# Some fixed point theorems and application to a nonlinear transport equation with delayed neutrons in slab geometry

Khalid Latrach, Ahmed Zeghal

► To cite this version:

Khalid Latrach, Ahmed Zeghal. Some fixed point theorems and application to a nonlinear transport equation with delayed neutrons in slab geometry. 2024. hal-04955759

HAL Id: hal-04955759

https://hal.science/hal-04955759v1

Preprint submitted on 19 Feb 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Some fixed point theorems and application to a nonlinear transport equation with delayed neutrons in slab geometry*

Khalid Latrachª, Ahmed Zeghalᵇ, †

ªUniversité Clermont Auvergne

CNRS, LMBP

F-63000 Clermont-Ferrand, France

E-mail addresses: khalid.latrach@uca.fr

ᵇ Université Abdelmalek Essaadi, Faculté des Sciences et Techniques

Laboratoire de Mathématiques et Applications

Tangier, Morocco

E-mail addresses: azeghal@uae.ac.ma

## Abstract

In this work we establish some fixed point theorems for a class of mappings on Banach spaces with the Dunford-Pettis property. Since $L^1$-spaces possess the Dunford-Pettis property, we apply our result to discuss existence of solutions to a nonlinear stationary transport equation with delayed neutrons and abstract boundary conditions in slab geometry; the space on the space $L^1(\Omega; dxd\xi)$ (cf. Section 4.1) which is the convenient setting of the problem.

**Key Words** Monoenergetic transport equation with delayed neutrons; Banach space with the Dunford-Pettis property; De Blasi measure of weak noncompactness; fixed point theorems

**2010 MR Subject Classification** 35E15; 47H09; 47H10

## 1 Introduction

In this paper we are concerned with existence of solutions to the following nonlinear boundary value problem which involves a one-dimensional monoenergetic neutron transport equation in multiplying media, that is,

<footnote>*The authors have no conflict of interest</footnote>

<footnote>†Corresponding author</footnote>
$$
\left\{
\begin{aligned}
& v\xi \frac{\partial f_0}{\partial x}(x, \xi) + v\sigma(x, \xi, f_0(x, \xi)) + \lambda f_0(x, \xi) \\
& \qquad = v \int_{-1}^{1} \kappa_0(x, \xi, \xi') \Theta_0(x, \xi', f_0(x, \xi')) d\xi' + \sum_{i=1}^{d} \lambda_i \beta_i(x, \xi) f_i(x, \xi), \\
& \lambda_i f_i(x, \xi) = v \int_{-1}^{1} \kappa_i(x, \xi, \xi') \Theta_i(x, \xi', f_0(x, \xi')) d\xi' \quad (1 \le i \le d), \\
& f_0^- = H(f_0^+)
\end{aligned}
\right.
\qquad (1)
$$

with general boundary conditions where $x \in [-a, a]$, for a parameter $0 < a < \infty$ and $\xi \in [-1, 1]$,
the variable $\xi$ may be thought of as the cosine of the angle between the velocity of neutrons and the
$x$-direction and $v$ denotes the neutrons velocity. This operator describes the transport of particles
(neutrons, photons, molecules of gas, etc.) in a plane parallel domain with a width of $2a$ mean free
paths. The function $f_0$ represents the number (or probability) density of gas particles having the
position $x$ and the direction cosine of propagation $\xi$. By $\Omega^-$ (resp. $\Omega^+$) we denote the incoming
(resp. outgoing) part of the boundary of the phase space $\Omega := [-a, a] \times [-1, 1]$, $f_0^- := f_{0|\Omega^-}$
(resp. $f_0^+ := f_{0|\Omega^+}$) stands for the restriction of the function $f_0$ to $\Omega^-$ (resp. $\Omega^+$), and $H$ is
a bounded linear operator from a suitable function space on $\Omega^+$ to a similar one on $\Omega^-$. The
nonnegative numbers $\lambda_i$ are the radioactive decay constants. The functions $\sigma(\cdot, \cdot, \cdot)$ and $\kappa_i(\cdot, \cdot, \cdot)$,
$i = 0, \dots, N$ are called, respectively, the collision frequency and the scattering kernels. For the
sake of simplicity, in the remainder of the paper, we shall take $v = 1$. Note that, assuming $v = 1$
is no loss of generality since it consists only in a suitable time scaling.

In general, to solve a stationary problem like (1) consists in transforming the problem to a fixed
point problem, like

$$
F(x) + G(x) = x, x \in K \tag{2}
$$

where $K$ is a closed convex subset of a Banach space $X$. It is well known that if $K$ is bounded, $F$
is continuous and compact, $G$ is a contraction and, for every $x, y \in K$, $F(x) + G(y) \in K$, then
the classical Krasnosel'skii point theorem permits to insure the existence of solutions to Problem
(2). Unfortunately, in many concrete problems in natural science (mechanic, physics, chemistry,
biology, economics, etc) we have a lack of compactness in fact, the involved operators are some
times weakly compact but not weakly continuous. This holds, in particular for neutron transport
equations on $L^1$-spaces. In the last two decades, much effort has been devoted to the study of
perturbed problems like (2) in the context of the weak topology where $K$ is a closed convex of a
Banach space $X$. The reader is referred, for example, to [16, 10, 11, 13] or [12] and the references
therein.

