we have
$\displaystyle\phi_{k}\to\phi_{\infty}\neq 0$ in $\displaystyle Y$, with
$\displaystyle H_{\infty}T\phi_{\infty}=\phi_{\infty}\in X$.

We show $\displaystyle\phi_{\infty}\neq\pm\psi_{\infty}$. Indeed, there is a
sequence of dual normalized eigenvectors $\displaystyle\phi_{k}^{*}\in
Y^{\ast}$, $\displaystyle\phi_{k}^{*}>0$,
$\displaystyle(H_{k}T)^{*}\phi_{k}^{*}=\lambda_{k}\phi_{k}^{*}$, which
converges to $\displaystyle\phi_{\infty}^{*}\in Y^{\ast}$ by an argument as in
Lemma 3. Also,
$\displaystyle(H_{\infty}T)^{*}\phi_{\infty}^{*}=\phi_{\infty}^{*}$,
$\displaystyle\phi_{\infty}^{*}\in K^{\ast}$,
$\displaystyle\phi_{\infty}^{\ast}\neq 0$. From Corollary 1 (i),
$\displaystyle\langle\psi_{k},\phi_{k}^{*}\rangle=0$ and by taking limits,
$\displaystyle\langle\psi_{\infty},\phi_{\infty}^{*}\rangle=0$. If
$\displaystyle\phi_{\infty}=\pm\psi_{\infty}$, then
$\displaystyle\langle\psi_{\infty},\phi_{\infty}^{*}\rangle\neq 0$ by
positivity. But then $\displaystyle\phi_{\infty}$ and
$\displaystyle\psi_{\infty}$ are independent eigenvectors, contradicting the
simplicity of $\displaystyle r(H_{\infty}T)$ (Proposition 1).

(ii) Take $\displaystyle\bf b$ such that item (i) holds. Write

$\displaystyle H_{k}Tw_{k}=w_{k}-c_{k}\ \phi(T)\ ,\quad w_{k}\in
W\setminus\\{0\\},\quad b_{k}=\|B_{k}\|\to 0\ .$

If $\displaystyle c_{k}=0$, one has $\displaystyle 1\in\sigma(H_{k}T)$ and,
from (i), $\displaystyle r(H_{k}T)=1$. From Proposition 1, we assume
$\displaystyle w_{k}>0$. By positivity, this contradicts $\displaystyle
w_{k}\in W=\operatorname{Ker}\phi(T^{\ast})$. Thus $\displaystyle c_{k}\neq
0$: rescale and consider $\displaystyle c_{k}=1$ for all $\displaystyle k$.

If $\displaystyle\ |w_{k}|$ is unbounded, normalize
$\displaystyle\hat{w}_{k}=w_{k}/|w_{k}|$. For a subsequence of
$\displaystyle\\{H_{k}\\}$, $\displaystyle
H_{k}{\stackrel{{\scriptstyle{\scriptstyle w}}}{{\longrightarrow}}\
}H_{\infty}\in{\mathscr{P}}^{b}$ and, from Lemma 3,
$\displaystyle\hat{w}_{k}\to\hat{w}_{\infty}\in W\setminus\\{0\\}$ and
$\displaystyle H_{\infty}T\hat{w}_{\infty}=\hat{w}_{\infty}$. Therefore,
$\displaystyle 1\in\sigma(H_{\infty}T)$ and $\displaystyle r(H_{\infty}T)=1$
by item (i). Again, take $\displaystyle\hat{w}_{\infty}>0$, contradicting
$\displaystyle\hat{w}_{\infty}\in W=\operatorname{Ker}\phi(T^{\ast})$.

Suppose instead that $\displaystyle|w_{k}|$ is bounded. For subsequences,
$\displaystyle w_{k}\rightharpoonup w_{\infty}$ in $\displaystyle Y$,
$\displaystyle H_{k}{\stackrel{{\scriptstyle{\scriptstyle
w}}}{{\longrightarrow}}\ }H_{\infty}$, and again from Lemma 3, $\displaystyle
w_{k}\to w_{\infty}\neq 0$ and $\displaystyle
w_{\infty}-H_{\infty}Tw_{\infty}=\phi(T)$.

Again by monotonicity, $\displaystyle 1<r(H_{k}T)\leq(1+b_{k})r(T)\to 1$ and
$\displaystyle r(H_{\infty}T)\in[a,1]$. If $\displaystyle r(H_{\infty}T)<1$,
$\displaystyle w_{\infty}=(I-H_{\infty}T)^{-1}\phi$ and by Theorem 4,
$\displaystyle w_{\infty}\in K$, contradicting $\displaystyle w_{\infty}\in
W\setminus\\{0\\}$. If $\displaystyle r(H_{\infty}T)=1$, we must have
$\displaystyle\phi\in\operatorname{Ran}(I-H_{\infty}T)=\ker\phi((H_{\infty}T)^{\ast})$
and again this cannot happen, as $\displaystyle\phi((H_{\infty}T)^{\ast})>0$.

(iii) Suppose $\displaystyle\lambda,\mu\in\sigma(HT)$ such that $\displaystyle
1<\lambda<\mu$ (recall that the top eigenvalue is simple, and the eigenvalues
are isolated). Now consider the operator $\displaystyle HT=HT/\lambda$, with
eigenvalues $\displaystyle 1/\lambda<1<\mu/\lambda$. The operator belongs to
an appropriate $\displaystyle{\mathscr{P}}^{b}$ (replace $\displaystyle S$ by
$\displaystyle S/\lambda$) and thus, if $\displaystyle 1\in\sigma(HT)$, we
must have $\displaystyle 1=r(HT)$, by (i), contradicting $\displaystyle
1<\mu/\lambda$.

(iv) Suppose $\displaystyle\epsilon_{k}\to 0$ and $\displaystyle v_{k}\in W$,
$\displaystyle t_{k}\in{\mathbb{R}}$ for which
$\displaystyle|(I-\Pi_{W}HT)v_{k}|\ <\ \epsilon_{k}\ |v_{k}|$. Thus
$\displaystyle v_{k}\neq 0$ and one may normalize, $\displaystyle
z_{k}=v_{k}/|v_{k}|$, so that $\displaystyle z_{k}-\Pi_{W}H_{k}Tz_{k}\to 0$.
By Proposition 1, for a subsequence, $\displaystyle
H_{k}{\stackrel{{\scriptstyle{\scriptstyle w}}}{{\longrightarrow}}\
}G_{\infty}\in{\mathscr{P}}^{b}$. As in Lemma 3, for a subsequence
$\displaystyle z_{k}\to z_{\infty}\in W$, $\displaystyle
z_{\infty}-\Pi_{W}G_{\infty}Tz_{\infty}=0$ so that $\displaystyle
z_{\infty}-G_{\infty}Tz_{\infty}=c\phi(T)$.

By item (ii), since $\displaystyle z_{\infty}\in W$, we must have
$\displaystyle c=0$ and then $\displaystyle z_{\infty}=0$. On the other hand,
since $\displaystyle z_{k}\to z_{\infty}=0$ and the $\displaystyle H_{k}$’s
are uniformly bounded,

$\displaystyle|H_{k}Tz_{k}|\to 0,\quad{\rm and}\quad|\Pi_{W}H_{k}Tz_{k}|\to
0,$

yielding a contradiction: $\displaystyle z_{k}-\Pi_{W}H_{k}Tz_{k}\to 0$ and
$\displaystyle|z_{k}|=1$.

###  2.3 Hypothesis (r-H): adapted coordinates

Let $\displaystyle T\in{{\cal B}(Y)}$ as in Theorem 3, basic eigenvalue
$\displaystyle r(T)$ and associated eigenfunctions
$\displaystyle\phi=\phi(T)\in Y$, $\displaystyle\phi^{\ast}=\phi(T^{\ast})\in
Y^{\ast}$. From Corollary 1, the decomposition $\displaystyle Y=W\oplus V$ for
$\displaystyle V=\langle\phi\rangle$ and $\displaystyle
W=\operatorname{Ker}\phi^{*}$ induces projections $\displaystyle\Pi_{W}:Y\to
W$ and $\displaystyle\Pi_{V}:Y\to V$. The horizontal hyperplane at height
$\displaystyle t$ is $\displaystyle W^{t}=W+t\phi$. Define the projected
restrictions of $\displaystyle F$,

$\displaystyle F^{t}:W\to W,\quad F^{t}(w)=w-\Pi_{W}P(T(w+t\phi)).$

The next proposition finesses some elliptic estimates employed in Theorem 2.

######  Proposition 3.

Assume the hypotheses of Theorem 3 and (r-H) for $\displaystyle{\bf b}$ such
that Proposition 2 holds. Then $\displaystyle F^{t}:W\to W$ are uniformly
Lipschitz coercive:

$\displaystyle\exists\ C>0:\ \ \forall\ w,\ v\in W,\ \forall\
t\in{\mathbb{R}},\quad|F^{t}(w)-F^{t}(v)|\ \geq\ C\ |w-v|\ ,$

and so are the maps $\displaystyle\Psi:W\oplus{\mathbb{R}}\to
W\oplus{\mathbb{R}},(w,t)\mapsto(z,t)=(F^{t}(w),t)$.

Proof:  Suppose by contradiction $\displaystyle\epsilon_{k}\to 0$ and
$\displaystyle w_{k},v_{k}\in W$, $\displaystyle t_{k}\in{\mathbb{R}}$ for
which

$\displaystyle|{F^{t_{k}}}(w_{k})-{F^{t_{k}}}(v_{k})|\ <\ \epsilon_{k}\
|w_{k}-v_{k}|.$

For the linearization $\displaystyle
G_{k}=G_{k}(T(w_{k}+t_{k}\phi),T(v_{k}+t_{k}\phi))\in{\mathscr{P}}^{b}$,

$\displaystyle|{F^{t_{k}}}(w_{k})-{F^{t_{k}}}(v_{k})|=|(w_{k}-v_{k})-\Pi_{W}\big{(}P(T(w_{k}+t_{k}\phi))-P(T(v_{k}+t_{k}\phi))\big{)}|$
$\displaystyle=|(w_{k}-v_{k})-\Pi_{W}G_{k}T(w_{k}-v_{k})|>C\ |w_{k}-v_{k}|.$

for the uniform Lipschitz bound $\displaystyle C$ in Proposition 2(iv).

As for $\displaystyle\Psi$, use the equivalent norm
$\displaystyle|||u|||=|||w+t\phi|||=|w|+|t|$.  For the rest of Section 2,
$\displaystyle b=\bf b$ is taken as in Proposition 2. Indeed, this is the
required $\displaystyle b>0$ in Theorem 3.

We recall two well known facts. Let $\displaystyle Z$ be a real Banach space
and $\displaystyle U\subset Z$ be an open subset. A continuous function
$\displaystyle f:U\to Z$ is compact if the image of bounded sets is relatively
compact.

######  Theorem (Banach-Mazur, (5.1.4) in [6]).

A continuous map $\displaystyle H:Z\to Z$ is a homeomorphism if and only if it
is proper and a local homeomorphism.

######  Theorem (Schauder Invariance of Domain, [34]).

Let $\displaystyle H:U\to Z$ be a continuous injective function of the form
$\displaystyle H(x)=x-Q(x)$ for a compact map $\displaystyle Q:U\to Z$. Then
$\displaystyle H(U)\subset B$ is an open set.

We introduce adapted coordinates ([8], [12]).

######  Proposition 4.

Assume the hypotheses of Proposition 3. For $\displaystyle
V\simeq{\mathbb{R}}$, the map

$\displaystyle\Psi:Y=W\oplus{\mathbb{R}}\to
Y=W\oplus{\mathbb{R}},(w,t)\mapsto(z,t)=(F^{t}(w),t)$

is a homeomorphism. For $\displaystyle F^{a}(z,t)=F\circ\Psi^{-1}$, the
following diagram commutes.

$\displaystyle\begin{array}[]{ccl}{(w,t)}&\stackrel{{\scriptstyle{\scriptstyle
F}}}{{\longrightarrow}}&{(F^{t}(w),\Pi_{V}F(w+t\phi))}\\\ \\\
{\scriptstyle\Psi^{-1}}\nwarrow&&\nearrow{\scriptstyle
F^{a}(z,t)=(z,h^{a}(z,t))=(z,\Pi_{V}F(\Psi^{-1}(z,t)))}\\\ &{(z,t)}&\\\
\end{array}$

Proof:  Write $\displaystyle y=w+t\phi=\Pi_{W}y+t\phi$. We apply the Banach-
Mazur theorem to

$\displaystyle\Psi(w,t)=F^{t}(w)+t\phi=(y-t\phi)-\Pi_{W}P(Ty)+t\phi\equiv
y-Q(y).$

The properness of $\displaystyle\Psi$ follows from Proposition 3. Indeed,
given sequences such that
$\displaystyle\Psi(w_{k},t_{k})=(z_{k},t_{k})\to(z_{\infty},t_{\infty})$, use
the equivalent norm $\displaystyle|||y|||=|w|+|t|$,

$\displaystyle|w_{n}-w_{m}|+|t_{n}-t_{m}|=|\Psi(w_{n},t_{n})-\Psi(w_{m},t_{m})|\geq
C(|w_{n}-w_{m}|+|t_{n}-t_{m}|)\ ,$

to conclude that the sequences $\displaystyle\\{(z_{k},t_{k})\\}$ and
$\displaystyle\\{(w_{k},t_{k})\\}$ are both Cauchy sequences, necessarily
convergent. The same estimate implies the injectivity of $\displaystyle\Psi$.
We verify local surjectivity: the continuity of the inverse again would follow
from Proposition 3. By the Banach-Mazur theorem, it suffices to show that
$\displaystyle\Psi$ is an open map, which implies local surjectivity: we use
Schauder’s theorem.

We show that, around each $\displaystyle(w,t)\in W\oplus V$, there is a closed
ball $\displaystyle D\subset Y$ whose image $\displaystyle Q(D)\subset Y$ is
relatively compact for $\displaystyle Q(y)=\Pi_{W}P(Ty)$. Since $\displaystyle
T$ is compact, and $\displaystyle P$ continuous (Lipschitz), $\displaystyle
P\circ T$ is a compact map, and the linear, bounded operators
$\displaystyle\Pi_{W}$ preserves compactness.

Thus $\displaystyle\Psi$ is a genuine change of variables. The diagram now is
immediate.

Thus, for each $\displaystyle t\in{\mathbb{R}}$ and $\displaystyle z\in W$,
there is a unique $\displaystyle w(z)\in W$ for which
$\displaystyle{F^{t}}(w(z))=z$. Said differently, the image under
$\displaystyle F$ of a horizontal hyperplane $\displaystyle W^{t}$ intercepts
the vertical line $\displaystyle L_{z}=\\{z+h\phi\\}$ at a single point: each
hyperplane $\displaystyle W^{t}$ contains a unique preimage of $\displaystyle
L_{z}$. We denote by $\displaystyle\Lambda_{z}=F^{-1}(L_{z})$ the fiber
associated with $\displaystyle z$. A fiber contains a unique point on each
$\displaystyle W^{t}$ and is parameterized by $\displaystyle{\mathbb{R}}$:
$\displaystyle\Lambda_{z}(t)=\\{y(t)=w(z,t)+t\phi,t\in{\mathbb{R}}\\}$.

