# Existence results for nonlinear mono-energetic singular transport equations:

## $L_1$-spaces

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal

### ► To cite this version:

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations:

$L_1$-spaces. Mediterranean Journal of Mathematics, 2019, 16 (1), pp.4. 10.1007/s00009-018-1282-x.
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# Existence results for nonlinear mono-energetic singular transport equations: $L^1$-spaces

Khalid Latrach<sup>a</sup> Hssaine Oummi<sup>b</sup> and Ahmed Zeghal<sup>b, *</sup>

<sup>a</sup>Université Clermont Auvergne

CNRS, LMBP

F-63000 Clermont-Ferrand, France

E-mail: khalid.latrach@uca.fr

<sup>b</sup>Laboratoire de Mathématiques et Applications

Université Sultan Moulay Slimane, Faculté des Sciences et Techniques

B.P. 523, 23000 Béni-Mellal, Morocco

E-mail: a.zeghal@fstbm.ac.ma

September 4, 2018

## Abstract

We prove some results regarding the existence of solutions on $L^1$-spaces to a nonlinear mono-energetic singular transport equation (i.e. transport equation with unbounded collision frequency and unbounded collision operator) in slab geometry. Our approach consists in rewriting the problem as a fixed point one involving a nonlinear ws-compact operator and we use a recent fixed point theorem for this class of operators (see Theorem 2.1) to derive existence results. We present a detailed analysis for the case where the boundary conditions are modeled by specular reflections while the problem with periodic boundary conditions is discussed succinctly because, except some minor modifications, the arguments of proofs are almost the same.

**Keywords:** nonlinear transport equation; fixed point theorems; existence results; weak compactness.

**AMS subject classifications:** 35F20; 45K05; 47H10

## 1 Introduction

In the works [8, 9] the authors discussed the well-posedness and the time asymptotic behavior of solutions of a mono-energetic singular transport equation in slab geometry. In this model, the density of population $\psi(x, \xi, t)$ is described by the following linear partial differential equation

$$ \frac{\partial \psi}{\partial t}(x, \xi, t) = -\xi \frac{\partial \psi}{\partial x}(x, \xi, t) - \sigma(\xi)\psi(x, \xi, t) + \int_{-1}^{1} \kappa(x, \xi, \xi')\psi(x, \xi', t)d\xi' \quad (1) $$

<footnote>*Corresponding author</footnote>
supplemented with specular reflection boundary conditions i.e.

$$ \begin{cases} \psi(t, a, \xi) = \psi(t, a, -\xi), & \xi \in (-1, 0) \\ \psi(t, -a, \xi) = \psi(t, -a, -\xi), & \xi \in (0, 1) \end{cases} \qquad (2) $$

on $L_p$ spaces with $1 \le p < \infty$ where $x \in (-a, a)$ for a parameter $0 < a < \infty$, $\xi \in (-1, 1)$. It describes the transport of particles (neutrons, photons, molecules of gas, etc.) in a slab with thickness $2a$. The function $\psi(x, \xi)$ represents the number density of gas particles having the position $x$ and the direction cosine of propagation $\xi$. (The variable $\xi$ may be thought of as the cosine of the angle between the velocity of particles and the $x$-direction). The functions $\sigma(\cdot)$ and $\kappa(\cdot, \cdot, \cdot)$ are measurable functions called, respectively, the collision frequency and the scattering kernel. The analysis was done under the following conditions:

(i) there exist a closed subset $O \subseteq (-1, 1)$ with zero Lebesgue measure and a constant $\sigma_0 > 0$ such that $\sigma(\cdot) \in L_{\text{loc}}^{\infty}((-1, 1)\setminus O)$, $\sigma(\xi) > \sigma_0$ a.e. on $(-1, 1)$ and $\sigma(\xi) = \sigma(-\xi) \quad \forall \xi \in (-1, 1)$;

(ii) the collision operator $K\psi(x, \xi) := \int_{-1}^{1} \kappa(x, \xi, \xi')\psi(x, \xi')d\xi'$ viewed as an operator from $L^p([-1, 1]; \sigma(\xi)d\xi)$ into $L^p([-1, 1]; d\xi)$ is assumed to be compact and positive (in the lattice sense).