This work is motivated by the paper [1] where problem (1) is solved in multi-dimensional ge-
ometry, that is, $x \in D$ where D is a smooth bounced subset of $\mathbb{R}^N$ where $N > 1$. Our purpose
here is to complete this analysis by discussing existence results for the boundary value problem
(1) in one-dimensional geometry where the boundary conditions are modeled by an abstract oper-
ator $H$ relating the outgoing flux to the incoming one. The known classical boundary conditions
(vacuum boundary, specular reflections, periodic, diffuse reflections, generalized and mixed type
boundary conditions) are special examples of our general framework. We seek solutions in the space $L^1(\Omega, dxd\xi)$ which is the convenient setting of the problem. After transforming the problem into a fixed point one like $F(\lambda)x + G(\lambda)x = x$, we observe that the operator $F(\lambda)$ is not compact but just weakly compact, Since $L^1(\Omega, dxd\xi)$ is a Banach space with the Dunford Pettis property, we establish some fixed point theorems for a class of mappings in Banach spaces with the Dunford Pettis property which can be applied to derive existence of solutions to Problem (1).

## 2 Preliminary results

Let $X$ be a Banach space. From now on we denote by $B(X)$ the collection of all nonempty bounded subsets of $X$, $W_r(X)$ the subset of $B(X)$ consisting of all nonempty relatively weakly compact subsets of $X$, while $W(X)$ stands for the collection of all nonempty weakly compact subsets of $X$. Denote by $B_r$ the closed ball in $X$ centred at 0 with radius $r$. Recall that the notion of the measure of weak noncompactness has been introduced by De Blasi [7]. It is a map $\omega: B(X) \to [0, \infty)$ defined in the following way

$$ \omega(M) = \inf\{r > 0 : \text{ there exists } W \in W(X) \text{ with } M \subseteq W + B_r\}, \quad (3) $$

for each $M \in B(X)$. For completeness we recall below some useful properties of the function $\omega(\cdot)$ needed below (for the proofs we refer to [7, 12]).

**Proposition 2.1.** Let $M, M_1, M_2 \in B(X)$, then we have:

(a) The family $\ker \omega := \{M \in B(X) \text{ such that } \omega(M) = 0\}$ is nonempty and $\ker \omega$ is contained in $W_r(X)$.

(b) $M_1 \subseteq M_2$ implies $\omega(M_1) \le \omega(M_2)$.

(c) $\omega(\overline{M^w}) = \omega(M)$, where $\overline{M^w}$ denotes the weak closure of $M$.

(d) $\omega(\lambda M) = \lambda \omega(M)$, for all $\lambda > 0$.

(e) $\omega(\mathrm{co}(M)) = \omega(M)$, where $\mathrm{co}(M)$ denotes the convex hull of $M$.

(f) $\omega(M_1 + M_2) \le \omega(M_1) + \omega(M_2)$.

(g) Generalized Cantor intersection theorem: If $(M_n)_{n \ge 1}$ is a sequence of nonempty, weakly closed subsets of $X$ with $M_1$ bounded and $M_1 \supseteq M_2 \supseteq \dots \supseteq M_n \supseteq \dots$ such that $\lim_{n \to \infty} \omega(M_n) = 0$, then $\bigcap_{n=1}^{\infty} M_n \ne \emptyset$ and $\omega(\bigcap_{n=1}^{\infty} M_n) = 0$.

The family of subsets $\ker \omega$ given in the first assertion is called the kernel of $\omega$ and the map $\omega(\cdot)$ is called the measure of weak noncompactness of De Blasi.
**Remark 2.1.** In [3], Appell and De Pascale showed that in $L_1$-spaces $\omega(\cdot)$ possesses the following simple form:

$$ \omega(M) = \limsup_{\epsilon \to 0} \left\{ \sup_{\psi \in M} \left[ \int_D |\psi(t)| dt : |D| \le \epsilon \right] \right\} \quad (4) $$

for all bounded subset $M$ of $L^1(\Omega)$, where $|D|$ denotes the Lebesgue measure of the set $D$. □

Let $X$ be a Banach space and let $\mathcal{J}: D(\mathcal{J}) \subset X \to X$ be a nonlinear mapping. We recall the following condition introduced in [16, p. 260]:

(H) For each weakly convergent sequence $(x_n)_{n \in \mathbb{N}}$ of $D(\mathcal{J})$, the sequence $(\mathcal{J}(x_n))_{n \in \mathbb{N}}$ has a weakly convergent subsequence.

Let $X, Y$ be two Banach spaces and $T \in \mathcal{L}(X, Y)$ where $\mathcal{L}(X, Y)$ denotes the set of all bounded linear operators from $X$ into $Y$. We recall that $T$ is called weakly compact if it carries norm bounded subsets of $X$ to relatively weakly compact subsets of $Y$. It is clear that if $X$ is a reflexive Banach space, then each operator of $\mathcal{L}(X, Y)$ is weakly compact.

A Banach space $X$ has the Dunford-Pettis property (DPP, in short) if, for any Banach space $Y$ every weakly compact operator $T: X \to Y$ transforms weakly compact sets in $X$ into norm compact sets of $Y$. The property DPP as introduced above was explicitly defined by A. Grothendieck [14]. It is well known that any $L^1$-space has the DPP [9]. Also, if $\Omega$ is a compact Hausdorff space, then $C(\Omega)$ has the DPP [8]. For further examples we refer to [9, p. 494, 497, 508 and 511] or [8]. Note that the DPP is not preserved under conjugation. However, if $X$ is a Banach space whose dual has the DPP, then $X$ has the DPP (see, for example, [14]. For more informations we refer to the paper by J. Diestel [8] which contains a survey and exposition on the DPP and related topics.

We recall the definition of $\Phi$-contraction mappings.

**Definition 2.1.** Let $X$ be a Banach space. A mapping $B: X \to X$ is said to be a $\Phi$-contraction if there exists a continuous function $\phi: \mathbb{R}^+ \to \mathbb{R}^+$ such that $\phi(r) < r$ for any $r > 0$ and, for all $x, y \in X$, we have

$$ \|B(x) - B(y)\| \le \phi(\|x - y\|). $$

The function $\phi$ is called the $\Phi$-function of $B$. Moreover, if the function $\phi$ is nondecreasing, then $B$ is called a nonlinear contraction.