The map $\displaystyle\Psi^{-1}$ takes each $\displaystyle W^{t}$ to itself
and vertical lines are taken to fibers. The function $\displaystyle t\mapsto
h(z,t)=\Pi_{V}F(y(t))=\langle\phi(T^{\ast}),F(y(t))\rangle$ takes the point at
height $\displaystyle t$ in the fiber $\displaystyle\Lambda_{z}$ to the point
at height $\displaystyle h(z,t)$ of the vertical line $\displaystyle L_{z}$.
Equivalently, $\displaystyle h^{a}(z,\cdot)$ takes $\displaystyle(z,t)$ to a
point at height $\displaystyle h(z,t)$ of the vertical line $\displaystyle
L_{z}$.

Remark For a given $\displaystyle z\in Y$, the solutions of $\displaystyle
F(y)=z$ lie in the fiber $\displaystyle\Lambda_{z}$ which, in principle, may
be searched by a (one dimensional) continuation method ([36]). Numerical
algorithms have been written exploiting this special feature ([12]).

######  Proposition 5.

Under the hypotheses of Theorem 3 (1), $\displaystyle F:Y\to Y$ is injective.

Proof:  For $\displaystyle y_{1},y_{2}\in Y$ for which $\displaystyle
F(y_{1})-F(y_{2})=0$, use linearizations,

$\displaystyle F(y_{1})-F(y_{2})=(y_{1}-y_{2})-G(Ty_{1},Ty_{2})\
T(y_{1}-y_{2})=0\ .$

From Proposition 1, $\displaystyle G(Ty_{1},Ty_{2})\in{\mathscr{P}}^{b}$, so
that $\displaystyle 1\in\sigma(G(Ty_{1},Ty_{2})T)$ and thus $\displaystyle
r(y_{1},y_{2})=1$, by Proposition 2, contradicting the hypothesis.

Proof of Theorem 3(1): From Proposition 5, $\displaystyle F$ is injective.
Local surjectivity is again a consequence of Schauder’s theorem, as in the
proof of Theorem 4. We consider the continuity of the (local) inverse. In the
notation of Proposition 4, it suffices to show that $\displaystyle
F^{a}:W\oplus{\mathbb{R}}\to W\oplus{\mathbb{R}},(z,t)\mapsto(z,h^{a}(z,t))$
is a local homeomorphism. The exercise in real analysis is left to the reader.

###  2.4 Hypothesis (r-Conv): simple maps

As in Theorem 3(2), suppose (r-H), (r-Conv) hold and $\displaystyle{\bf b}$ is
taken such that Proposition 2 is true. Set $\displaystyle
r(G(Ty_{1},Ty_{2})T)\equiv r(y_{1},y_{2})$ and $\displaystyle
r(G(Ty,0)T)\equiv r(y)$.

######  Proposition 6.

If $\displaystyle F(y_{1})=F(y_{2})$, then trichotomy holds: either
$\displaystyle y_{1}>y_{2}$ or $\displaystyle y_{1}=y_{2}$ or $\displaystyle
y_{1}<y_{2}$. For $\displaystyle y\in Y$, the equation $\displaystyle
F(y_{1})=y$ has at most two solutions.

Proof:  As in Proposition 5, if $\displaystyle F(y_{1})-F(y_{2})=0$,
$\displaystyle G(Ty_{1},Ty_{2})\in{\mathscr{P}}^{b}$ and $\displaystyle
r(y_{1},y_{2})=1$. From Theorem 4, for a choice of sign,
$\displaystyle\pm(y_{1}-y_{2})$ is a quasi-interior eigenvector of
$\displaystyle G(Ty_{1},Ty_{2})T\in{{\cal B}(Y)}$: trichotomy holds.

Suppose that $\displaystyle z\in Y$ has three preimages, $\displaystyle
z=F(y_{1})=F(y_{2})=F(y_{3})$ for which $\displaystyle y_{1}<y_{2}<y_{3}$. As
before, $\displaystyle r(y_{3},y_{2})=r(G(Ty_{2},Ty_{1})T)=1$. But from
(r-Conv), since $\displaystyle Ty_{1}<Ty_{2}<Ty_{3}$ (as $\displaystyle T$ is
r-special), we must have $\displaystyle G(Ty_{3},Ty_{2})>G(Ty_{2},Ty_{1})$ and
thus $\displaystyle r(y_{3},y_{2})>r(y_{2},y_{1})$, from the strict
monotonicity of the spectral radius (Lemma 1), a contradiction.

If (r-Conv) is not satisfied, the equation $\displaystyle F(y)=z$ for a fixed
$\displaystyle z\in Y$ may have a number of solutions, but their graphs still
sit one on top of the other.

Proof of Theorem 3 (2): The homeomorphism $\displaystyle\Psi:W\oplus V\to
W\oplus V$ in Proposition 4 preserves the vertical component: it suffices to
prove that $\displaystyle F^{a}:W\oplus V\to W\oplus V$ is a simple map. This
is immediate from Proposition 6: notice that all preimages of a point
$\displaystyle z\in Y$ necessarily belong to $\displaystyle F^{a}(L_{z})$,
where $\displaystyle L_{z}\subset Y$ is the vertical line through
$\displaystyle z$.

A finer geometric description of the map $\displaystyle F:Y\to Y$ follows from
the asymptotic behavior of the images of its fibers, as in Theorem 5 below.

###  2.5 Nemitskii maps: proof of Theorem 2

At this point, standard techniques obtain from Theorem 3 (2) an extension of
Theorem 2. We resume the proof of Theorem 3 in Section 2.7.

Let $\displaystyle Y$ be a real, separable, reflexive Banach space of
functions containing the constants, for which $\displaystyle K$, the cone of
pointwise non-negative functions, is normal and generating. We assume that
multiplication $\displaystyle M_{q}$ by bounded functions $\displaystyle
q:{\mathbb{R}}\to{\mathbb{R}}$ be bounded,
$\displaystyle\|M_{q}\|\leq\sup|q|$. As in Theorem 3, $\displaystyle
T\in{{\cal B}(Y)}$ is an r-special operator for which $\displaystyle
r(T)=1,T\phi=\phi>0,T^{\ast}\phi^{\ast}=\phi^{\ast}>0,\langle\phi^{\ast},\phi\rangle=1$.

######  Theorem 5.

Let $\displaystyle P(u)=f(u)$ for a strictly convex function $\displaystyle
f:{\mathbb{R}}\to{\mathbb{R}}$ such that

$\displaystyle 0<a=\inf_{r\neq s}q(r,s)<1<1+b=\sup_{r\neq s}q(r,s)\ .$

Then, for some $\displaystyle b>0$, $\displaystyle F=I-PT:Y\to Y$ folds
downwards with respect to $\displaystyle\phi$.

Here, the Newton quotient $\displaystyle q(r,s)$ is $\displaystyle
q(r,s)=(q(r)-q(s))/(r-s)$ for $\displaystyle r\neq s$.

Proof:  We check the hypotheses of Theorem 3 (2). As in [2], use
linearizations $\displaystyle G(u,v)=M_{q(u,v)}$, where $\displaystyle
M_{q(u,v)}$ is the multiplication operator by the bounded function
$\displaystyle q(u,v)(x)\ =\ q(u(x),v(x))$ for $\displaystyle u(x)\neq v(x)$
and $\displaystyle q(u,v)(x)=a$ otherwise.

Clearly $\displaystyle P,G(u,v):Y\to Y$ are well defined and continuous, as
$\displaystyle f$ is Lipschitz. Moreover, $\displaystyle
M_{q(u,v)}=G(u,v)\in{\mathscr{P}}^{b}$ for $\displaystyle S=aT$,
$\displaystyle R=1+b$. Indeed,

$\displaystyle
q(u,v)=\alpha+\beta=\min\\{q(u,v),1\\}+(q(u,v)-\min\\{q(u,v),1\\}),M_{q(u,v)}=M_{\alpha}+M_{\beta}\
,$

and then $\displaystyle S=aT\leq M_{\alpha}T\leq T$,
$\displaystyle\|M_{\beta}\|\leq b$. This settles (r-H).

For (r-Conv), consider pointwise estimates for $\displaystyle x\in\Omega$, the
domain of the functions in $\displaystyle Y$: as $\displaystyle f$ is strictly
convex, for $\displaystyle y_{1},y_{2},z_{1},z_{2}\in Y$ as in (m-Conv),

$\displaystyle\frac{f(y_{1}(x))-f(z_{1}(x))}{y_{1}(x)-z_{1}(x)}>\frac{f(y_{2}(x))-f(z_{2}(x))}{y_{2}(x)-z_{2}(x)}\
.$

Thus $\displaystyle G(y_{1},z_{1})-G(y_{2},z_{2})\in{{\cal B}(Y)}$ is strictly
positive. Now use Theorem 3 (2) to conclude that $\displaystyle F$ is a simple
map.

From the hypotheses on $\displaystyle f$, there are lines
$\displaystyle\ell_{\pm}(t)=\alpha_{\pm}t+\beta_{\pm}$ such that
$\displaystyle f(t)>\ell_{\pm}(t)$ with
$\displaystyle\alpha_{-}<1=r(T)<\alpha_{+}$ ([35], for example).

We compute the height of the images of the fiber $\displaystyle
y(t)=w(t)+t\phi,w(t)\in W$:

$\displaystyle
h(F(y(t)))=\langle\phi^{\ast},F(y(t))\rangle=\langle\phi^{\ast},w(t)+t\phi-P(Ty(t))\rangle=t-\langle\phi^{\ast},f(Ty(t))\rangle\
.$ $\displaystyle<\
t-\langle\phi^{\ast},\alpha_{+}T(w(t)+t\phi)+\beta_{+}\rangle=(1-\alpha_{+})t-\langle\phi^{\ast},\beta_{+}\rangle,$

(recall $\displaystyle\beta_{+}\in Y$). As $\displaystyle t\to\infty$, we have
$\displaystyle h(F(y(t)))\to-\infty$ since $\displaystyle 1-\alpha_{+}<0$. A
similar estimate using the line $\displaystyle\ell_{-}$ obtains $\displaystyle
h(F(y(t)))\to-\infty$ also for $\displaystyle t\to-\infty$.

By definition, the image $\displaystyle F(y(t))$ of the fiber $\displaystyle
y(t)$ lies in a vertical line of $\displaystyle Y$. As $\displaystyle
t\to\pm\infty$, $\displaystyle h(F(y(t)))\to-\infty$: $\displaystyle F$ folds
downwards.

The set $\displaystyle{\mathscr{P}}^{b}$ is not necessary in this case: the
implicit set in the arguments is

$\displaystyle{\mathscr{Q}}^{ab}=\\{M_{q}\in{\cal B}(Y):\ q\in
L^{\infty}(\Omega)\ ,\ a\leq q(x)\leq 1+b\ \ {\rm a.e.}\\}\ ,$

which is convex and sequentially compact with respect to the weak operator
topology of $\displaystyle{\cal B}(Y)$ from the Banach-Alaoglu Theorem.
Clearly, $\displaystyle{\mathscr{Q}}^{ab}\subset{\mathscr{P}}^{b}$.

Proof of Theorem 2

The space $\displaystyle Y=L^{n}(\Omega)$ clearly satisfies the hypotheses of
Theorem 5, as does $\displaystyle K\subset Y$, the cone of nonnegative
functions (notice that nonnegative functions in $\displaystyle
X=W^{2,n}(\Omega)\cap W^{1,n}_{0}(\Omega)$ do not form a normal cone). Without
loss, $\displaystyle\lambda_{m}=\lambda_{m}({\bf L})=0$ (simply replace
$\displaystyle{\bf L}$ by $\displaystyle{\bf L}-\lambda_{m}$). Denote by
$\displaystyle{\mathbb{C}}^{+}=\\{z\in{\mathbb{C}},\Re z>0\\}$ the open right
half-plane: by hypothesis, $\displaystyle\sigma({\bf
L})\subset{\mathbb{C}}^{+}\cup\\{0\\}$.

Recall $\displaystyle X=W^{2,n}\cap W^{1,n}_{0}\subset C^{0}$. The operator
$\displaystyle{\bf L}+\gamma I:X\to Y$ is invertible for
$\displaystyle\gamma>0$. Denote the compact inclusion of $\displaystyle X$ in
$\displaystyle Y$ by $\displaystyle\iota:X\to Y$. Given the conformal mapping
$\displaystyle z\mapsto\Gamma_{\gamma}(z)=\gamma/(z+\gamma)$,
$\displaystyle\gamma>0$, define the operator $\displaystyle
T=\iota\circ\gamma({\bf L}+\gamma I)^{-1}\in{{\cal B}(Y)}$. As
$\displaystyle\Gamma_{\gamma}$ takes $\displaystyle{\mathbb{C}}^{+}$ to the
open disk $\displaystyle D=\\{|z-1/2|<1/2\\}$, $\displaystyle\sigma(T)\subset
D\cup\\{1\\}$ (recall the Dunford-Schwartz functional calculus for closed
operators $\displaystyle L:\operatorname{Dom}(L)\to Y$ [19], p. 599). Also,
$\displaystyle\phi_{m}>0$ is an eigenvector of both $\displaystyle{\bf L}$ and
$\displaystyle T$ for eigenvalues $\displaystyle 0$ and $\displaystyle r(T)=1$
respectively.

The hypotheses of Theorem 3 for $\displaystyle T$ are satisfied: from well
known properties of the operator $\displaystyle{\bf L}:X\to Y$ ([5], [4]),
$\displaystyle T\in{{\cal B}(Y)}$ is r-special. Also, $\displaystyle
r(T)=1,T\phi_{m}=\phi_{m}\in Y,\phi_{m}>0$ and similar hypotheses hold for
$\displaystyle\phi^{\ast}_{m}$, since $\displaystyle T$ is r-special.

The map $\displaystyle{\bf P}:Y\to Y,\ {\bf P}(u)={\bf f}(u)$ is associated
with a strictly convex function $\displaystyle{\bf
f}:{\mathbb{R}}\to{\mathbb{R}}$ for which $\displaystyle{\bf a}=\inf q_{{\bf
f}}(x,y)<0<{\bf b}=\sup q_{{\bf f}}(x,y)$. Set $\displaystyle P=I+{\bf
P}/\gamma$, so that $\displaystyle P(y)=f(y)$ for $\displaystyle f=1+{\bf
f}/\gamma$ and

$\displaystyle 0<a=1+{\bf a}/\gamma<r(\tilde{T})=1<1+b=1+{\bf b}/\gamma\ .$

As in the proof of Theorem 5, $\displaystyle P$ satisfies (r-H) and (r-Conv).
Now take $\displaystyle\gamma$ so large that $\displaystyle b>0$ complies with
the hypothesis of Theorem 5. Thus $\displaystyle F=I-PT:Y\to Y$ folds
downwards, and the same must happen to $\displaystyle{\bf F}={\bf L}-{\bf
P}:X\to Y$: Theorem 2 holds for the appropriate $\displaystyle{\bf b}=\gamma
b$.