These assumptions were first made by M. Chabi and M. Mokhtar-Kharroubi [20, chapter 9]. It should be noticed that pioneering work in this direction concerning multi-dimensional bounded geometry in $L^1$ spaces, motivated by free gas models (see [10, 21]), was accomplished by M. Chabi in his Phd thesis [7]. Further developments were given in [18].

In this paper, we are concerned with existence of solutions to the nonlinear stationary problem

$$ \xi \frac{\partial \psi}{\partial x}(x, \xi) + \sigma(\xi)\psi(x, \xi) + \lambda\psi(x, \xi) = \int_{-1}^{1} \kappa(x, \xi, \xi')f(x, \xi', \psi(x, \xi'))d\xi', \qquad (3) $$

supplemented with the boundary conditions (2) where the function $\sigma(\cdot)$ satisfies the condition (i) above. However, unlike the works [7, 8, 9], here the kernel $\kappa(\cdot, \cdot, \cdot)$ defines a bounded linear operator from $L^1([-a, a] \times [-1, 1]; \sigma(\xi)dxd\xi)$ into $L^1([-a, a] \times [-1, 1]; dxd\xi)$ but its restriction to $L^1([-1, 1]; \sigma(\xi)d\xi)$ is a weakly compact operator from $L^1([-1, 1]; \sigma(\xi)d\xi)$ into $L^1([-1, 1]; d\xi)$. Note that the assumption about $\sigma(\cdot)$ means that the singularities of the collision frequency are contained in a set of zero Lebesgue measure. In fact, unbounded and nonnegative collision frequencies act as strong absorptions which allow the unboundedness of the collision operator. The map $f(\cdot, \cdot, \cdot)$ is a Carathéodory function on $[-a, a] \times [-1, 1] \times \mathbb{C}$ which defines a continuous Nemytskii (or superposition) operator (cf. Definition 2.1) which maps bounded sets into bounded sets. It is a nonlinear function of the number density of gas particles $\psi$. By $\lambda$ we denote a complex number.

The outline of this work is as follows. In Section 2, we introduce the functional setting of the problem and we establish some preliminary results required in the rest of the paper. In Section 3, we present our existence results in the case of specular reflecting boundary conditions, Theorem 3.1, Corollary 3.1 and 3.2. Section 4 deals with the proofs of these results. Our strategy consists in writing Problem (3)-(2) as a fixed point problem in the form $\psi = U(\lambda)(\psi)$ where, for suitable
complex numbers $\lambda$, $\mathcal{U}(\lambda)$ is a nonlinear continuous weakly compact operator and we show that it satisfies the hypotheses of Theorem 2.1 (a new Schauder fixed point theorem type for the weak topology). In Section 5, we discuss the Problem (3) subjected to periodic boundary conditions (18). Although the shape of the operator $\mathcal{U}(\lambda)$ for periodic boundary conditions is not the same as for the boundary conditions (2), the ideas of proofs of existence results for Problem (3)-(18) use Theorem 2.1 and are similar to those of Theorem 3.1. So, they are succinctly discussed.

## 2 Notations and Preliminaries

In this Section we collect notations and preliminary facts connected to the problem. Let

$$ \mathcal{X} = L^1(\Omega; dxd\xi) $$

where $\Omega = [-a, a] \times [-1, 1]$, $(a > 0)$. Define the following sets representing the incoming and the outgoing boundaries of the phase space $\Omega$:

$$ \Omega^i = \Omega_1^i \cup \Omega_2^i = \{-a\} \times [0, 1] \cup \{a\} \times [-1, 0], $$

$$ \Omega^o = \Omega_1^o \cup \Omega_2^o = \{-a\} \times [-1, 0] \cup \{a\} \times [0, 1]. $$