# 3 Fixed point results

In this section we will establish some fixed points theorems involving two or three mappings in Banach spaces with the DPP.

**Proposition 3.1.** Let $X$ be a Banach space with the DPP and let $K$ be a nonempty closed convex subset of $X$. Let $f: X \to X$ and let $g: K \to X$ be continuous maps. Set $F = f \circ g$. Assume that
(a) $f$ is a linear weakly compact operator,

(b) $g$ satisfies (H) and $g(K)$ is bounded.

Then $\mathrm{Fix}(F) \neq \emptyset$.

**Proof** Let $C = \overline{\mathrm{co}}(F(K))$. Since $K$ is a closed convex subset of $X$ satisfying $F(K) \subseteq K$, then $C \subseteq K$ and therefore $F(C) \subseteq F(K) \subseteq \overline{\mathrm{co}}(F(K)) = C$. This shows that $F$ maps $C$ into itself. Since $g(K)$ is bounded, the use of assumption (a) shows that $F(K)$ is relatively weakly compact. Hence, by Krein-Šmulian theorem [9, p. 434], $C$ is weakly compact too. Since $g$ satisfies (H), we conclude that $g(C)$ is also weakly compact. Because $X$ has the DPP, it follows that the set $F(C) = f(g(C))$ is compact. To derive the result, it suffices to apply Schauder's fixed point theorem to the map $F : C \to C$. QED

**Theorem 3.1.** Let $X$ be a Banach space with the DPP and let $K$ be a nonempty closed convex subset of $X$. Let $f : X \to X$, $g : K \to X$ and $B : K \to X$ be continuous maps and set $F = f \circ g$. Suppose that

(a) $f$ is a linear weakly compact operator,

(b) $g$ satisfies (H) and $g(K)$ is bounded,

(c) $B$ is a nonlinear contraction with $\Phi$-function $\phi$ and satisfies (H),

$$ (d) \quad (x = B(x) + F(y), y \in K) \implies x \in K. $$

Then $\mathrm{Fix}(F+B) \neq \emptyset$.

**Proof** Since $B$ is a nonlinear contraction with $\Phi$-function $\phi$, it follows from [12, Lemma 7.12.2, p. 359] that $(I-B)$ is a homeomorphism from $K$ onto $(I-B)(K)$. Let $y$ be a point in $K$ and define the map $h_y$ by $K \ni x \mapsto B(x)+F(y)$. It is clear that $h_y$ is a nonlinear contraction with $\Phi$-function $\phi$. Hence, according to [12, Theorem 6.3.1, p. 261] the equation $z = B(z) + F(y)$ has a unique solution $x \in X$. Using hypothesis (d), we infer that $x \in K$ and therefore $x = (I-B)^{-1}F(y) \in K$. Hence

$$ (I - B)^{-1} F(K) \subset K. \qquad (5) $$

Set $\mathbf{H} := (I - B)^{-1}F$ and let

$$ C := \overline{\mathrm{co}}(\mathbf{H}(K)). $$

The map $\mathbf{H}$ transforms $C$ into itself. Indeed, since $K$ is closed and convex, it follows from (5) that $C \subset K$ and therefore

$$ \mathbf{H}(C) \subset \mathbf{H}(K) \subset \overline{\mathrm{co}}((\mathbf{H}(K))) = C. $$

We claim that the subset $C$ is weakly compact. We suppose that $C$ is not weakly compact (that is $\omega(C) > 0$). We note first that elementary calculations show that

$$ (I - B)^{-1} F = F + B(I - B)^{-1} F. \qquad (6) $$
Since $g(K)$ is bounded, using assumption (a) and arguing as in the proof of Proposition 3.1 we conclude that $F(K)$ is relatively weakly compact. Now the use of (6) and the properties of $\omega(\cdot)$ gives

$$ 
\begin{aligned}
\omega(C) &= \omega(\overline{co}((\mathbf{H}(K)))) = \omega(\mathbf{H}(K)) \\
&\leq \omega(F(K)) + \omega(B\mathbf{H}(K)) \\
&\leq \omega(B\mathbf{H}(K)).
\end{aligned}
$$

Now, according to [12, Proposition 7.2. (b), p. 340], we have

$$ \omega(B(I-B)^{-1}F(K)) \leq \phi(\omega((I-B)^{-1}F(K))) = \phi(\omega(C)) \quad (7) $$

and therefore

$$ \omega(C) \leq \phi(\omega(C)) < \omega(C), $$

which is a contradiction. Hence $\omega(C) = 0$ which proves that $C$ is relatively weakly compact.

Let $U$ be a subset of $C$. It is clear that $\bar{U}$ is a weakly compact subset of $X$. Since $g$ satisfies (H), $g(\bar{U})$ is weakly compact too. Moreover, since $X$ has the DPP and $f$ is a linear weakly compact operator, we infer that $F(\bar{U}) = f(g(\bar{U}))$ is compact. The continuity of $(I - B)^{-1}$ yields that $(I - B)^{-1}F(\bar{U})$ is also compact. Hence, $(I - B)^{-1}F$ is a continuous compact map from $C$ into itself. Now applying Schauder's fixed point theorem we conclude that $(I - B)^{-1}F$ has a fixed point in $C$ and therefore $\mathrm{Fix}(F + B) \neq \emptyset$ because $C \subset K$. QED

Let $X$ be a Banach space. It is well known that every bounded linear operator $T$ on $X$ is $\|T\|$-
lipschitzian where $\|T\|$ denotes the operator norm of $T$. Moreover, it is weakly continuous, and si
it satisfies condition (H). Thus, according to [12, Proposition 7.8.2 (a)], we have

$$ \omega(T(M)) \leq \|T\| \omega(M), \quad \text{for all } M \in \mathcal{B}(X). \quad (8) $$

□

As an immediate consequence of Theorem 3.1 and the fact that any $k$-contrative mapping is a nonlinear contraction, we have:

**Corollary 3.1.** Let $X$ be a Banach space with the DPP and let $K$ be a nonempty closed convex subset of $X$. Let $f: X \to X$, $g: K \to X$ and $B: K \to X$ be continuous maps. Set $F = f \circ g$. Suppose that

(a) $f$ is a linear weakly compact operator,

(b) $g$ satisfies (H) and $g(K)$ is bounded,

(c) $B$ is $k$-contractive for some $k \in (0, 1)$ and satisfies (H),

(d) $(x = B(x) + F(y), y \in K) \implies x \in K$.