We consider extensions of Theorem 2, by replacing $\displaystyle{\bf L}$ with
an appropriate $\displaystyle g({\bf L})$ which still induces an r-special
operators $\displaystyle T$ through the conformal mapping
$\displaystyle\Gamma$. We assume $\displaystyle\lambda_{m}=0$.

As a simple example, for $\displaystyle{\bf L}$ as above, powers
$\displaystyle{\bf L}^{k}$, for some $\displaystyle k\in{\mathbb{N}}$ still
yield r-special operators, provided $\displaystyle\sigma({\bf L}^{k})$ lies in
the closed half-plane. Clearly, this is the case if $\displaystyle\sigma({\bf
L})\subset[0,\infty]$. No other eigenvalue of $\displaystyle{\bf L}^{k}$ is
sent to zero and thus $\displaystyle T=\Gamma_{\gamma}({\bf L}^{k})$ still has
a simple spectral radius 1. Compactness and preservation of the cone
$\displaystyle K$ are immediate.

One might consider more complicated functions $\displaystyle g$, at the cost
of additional hypothesis (a decay rate of the resolvent of $\displaystyle L$
would be natural). For example, sectorial properties of
$\displaystyle\sigma({\bf L})$ might lead to fractional powers for which the
associated $\displaystyle T$ is still r-special. For a related situation, see
Proposition 18.

###  2.6 Consequences of properness: Theorem 3 (3)

The function below is already given in adapted coordinates,

$\displaystyle(x,y)\mapsto(x,(1-xy)y)\ .$

Vertical lines in the right (resp. left) half-plane fold downwards (resp.
upward), the vertical axis stays fixed. This map is not proper. Under
properness, the height functions of the image of all vertical lines have the
same asymptotic behavior, as shown in the proposition below. (Proposition 10
of [35], in the current notation).

######  Proposition 7.

Let $\displaystyle Z$ be a real Banach space. A proper map $\displaystyle
A:Z\oplus{\mathbb{R}}\to Z\oplus{\mathbb{R}}$ of the form $\displaystyle
A(z,t)=\big{(}z,\alpha(z,t)\big{)}$ for which no point has more than two
preimages is either a homeomorphism or folds vertically.

Proof of Theorem 3 (3): Combine Theorem 3 (2) and Proposition 7.

For a fiber $\displaystyle\Lambda=\\{u(t),t\in{\mathbb{R}}\\}$, asymptotic
information of the height $\displaystyle h(t)$ of its image $\displaystyle
F(u(t))=z+h(t)\phi$ determines if $\displaystyle F$ is a homeomorphism or a
fold, as well as the type of the fold. By Proposition 7, if $\displaystyle F$
is proper, such limits are independent of the fiber. One approach to the
asymptotic limits has been used at the end of the proof of Theorem 5: the
hypothesis on $\displaystyle f$ is so stringent that the images of fibers and
vertical lines in the domain have the same asymptotic behavior. We present an
alternative approach in Section 2.7.

###  2.7 Hypotheses (r-Hs), (r-Convs) and (r-Crit): smoothness

When $\displaystyle F$ is a $\displaystyle C^{1}$ map we use different tools.
From Proposition 7, the proof of Theorem 3 (4) is complete once we establish
that $\displaystyle F$ is proper (Proposition 10) and $\displaystyle F$ folds
downwards (Proposition 13 (3)).

Throughout this section, we assume the hypotheses of Theorem 3(4): (r-Hs),
(r-Convs) and (r-Crit) and the usual $\displaystyle{\bf b}>0$ for which
Proposition 2 holds. For $\displaystyle F(y)=y-P(Ty):Y\to Y$, we have
$\displaystyle DF(y)z=z-DP(Ty)T$. From (r-Crit), there is $\displaystyle
y_{c}\in Y$ for which $\displaystyle DF(y_{c})$ is not invertible. Without
loss,

$\displaystyle y_{c}=0\ \ \quad F(0)=0$

(compose $\displaystyle F$ with translations in domain and counterdomain —
hypotheses (r-Hs) and (r-Convs) are invariant under these operations).

We take the linearization $\displaystyle G(y,z)\in{\cal B}(Y)$ to be

$\displaystyle P(y)-P(z)=\int_{0}^{1}DP(z+s(y-z))\ ds\ (y-z)=:G(y,z)(y-z)\ ,$

so that $\displaystyle G(y,y)=J(y)=DP(y)$. Set $\displaystyle G(Ty)=G(Ty,0)$.
As $\displaystyle F(0)=0$ we have $\displaystyle P(Ty)=G(Ty)Ty$ with
$\displaystyle G(Ty)T\in{{\cal B}(Y)}$. Also, $\displaystyle F(y)=y-G(Ty)Ty$.

######  Proposition 8.

For $\displaystyle y,z\in Y$, $\displaystyle J(y),G(y,z)\in{\mathscr{P}}^{b}$.
The maps

$\displaystyle y,z\mapsto G(y,z)\in{{\cal B}(Y)}\quad\hbox{and}\quad
y,z\mapsto r(G(y,z)T)\in{\mathbb{R}}$

are continuous. The linearizations $\displaystyle G(y,z)$ satisfy (r-H) and
(r-Conv).

Proof:  From (r-Hs), $\displaystyle DP(y),DP(z)\in{\mathscr{P}}^{b}$. From
Lemma 2, convex combinations of Jacobians, as well as integrals along a
segment, also belong to $\displaystyle{\mathscr{P}}^{b}$ and satisfy (r-H) and
(r-Conv). The map $\displaystyle y,z,\mapsto G(y,z)\in{{\cal B}(Y)}$ is
continuous as $\displaystyle P:Y\to Y$ is $\displaystyle C^{1}$. Lemma 1
implies the continuity of $\displaystyle r(G(y,z)T)$.

The critical set $\displaystyle{\mathcal{C}}$ of $\displaystyle F$ consists of
the points in $\displaystyle Y$ for which $\displaystyle DF$ is not
invertible. From Propositions 1 and 2,

$\displaystyle{\mathcal{C}}=\\{y\in Y\ |\ 0\in\sigma(DF(y))\\}=\\{y\in Y\ |\
r(y)=r(J(Ty)T)=1\\}\ .$

Our next step is Proposition 9: critical points are abundant. We obtain
(exactly) one for each line
$\displaystyle\\{y+s\psi,t\in{\mathbb{R}},\psi>0\\}$. Set $\displaystyle
B_{\delta}^{Y}=\\{z\in Y,|z|\leq\delta\\}$.

######  Lemma 4.

Let $\displaystyle y,z,\psi\in Y,\psi>0,\delta>0$. Then there are
$\displaystyle s^{\ast}\in{\mathbb{R}}$, $\displaystyle k\in K$ and
$\displaystyle z_{\delta}\in B_{\delta}^{Y}$ such that $\displaystyle
y+s^{\ast}\psi=z+z_{\delta}+k$.

Let $\displaystyle z\in L^{2}(0,1)$ not bounded from below, $\displaystyle
y=0,\psi\equiv 1$. Then no constant $\displaystyle s$ gives $\displaystyle
0<s\psi=z+k$: the small perturbation $\displaystyle z_{\delta}$ is needed.

Proof:  If one cannot obtain $\displaystyle-z+y+s^{\ast}\psi=k+z_{\delta}$,
the line $\displaystyle L=\\{-z+y+s\psi,s\in{\mathbb{R}}\\}$ does not
intercept $\displaystyle K_{\delta}$, the (closed) convex span of
$\displaystyle K$ and $\displaystyle B_{\delta}^{Y}$, a set having
$\displaystyle\psi\in K$ in its interior. By the Hahn-Banach theorem,
$\displaystyle L$ and $\displaystyle K_{\delta}$ are separated by
$\displaystyle H=\\{z\in Y,\ \langle\ell,z\rangle=c\\}$, the level
$\displaystyle c\in{\mathbb{R}}$ of a functional $\displaystyle\ell\in
Y^{\ast}\setminus\\{0\\}$: $\displaystyle\ell|_{{L}}\leq
c\leq\ell|_{{K_{\delta}}}$.

We first show that $\displaystyle\ell\in K^{\ast}\setminus\\{0\\}$. Indeed, if
there is $\displaystyle k\in K$ such that
$\displaystyle\langle\ell,k\rangle<0$, then for large positive $\displaystyle
t$ (so that $\displaystyle tk\in K$),
$\displaystyle\langle\ell,t\,k\rangle<c$, a contradiction. Also, since
$\displaystyle\ell(y+s\psi)=\ell(y)+s\ell(\psi)\leq c$ for all $\displaystyle
s\in{\mathbb{R}}$, we must have $\displaystyle\ell(\psi)=0$. But, as
$\displaystyle\ell\in K^{\ast}\setminus\\{0\\}$ and $\displaystyle\psi>0$, we
must have $\displaystyle\ell(\psi)>0$.

As usual, $\displaystyle T\in{{\cal B}(Y)}$ is r-special, $\displaystyle
r(T)=1$. Set $\displaystyle r(y)=r(G(Ty)T)$.

######  Proposition 9.

Let $\displaystyle y\in Y$, $\displaystyle\psi>0$.

  1. (i)

The map $\displaystyle s\in{\mathbb{R}}\to r(y+s\psi)=r(G(T(y+s\psi),0)T)$ is
strictly increasing.
For some $\displaystyle s_{0}\in{\mathbb{R}}$, $\displaystyle
r(y+s_{0}\psi)=1$.

  2. (ii)

For all increasing sequences $\displaystyle s_{k}\to\infty$, the operators
$\displaystyle\\{G(Ts_{k}\psi)\\}\subset{\mathscr{P}}^{{\bf b}}$ have a common
weak limit $\displaystyle G(Ts_{k}\psi){\stackrel{{\scriptstyle{\scriptstyle
w}}}{{\longrightarrow}}\ }H_{\infty}\in{\mathscr{P}}^{{\bf b}}$. An analogous
result holds for decreasing sequences $\displaystyle s_{k}\to-\infty$.

Proof:  (i): By Proposition 1, $\displaystyle G(T(y+s\psi))T\in{{\cal B}(Y)}$
is r-special and the eigenvalue $\displaystyle r(y+s\psi)$ is basic. The map
$\displaystyle s\mapsto r(y+s\psi)$ is a continuous, strictly increasing
function, from Lemma 1. We search for $\displaystyle s_{+}$ for which
$\displaystyle r(y+s_{+}\psi)>1$.

From (r-Crit), $\displaystyle 0\in{\mathcal{C}}$, so that $\displaystyle
r(0)=r(G(0,0)T)=1$. By monotonicity (Lemma 1), for $\displaystyle
z_{+}=p\psi,p>0$, we have $\displaystyle r(z_{+})=r(G(Tz_{+})T)>1$. By
continuity, there is $\displaystyle\delta>0$ for which $\displaystyle
r(z_{+}+z_{\delta})>1$ for $\displaystyle z_{\delta}\in B_{\delta}^{Y}$.

Apply Lemma 4 for $\displaystyle z=z_{+}$: we obtain $\displaystyle
s^{\ast}\in{\mathbb{R}}$, $\displaystyle k\in K$, and $\displaystyle
z_{\delta}\in B_{\delta}^{Y}$ such that $\displaystyle
y+s^{\ast}\psi=z_{+}+k+z_{\delta}$. We compare the spectral radius at
different points:

$\displaystyle r(y+s^{\ast}\psi)=r(z_{+}+k+z_{\delta})\geq
r(z_{+}+z_{\delta})>1.$

Set $\displaystyle s_{+}=s^{\ast}$. Simple modifications of Lemma 4 obtain
$\displaystyle s_{-}$ for which $\displaystyle r(y+s_{-}\psi)<1$. By
continuity and monotonicity, there is a single $\displaystyle s_{0}$ such that
$\displaystyle r(y+s_{0}\psi)=1$.

(ii): From (r-Convs), as $\displaystyle\psi>0$, the linearizations
$\displaystyle G(Ts\psi)$ increase with $\displaystyle s$. Increasing
sequences $\displaystyle\\{s^{1}_{k}\\}$ and $\displaystyle\\{s^{2}_{k}\\}$
yield limits $\displaystyle H_{1}$ and $\displaystyle H_{2}$ by Lemma 2. For
$\displaystyle y^{\ast}\in K^{\ast},\ z\in K,$ the sequences
$\displaystyle\tau_{k}^{1}=\langle y^{\ast},G(Ts^{1}_{k}\psi)\ z\rangle$ and
$\displaystyle\tau_{k}^{2}=\langle y^{\ast},G(Ts^{2}_{k}\psi)\ z\rangle$
increase and interlace: for every $\displaystyle N\in{\mathbb{N}}$, there are
$\displaystyle m,n>N$ such that $\displaystyle\tau_{m}^{1}>\tau_{n}^{2}$ and
other $\displaystyle m,n>N$ with $\displaystyle\tau_{m}^{1}<\tau_{n}^{2}$.
Thus $\displaystyle H_{1}=H_{2}:=H_{\infty}$.

######  Corollary 2.

Every vertical line $\displaystyle\\{w+t\phi(T),t\in{\mathbb{R}}\\}$ in the
domain $\displaystyle Y$ of $\displaystyle F$ contains a critical point of
$\displaystyle F$. The same holds for lines
$\displaystyle\\{t\psi,\psi>0,t\in{\mathbb{R}}\\}$.

######  Proposition 10.

$\displaystyle F:Y\to Y$ is proper.

The argument follows in spirit the proof of the properness of $\displaystyle
F$ in [2].

Proof:  Let $\displaystyle\\{y_{k}\\}\subset Y$ be a sequence for which
$\displaystyle F(y_{k})=g_{k}\to g_{\infty}\in Y$. Suppose first that
$\displaystyle\\{y_{k}\\}$ is bounded in $\displaystyle Y$. Since the
linearizations $\displaystyle G(y,z),\ G(y)$ are uniformly bounded and
$\displaystyle P(0)=0$,

$\displaystyle|P(Ty_{k})|=|G(Ty_{k})y_{k}|\leq C|y_{k}|$

is also bounded in $\displaystyle Y$, and the sequence $\displaystyle
y_{k}=P(Ty_{k})+g_{k}$ is bounded. Therefore, for some subsequence,
$\displaystyle y_{k}\rightharpoonup y_{\infty}$. By the compactness of
$\displaystyle T$ (and the fact that $\displaystyle P$ is Lipschitz), again up
to a subsequence, $\displaystyle Ty_{k}\to Ty_{\infty}$ and thus
$\displaystyle P(Ty_{k})\to P(Ty_{\infty})$. Therefore, $\displaystyle
y_{k}\to y_{\infty}=P(Ty_{\infty})+g_{\infty}$.