Moreover, we introduce the following boundary spaces

$$ \begin{aligned} L^i &:= L^1(\Omega^i, |\xi|d\xi) \sim L^1(\Omega_1^i, |\xi|d\xi) \oplus L^1(\Omega_2^i, |\xi|d\xi) \\ &= L_1^i \oplus L_2^i, \end{aligned} $$

endowed with the norm

$$ \|\psi^i\|_{L^i} = \|\psi_1^i\|_{L_1^i} + \|\psi_2^i\|_{L_2^i} = \int_0^1 |\psi(-a, \xi)||\xi|d\xi + \int_{-1}^0 |\psi(a, \xi)||\xi|d\xi, $$

and

$$ \begin{aligned} L^o &:= L^1(\Omega^o, |\xi|d\xi) \sim L^1(\Omega_1^o, |\xi|d\xi) \oplus L^1(\Omega_2^o, |\xi|d\xi) \\ &:= L_1^o \oplus L_2^o, \end{aligned} $$

endowed with the norm

$$ \|\psi^o\|_{L^o} = (\|\psi_1^o\|_{L_1^o} + \|\psi_2^o\|_{L_2^o}) = \int_{-1}^{0} |\psi(-a, \xi)||\xi|d\xi + \int_{0}^{1} |\psi(a, \xi)||\xi|d\xi; $$

where $\sim$ means the natural identification of these spaces.

The boundary conditions (2) may be written abstractly as an operator $H$ relating the incoming and the outgoing fluxes, namely

$$ \left\{ \begin{array}{l} H : L_1^o \oplus L_2^o \to L_1^i \oplus L_2^i \\ H \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} := \begin{pmatrix} H_{11} & 0 \\ 0 & H_{22} \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \end{array} \right. $$
where

$$ \begin{cases} H_{11} : L_1^o \to L_1^i \\ \psi(-a, \xi) \to \psi(-a, -\xi), \quad -1 < \xi < 0; \end{cases} \quad \begin{cases} H_{22} : L_2^o \to L_2^i \\ \psi(a, \xi) \to \psi(a, -\xi), \quad 0 < \xi < 1. \end{cases} $$

**Remark 2.1.** Clearly $H$ is invertible. On the other hand, if $(\begin{matrix} u_1 \\ u_2 \end{matrix}) \in L^o$, then

$$ \begin{aligned} \|Hu\|_{L^i} &= \|H_{11}u_1\|_{L_1^i} + \|H_{22}u_2\|_{L_2^i} \\ &= \int_0^1 |u_1(-a, -\xi)||\xi|d\xi + \int_{-1}^0 |u_2(a, -\xi)||\xi|d\xi \\ &= \|u_1\|_{L_1^o} + \|u_2\|_{L_2^o} = \|u\|_{L^o}. \end{aligned} $$

Thus, $H$ is a perfect reflecting boundary operator.

<figure><img src="image_2.png" /></figure>

Throughout this paper, the collision frequency $\sigma(\cdot)$ will be assumed to satisfy the assumption:

(A1) there exist a closed subset $O \subset [-1, 1]$ with **zero Lebesgue measure** and a constant $\sigma_0 > 0$ such that $\sigma(\cdot) \in L_{\text{loc}}^{\infty}([-1, 1]\O)$, $\sigma(\xi) > \sigma_0$ a.e. on $[-1, 1]$ and

$$ \sigma(\xi) = \sigma(-\xi) \quad \forall \xi \in [-1, 1]. $$

We need also the following functional spaces

$$ X^{\sigma} := L^{1}([-a, a] \times [-1, 1]; \sigma(\xi)dxd\xi), $$

$$ L^{\sigma} := L^{1}([-1, 1]; \sigma(\xi)d\xi), $$

$$ L := L^{1}([-1, 1]; d\xi). $$

Next we introduce the following partial Sobolev space

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

It should be noticed that elementary calculations using the estimate $\sigma(\xi) > \sigma_0$ (see (A1)) show that

$$ X^{\sigma} \subset X $$

and therefore

$$ W \subset \{ \psi \in X \text{ such that } \xi \frac{\partial \psi}{\partial x} \in X \}. $$