Then $\mathrm{Fix}(F + B) \neq \emptyset$.
The next result is a consequence of the Corollary 3.1.

**Corollary 3.2.** Let $X$ be a Banach space with the DPP and let $K$ be a nonempty closed convex subset of $X$. Let $f: X \to X$, $g: K \to X$ and $B: K \to X$ be continuous maps and set $F = f \circ g$. Suppose that

(a) $f$ is a linear weakly compact operator,

(b) $g$ is weakly sequentially continuous satisfying (H) and $g(K)$ is bounded,

(c) $B$ is nonexpansive, weakly sequentially continuous and satisfies (H),

(d) if $(x_n)_{n \in \mathbb{N}}$ is a sequence of elements of $K$ such that $(I - B)(x_n) \to y$, then $(x_n)_{n \in \mathbb{N}}$ has a weakly convergent subsequence,

(e) if $\lambda \in (0, 1)$ and $x = \lambda B(x) + F(y)$ where $y \in K$, then $x \in K$.

Then $\mathrm{Fix}(F+B) \neq \emptyset$.

**Proof** For each $\lambda \in (0, 1)$, the operators $F$ and $\lambda B$ satisfy the conditions of Corollary 3.1 and then $F + \lambda B$ has a fixed point in $K$, say $x_\lambda$. Let $(\lambda_n)_{n \in \mathbb{N}}$ be a sequence of $(0, 1)$ such that $\lambda_n \to 1$ as $n \to +\infty$ and let $(x_n)_{n \in \mathbb{N}}$ be the sequence of points of $K$ such that

$$F(x_n) + \lambda_n B(x_n) = x_n. \qquad (9)$$

Using the hypothesis (b) (passing eventually to a subsequence), we may assume that $(F(x_n))_{n \in \mathbb{N}}$ converges weakly to some $y \in K$. Accordingly,

$$(I - \lambda_n B)(x_n) \to y.$$

Since $K$ is bounded, the set $\{x_n : n \in \mathbb{N}\}$ is bounded and then $\{B(x_n) : n \in \mathbb{N}\}$ is also bounded. Hence

$$\|(x_n - B(x_n)) - (x_n - \lambda_n B(x_n))\| = (1 - \lambda_n) \|B(x_n)\| \to 0 \quad \text{as } n \to \infty.$$

This implies that $x_n - B(x_n) \to y$. Next, by (d), the sequence $(x_n)_{n \in \mathbb{N}}$ has a subsequence $(x_{n_k})_{k \in \mathbb{N}}$ which converges weakly to some $x \in K$. Now, remember that $f$ is a continuous linear operator, so it is weakly continuous and therefore weakly sequentially continuous. Finally, because the mappings $g$ and $B$ are weakly sequentially continuous, it follows from (9) that $F(x) + B(x) = x$, which ends the proof.

**QED**

The following result can be obtained as a corollary of [1, Theorem 3.7]; for completeness we provide a proof.

**Theorem 3.2.** Let $X$ be a Banach space with the DPP and let $K$ be a nonempty closed convex bounded subset of $X$. Let $f: X \to X$, $g: K \to X$ and $B: K \to X$ be continuous maps and set $F = f \circ g$. Suppose that
(a) $f$ is a linear weakly compact operator,

(b) there exists $\gamma \in (0, 1)$ such that $\omega(F(S) + B(S)) \le \gamma \omega(S)$ for all $S \subset K$ with $\omega(S) > 0$,

(c) $g$ satisfies (H) and $g(K)$ is bounded,

(d) $B$ is a nonlinear contraction with $\Phi$-function $\phi$,

(e) $(x = B(x) + F(y), y \in K) \implies x \in K$.

Then $\text{Fix}(F + B) \ne \emptyset$.

**Proof** Since $B$ is a nonlinear contraction, arguing as in the beginning of the proof of Theorem 3.1 we derive the inclusion

$$ (I - B)^{-1} F(K) \subset K. \qquad (10) $$

We define a sequence $(K_n)_{n \in \mathbb{N}}$ of subsets of $K$ by

$$ K_0 = K \text{ and } K_{n+1} = \overline{co}((I - B)^{-1}F(K_n)). \qquad (11) $$

It is clear that $(K_n)_{n \in \mathbb{N}}$ is a sequence of nonempty closed convex subsets of $K$. Moreover, the inclusion (10) shows that $(K_n)_{n \in \mathbb{N}}$ is decreasing (in the sense of the inclusion). Furthermore, using equation (6) we get

$$ (I - B)^{-1} F(K_n) \subset F(K_n) + B(\overline{co}((I - B)^{-1} F(K_n)) \subset F(K_n) + B(K_{n+1}). $$

Since $(K_n)_{n \in \mathbb{N}}$ is a decreasing sequence, we get $(I - B)^{-1} F(K_n) \subset F(K_n) + B(K_n)$. Thus, using the boundedness of $K$ and assumption (a), one sees that $F(K_n)$ is weakly compact and therefore, condition (b) implies

$$ \omega(K_{n+1}) \le \omega(F(K_n) + B(K_n)) \le \gamma \omega(K_n). $$

Proceeding by induction we get $\omega(K_{n+1}) \le \gamma^n \omega(K)$. The boundedness of $K$ together with the fact that $\gamma \in (0, 1)$ yields $\lim_{n \to +\infty} \omega(K_n) = 0$. Hence, by Proposition 2.1 (g), we conclude that $K_\infty = \bigcap_{n \ge 0} K_n$ is a nonempty convex weakly compact subset of $K$. Moreover, we have $(I - B)^{-1} F(K_\infty) \subset K_\infty$.