Consider now $\displaystyle\\{y_{k}\\}$ unbounded in $\displaystyle Y$, for
strictly increasing $\displaystyle|y_{k}|\to\infty$. Normalize
$\displaystyle\hat{y}_{k}=y_{k}/|y_{k}|$ and use the linearization
$\displaystyle G_{k}=G(Ty_{k})$:

$\displaystyle\hat{y}_{k}-P(Ty_{k})/|y_{k}|=\hat{y}_{k}-G_{k}\hat{y}_{k}=g_{k}/|y_{k}|\to
0.$

As usual, since $\displaystyle G_{k}\in{\mathscr{P}}^{b}$, up to subsequences,
$\displaystyle G_{k}{\stackrel{{\scriptstyle{\scriptstyle
w}}}{{\longrightarrow}}\ }G_{\infty}$ for some $\displaystyle
G_{\infty}\in{\mathscr{P}}^{b}$, and from Lemma 3,

$\displaystyle\hat{y}_{k}\to\hat{y}_{\infty}\neq 0,\ \hbox{and}\
G_{\infty}T\hat{y}_{\infty}=\hat{y}_{\infty}.$

By Propositions 1 and 2, $\displaystyle r(G_{\infty}T)=1$ is a basic
eigenvalue and either $\displaystyle\hat{y}_{\infty}$ or
$\displaystyle-\hat{y}_{\infty}$ is a quasi-interior point. Say
$\displaystyle\hat{y}_{\infty}>0$.

Consider now the projections $\displaystyle y_{k}^{p}=|y_{k}|\hat{y}_{\infty}$
along the ray $\displaystyle\\{t\hat{y}_{\infty},t>0\\}$. Write the equation
$\displaystyle y_{k}-P(Ty_{k})=g_{k}$ as

$y_{k}^{p}-P(Ty_{k}^{p})=P(Ty_{k})-P(Ty_{k}^{p})+g_{k}+y_{k}^{p}-y_{k}\
.\quad\quad$ (1)

The right hand side is $\displaystyle o(|y_{k}|)$. This is clear for the
bounded sequence $\displaystyle\\{g_{k}\\}$ and for
$\displaystyle(y_{k}^{p}-y_{k})=|y_{k}|(\hat{y}_{k}-\hat{y}_{\infty})$, since
$\displaystyle\hat{y}_{k}\to\hat{y}_{\infty}\in Y$. Moreover,

$\displaystyle|P(Ty_{k})-P(Ty_{k}^{p})|\leq
C|y_{k}-y_{k}^{p}|=C|y_{k}||\hat{y}_{k}-\hat{y}_{\infty}|=o(|y_{k}|).$

Set $\displaystyle P(Ty_{k}^{p})=G_{k}^{p}\ Ty_{k}^{p}$. By Lemma 2, for a
subsequence, $\displaystyle G_{k}^{p}{\stackrel{{\scriptstyle{\scriptstyle
w}}}{{\longrightarrow}}\ }G_{\infty}^{p}\in{\mathscr{P}}^{b}$.

Divide equation (1) by $\displaystyle|y_{k}|$ to obtain

$\displaystyle\lim\
\big{(}y_{k}^{p}/|y_{k}|-P(Ty_{k}^{p})/|y_{k}|\big{)}=\hat{y}_{\infty}-G_{\infty}^{p}\
T\hat{y}_{\infty}=0,$

and, as $\displaystyle\hat{y}_{\infty}>0$, $\displaystyle
1\in\sigma(G_{\infty}^{p}T)$ and, by Theorem 4, $\displaystyle
r(G_{\infty}^{p}T)=1$. This contradicts Proposition 9: the points
$\displaystyle y_{k}^{p}$ belong to a common ray
$\displaystyle\\{s\hat{u}_{\infty}=s\psi,s\in{\mathbb{R}}^{+}\\}$. Combining
items (i) and (ii), $\displaystyle r(G_{k}^{p}T)$ increases and takes a value
larger than one: the limit cannot be one.

Minor adjustments handle also the case $\displaystyle\hat{y}_{\infty}<0$.

When $\displaystyle F:Y\to Y$ is a $\displaystyle C^{1}$ map, the adapted
coordinates in Proposition 4 are obtained from $\displaystyle C^{1}$ changes
of variables $\displaystyle\Psi:Y\to Y$. The inverses of vertical lines
$\displaystyle L_{z}=\\{z+h\phi,h\in{\mathbb{R}}\\}$ under $\displaystyle F$,
the fibers

$\displaystyle\Lambda_{z}=\\{y(t)=w(z,t)+t\phi,w(z,t)\in W,\
t\in{\mathbb{R}}\\}\ ,$

have $\displaystyle C^{1}$ parameterizations. Fix $\displaystyle z\in W$, set
$\displaystyle F(y(t))=z+h(t)\phi$, where again $\displaystyle
h(\cdot):{\mathbb{R}}\to{\mathbb{R}}$ is a $\displaystyle C^{1}$ map. Let
$\displaystyle\lambda(y(t))=1-r(DP(Ty(t)T))$.

######  Lemma 5.

For a continuous strictly positive
$\displaystyle\alpha:{\mathbb{R}}\to{\mathbb{R}}^{+}$, $\displaystyle
h^{\prime}(t)=\lambda(y(t))\alpha(t)$.

Part of this lemma is Proposition 2.6 in [14].

Proof:  Differentiate $\displaystyle F(y(t))=z+h(t)\phi$ along a fiber
$\displaystyle y(t)$ to obtain

$\displaystyle DF(y(t))y^{\prime}(t)=h^{\prime}(t)\phi\ .$

Since $\displaystyle\Pi_{W}DF(y(t)):W\to W$ is always invertible, at a
critical point $\displaystyle y(t_{c})$, we must have $\displaystyle
h^{\prime}(t_{c})=0$ and $\displaystyle\phi\notin DF(y(t))$. Thus
$\displaystyle y^{\prime}(t_{c})$ is a generator of
$\displaystyle\operatorname{Ker}DF(y(t_{c}))$, and either $\displaystyle
y^{\prime}(t_{c})>0$ or $\displaystyle-y^{\prime}(t_{c})>0$. As $\displaystyle
y^{\prime}(t_{c})=w^{\prime}(t_{c})+\phi$ and $\displaystyle
W=\operatorname{Ker}\phi^{\ast}$,

$\displaystyle\langle\phi^{\ast},y^{\prime}(t_{c})\rangle=\langle\phi^{\ast},w^{\prime}(t_{c})+\phi\rangle=\langle\phi^{\ast},\phi\rangle>0$

and $\displaystyle y^{\prime}(t_{c})>0$. Conversely, if $\displaystyle
h^{\prime}(\bar{t})=0$, as $\displaystyle
y^{\prime}(\bar{t})=w^{\prime}(\bar{t})+\phi\neq 0$, we must have that
$\displaystyle DF(y(\bar{t}))$ is not invertible and hence $\displaystyle
y(\bar{t})=y_{c}$. Adding up, $\displaystyle y(t)$ is a critical point of
$\displaystyle F$ if and only if $\displaystyle h^{\prime}(t)=0$.

As $\displaystyle DF(y(t))=I-H(y(t))T$ for $\displaystyle
H(y(t))=DP(Ty(t))\in{\mathscr{P}}^{b}$, eigenvectors
$\displaystyle\phi(y(t))>0$ and $\displaystyle\phi^{\ast}(y(t))>0$ are
associated with $\displaystyle r(H(y(t)))$. Use brackets for evaluation of
dual vectors,

$\displaystyle\langle\phi^{\ast}(y(t)),DF(y(t))y^{\prime}(t)\rangle=h^{\prime}(t)\langle\phi^{\ast}(y(t)),\phi\rangle$

and $\displaystyle h^{\prime}(t)\langle\phi^{\ast}(y(t)),\phi\rangle=\langle
DF(y(t))^{\ast}\phi^{\ast}(y(t)),y^{\prime}(t)\rangle=\lambda(y(t))\langle\phi^{\ast}(y(t)),y^{\prime}(t)\rangle$.

As $\displaystyle\langle\phi^{\ast}(y(t)),\phi\rangle>0$,
$\displaystyle\langle\phi^{\ast}(y(t)),y^{\prime}(t)\rangle=0$ if and only if
$\displaystyle h^{\prime}(t)=0$. From the first part of the proof,
$\displaystyle t=t_{c}$ for a critical point $\displaystyle y(t_{c})$ and then
$\displaystyle\langle\phi^{\ast}(y(t_{c})),y^{\prime}(t_{c})\rangle=0$, which
cannot happen because $\displaystyle y^{\prime}(t_{c})>0$. Thus
$\displaystyle\langle\phi^{\ast}(y(t)),y^{\prime}(t)\rangle>0$ and the
quotient
$\displaystyle\alpha(t)={\langle\phi^{\ast}(y(t)),y^{\prime}(t)\rangle}/{\langle\phi^{\ast}(y(t)),\phi\rangle}$
is strictly positive.

######  Proposition 11.

Every fiber contains points $\displaystyle y^{-}$, $\displaystyle y^{0}$ and
$\displaystyle y^{+}$ for which $\displaystyle r(y^{-})<r(y^{0})=1<r(y^{+})$,
or equivalently,
$\displaystyle\lambda(y^{-})>\lambda(y^{0})=0>\lambda(y^{+})$.

Proof:  Let $\displaystyle z^{0}\in Y$ be critical. For $\displaystyle\psi>0$,
set $\displaystyle z^{\pm}=z^{0}+s^{\pm}\psi$, where $\displaystyle
s^{-}<0<s^{+}$. By monotonicity (Lemma 1), $\displaystyle
r(z^{-})<r(z^{0})=1<r(z^{+})$. From Theorem 3 (2), the height function
$\displaystyle h(t)$ of each fiber is either strictly increasing or unimodal.
From properness (Proposition 10), all functions $\displaystyle h(t)$ have the
same asymptotic behavior (Proposition 7).

If $\displaystyle h$ along a fiber is strictly decreasing (hence along all
fibers), for example, we must have $\displaystyle h^{\prime}(t)\leq 0$. But
from Proposition 5, $\displaystyle\lambda(y(t))\leq 0$, or $\displaystyle
r(y(t))\geq 1$, and this violates the existence of a point $\displaystyle
z^{-}$ for which $\displaystyle r(z^{-})<1$. Similarly $\displaystyle h$
cannot be strictly increasing. We are left with folds, implying the result.

We extend Proposition 6. A point $\displaystyle y\in Y$ is regular if it is
not critical.

######  Proposition 12.

Let $\displaystyle y_{c}\in Y$ be a critical point of $\displaystyle F$,
$\displaystyle r(y_{c},y_{c})=r(J(Ty_{c})T)=1$. Then $\displaystyle F(y_{c})$
has a single preimage (clearly $\displaystyle y_{c}$ itself). For a regular
point $\displaystyle y\in Y$, the point $\displaystyle F(y)$ only has regular
preimages.

Proof:  Suppose $\displaystyle F(y)=F(y_{c})$, with $\displaystyle y\neq
y_{c}$. Then, as in Proposition 6, trichotomy holds: $\displaystyle y>y_{c},$
or $\displaystyle y<y_{c}$ and $\displaystyle r(y,y_{c})=1$. Since
$\displaystyle T$ is ergodic (r-special), either $\displaystyle Ty>Ty_{c}$ or
$\displaystyle Ty<Ty_{c}$. Since by hypothesis $\displaystyle
r(y_{c},y_{c})=1$, from (r-Convs), in both cases $\displaystyle r(y,y_{c})\neq
1$. The second statement now is obvious.

From Proposition 10, the map $\displaystyle F=I-PT:Y\to Y$ is proper and
clearly $\displaystyle PT:Y\to Y$ is a compact map. Moreover, any point has at
most two preimages (Proposition 6). The degree $\displaystyle\deg(F)$ is a
well defined map, and we compute it by considering indices at regular points.
More explicitly, for appropriate $\displaystyle y\in Y$, set
$\displaystyle\operatorname{ind}(F,y)=\deg(F,B_{\epsilon}(y),F(y))$, for small
$\displaystyle\epsilon>0$.

######  Proposition 13.

  1. (i)

For $\displaystyle\lambda(y)=1-r(y)$,
$\displaystyle\operatorname{ind}(F,y)=\operatorname{sgn}\lambda(y)$.

  2. (ii)

The function $\displaystyle F:Y\to Y$ is not injective. Also,
$\displaystyle\deg(F)=0$.

  3. (iii)

The function $\displaystyle F:Y\to Y$ folds downwards.

Proof:  (i) From a standard argument using the inverse function theorem, at a
regular point $\displaystyle y\in Y$,
$\displaystyle\operatorname{ind}(F,y)=\operatorname{ind}(F^{L},y)$, where

$\displaystyle F^{L}(z)=F(y)+DF(y)(z-y)=F(y)+(z-y)-J(Ty)T(z-y)\ .$

Recall that, for a compact operator $\displaystyle K$, the degree of
$\displaystyle I-K\in{{\cal B}(Y)}$ is $\displaystyle\operatorname{par}(K)$,
the parity of the number of strictly negative eigenvalues of $\displaystyle
I-K$, or equivalently, the parity of the number of eigenvalues of
$\displaystyle K$ which are larger than 1. Thus, for the restricted affine map
$\displaystyle F^{L}$,
$\displaystyle\deg(F^{L},B_{\epsilon}(y),F^{L}(y))=\operatorname{par}(J(Ty)T)$.
By Proposition 2, $\displaystyle J(Ty)T$ has zero or one eigenvalue larger
than 1, depending if $\displaystyle r(y,y)<1$ or $\displaystyle r(y,y)>1$ and
then $\displaystyle\operatorname{ind}(F,y)=\operatorname{par}(J(Ty)T)=1$ or
$\displaystyle-1$, always $\displaystyle\operatorname{sgn}\lambda(y)$.

(ii) Let $\displaystyle y_{c}\in Y$ be critical. For $\displaystyle\psi>0$,
set $\displaystyle y^{\pm}=y_{c}+s^{\pm}\psi$, where $\displaystyle
s^{-}<0<s^{+}$. By monotonicity (Lemma 1), the sign of $\displaystyle s^{\pm}$
equals the sign of $\displaystyle 1-r(J(Ty^{\pm})T)$, so that
$\displaystyle\operatorname{ind}(F,y^{-})=1$, and
$\displaystyle\operatorname{ind}(F,y^{+})=-1$. By Propositions 6 and 12, the
image $\displaystyle F(y)$ of any regular point $\displaystyle y\in Y$ has at
most two regular preimages, one being $\displaystyle y$. Call the preimages of
$\displaystyle F(y^{\pm})$ by $\displaystyle y_{1}^{\pm}$ and $\displaystyle
y_{2}^{\pm}$, where, say, $\displaystyle y_{1}^{\pm}=y^{\pm}$. Then

$\displaystyle\deg(F)=\operatorname{ind}(F,y_{1}^{-})+\operatorname{ind}(F,y_{2}^{-})=\operatorname{ind}(F,y_{1}^{+})+\operatorname{ind}(F,y_{2}^{+})$
$\displaystyle=1+\operatorname{ind}(F,y_{2}^{-})=\operatorname{ind}(F,y_{2}^{+})-1,$

and each unknown index is either 1 or $\displaystyle-1$. Then necessarily

$\displaystyle\deg(F)=0,\quad\operatorname{ind}(F,y_{2}^{-})=-1,\quad\operatorname{ind}(F,y_{2}^{+})=1\
.$

(iii) Since $\displaystyle F$ is not injective, consider $\displaystyle
y^{-}\neq y^{+}$ such that $\displaystyle F(y^{+})=F(y^{-})$. Both points are
regular (Proposition 12) and clearly belong to the same fiber, and we write
$\displaystyle y^{+}=y(t^{+})=w^{+}+t^{+}\phi,y^{-}=y(t^{-})=w^{-}+t^{-}\phi$,
with $\displaystyle t^{+}>t^{-}$. From (ii), we must have that
$\displaystyle\operatorname{ind}(F,y^{+})$ and
$\displaystyle\operatorname{ind}(F,y^{-})$ are opposite, so that
$\displaystyle\deg(F)=0$, and then $\displaystyle r(y(t^{+}))$ and
$\displaystyle r(y(t^{-}))$ re at opposite sides of $\displaystyle 1$.