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

Now we are in position to define the streaming operator $T_H$
$$
\left\{
\begin{array}{l}
T_H : D(T_H) \subseteq \mathbf{X}^{\sigma} \to \mathbf{X} \\
\psi \to T_H \psi(x, \xi) = -\xi \frac{\partial \psi}{\partial x}(x, \xi) - \sigma(\xi) \psi(x, \xi) \\
D(T_H) = \{ \psi \in \mathbf{W} \text{ such that } \psi^i = H(\psi^o) \},
\end{array}
\right.
$$

where $\psi^o = (\psi_1^o, \psi_2^o)^T$ and $\psi^i = (\psi_1^i, \psi_2^i)^T$ with $\psi_1^o, \psi_2^o, \psi_1^i$ and $\psi_2^i$ are given by

$$
\left\{
\begin{array}{ll}
\psi_1^i(\xi) = \psi(-a, \xi), & \xi \in (0, 1), \\
\psi_2^i(\xi) = \psi(a, \xi), & \xi \in (-1, 0), \\
\psi_1^o(\xi) = \psi(-a, \xi), & \xi \in (-1, 0), \\
\psi_2^o(\xi) = \psi(a, \xi), & \xi \in (0, 1).
\end{array}
\right.
$$

Let $\varphi \in \mathbf{X}$ and consider the resolvent equation for the operator $T_H$

$$
(\lambda - T_H)(\psi) = \varphi \tag{4}
$$

where $\lambda$ is a complex number and the unknown $\psi$ must be sought in $D(T_H)$. For $\mathrm{Re}\lambda + \sigma_0 > 0$, the
solution of (4) is formally given by

$$
\psi(x, \xi) =
\begin{cases}
\psi(-a, \xi) \exp \left( - \frac{(\lambda + \sigma(\xi))|a+x|}{|\xi|} \right) + \frac{1}{|\xi|} \int_{-a}^{x} \exp \left( - \frac{(\lambda + \sigma(\xi))|x-y|}{|\xi|} \right) \varphi(y, \xi) dy, & \text{for } \xi \in (0, 1) \\
\\
\psi(a, \xi) \exp \left( - \frac{(\lambda + \sigma(\xi))|a-x|}{|\xi|} \right) + \frac{1}{|\xi|} \int_{x}^{a} \exp \left( - \frac{(\lambda + \sigma(\xi))|x-y|}{|\xi|} \right) \varphi(y, \xi) dy, & \text{for } \xi \in (-1, 0).
\end{cases}
\tag{5}
$$

For $x = a$, we get for $\xi \in (0, 1)$

$$
\psi(a, \xi) = \psi(-a, \xi) \exp \left( -2a \frac{(\lambda + \sigma(\xi))}{|\xi|} \right) + \frac{1}{|\xi|} \int_{-a}^{a} \exp \left( -\frac{(\lambda + \sigma(\xi))|a-x|}{|\xi|} \right) \varphi(x, \xi) dx, \quad (6)
$$

and for $x = -a$, we get for $\xi \in (-1, 0)$

$$
\psi(-a, \xi) = \psi(a, \xi) \exp \left( -2a \frac{(\lambda + \sigma(\xi))}{|\xi|} \right) + \frac{1}{|\xi|} \int_{-a}^{a} \exp \left( -\frac{(\lambda + \sigma(\xi))|a+x|}{|\xi|} \right) \varphi(x, \xi) dx. \quad (7)
$$

Set

$$
\mathbf{m}(\lambda, \tau, r) = \exp \left( -2a \frac{(\lambda + \tau)}{|r|} \right), \quad \tau \in \mathbb{R}, \ r > 0.
$$