Let $U$ be a subset of $K_\infty$. It is clear that $\bar{U}$ is a weakly compact subset of $X$. Since $g$ satisfies (H), the set $g(\bar{U})$ is weakly compact too. Because $X$ has the DPP, it follows that $F(\bar{U}) = f(g(\bar{U}))$ is compact. By the continuity of $(I - B)^{-1}$ we infer that $(I - B)^{-1} F : K_\infty \to K_\infty$ is a continuous compact map. Next, applying Schauder's fixed point theorem we conclude that $(I - B)^{-1} F$ has a fixed point $z \in K_\infty \subset K$ and therefore $\text{Fix}(F + B) \ne \emptyset$. QED
# 4 Application to a transport equation with delayed neutrons in slab geometry

## 4.1 Basic facts related to the problem

In this section, we shall gather some definitions and results related to monoenergetic neutron transport equation with delayed neutrons in slab geometry required in the sequel. Let $\Omega := [-a, a] \times [-1, 1]$ and set

$$X = L^1(\Omega; dxd\xi).$$

Define the following sets representing the incoming and the outgoing boundaries of the phase space $\Omega$:

$$\Omega^{-} = \Omega_{1}^{-} \cup \Omega_{2}^{-} = \{-a\} \times [0, 1] \cup \{a\} \times [-1, 0],$$

$$\Omega^{+} = \Omega_{1}^{+} \cup \Omega_{2}^{+} = \{-a\} \times [-1, 0] \cup \{a\} \times [0, 1],$$

and define the boundary spaces $L_1^{-}$ and $L_1^{+}$ by

$$L_1^{-} := L^1(\Omega^{-}; |\xi|d\xi) \sim L^1(\Omega_1^{-}; |\xi|d\xi) \oplus L^1(\Omega_2^{-}; |\xi|d\xi) \\ = L_{1,1}^{-} \oplus L_{2,1}^{-},$$

endowed with the norm

$$\|f^{-}\|_{L_{1}^{-}} = \|f_{1}^{-}\|_{L_{1,1}^{-}} + \|f_{2}^{-}\|_{L_{2,1}^{-}} = \int_{0}^{1} |f(-a, \xi)||\xi|d\xi + \int_{-1}^{0} |f(a, \xi)||\xi|d\xi$$

and

$$L_1^+ := L^1(\Omega^+; |\xi|d\xi) \sim L^1(\Omega_1^+; |\xi|d\xi) \oplus L^1(\Omega_2^+; |\xi|d\xi) \\ := L_{1,1}^+ \oplus L_{2,1}^+,$$

endowed with the norm

$$\|f^{+}\|_{L_{1}^{+}} = \|f_{1}^{+}\|_{L_{1,1}^{+}} + \|f_{2}^{+}\|_{L_{2,1}^{+}} = \int_{-1}^{0} |f(-a, \xi)||\xi|d\xi + \int_{0}^{1} |f(a, \xi)||\xi|d\xi.$$

Next, we introduce the following partial Sobolev space

$$W := \{ f \in X \text{ such that } \xi \frac{\partial f}{\partial x} \in X \}.$$

Accordingly, any function $f \in W$ possesses traces on $\Omega^{-}$ and $\Omega^{+}$ belonging to the spaces $L_1^{-}$ and $L_1^{+}$ (see, for instance, [5, 6] or [4]). They are denoted, respectively, by $f^{-} := f|_{\Omega^{-}}$ and $f^{+} := f|_{\Omega^{+}}$ and represent the incoming and the outgoing fluxes.

The boundary conditions $f_0^{-} = H(f_0^{+})$ appearing in Problem (1) may be written abstractly as an operator $H$ relating the incoming and the outgoing fluxes where the boundary operator $H$ is defined by

$$\left\{ \begin{array}{l} H : L_{1,1}^{+} \oplus L_{2,1}^{+} \rightarrow L_{1,1}^{-} \oplus L_{2,1}^{-} \\ H \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} := \begin{pmatrix} H_{11} & H_{12} \\ H_{21} & H_{22} \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \end{array} \right.$$
where $H_{j,k}: L_{k,1}^+ \to L_{j,1}^-, H_{j,k} \in \mathcal{L}(L_{k,1}^+, L_{j,1}^-)$, $j, k = 1, 2$.

Next we consider the free streaming operator $T$ defined by

$$ \left\{ \begin{array}{l} T: D(T) \subseteq X \to X \\ f \to Tf(x, \xi) = -\xi \frac{\partial f}{\partial x}(x, \xi), \\ D(T) = \{ f \in W \text{ such that } f^- = H(f^+) \}. \end{array} \right. $$

Let $\tau$ be a real number. In the following we shall use the notation:

$$ \mathbb{C}_{\tau} = \{ \lambda \in \mathbb{C} \text{ such that } \mathrm{Re}\lambda > \tau \}. $$

The operator $T$ is a closed densely defined linear operator. Its resolvent set, $\rho(T)$, contains the half plane $\mathbb{C}_0$ (see, for example, [15]).