From Proposition 8, $\displaystyle(y^{+}-y^{-})-G(y^{+},y^{-})(y^{+}-y^{-})=0$
and either $\displaystyle y^{+}-y^{-}>0$ or $\displaystyle(y^{+}-y^{-})<0$.
But
$\displaystyle\langle\phi^{\ast},y^{+}-y^{-}\rangle=\langle\phi^{\ast},(t^{+}-t^{-})\phi\rangle>0$,
so that $\displaystyle y^{+}-y^{-}>0$. By (r-Convs), $\displaystyle
r(y(t^{+}))>r(y(t^{-}))$, so that $\displaystyle r(y(t^{+}))>1>r(y(t^{-}))$.
From Theorem 3 (3) and $\displaystyle F$ s not injective (item (ii)),
$\displaystyle h$ is a (strictly) unimodal function. From Lemma 5,
$\displaystyle F$ folds downwards.

Prof of Theorem 3 (4): Combine Propositions 10 and 13 (3), as stated in the
beginning of the section.

######  Corollary 3.

The critical points of $\displaystyle F:Y\to Y$ are topological folds.

Proof:  It suffices to show that the critical points of $\displaystyle
F^{a}:W\oplus{\mathbb{R}}\to W\oplus{\mathbb{R}}$ are folds. Indeed, when
restricted to each vertical line of the domain, $\displaystyle F^{a}$ is a
proper, unimodal function folding downwards. The construction of local charts
leading to the normal form of a (topological) fold is left to the reader.

###  2.8 New folds from regularity

Functions $\displaystyle{\bf F}=L-P:X\to Y$ between Sobolev spaces sometimes
restrict to functions $\displaystyle{\bf G}:R\to Z$ between Hölder spaces.
This is the case for the Dirichlet Laplacian $\displaystyle L=-\Delta_{D}$
acting on a bounded domain $\displaystyle\Omega\subset{\mathbb{R}}^{n}$ and a
Nemitskii map $\displaystyle P(u)=f(u)$ for a smooth function $\displaystyle
f:{\mathbb{R}}\to{\mathbb{R}}$ ([2], [8]). Here, $\displaystyle n>1$, and
$\displaystyle X=W^{2,n}\cap
W^{1,n}_{0},Y=L^{n},R=C^{2,\alpha}_{0},Z=C^{0,\alpha}$, where we drop the
reference to the bounded set $\displaystyle\Omega\subset{\mathbb{R}}^{n}$. In
both scenarios, $\displaystyle{\bf F}$ folds downwards.

Hölder spaces are not reflexive, but there Nemitskii operators have better
differentiability properties. The non-normal cone of nonnegative continuous
functions in $\displaystyle C^{0,\alpha}(\Omega)$ has a nonempty interior.

There is a feature in the example which was not explored in Theorem 3 (4). As
in Section 2.5, from $\displaystyle L:X\to Y$ we define $\displaystyle
T=\iota\gamma(L+\gamma I)^{-1}$, and then systematically operate with the
composition $\displaystyle y\mapsto PT(y)$ of functions from $\displaystyle Y$
to itself. Instead, we interpret $\displaystyle PT$ as the composition
$\displaystyle\tilde{P}:X\to Y$ and $\displaystyle\tilde{T}:Y\to X$:
$\displaystyle\tilde{P}$ may have better smoothness properties than
$\displaystyle P$.

The following amplification of Theorem 3 (4) explores this feature.

######  Theorem 6.

Suppose $\displaystyle Y$ and $\displaystyle K$ as above,
$\displaystyle\iota:X\to Y$ an inclusion. For some
$\displaystyle\tilde{T}\in{\cal B}(Y,X)$, let $\displaystyle
T=\iota\tilde{T}\in{{\cal B}(Y)}$ an r-special operator for which
$\displaystyle r(T)=1,T\phi=\phi>0$. Suppose $\displaystyle\tilde{P}:X\to Y$
satisfies the following properties, for some $\displaystyle{\bf b}>0$.

  1. (r-Hs)

$\displaystyle\tilde{P}:X\to Y$ is a $\displaystyle C^{1}$ map such that
$\displaystyle\tilde{J}(x)=D\tilde{P}(x)\in{\cal B}(X,Y)$ admits an extension
$\displaystyle J(x)\in{\mathscr{P}}^{b}$.

  2. (r-Convs)

Let $\displaystyle y,z\in Y$, $\displaystyle y>z$. Then $\displaystyle
J(\tilde{T}y)-J(\tilde{T}z)\in{{\cal B}(Y)}$ is strictly positive.

  3. (r-Crit)

$\displaystyle PT=\tilde{P}\tilde{T}:Y\to Y$ is $\displaystyle C^{1}$ and some
$\displaystyle y\in Y$ is critical: $\displaystyle 0\in\sigma(DF(y))$.

Then $\displaystyle F:Y\to Y,y\mapsto y-P(Ty)$ folds downwards.

The proof of Theorem 6 is identical to the proof of Theorem 3 (4).

Theorem 7 provides a situation in which one fold gives rise to another. We
present it in the geometric context of Theorem 3: say $\displaystyle F:Y\to Y$
and $\displaystyle G:Z\to Z$ admit a common expression – can they share
geometric properties?

######  Theorem 7.

Let $\displaystyle Y$ and the map $\displaystyle F=I-PT:Y\to Y$ be as in
Theorem 6, so that $\displaystyle F$ folds downwards with respect to
$\displaystyle\phi=\phi(T)\in Y$. Let $\displaystyle Z$ be a Banach space,
with $\displaystyle\upsilon:Z\to Y$ be a dense inclusion and $\displaystyle
G:Z\to Z$ be a $\displaystyle C^{1}$ restriction of $\displaystyle F$ such
that $\displaystyle F\circ\upsilon=\upsilon\circ G$. Moreover, assume that (i)
$\displaystyle TY\subset Z$, (ii) $\displaystyle P:Z\to Z$, (iii) For
$\displaystyle z\in Z$, the maps $\displaystyle DG(z):Z\to Z$ are Fredholm
operators of index 0.

Then $\displaystyle G$ folds downwards along $\displaystyle\phi$.

In the example above, $\displaystyle X=W^{2,n}\cap W^{1,n}_{0}\hookrightarrow
C^{0,\alpha}=Z$, for $\displaystyle\alpha\in(0,1)$. For a Nemitskii map
$\displaystyle P$ associated with a smooth function $\displaystyle f$,
$\displaystyle\tilde{P}:X\to Z$ is $\displaystyle C^{1}$. The remaining
hypotheses are standard.

Proof:  By (i), $\displaystyle\phi\in Z$. From (ii), for $\displaystyle z\in
Z$, the solutions $\displaystyle y$ of $\displaystyle F(y)=z\in Z$ are
necessarily in $\displaystyle Z$. Indeed, write $\displaystyle F(y)=z$ as
$\displaystyle y=P(Ty)+z$. As $\displaystyle T:Y\to Z$, by (i), the right hand
side is necessarily in $\displaystyle Z$ and $\displaystyle y\in Z$.

Moreover, $\displaystyle V=\langle\phi\rangle\subset Z\subset Y$ and
$\displaystyle\phi^{\ast}\in Y^{\ast}\subset Z^{\ast}$. Decompose
$\displaystyle Y=W_{Y}\oplus V$, for $\displaystyle W_{Y}=\ker\phi^{\ast}$ (as
$\displaystyle\phi^{\ast}\in Y^{\ast}$) and $\displaystyle Z=W_{Z}\oplus V$,
for $\displaystyle W_{Z}=\ker\phi^{\ast}$ (as $\displaystyle\phi^{\ast}\in
Z^{\ast}$). The four projections are continuous and, for $\displaystyle z\in
Z\subset Y$, both decompositions coincide: if $\displaystyle
z=w_{Z}+t_{Z}\phi=w_{Y}+t_{Y}\phi$, then $\displaystyle w_{Z}=w_{Y}$ and
$\displaystyle t_{Z}=t_{Y}$.

Vertical lines $\displaystyle L_{y}=\\{y+s\phi,s\in{\mathbb{R}}\\}\subset Y$
either lie in $\displaystyle Z$ or do not intercept it. Indeed, if
$\displaystyle y\in Z$, then $\displaystyle y+s\phi\in Z$, by hypothesis (i).
The same happens to fibers: if some point of the fiber
$\displaystyle\Lambda_{y}=F^{-1}(L_{y})$ lies in $\displaystyle Z$, then the
whole fiber does. Indeed, if $\displaystyle F^{-1}(y+h_{0}\phi)\in Z$, we have
$\displaystyle y\in Z$, from the implication of (ii) in the beginning of the
proof, and then $\displaystyle F^{-1}(y+h\phi)\in Z$ for all $\displaystyle
h\in{\mathbb{R}}$. Thus, for $\displaystyle y\in Z$, the fiber
$\displaystyle\Lambda_{y}$ is the same for $\displaystyle F$ and
$\displaystyle G$.

By hypothesis, $\displaystyle Y$ is foliated by its fibers
$\displaystyle\Lambda_{y}$ and, from the previous fact, the same occurs to
$\displaystyle Z$. Continuity is missing: we prove that fibers in
$\displaystyle Z$ are curves.

From Proposition 4, the map $\displaystyle\Psi_{F}:W_{Y}\oplus{\mathbb{R}}\to
W_{Y}\oplus{\mathbb{R}},(w,t)\mapsto(F^{t}(w),t)$ is a diffeomorphism. Its
restriction $\displaystyle\Psi_{G}:W_{Z}\oplus{\mathbb{R}}\to
W_{Y}\oplus{\mathbb{R}},(w,t)\mapsto(G^{t}(w),t)\in Z$ is clearly a
$\displaystyle C^{1}$ map. It is also a bijection, since inverses of points
$\displaystyle z\in Z$ under $\displaystyle F$ lie in $\displaystyle Z$. It is
a global diffeomorphism: it suffices to show it is locally so. We proceed to
apply the inverse function theorem.

Clearly, it is a matter of verifying the invertibility of $\displaystyle
DG^{t}:W_{Z}\to W_{Z}$ for each $\displaystyle t$ fixed. Explicitly,
$\displaystyle DG^{t}(w)v=\Pi(v-DP(Tw+tT\phi)Tv)$. Consider the inclusion
$\displaystyle\iota:W_{Z}\to Z$, and write $\displaystyle DG^{t}(w)=\Pi\circ
DG(w+t\phi)\circ\iota$, so that, by (iii), $\displaystyle DG^{t}(w)$ is a
Fredholm operator of index 0. Thus, $\displaystyle DG^{t}(w)$ is not
invertible if and only if it has a nontrivial kernel: let
$\displaystyle\psi\in\operatorname{Ker}DG^{t}(w),\psi\neq 0$. As
$\displaystyle DG^{t}(w)$ is a restriction of $\displaystyle DF^{t}(w)$, we
must have $\displaystyle\psi\in\operatorname{Ker}DF^{t}(w)$, contradicting the
fact that $\displaystyle\Psi_{F}$ is a diffeomorphism. Thus,
$\displaystyle\Psi_{G}$ is a local, hence global diffeomorpshim.

The fibers of $\displaystyle G$ are those $\displaystyle F$, which folds
downwards: $\displaystyle G$ does too.

##  3 The effect of self-adjointness

Let $\displaystyle H=L^{2}(M,d\nu)$ be a real Hilbert space with norm
$\displaystyle|\cdot|$ for a $\displaystyle\sigma$-finite measure space
$\displaystyle(M,\nu)$. Suppose $\displaystyle L:D\subset H\to H$ is a self-
adjoint operator. We make no reference to the complexifications required for
the underlying spectral theory.

The concepts related to positivity in both sections overlap, but we keep the
more familiar terms in each context. A function is positive, $\displaystyle
u\geq 0$ if $\displaystyle u(x)\geq 0\ a.e.$ and $\displaystyle u\neq 0$ and
strictly positive, $\displaystyle u>0$ if $\displaystyle u(x)>0\ a.e.$. The it
positive cone $\displaystyle K\subset H$ is the set containing the positive
functions and zero. A bounded operator $\displaystyle A:H\to H$ is positivity
preserving if $\displaystyle Au\geq 0$ for all $\displaystyle u\geq 0$. It is
positivity improving if for any $\displaystyle u\geq 0$, we have
$\displaystyle Au>0$ and positively stable if $\displaystyle Au>0$ whenever
$\displaystyle u>0$.

For appropriate self-adjoint operators $\displaystyle L:D\subset H\to H$ and
maps $\displaystyle P:H\to H$, we consider counterparts to Theorem 3 for
$\displaystyle F:D\to H,F(u)=Lu-P(u)$.

###  3.1 m-special operators and fine perturbations

Let $\displaystyle L:D\subset H\to H$ be a self-adjoint operator. An
eigenvalue $\displaystyle\lambda\in\sigma(L)$ of is basic if
$\displaystyle\lambda\in{\mathbb{R}}$ is an isolated point of
$\displaystyle\sigma(L)$ and the associated invariant eigenspace is spanned by
a strictly positive eigenvector $\displaystyle\phi\in D,\phi>0$.

Set $\displaystyle\lambda_{m}=\min\ \sigma(L)$. A self-adjoint operator
$\displaystyle L:D\to H$ is m-special if

  1. (m-S)

$\displaystyle\lambda_{m}$ is a basic eigenvalue; if $\displaystyle
0\in\sigma(L)$ is an eigenvalue, then $\displaystyle 0=\lambda_{m}$.

  2. (m-PI)

$\displaystyle(L-\mu)^{-1}$ is positivity improving for all
$\displaystyle\mu<\lambda_{m}$.

The operators $\displaystyle L$ or $\displaystyle(L-\mu)^{-1}$ are not
required to be compact.

Set $\displaystyle\mu_{m}=\inf\sigma(L)\setminus\\{\lambda_{m}\\}$. An
operator $\displaystyle A\in{\cal B}(H)$ is a fine perturbation of
$\displaystyle L$ if, for some $\displaystyle b,p\in{\mathbb{R}}$, it belongs
to

$\displaystyle{\mathscr{P}}^{b,p}=\\{A\in{{\cal B}(H)}\ \hbox{is symmetric},\
\|A\|\leq b<\mu_{m},\ \hbox{and}$ $\displaystyle A+pI\in{{\cal B}(H)}\
\hbox{is positive preserving.}\\}$

We prove that, for a fine perturbation $\displaystyle A$ of the m-special
operator $\displaystyle L$, the operator $\displaystyle L-A$ is m-special. We
quote Theorem XIII.44 of [28] and the m-special case of a result by Faris [21]
(also exercise 91, Chapter XIII of [28]).

######  Lemma 6.

Let $\displaystyle L:D\to H$ be a self-adjoint operator such that
$\displaystyle\lambda_{m}=\min\sigma(L)$ is an eigenvalue and $\displaystyle
e^{-tL}$ is positivity preserving for $\displaystyle t>0$. Then
$\displaystyle\lambda_{m}$ is a basic eigenvalue if and only if $\displaystyle
L$ satisfies (m-PI), or equivalently,

For all $\displaystyle t>0$, the operators $\displaystyle e^{-tL}:H\to H$ are
positivity improving.

######  Lemma 7.

Let $\displaystyle L:D\to H$ be a self-adjoint operator for which (m-PI) holds
and let $\displaystyle A:H\to H$ be a positivity preserving, bounded,
symmetric operator. Then (m-PI) (hence (mPI2)) holds for $\displaystyle
L-A:D\to H$.

######  Proposition 14.

Let $\displaystyle L:D\to H$ be an m-special operator and let $\displaystyle
A\in{\mathscr{P}}^{b,p}$ be a fine perturbation of $\displaystyle L$. Then
$\displaystyle S=L-A:D\to H$ is $\displaystyle m$-special.

Proof:  By the Kato-Rellich theorem, $\displaystyle S:D\to H$ is self-adjoint.
Since $\displaystyle\min\ \sigma(L)$ is basic, a standard argument shows that
$\displaystyle\sigma_{m}=\inf\sigma(S)$ is necessarily a simple eigenvalue
associated with a normal eigenvector $\displaystyle\psi_{m}$. We now prove
that if $\displaystyle 0$ is an eigenvalue of $\displaystyle S$, and then then
$\displaystyle 0=\lambda_{m}$. Consider the quadratic form $\displaystyle
Q(v)=\langle Sv,v\rangle$ for a normal $\displaystyle v\in D$. We must have
$\displaystyle Q(\psi_{m})=0$. If $\displaystyle\sigma_{m}<0$ there is a
normal $\displaystyle\psi\in D$ for which $\displaystyle Q(\psi)<0$ and then
$\displaystyle Q\leq 0$ in the plane $\displaystyle E$ spanned by
$\displaystyle\psi_{m}$ and $\displaystyle\psi$. For a normal $\displaystyle
v\in W_{D}=\phi_{m}^{\perp}\subset D$, $\displaystyle\langle
Lv,v\rangle\geq\mu_{m}$ and $\displaystyle Q(v)\geq\mu_{m}-\sup\sigma(A)>0$.
But $\displaystyle E$ must intersect the (closed) codimension one subspace
$\displaystyle W_{D}$, a contradiction. This settles (m-S) for $\displaystyle
S$.

We now prove (m-PI2). For $\displaystyle t>0$, $\displaystyle e^{-tT}$ is
positivity improving by (m-PI) for $\displaystyle L$. For some $\displaystyle
p$, $\displaystyle pI+A$ is positive preserving. From Lemma 6, $\displaystyle
e^{-tS}=e^{-tpI}e^{-t(S-pI)}=e^{-tp}e^{-t(L-(A+pI))}$ is positivity improving.

###  3.2 Counterparts to Theorem 3

Let $\displaystyle\phi_{m}>0$ be the normal eigenvector associated with
$\displaystyle\lambda_{m}=\lambda_{m}(L)$. Set $\displaystyle
V=\\{t\phi_{m},t\in{\mathbb{R}}\\}$ and consider the orthogonal decompositions
$\displaystyle H=W\oplus V,D=W_{D}\oplus V$, for $\displaystyle W_{D}=(W\cap
D)$. Let $\displaystyle\Pi$ be the orthogonal projection
$\displaystyle\Pi:H\to W$.

As in the previous section, a linearization of $\displaystyle F=L-P$ is a
bounded, symmetric operator $\displaystyle G(u,v)\in{{\cal B}(H)}$ such that,
for $\displaystyle u,v\in D\subset H$, $\displaystyle
F(v)-F(u)=(L-G(v,u))(v-u)$. Again, if $\displaystyle P:H\to H$ is a
$\displaystyle C^{1}$ map, linearizations are obtained from Jacobians
$\displaystyle J(u)=DP(u):H\to H\in{{\cal B}(H)}$,

$F(v)-F(u)=\big{(}L-\int_{0}^{1}DP(u+t(v-u))\ dt\big{)}(v-u):=(L-G(v,u))(v-u)\
.$ (2)

Consider the following properties for the Lipschitz map $\displaystyle P:H\to
H$.

  1. (m-H)

There are $\displaystyle b,p\in{\mathbb{R}}$ such that, for $\displaystyle
u,v\in D$, $\displaystyle\Pi G(u,v)\in{\mathscr{P}}^{b,p}$.

  2. (m-Hs)

There are $\displaystyle b,p\in{\mathbb{R}}$ such that, for $\displaystyle
u\in D$, $\displaystyle\Pi DP(u)\in{\mathscr{P}}^{b,p}$.

  3. (m-Conv)

If $\displaystyle u<v<w$, then $\displaystyle\langle
w-v,F(v)-F(u)\rangle<\langle v-u,F(w)-F(v)\rangle$ or, equivalently,
$\displaystyle\langle v-u,P(w)\rangle+\langle w-v,P(u)\rangle+\langle
u-w,P(v)\rangle>0$.

  4. (m-Convs)

$\displaystyle P:H\to H$ is a $\displaystyle C^{1}$ map, $\displaystyle u,v\in
H$, $\displaystyle v>u$. Then $\displaystyle DP(v)-DP(u)\in{{\cal B}(H)}$ is
positivity preserving.

  5. (m-Crit)

$\displaystyle P:H\to H$ is $\displaystyle C^{1}$ and some $\displaystyle u\in
H$ is critical: $\displaystyle 0\in\sigma(DF(y))$.

Again,(m-Hs) and (m-Convs) imply (m-H) and (m-Conv) respectively.

######  Theorem 8.

Suppose $\displaystyle H$ as above, $\displaystyle L:D\subset H\to H$ an
m-special operator, and $\displaystyle P:H\to H$ a Lipschitz (or
$\displaystyle C^{1}$) map admitting linearizations $\displaystyle G(u,v)$.
Define the function $\displaystyle F:D\to H,u\mapsto Lu-P(u)$.

(1) If $\displaystyle\ [-\sup_{u,v}\|G(u,v)\|,\ \sup_{u,v}\|G(u,v)\|\ ]\ \cap\
\sigma(L)=\emptyset$, then $\displaystyle F$ is a homeomorphism.

(2) Suppose that (m-H), (m-Conv) hold. Then $\displaystyle F$ is a simple map.

(3) If (m-H), (m-Conv) hold and $\displaystyle F:H\to H$ is proper, then it is
a homeomorphism or folds vertically.

(4) Suppose that (m-Hs), (m-Convs) and (m-Crit) hold. Then $\displaystyle F$
folds downwards.

There are no restrictions on $\displaystyle b$ besides $\displaystyle
b<\mu_{m}$: Theorem 3 is more demanding.

The proof follows closely the arguments of the previous section. For
$\displaystyle t\in{\mathbb{R}}$, set $\displaystyle P^{t}:W\to W,\
P^{t}(w)=\Pi P(w+t\phi_{m})$ and $\displaystyle F^{t}:W_{D}\to W,F^{t}(w)=\Pi
F(w+t\phi_{m})$.

######  Proposition 15.

Suppose that (m-H) holds. Then, for each $\displaystyle t\in{\mathbb{R}}$, the
maps $\displaystyle F^{t}:W_{H}\to W$ and
$\displaystyle\Psi:W\oplus{\mathbb{R}}\to
W\oplus{\mathbb{R}},\Psi(w,t)=(F^{t}(w),t)$ are bilipschitz homeomorphisms.
Thus

$\displaystyle F^{a}(z,t)=F\circ\Psi^{-1}(z,t)=(z,h^{a}(z,t))\ ,$

for a Lipschitz height function $\displaystyle h^{a}:Y=W\oplus
V\to{\mathbb{R}}$. The map $\displaystyle F$ admits fibers
$\displaystyle\Lambda_{z}=\\{u(t)=w(t)+t\phi_{m},t\in{\mathbb{R}}\\}=F^{-1}(L_{z})$
and $\displaystyle F^{a}(L_{z})=\Lambda_{z}$.

The argument is standard ([8], [11], [13], [27], [29], [30] [38]). The
positivity of the eigenfunction associated with $\displaystyle\lambda_{m}$ is
not required, nor the existence of positive maps $\displaystyle A+pI$ in the
definition of $\displaystyle{\mathscr{P}}^{b,p}$.

Proof:  Since $\displaystyle L$ is m-special and $\displaystyle
W_{D}=V^{\perp}$, the restriction $\displaystyle L_{W}:W_{D}\to W$ is
invertible: $\displaystyle\sigma(L_{W})=\sigma(L)\setminus\\{\lambda_{m}\\}$.
To show the invertibility of $\displaystyle F^{t}:W\to W$ for each
$\displaystyle t\in{\mathbb{R}}$, we solve $\displaystyle F^{t}(w)=\Pi
L(w+t\phi_{m})-\Pi P(w+t\phi_{m})=z$, for $\displaystyle z\in W$, or

$\displaystyle y=P^{t}(L_{W}^{-1}y)+z\quad\hbox{for}\quad y=L_{W}w\in W\ .$

From (m-H), the map $\displaystyle K^{t}(y)=P^{t}(L_{W}^{-1}y)$ is a
contraction, with constant $\displaystyle c=b/\mu_{m}$ independent of
$\displaystyle t$. Indeed,

$\displaystyle|K^{t}(w_{1})-K^{t}(w_{2})|=|\Pi\big{(}P(L_{W}^{-1}(w_{1})+t\phi_{m})-P(L_{W}^{-1}(w_{2})+t\phi_{m})\big{)}|$
$\displaystyle=|\Pi
G(L_{W}^{-1}w_{1}+t\phi_{m},L_{W}^{-1}w_{2}+t\phi_{m})L_{W}^{-1}(w_{1}-w_{2})|\leq\frac{b}{\mu_{m}}|w_{1}-w_{2}|,$

as $\displaystyle\|L_{W}^{-1}\|=1/\mu_{m}$ and, by (m-H), $\displaystyle\|\Pi
G\|\leq b<\mu_{m}$: $\displaystyle K^{t}$ is a contraction. Set $\displaystyle
c=b/\mu_{m}$.

Hence the maps $\displaystyle I-K^{t}:W\to W$ are (uniform) Lipschitz
bijections. The inverses $\displaystyle(I-K^{t})^{-1}:W\to W$ are also
uniformly Lipschitz with constant $\displaystyle 1/(1-c)$: for $\displaystyle
z_{i}=(I-K^{t})^{-1}(y_{i}),\ i=1,2$,

$\displaystyle|z_{1}-z_{2}|\leq|y_{1}-y_{2}|+|K^{t}(z_{1})-K^{t}(z_{2})|\leq|y_{1}-y_{2}|+c|z_{1}-z_{2}|\
.$

Thus $\displaystyle F^{t}=(I-K^{t})\circ L_{W}$ are also uniformly bilipschitz
homeomorphisms. The map $\displaystyle\Psi=\left(F^{t},\ \hbox{Id}\right):D\to
H$ is a homeomorphism because of the continuous dependence of the fixed point
with respect to $\displaystyle t$.

Remarks

1\. To use Proposition 15 with the hypotheses of Theorem 1, translate
$\displaystyle F={\bf F}-\gamma I$, $\displaystyle P={\bf P}-\gamma I$ with
$\displaystyle\gamma=(a+b)/2$.

2\. In Theorem 1, $\displaystyle P(u)=f(u)$ and the values of $\displaystyle
f^{\prime}$ are close to $\displaystyle\lambda_{m}$: $\displaystyle P(u)$ is
roughly a multiplication by $\displaystyle\lambda_{m}$ (plus a constant). For
a general $\displaystyle P$, once the projected maps $\displaystyle\Pi P^{t}$
are appropriately bounded, adapted coordinates apply, the behavior of the map
$\displaystyle F$ along the vertical axis may be different.

3\. Under the hypotheses of Theorem 8 (1), the estimates in the proof apply to
$\displaystyle F:D\to H$ itself: $\displaystyle F$ is a bilipschitz
homeomorphism. This is the usual Dolph-Hammerstein theorem ([18], [23]).

From Proposition 15, vertical lines $\displaystyle
L_{z}=\\{z+h\phi_{m},h\in{\mathbb{R}}\\}\subset H$, when inverted by
$\displaystyle F$, give rise to fibers
$\displaystyle\Lambda_{z}=\\{u(t)=w(t)+t\phi_{m},w(t)\in
W_{D},t\in{\mathbb{R}}\\}$.

Without loss, after a translation in the range $\displaystyle H$, we assume
$\displaystyle P(0)=F(0)=0$. Indeed, hypotheses (m-H) and (m-Conv) are
invariant under this operation.

######  Corollary 4.

Let $\displaystyle\Lambda_{z}$ be a fiber. For some $\displaystyle
C\in{\mathbb{R}}$, $\displaystyle|w(t)|\leq C(|z|+|\Pi P(t\phi_{m})|)$.

Proof:  Since $\displaystyle F^{t}:W_{D}\to W$ are bilipschitz maps and
$\displaystyle F(u(t))=z+h(t)\phi_{m}$,

$\displaystyle|w(t)-0|\leq C|F^{t}(w(t))-F^{t}(0)|$ $\displaystyle=C|\Pi
F(w(t)+t\phi_{m})-\Pi F(t\phi_{m})|=C|z-\Pi P(t\phi_{m})|$

and the result follows.

Adapted coordinates provide examples. For
$\displaystyle-\Delta:H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\to H^{0}(\Omega)$,
with smallest eigenvalue $\displaystyle\lambda_{m}$ and ground state
$\displaystyle\phi_{m}$, consider, for $\displaystyle
t=\langle\phi_{m},u\rangle$,

$\displaystyle F(u)=Lu-P(u)=(-\Delta-\lambda_{m})u+(I-\Pi)(\sin t)u\ ,$

The change of variables $\displaystyle z=(-\Delta-\lambda_{m})w$ yields
$\displaystyle F^{a}(z,t)=(z,t\sin t)$, an example of a function
$\displaystyle F$ with points with infinitely many preimages.

######  Proposition 16.

If $\displaystyle F(u)=F(v)$ and $\displaystyle P$ satisfies (m-Conv), then
either $\displaystyle u>v$ or $\displaystyle u=v$ or $\displaystyle u<v$. If
(m-Conv) also holds, $\displaystyle F(u)=g$ has at most two solutions.