We recall the following result estimate established in [15, p. 47]. If $\|H\| \le 1$, then, for all $\lambda \in \mathbb{C}_0$, we have

$$ \|(\lambda - T)^{-1}\|_{\mathcal{L}(X)} \le \frac{1}{\mathrm{Re}\lambda}. \qquad (12) $$

We note that the hypothesis $\|H\| \le 1$ is made in order to ensure the dissipativity of the $T_H$

The physical collision operators, used in nuclear reactor theory for all types of moderators (gas, liquid or solid), are bounded operators and most of them are (in slab geometry) of the form

$$ X \ni f \mapsto \int_{-1}^{1} \kappa(x, \xi, \xi') f(x, \xi') d\xi' \in X, \qquad (13) $$

where $k(\cdot, \cdot, \cdot)$ is a measurable function and, in slab geometry, they are compact with respect to the variable $\xi$ in $L^1$-spaces (see, [15] and the references therein).

**Definition 4.1.** A bounded collision operator $K$ in the form (13) is said to be regular on $X$ if, for almost all $x \in [-a, a]$, its restriction to the space $L^1([-1, 1], d\xi)$ is compact.

**Remark 4.1.** Note that the fact that $K$ is regular does not imply the compactness of $K$ on $X$. To see this, suppose that $\kappa(x, \xi, \xi') = f(\xi)g(\xi')$ where $f \in L^1([-1, 1], d\xi)$ and $g \in L^\infty([-1, 1], d\xi)$. Thus $K$ is regular because it is of rank one on $L^1([-1, 1], d\xi)$ but it is not compact on $X$ because its spectrum $\sigma(K) = \{f(\xi), -1 \le \xi \le 1\}$ is not necessary discrete. $\square$

We recall the following result established as Theorem 2.1 in [15].

**Proposition 4.1.** If $K$ is a regular collision operator on $X$, then, for all $\lambda \in \mathbb{C}_0$, the operator $(\lambda - T)^{-1}K$ is weakly compact on $X$.

Now we shall recall some facts concerning superposition operators required below. Let $\Sigma$ be a mesurable subset of $\mathbb{R}^n$. A function $g: \Sigma \times \mathbb{C} \to \mathbb{C}$ is said to be a Carathéodory function on $\Sigma \times \mathbb{C}$ if
$$ \begin{cases} t \to g(t, u) & \text{is measurable on } \Sigma \text{ for all } u \in \mathbb{C}, \\ u \to g(t, u) & \text{is continuous on } \mathbb{C} \text{ for almost all } t \in \Sigma. \end{cases} $$

If $g$ is a Carathéodory function, we can define the operator $N_g$ on the set of functions $\psi : \Sigma \to \mathbb{C}$ by

$$ (N_g\psi)(z) := g(z, \psi(z)) $$

for a.e. $z \in \Sigma$. The operator $N_g$ is called the Nemytskii operator generated by $g$. In $L_p$-spaces the Nemytskii operator has been extensively investigated (see for example [2] and the references therein). However, we recall the following result due to Krasnosel'skii which states a basic fact for the theory of these operators on $L^p$-spaces. Here we state Krasnosel'skii's result in the context of $L^1$-spaces.

**Lemma 4.1.** Assume that $g$ satisfies the Carathéodory conditions. If the operator $N_g$ acts from $L^1(\Sigma)$ into $L^1(\Sigma)$, then $N_g$ is continuous and takes bounded sets into bounded sets. Moreover, there is a constant $\eta > 0$ and a function $h(\cdot) \in L^{1,+}(\Sigma)$ (the positive cone of $L^1(\Sigma)$) such that

$$ |g(x, y)| \leq h(x) + \eta|y| \quad \text{a.e. } x \in \Sigma, \text{ for all } y \in \mathbb{C}. $$

**Proof** We refer, for example, to [2].

QED

## 4.2 Existence results

Hereafter we shall apply the results of Section 3 to discuss existence of solutions to the boundary value problem (1) on the space $X$.

For $i \in \{1, 2, \dots, d\}$, let $\Lambda_i$ denote the bounded multiplication operators from $X$ into itself defined by $\Lambda_i f_i = \beta_i(x, v)f_i$. Since the functions $\beta_i(\cdot, \cdot)$, $i = 1, \dots, d$, belong to $L^\infty(\Omega; dxd\xi)$, the operators $\Lambda_i$, $i = 1, \dots, d$ are bounded. We also denote by $K_i$, $0 \leq i \leq d$, the operator

$$ X \ni f \mapsto K_i f(x, \xi) = \int_{-1}^{1} \kappa_i(x, \xi, \xi') f_i(x, \xi') d\xi \in X $$

where $\kappa_i : [-a, a] \times [-1, 1] \times [-1, 1] \to \mathbb{R}$ is a measurable function. These operators are assumed to be bounded. Moreover, $N_g$ stands for the Nemytskii operator generated by the function $g$ (for $g = -\sigma, \Theta_0, \dots, \Theta_d$).

Now Problem (1) (with $v = 1$) may be written abstractly as

$$ \left\{ \begin{array}{l} (\lambda - T)f_0 = K_0 N_{\Theta_0} f_0 + \sum_{i=1}^{d} \lambda_i \Lambda_i f_i(x, \xi) + N_{-\sigma} \\ \lambda_i f_i(x, \xi) = K_i N_{\Theta_i} f_0 \quad (1 \leq i \leq d). \end{array} \right. \qquad (14) $$
Since, for $i = 1, \cdot, \cdot, \cdot, d$, the operator $\Lambda_i$ is bounded, we can replace $\lambda_i f_i(x, \xi)$ of the first equation of (14) by $K_i \mathcal{N}_{\Theta_i} f_0$. Moreover, as indicated above, each $\lambda \in \mathbb{C}_0$ belongs to $\rho(T)$. So, for such $\lambda$, Problem (1) reduces to the following fixed point problem

$$f_0 = F(\lambda)f_0 + B(\lambda)f_0, \quad f_0^- = H(f_0^+), \qquad (15)$$

where

$$\begin{cases} F(\lambda) = (\lambda - T)^{-1} K_0 \mathcal{N}_{\Theta_0} \quad \text{and} \\ B(\lambda) = (\lambda - T)^{-1} \mathcal{N}_{-\sigma} + \sum_{i=1}^{d} (\lambda - T)^{-1} \Lambda_i K_i \mathcal{N}_{\Theta_i}. \end{cases}$$

Since, for each $i \in \{1, \cdot, \cdot, d\}$, we have $f_i = \frac{K_i \mathcal{N}_{\Theta_i} f_0}{\lambda_i}$, hence to solve Problem (1), it suffices to search the solutions of the fixed point problem (15).