Proof:  An integral of fine perturbations as in Equation 2 for $\displaystyle
G(u,v)$, is also fine. By Proposition 14, $\displaystyle L-G(u,v)$ is
m-special. If $\displaystyle F(u)=F(v)$, then
$\displaystyle\pm(u-v)\in\operatorname{Ker}(L-G(u,v))$ is a positive
eigenfunction or 0, by (m-S): trichotomy holds.

Suppose $\displaystyle F(u)=F(v)=F(w)=g$. By the previous paragraph, we may
assume $\displaystyle u<v<w$, and then the first form of (m-Conv) is clearly
violated.

Proof of Theorem 8: Item (1) is a remark after Proposition 15. Proposition 15
showed that no additional restrictions on $\displaystyle b>0$ are necessary.
Proposition 16, in turn, employed (m-Conv) exactly as Proposition 6 employed
(r-Conv). The proof now follows the arguments in the previous section.

###  3.3 Asymptotics of heights: properness

Informally, if (m-H) holds, the properness of $\displaystyle F$ depends on its
behavior along the vertical axis $\displaystyle V=\langle\phi_{m}\rangle$.
Assume $\displaystyle P:H\to H$ Lipschitz and recall $\displaystyle F(0)=0$.
Set

$\displaystyle F(t\phi_{m})=z_{v}(t)+h_{v}(t)\phi_{m}=\Pi
P(t\phi_{m})+\langle\phi_{m},F(t\phi_{m})\rangle\phi_{m}\ ,\ \hbox{for}\ \
z_{v}\in W\ .$

######  Proposition 17.

Suppose the function $\displaystyle F:D\to H$ satisfies (m-H) and

  1. (m-hor)

$\displaystyle\lim_{|t|\to\infty}|z_{v}(t)|_{H}\ /\ t=0$.

  2. (m-ver)

For some $\displaystyle c>0$ and large $\displaystyle|t|$,
$\displaystyle|h_{v}(t)|\geq c|t|$.

Then $\displaystyle F$ is proper.

From Corollary 4, $\displaystyle|w(t)|\leq C(|z|+|z_{v}(t)|)$. Hypothesis
(m-hor) implies that fibers $\displaystyle w(t)+t\phi_{m}$ are asymptotically
vertical: indeed, $\displaystyle w(t)/t\to 0$ for $\displaystyle|t|\to\infty$.
Hypothesis (m-ver) imposes a rate of growth for the height $\displaystyle
h_{v}(t)$.

Proof:  For $\displaystyle z\in W$, consider the fiber
$\displaystyle\Lambda_{z}=\\{u(t)=w(t)+t\phi_{m}\\}$, and its image
$\displaystyle F(u(t))=z+h_{z}(t)\phi_{m}$. As $\displaystyle LW_{D}\subset
W$,

$\displaystyle
h_{z}(t)=\langle\phi_{m},L(w(t)+t\phi_{m})-P(w(t)+t\phi_{m})\rangle=\lambda_{m}t-\langle\phi_{m},P(w(t)+t\phi_{m})\rangle$
$\displaystyle=\lambda_{m}t-\langle\phi_{m},(P(w(t)+t\phi_{m})-P(t\phi_{m}))\rangle-\langle\phi_{m},P(t\phi_{m})\rangle$

Since $\displaystyle P:H\to H$ is Lipschitz, from Corollary 4,

$\displaystyle|P(w(t)+t\phi_{m})-P(t\phi_{m})|/t\leq C_{1}|w(t)/t|\leq
C_{2}|\Pi P(t\phi_{m})/t|+o(1)\ ,$

for constants $\displaystyle C_{i}$ locally uniformly in $\displaystyle z$.
From (m-hor), for large $\displaystyle|t|$,

$\displaystyle
h_{z}(t)/t=\lambda_{m}-\langle\phi_{m},P(t\phi_{m})\rangle/t+o(1)=h_{v}(t)/t+o(1)\
.$

Thus $\displaystyle h_{v}$ and $\displaystyle h_{z}$ have the same rate of
growth, given by (m-ver).

We prove the equivalent properness of $\displaystyle F^{a}=F\circ\Psi^{-1}$.
The estimates for $\displaystyle h_{z}$ translate to $\displaystyle
h_{z}^{a}=h_{z}\circ\Psi^{-1}$, as
$\displaystyle\Psi:W_{D}\oplus{\mathbb{R}}\to W\oplus{\mathbb{R}}$ in
Proposition 15 leaves the vertical component unaltered. For a convergent
sequence in the image of $\displaystyle F^{a}$,
$\displaystyle(z_{n},s_{n})=(z_{n},h^{a}_{z}(z_{n},t_{n}))\to(z_{\infty},s_{\infty})$.
The estimates for $\displaystyle h^{a}$, being locally uniform in
$\displaystyle z$, imply the boundedness of $\displaystyle\\{t_{n}\\}$:
$\displaystyle F^{a}$ is proper.

Under the hypotheses of Theorem 8, if the asymptotic signs of $\displaystyle
h_{v}(t)$ are different then $\displaystyle F$ is a homeomorphism. If they are
equal, $\displaystyle F$ is a global fold.

####  3.3.1 Nemitskii maps yield properness and folds

This natural extension of Theorem 1 is a counterpart of Theorem 5.

######  Theorem 9.

Suppose $\displaystyle H$ as above, $\displaystyle L:D\subset H\to H$ an
m-special operator such that $\displaystyle\phi_{m}\in L^{1}$. Let
$\displaystyle P:H\to H$ be a Nemitskii map $\displaystyle P(u)=f(u)$ for a
strictly convex function $\displaystyle f:{\mathbb{R}}\to{\mathbb{R}}$ whose
Newton quotient satisfies

$\displaystyle\ -\infty<a=\inf_{r\neq s}q(r,s)<\lambda_{m}<b=\sup_{r\neq
s}q(r,s)<\mu_{m}\ .$

Then the function $\displaystyle F:D\to H,u\mapsto Lu-P(u)$ folds downwards

The functions in $\displaystyle H$ do not have to be defined on bounded
domains.

We obtain the properness of $\displaystyle F$ from Proposition 17.

######  Lemma 8.

(i) $\displaystyle P(u)=f(u)-(b+a)u/2$ satisfies (m-H) and (m-Conv).

(ii) (m-hor) holds.

(iii) (m-ver) also holds:
$\displaystyle\lim_{t\to-\infty}h_{v}(t)/t=\lambda_{m}-a>0,\
\lim_{t\to\infty}h_{v}(t)/t=\lambda_{m}-b<0$.

Proof:  (i) The verification of properties (m-H) and (m-Conv) is by now
familiar. (ii) For $\displaystyle v\in W_{D}$ of norm one and compact support
(and then $\displaystyle\Pi v=v$),

$\displaystyle\|\Pi P(t\phi_{m})/t\|_{H}=\sup_{|v|=1}\ \left|\langle v,\Pi
P(t\phi_{m})/t\rangle\right|=\sup_{|v|=1}\ \left|\langle
v,f(t\phi_{m})/t\rangle\right|\ .$

We use the dominated convergence theorem. We consider $\displaystyle
t\to\infty$, the other case is similar. By the convexity of $\displaystyle f$,
for $\displaystyle x>0$, $\displaystyle f(tx)/t\to bx$ if $\displaystyle
t\to\infty$, and the integrand $\displaystyle|vf(t\phi_{m})/t|$ is pointwise
bounded by an $\displaystyle L^{1}$ function. Taking limits,

$\displaystyle\|\lim_{t\to\infty}\Pi P(t\phi_{m})/t\|_{H}=\sup_{|v|=1}\
\big{|}\int_{\Omega}\lim_{t\to\infty}vf(t\phi_{m})/t\big{|}=\sup_{|v|=1}\
\big{|}b\int_{\Omega}v\ \phi_{m}\big{|}\ =0,$

since $\displaystyle v\in W_{D}=V^{\perp}$. Statement (iii) also follows from
dominated convergence.

Proof of Theorem 9: Combine Theorem 8 (3) with Lemma 8 (iii).

A Lipschitz function $\displaystyle f$ induces a Lipschitz Nemitskii map
$\displaystyle P:L^{2}(\Omega)\to L^{2}(\Omega)$ on unbounded sets
$\displaystyle\Omega\subset{\mathbb{R}}^{n}$ if $\displaystyle f(0)=0$. As an
example of application of Theorem 9, consider the m-special (see Proposition
3.4.1) Hamiltonian for the hydrogen atom in $\displaystyle{\mathbb{R}}^{3}$,
$\displaystyle Lu=-\Delta u-u/r$, with basic eigenvalue
$\displaystyle\lambda_{m}(L)$ associated with a normal, positive ground state
$\displaystyle\phi_{m}\in L^{1}\cap L^{2}$.

###  3.4 Some examples

Folds related to self-adjoint elliptic operators different from the Dirichlet
Laplacian are known ([4]): we present some more. The larger class of
$\displaystyle m$-special operators $\displaystyle L:D\to H$ includes the
finite dimensional case $\displaystyle D=H={\mathbb{R}}^{n}$ when
$\displaystyle\nu$ is a finite collection of deltas and the operators
described in Proposition 18.

####  3.4.1  m-special operators $\displaystyle L$

The identification of operators $\displaystyle L$ generating positivity
preserving semigroups (hypothesis (m-PI2)) is by itself a field of
mathematics. Arguments in the spirit of Bochner’s theorem on distributions of
positive type and the Levy-Khintchine formula (Appendix 2, [28], vol. IV) lead
to a wealth of examples.

######  Proposition 18.

The following operators are $\displaystyle m$-special: I.
$\displaystyle-\Delta$ for Dirichlet, Neumann, periodic or mixed boundary
conditions on bounded smooth domains; II. The hydrogen atom in
$\displaystyle{\mathbb{R}}^{3}$, $\displaystyle Lu=-\Delta u-u/r$; III. The
quantum oscillator $\displaystyle Lu(x)=-u^{\prime\prime}(x)+x^{2}u(x)$; IV.
Fractional powers $\displaystyle L^{s},s\in(0,1)$ of positive $\displaystyle
m$-special operators.

Proof:  Hypothesis (m-S) is familiar for all examples, we check (m-PI). For
(I), see Sections 8.1 and 8.2 of [3]. For (II), take $\displaystyle
L_{0}=-\Delta$ with $\displaystyle D=H^{2}({\mathbb{R}}^{3})$ and define
$\displaystyle L=L_{0}+V$ for the potential $\displaystyle V=-1/r$. We prove
(m-PI) for $\displaystyle L$ using Theorem XIII.45, vol. IV of [28]. Define
the bounded truncations $\displaystyle V_{n}$ which coincide with
$\displaystyle V$ for $\displaystyle|x|>1/n$ and are zero otherwise. Set
$\displaystyle q_{n}=V-V_{n}$. Both $\displaystyle L_{0}$ and $\displaystyle
L$ are bounded from below. Comparing quadratic forms,

$\displaystyle L\leq L_{0}+V_{n}\quad\hbox{and}\quad L_{0}\leq L-V_{n}\ ,$

so that $\displaystyle L_{0}+V_{n}$ and $\displaystyle L-V_{n}$ are uniformly
bounded from below. We are left with showing that $\displaystyle
L_{0}+V_{n}\to L$ and $\displaystyle L-V_{n}\to L_{0}$ in the strong resolvent
sense. By Theorem VIII.25, vol. I of [28] it suffices to show that, for a
given $\displaystyle u\in H^{2}$,

$\displaystyle q_{n}\ u\to 0\quad\hbox{in}\ L^{2}({\mathbb{R}}^{3})\
,\quad\hbox{i.e.}\quad\lim_{\epsilon\to 0}\
\int_{|x|\leq\epsilon}\frac{u^{2}(x)}{|x|^{2}}\ dx\ =\ 0\ ,$

which is true, since $\displaystyle H^{2}({\mathbb{R}}^{3})$ consists of
bounded, continuous functions. The proof of (III) is similar. For (IV), use
the arguments with Laplace transforms in Section IX.11 of [41] (see also
[37]).

An m-special operator $\displaystyle L:D\to H$ with
$\displaystyle\lambda_{m}>0$ yields m-special operators $\displaystyle
L\otimes L,L\wedge L,L\otimes I+I\otimes L$ in appropriate domains.

There are natural m-special operators associated to systems. For a finite
measure space $\displaystyle(M,d\nu)$, let $\displaystyle(\tilde{M},d\nu)$ be
the disjoint union of $\displaystyle k$ copies of $\displaystyle(M,d\nu)$.
Clearly, for $\displaystyle H=L^{2}(M,d\mu)$, we have
$\displaystyle\tilde{H}=L^{2}(\tilde{M},d\nu)\simeq H^{k}$. Inequalities
between vectors must hold componentwise. We consider a simple example.

Let $\displaystyle L:D\to H$ be m-special,
$\displaystyle\lambda_{m}(L)=\min\sigma(L)$ associated with
$\displaystyle\phi_{m}(L)>0$,
$\displaystyle\mu_{m}(L)=\inf\sigma(L)\setminus\lambda_{m}(L)$. Then, for
$\displaystyle\alpha>0$ with $\displaystyle-\lambda_{m}(L)<\alpha<\mu_{m}(L)$,

$\displaystyle\tilde{L}:\tilde{D}=D\oplus D\mapsto\tilde{H}=H\oplus H,\
(u_{1},u_{2})\mapsto(Lu_{1}-\alpha\ u_{2},Lu_{2}-\alpha\ u_{1})$

is m-special. We have
$\displaystyle\tilde{\lambda}_{m}=\min\sigma(\tilde{L})=\lambda_{m}(L)-\alpha$
and ground state $\displaystyle\sqrt{2}\phi_{m}=(\phi_{m}(L),\phi_{m}(L))$.
Also,
$\displaystyle\tilde{\mu}_{m}=\inf(\sigma(\tilde{L})\setminus\\{\tilde{\lambda}_{m}\\})=\min\\{\lambda_{m}(L)+\alpha,\mu_{m}(L)-\alpha\\}>0$,
from which (m-S) follows; $\displaystyle\alpha>0$ in turn implies (m-PI).

####  3.4.2 Compatible maps $\displaystyle P$

From Proposition 14, we must find maps $\displaystyle P(u)$ admitting
linearizations which are fine perturbations of an m-special operator
$\displaystyle L$. We present two examples.

Consider maps $\displaystyle P(u)$ which are gradients of
$\displaystyle\Psi:D\to{\mathbb{R}}$, so that the Jacobian $\displaystyle DP$
is a Hessian, hence symmetric with appropriate smoothness. Let
$\displaystyle\Omega\subset{\mathbb{R}}^{n}$ bounded, $\displaystyle
H=L^{2}(\Omega,dx)$, $\displaystyle B,A\in{\cal B}(H)$ and $\displaystyle{\bf
f}$ a primitive of $\displaystyle f,{\bf f}^{\prime}=f$, set

$\displaystyle\Psi:H\to{\mathbb{R}}\ ,\quad u\mapsto\int_{\Omega}B{\bf f}(Au)\
dx$

for which, at least formally (here, $\displaystyle 1\in H$ is the constant
function),

$\displaystyle P(u)=\nabla\Psi(u)=A^{T}f(Au)(B^{T}1)\ .$

The usual Nemitskii map is the case $\displaystyle B=A=I$.

######  Proposition 19.

Suppose $\displaystyle A\in{{\cal B}(H)}$ is positively stable, $\displaystyle
g=B^{T}1>0$, $\displaystyle f$ as in Theorem 9, $\displaystyle
P(u)=A^{T}gf(Au)$. Then, if
$\displaystyle\|A\|^{2}\|gf^{\prime}\|_{\infty}\leq b<\mu_{m}$, (m-H) and
(m-Conv) hold and thus $\displaystyle F=L-P$ is a simple map.

As $\displaystyle f$ is Lipschitz, $\displaystyle f^{\prime}$ exists a.e. A
finer result may be obtained assuming smoothness, as in Proposition 10, but we
give no details.

Proof:  Take for linearizations the maps $\displaystyle
G(u,v)=A^{T}M_{gq(u,v)}A$, where, as usual, $\displaystyle
M_{gq(u,v)}\in{{\cal B}(H)}$ is multiplication by $\displaystyle gq(u,v)\in
L^{\infty}(\Omega)$ and $\displaystyle q$ is the usual Newton quotient
associated with $\displaystyle f$. The hypotheses immediately yield (m-H).

We consider (m-Conv). From the convexity of $\displaystyle f$, for
$\displaystyle u<v<w$,

$\displaystyle(w-v)(f(v)-f(u))<(v-u)(f(w)-f(v))\ .$

As $\displaystyle A$ is positively stable, $\displaystyle Au<Av<Aw$ and

$\displaystyle(Aw-Av)g(f(Av)-f(Au))<(Av-Au)g(f(Aw)-f(Av))$

pointwise (for all $\displaystyle x\in\Omega$). Integrate over
$\displaystyle\Omega$,

$\displaystyle\langle w-v,A^{T}g(f(Av)-f(Au))\rangle<\langle
v-u,A^{T}g(f(Aw)-f(Av))\rangle,$

so that $\displaystyle\langle w-v,P(v)-P(u)\rangle<\langle
v-u,P(w)-P(v)\rangle$.

We consider invariant maps. For an orthogonal projection
$\displaystyle\pi:H\to H$, split

$\displaystyle H=H_{k}\oplus H,\quad\hbox{for }\ H_{k}=\operatorname{Ker}\pi,\
H=\operatorname{Ran}\pi\ ,\quad u=u_{k}+u\ .$

Suppose $\displaystyle L:D\subset H\to H$ is an invertible operator commuting
with $\displaystyle\pi$ and $\displaystyle P:H\to H$ is a map keeping
$\displaystyle H$ invariant (as a subspace) for which $\displaystyle P(0)=0$.
There are two maps to consider: $\displaystyle F:D\to H,F(u)=Lu-\pi\circ
P\circ\pi$ and its restriction $\displaystyle F_{\pi}:D\cap H\to H$, which is
necessarily of the form $\displaystyle F_{\pi}(u)=Lu-P(u)$.

If the restriction $\displaystyle L_{\pi}:D\cap H\to H$ is m-special and
$\displaystyle P_{\pi}:H\to H$ is $\displaystyle L$-compatible with
$\displaystyle L$, Theorem 8 applies to $\displaystyle F$. We now consider
$\displaystyle F$: in order to solve $\displaystyle F(u)=g$, decompose the
equation as

$\displaystyle Lu-P(u)=g,\quad Lu_{k}=g_{k}$

and the invertibility of $\displaystyle L$ implies that $\displaystyle
F=F\oplus L:H\oplus H_{k}\to H\oplus H_{k}$, so that the implications of
Theorem 8 and Proposition 17 are common to $\displaystyle F$ and
$\displaystyle F_{\pi}$.

As a simple example, let $\displaystyle\pi$ be the radial projection on
functions defined in a ball, $\displaystyle L=-\Delta$ with Dirichlet
conditions and $\displaystyle P$ a Nemitskii map as in Theorem 1.

##  4 Appendix: Eigenvectors on cones, after Marek

We will use a standard version of the Krein-Rutman theorem ([17]).

######  Theorem (Krein-Rutman).

Let $\displaystyle K$ be a reproducing cone of a Banach space $\displaystyle
Y$, $\displaystyle S$ a compact operator for which $\displaystyle SK\subset K$
and $\displaystyle r(S)>0$. Then $\displaystyle\lambda(S)=r(S)$ is an
eigenvalue associated with an eigenvector $\displaystyle\phi\in K$.

Theorem 10 below implies Theorem 4 in Section 2.1. Theorem 10 is a subset of
the statements in Theorems 2.1, 2.2 and 2.3 in [26], inspired by ideas in
[24], [10], [31], [32] and [33]. Nonsupporting operators and $\displaystyle
B$-cones will not be used in this text, and are defined in [26], [33].

######  Theorem 10.

[Bonsall-Marek-Sawashima-Schaefer] Let $\displaystyle Y$ be a real Banach
space and $\displaystyle K\subset Y$ be a normal B-cone of $\displaystyle Y$.
Let $\displaystyle S\in{{\cal B}(Y)}$ be a positive, nonsupporting operator
and suppose that the spectral radius $\displaystyle r(S)$ is a pole of the
resolvent map $\displaystyle\lambda\mapsto R(\lambda,S)=(\lambda I-S)^{-1}$.
Then the following properties hold.

  1. (i)

There exists an eigenvector $\displaystyle\phi\in K$ associated with
$\displaystyle r(S)$, $\displaystyle\phi>0$.

  2. (ii)

The spectral radius $\displaystyle r(S)$ is a simple pole of $\displaystyle
R(\lambda,S)$.

  3. (iii)

Every eigenvector of $\displaystyle S$ in $\displaystyle K$ is a multiple of
$\displaystyle\phi$.

  4. (iv)

For an eigenvalue $\displaystyle\lambda\in\sigma(S)$ different from
$\displaystyle r(S)$, $\displaystyle|\lambda|<r(S)$.

  5. (v)

There is an eigenvector $\displaystyle\phi^{\ast}\in K^{\ast}$ of
$\displaystyle S^{\ast}$ associated with $\displaystyle r(S)$,
$\displaystyle\phi^{\ast}>0$.

  6. (vi)

If $\displaystyle\mu>r(S)$, then $\displaystyle(\mu I-S)^{-1}\in{{\cal B}(Y)}$
and $\displaystyle(\mu I-S)^{-1}K\subset K$.

We prove Theorem 4 from this theorem. In Section 2.1, $\displaystyle Y$ is a
real Banach space, $\displaystyle K\subset Y$ is a normal, reproducing cone. A
reproducing cone is unflattened (from the Krein-Smulian theorem, [39]) and
unflattened cones are B-cones ([33]). Also, $\displaystyle S\in{{\cal B}(Y)}$
is compact and ergodic with $\displaystyle r(S)>0$. Ergodicity implies that
$\displaystyle S$ is positive, nonsupporting. From the Krein-Rutman theorem,
$\displaystyle r(S)>0$ and the spectral theory of compact operators implies
that $\displaystyle r(S)$ is a pole of the resolvent map. Hence, the
hypotheses of Theorem 4 imply those of Theorem 10.

We show that $\displaystyle r$ is a basic eigenvalue of $\displaystyle S$.
From (1) and (5), $\displaystyle S$ and $\displaystyle S^{\ast}$ have
eigenvectors $\displaystyle\phi_{m}>0$ and $\displaystyle\phi_{m}^{\ast}>0$
associated with $\displaystyle r(S)$: this is property (b-1). By the
compactness of $\displaystyle S$ and $\displaystyle S^{\ast}$, the invariant
subspaces $\displaystyle V_{r}$ and $\displaystyle V^{\ast}$ associated to
$\displaystyle r(S)$ are finite dimensional and it is easy to see that
$\displaystyle\dim V=\dim V^{\ast}$. The Jordan theorem for matrices applied
to the restriction of $\displaystyle S$ in $\displaystyle V$ implies that the
resolvent map $\displaystyle\lambda\to(S-\lambda I)^{-1}$ has a pole at
$\displaystyle r(S)$ of multiplicity given by the dimension of $\displaystyle
V$. By (2), then, $\displaystyle V$ is one dimensional. This proves properties
(b-2) and (b-3): $\displaystyle r(S)$ is a basic eigenvalue of $\displaystyle
S$. The remaining claims of Theorem 4 are explicit consequences of Theorem 10.

## References

  * [1] A. Ambrosetti, Osservazioni sui teoremi di inversione globale, Rend. Lincei Mat. Appl. 22 (2011) 3–15.
  * [2] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1972) 231-246.
  * [3] W. Arendt, Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates, Handbook Differential Equations, Eds. C.M. Dafermos and E. Feireisl, vol. I (2004), Elsevier, Amsterdam, 1-86.
  * [4] H. Berestycki, Le nombre des solutions de certain problèmes sémi-linéaires elliptiques, J. Funct. Anal. 40 (1981) 1-29.
  * [5] H. Berestycki, L. Nirenberg and S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47-92.
  * [6] M.S. Berger, Nonlinearity and Functional Analysis, Lectures on Nonlinear Problems in Mathematical Analysis, Academic Press (1997).
  * [7] M.S. Berger and P.T. Church, Complete integrability and perturbation of a nonlinear Dirichlet problem. I., Indiana Univ. Math. J. 28 (1979) 935-952.
  * [8] M.S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J. 24 (1974) 837-846.
  * [9] I. Birindelli, Hopf’s lemma and Anti-maximum Principle in General Domains, J. Diff. Eqs. 119 (1995) 450-472.
  * [10] F. F. Bonsall, Endomorphisms ofpartially ordered vector spaces, J. London Math. Soc. 30 (1955) 133-144.
  * [11] H. Brezis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Normale Sup. Pisa, Classe di Scienze 4a serie, 5, 2 (1978), 225–326.
  * [12] J.T. Cal Neto and C. Tomei, Numerical analysis of semilinear elliptic equations with finite spectral interaction, J.Math.Anal.Appl. 395 (2012) 63-77.
  * [13] M. Calanchi, C. Tomei and A. Zaccur, Fibers and global geometry of functions, PNLDE 86, Springer, Cham (2015) 55-75.
  * [14] M. Calanchi, C. Tomei and A. Zaccur, Cusps and a converse to the Ambrosetti-Prodi theorem, Ann. Sc. Norm. Sup. Pisa (5) XVIII (2018) 483–507.
  * [15] P.T. Church and J.G. Timourian, Global fold maps in differential and integral equations, Nonlinear Anal. 18 (1992) 743-758.
  * [16] P.T. Church and J.G. Timourian, Global Structure for Nonlinear Operators in Differential and Integral Equations I. Folds, Topological Nonlinear Analysis, II, PNLDE 27, Birkhäuser, Boston, MA, (1997) 109-160.
  * [17] K. Deimling, Nonlinear Functional Analysis, Dover, New York (2010).
  * [18] C.L. Dolph, Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc. 66 (1949) 289-307.
  * [19] N. Dunford and J. Schwartz, Linear Operators I, Interscience, New York (1957).
  * [20] N. Eldredge, Compactness of unit ball in WOT of $\displaystyle{\cal B}(X)$, available at https://math.stackexchange.com/questions/1187844/
  * [21] W. Faris, Invariant cones and uniqueness of the ground state for Fermion systems, J. Math. Phys 13 (1972) 1285-1290.
  * [22] D. Guo, Y. Je Cho, J. Zhu, Partial ordering methods in nonlinear problems Nova Science Publishers, Inc., Hauppauge, NY (2004).
  * [23] A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math. 54 (1929) 117-176.
  * [24] S. Karlin, Positive Operators, J. Math. Mech. 8 (1959) 907-937.
  * [25] A. Manes and A.M. Micheletti, Un’estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Unione Mat. Ital. 7 (1973) 285-301.
  * [26] I. Marek, Frobenius theory of positive operators: comparison theorems and applications, SIAM J. Appl. Math. 19 (1970) 607–628.
  * [27] I. Malta, N. Saldanha and C. Tomei, Morin singularities and global geometry in a class of ordinary differential operators, Topol. Meth. Nonlin. Anal. 10 (1997) 137-169.
  * [28] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vols. I-IV, Academic Press, New York (1975-78).
  * [29] B. Ruf, Singularity theory and the geometry of a nonlinear elliptic equation, Ann. Scuola Norm. Sup. Pisa C1. Sc., (2), 17 (1990) 1-33.
  * [30] B. Ruf, Singularity theory and bifurcation phenomena in differential equations, Topological Nonlinear Analysis, II, Progr. Nonlinear Differential Equations Appl., 27, Birkhäuser, Boston, MA, (1997) 315-395.
  * [31] I. Sawashima, On spectral properties of some positive operators, Natur. Sci. Rep. Ochanomizu Univ., 151 (1964) 55–64.
  * [32] H.H. Schaefer, On the singularities of an analytic function with values in a Banach space, Arch. Math., 11 (1960) 40–43.
  * [33] H.H. Schaefer, Some spectral properties of positive linear operators, Pacific Journal of Mathematics, Vol. 10, No. 3 (1960), 1009–1019.
  * [34] J. Schauder, Invarianz des Gebietes in Funktionalräumen, Studia Math., 1 (1929)123-139.
  * [35] B. Sirakov, C. Tomei and A. Zaccur, Results of Ambrosetti-Prodi type
for non-self-adjoint elliptic operators, Annales de l’Institut Henri Poincaré,
AN 35(2018), 1757-1772.

  * [36] M.W. Smiley, A finite element method for computing the bifurcation function for semilinear elliptic BVPs, J. Comp. Appl. Math., 70 (1996) 311- 327.
  * [37] M. Taylor, Remarks on Fractional Diffusion Equations, www.unc.edu/ math/Faculty/met/fdif.pdf .
  * [38] C. Tomei and A. Zaccur, Geometric aspects of Ambrosetti-Prodi operators with Lipschitz nonlinearities, Analysis and topology in nonlinear differential equations, PNLDE 85, Springer, Cham, (2014) 445-456.
  * [39] B. Z. Vulikh, Cone, Encyclopedia of Mathematics, Springer.
  * [40] E. Wofsey, Sequential compactness of unit ball in $\displaystyle{\cal B}(X)$, available at https://math.stackexchange.com/questions/3344475/.
  * [41] K. Yosida, Functional Analysis, Springer, Berlin (1974).
  * [42] E. Zeidler, Nonlinear Functional Analysis and its Applications, IV: Applications to Mathematical Physic, Springer (1988), New York.

Marta Calanchi, Dipartimento di Matematica, Università degli Studi di Milano,
Via Saldini, 50 20147 Milano, Italy<EMAIL_ADDRESS>

Carlos Tomei, Departamento de Matemática, PUC-Rio R. Mq. S. Vicente 225, Rio
de Janeiro 22451-900, Brazil<EMAIL_ADDRESS>