Let $r > 0$ and let $\mathbb{B}_r$ denote the ball defined by

$$\mathbb{B}_r := \{ f \in \mathcal{X} \text{ such that } \|f\| \le r \}.$$

We now introduce the following assumptions

(A1) For each $i \in \{0, 1, \cdot, \cdot, \cdot, d\}$, the map $\Theta_i(\cdot, \cdot, \cdot, \cdot)$ is a Carathéodory function and $\mathcal{N}_{\Theta_i}$ acts from $\mathcal{X}$ into itself.

(A2) For each $r > 0$, the function $\sigma(\cdot, \cdot, \cdot)$ satisfies

$$|\sigma(x, \xi, \psi_1(x, \xi)) - \sigma(x, \xi, \psi_2(x, \xi))| \le \sigma_0(x, \xi) |\psi_1 - \psi_2|$$

$\forall \psi_1, \psi_2 \in \mathbb{B}_r$, $\sigma(\cdot, \cdot, \cdot)$ is a Carathéodory function and $\mathcal{N}_{-\sigma}$ acts from $\mathcal{X}$ into itself, where $\sigma_0(\cdot, \cdot) \in L^\infty(\Omega; dxd\xi)$.

(A3) For each $r > 0$ and each $i \in \{1, 2, \cdot, \cdot, d\}$, the function $\Theta_i(\cdot, \cdot, \cdot)$ satisfies

$$|\Theta_i(x, \xi, \psi_1(x, \xi)) - \Theta_i(x, \xi, \psi_2(x, \xi))| \le \rho_i(x, \xi) |\psi_1 - \psi_2|$$

$\forall \psi_1, \psi_2 \in \mathbb{B}_r$, where $\rho_i(\cdot, \cdot) \in L^\infty(\Omega; dxd\xi)$.

**Theorem 4.1.** Let $K_0$ be a regular collision operator on $\mathcal{X}$ and let $H$ be a boundary operator satisfying $\|H\| \le 1$. Assume further that hypotheses (A1)-(A3) hold true. Then, for each $r > 0$, there exists $\lambda_r$ such that, for all $\lambda$ satisfying $\text{Re}\lambda > \lambda_r$, the boundary value problem (1) has at least one solution with $f_0 \in \mathbb{B}_r$.

**Remark 4.2.** In contrast to the case of multidimensional geometry, in slab geometry the boundary operator $H$ needs not to be weakly compact, it suffices to suppose that $H$ is bounded and $\|H\| \le 1$.
**Proof** Let $\lambda \in \mathbb{C}_0$ and $r > 0$. We shall show that the operators $F(\lambda)$ and $B(\lambda)$ appearing in (15) satisfy the hypotheses of Theorem 3.2.

It is clear that the operators $B(\lambda)$ and $F(\lambda)$ are well defined and continuous on $X$.

Our next objective is to show that $B(\lambda)$ is a contraction on $X$. Indeed, let $\psi_1, \psi_2 \in X$ and let $\lambda \in \mathbb{C}_0$. Using (12) and (A3)-(A4), we obtain

$$ \|B(\lambda)\psi_1 - B(\lambda)\psi_2\| \le \frac{1}{\text{Re}\lambda} \left( \|\sigma_0\|_{\infty} + \sum_{i=1}^{d} \|\beta_i\|_{\infty} \|K_i\| \|\rho_i\|_{\infty} \right) \|\psi_1 - \psi_2\|. $$

Let $\tau_1$ be a real number such that

$$ k := \frac{1}{\tau_1} (\|\sigma_0\|_{\infty} + \sum_{i=1}^{d} \|\beta_i\|_{\infty} \|K_i\| \|\rho_i\|_{\infty}) < 1. $$

Hence, for each $\lambda$ in $\mathbb{C}_{\tau_1}$, $B(\lambda)$ is $k$-contractive.

Now let $\varphi, \psi \in \mathbb{B}_r$. For any $\lambda \in \mathbb{C}_0$, we have

$$ \|F(\lambda)\varphi + B(\lambda)\psi\| \le \frac{1}{\text{Re}\lambda} \left( \|K_0\| \mathcal{M}_{r,0} + \mathcal{M}_{r,\sigma} + \sum_{i=1}^{d} \|\beta_i\|_{\infty} \|K_i\| \mathcal{M}_{r,i} \right), $$

where $\mathcal{M}_{r,i}$ (resp. $\mathcal{M}_{r,\sigma}$) denotes the upper bound of $\mathcal{N}_{\Theta_i}$ (resp. $\mathcal{N}_{-\sigma}$) on $\mathbb{B}_r$ (these bounds exist because the operators $\mathcal{N}_{-\sigma}$ and $\mathcal{N}_{\Theta_i}$, $0 \le i \le d$, are continuous on $\mathbb{B}_r$).

Let $\mathcal{M}_r = \max(\mathcal{M}_{r,0}, \dots, \mathcal{M}_{r,d}, \mathcal{M}_{r,\sigma})$ and let $\tau_2$ be a positive real such that

$$ \frac{1}{\tau_2} \left( 1 + \|K_0\| + \sum_{i=1}^{d} \|\beta_i\|_{\infty} \|K_i\| \right) \le \frac{r}{\mathcal{M}_r}. $$

Hence, for each $\lambda$ in $\mathbb{C}_{\tau_2}$, we have

$$ \|F(\lambda)\varphi + B(\lambda)\psi\| \le r, $$

and therefore $F(\lambda)(\mathbb{B}_r) + B(\lambda)(\mathbb{B}_r) \subset \mathbb{B}_r$.

Let $S$ be a subset of $\mathbb{B}_r$. Since the maps $\Theta_i$, $i = 0, \dots, d$, satisfy the condition (A1), by Lemma 4.1, there exist $\eta_i > 0$ and $h_i(\cdot) \in X^+$ such that

$$ |\Theta_i(x, \xi, f(x, \xi))| \le h_i(x, \xi) + \eta_i |f(x, \xi)|, $$

for all $f \in S$ and for almost all $(x, \xi) \in \Omega$. Accordingly,

$$ \iint_E |(\mathcal{N}_{\Theta_i} f)(x, \xi)| dxd\xi \le \int_E h_i(x, \xi) dxd\xi + \eta_i \iint_E |f(x, \xi)| dxd\xi, \quad (16) $$

for all measurable subset $E$ of $\Omega$ and $0 \le i \le d$. Since the operators $F(\lambda)$ and $B(\lambda)$ satisfy the condition (H), the use of (8) implies
$$
\begin{align*}
\omega(F(\lambda)(S) + B(\lambda)(S)) &\le \omega(F(\lambda)(S)) + \omega(B(\lambda)(S)) \\
&\le \|(\lambda - T_H)^{-1}\| (\|K_0\| \omega(\mathcal{N}_{\Theta_0}S) + \omega(\mathcal{N}_{-\sigma}S) + \sum_{i=1}^d \|\Lambda_i K_i\| \omega(\mathcal{N}_{\Theta_i}S)) \\
&\le \frac{1}{\mathrm{Re}\lambda} (\eta_0 \|K_0\| + \eta_\sigma + \sum_{i=1}^d \eta_i \|\beta_i\|_\infty \|K_i\|) \omega(S).
\end{align*}
$$

Now, let $\tau_3$ be a real number such that

$$
\gamma := \frac{1}{\tau_3} (\eta_0 \|K_0\| + \eta_\sigma + \sum_{i=1}^d \eta_i \|\beta_i\|_\infty \|K_i\|) < 1.
$$

Hence, for each $\lambda$ in $C_{\tau_3}$, we have

$$
\omega(F(\lambda)(S) + B(\lambda)(S)) \leq \gamma\omega(S),
$$

for all mesurable subset $S \subset \mathbb{B}_r$.

Next, since $\Theta_0$ satisfies (A3), it follows from Lemma 3.3 in [16] that $\mathcal{N}_{\Theta_0}$ satisfies condition (H).

Set $\lambda_r = \max(\tau_1, \tau_2, \tau_3)$. The steps above show that, for all $\lambda \in C_{\lambda_r}$, the assumptions of Theorem 3.2 are satisfied and so the fixed point Problem (15) has at last a solution $f_0 \in \mathbb{B}_r$. Therefore, the boundary value problem (1) has at least one solution $(f_0, \dots, f_d)$ with $f_0 \in \mathbb{B}_r$. QED

References

[1] Abdallah, M. Y., Latrach, K., Zeghal, A. Some fixed point theorems in Banach spaces and application to a transport equation with delayed neutrons. Mediterr. J. Math.18(2021), no.4, Paper No. 154, 23 pp.

[2] Appell, J. The superposition operator in function spaces - A survey. Expo. Math. **6** 209–270 (1988)

[3] Appell, J., De Pascale E. Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili. Boll. Unione Mat. Ital. B **3** (6), 497–515 (1984)

[4] Beals, R., Protopopescu, V. Abstract time-dependent transport equations. J. Math. Anal. Appl. **121**, 370–405 (1987)

[5] Cessenat, M. Théorèmes de traces $L_p$ pour des espaces de fonctions de la neutronique. C. R. Acad. Sci. Paris sér. I Math. **299**, 831–834 (1984)

[6] Cessenat, M. Théorèmes de traces pour des espaces de fonctions de la neutronique. C. R. Acad. Sci. Paris sér. I Math. **300**, 89–92 (1985)
[7] De Blasi, F.S. On a property of the unit sphere in a Banach space. Bull. Math. Soc. Sci. Math. R.S. Roumanie **21**, 259–262 (1977)

[8] Diestel, J.: A survey of results related to the Dunford Pettis property. Contemporary Math. **2**, 15–60 (1980)

[9] Dunford, N., Schwartz, J.T. Linear Operators, Part 1. Interscience Publishers Inc., New York (1958)

[10] Garcia-Falset, J.: Existence of fixed points and measure of weak noncompactness. Nonlinear Anal. **71**, 2625–2633 (2009)

[11] Garcia-Falset, J., Latrach, K. Krasnoselskii-type fixed-point theorems for weakly sequentially continuous mappings. Bull. Lond. Math. Soc. **44**, 25–38 (2012)

[12] Garcia-Falset, J., Latrach, K. Nonlinear Functional Analysis and Applications. Series in Nonlinear Analysis and Applications, vol. 41, De Gruyter (2023).

[13] Garcia-Falset, J., Latrach, K., Moreno-Gálvez, E., Taoudi, M.A. Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness. J. Differential Equations **252**, 3436–3452 (2012)

[14] Grothendieck, A. Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$. Canad. J. Math. **5**, 129–173 (1953)

[15] Latrach, K. Compactness properties for linear transport operators with abstract boundary conditions in slab geometry. Transport Theory Statist. Phys. **22**, 39–64 (1993)

[16] Latrach, K., Taoudi, M.A., Zeghal, A. Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations. J. Differential Equations **221**, 256–271 (2006)