Using (5)-(7) and taking into account of the boundary conditions, simple calculations show that the
solution of (4) writes
$$
\begin{align*}
\psi(x, \xi) = & \frac{1}{(1 - (\mathbf{m}(\lambda, \sigma(\xi), \xi))^2)|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(4a + x - y)}{|\xi|}\right) \varphi(y, \xi)dy \\
& + \frac{1}{(1 - (\mathbf{m}(\lambda, \sigma(\xi), \xi))^2)|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a + x + y)}{|\xi|}\right) \varphi(y, -\xi)dy \\
& + \frac{1}{|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(x - y)}{|\xi|}\right) \varphi(y, \xi)dy
\end{align*}
\qquad \text{for } \xi \in (0, 1),
$$

and

$$
\begin{align*}
\psi(x, \xi) = & \frac{1}{(1 - (\mathbf{m}(\lambda, \sigma(\xi), \xi))^2)|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(4a - x + y)}{|\xi|}\right) \varphi(y, \xi)dy \\
& + \frac{1}{(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a - x - y)}{|\xi|}\right) \varphi(y, -\xi)dy \\
& + \frac{1}{|\xi|} \int_{x}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(y - x)}{|\xi|}\right) \varphi(y, \xi)dy
\end{align*}
\qquad \text{for } \xi \in (-1, 0).
$$

Let $\lambda \in \mathbb{C}$ be such that $\text{Re}\lambda > -\sigma_0$ and define the operators

$$
\left\{
\begin{array}{ll}
\mathcal{P}_{\lambda} : \mathbf{X} \rightarrow \mathbf{X}^{\sigma}, & \\
\mathcal{P}_{\lambda}\varphi := \chi_{(-1,0)}(\xi)\mathcal{P}_{\lambda}^{-}\varphi + \chi_{(0,1)}(\xi)\mathcal{P}_{\lambda}^{+}\varphi, & \text{where} \\
(\mathcal{P}_{\lambda}^{+}\varphi)(x, \xi) := \frac{1}{(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(4a + x - y)}{|\xi|}\right) \varphi(y, \xi)dy, & \\
\mathcal{P}_{\lambda}^{-}\varphi(x, \xi) := \frac{1}{(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(4a - x + y)}{|\xi|}\right) \varphi(y, \xi)dy, & \text{for } \xi \in (-1, 0),
\end{array}
\right.
$$

$$
\left\{
\begin{array}{ll}
S_{\lambda} : X \to X^{\sigma}, & \\
S_{\lambda}\varphi := \chi_{(-1,0)}(\xi)S_{\lambda}^{-}\varphi + \chi_{(0,1)}(\xi)S_{\lambda}^{+}\varphi, & \text{where} \\
(S_{\lambda}^{+}\varphi)(x, \xi) := \frac{1}{(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a + x + y)}{|\xi|}\right) \varphi(y, -\xi)dy, & \\
(S_{\lambda}^{-}\varphi)(x, \xi) := \frac{1}{(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a - x - y)}{|\xi|}\right) \varphi(y, -\xi)dy, & \text{for } \xi \in (-1, 0),
\end{array}
\right.
$$

and
$$
\begin{cases}
Q_{\lambda}: X \to X^{\sigma}, \\
Q_{\lambda\varphi} := \chi_{(-1,0)}(\xi) Q_{\lambda}^{-} \varphi + \chi_{(0,1)}(\xi) Q_{\lambda}^{+} \varphi, \quad \text{where} \\
(Q_{\lambda}^{+} \varphi)(x, \xi) := \frac{1}{|\xi|} \int_{-a}^{x} \exp\left(-\frac{(\lambda + \sigma(\xi))|x-y|}{|\xi|}\right) \varphi(y, \xi)dy, \quad \xi > 0, \\
(Q_{\lambda}^{-} \varphi)(x, \xi) := \frac{1}{|\xi|} \int_{x}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))|x-y|}{|\xi|}\right) \varphi(y, \xi)dy, \quad \xi < 0,
\end{cases}
$$

where $\chi_{(-1,0)}$, $\chi_{(0,1)}$ denote, respectively, the characteristic functions of the intervals $(-1,0)$ and $(0,1)$.

The following lemma was established in [8].

**Lemma 2.1.** Let $\lambda$ be a complex number such that $\text{Re}\lambda + \sigma_0 > 0$ and assume that the condition (A1) holds true. Then the operators $\mathcal{P}_\lambda$, $\mathcal{S}_\lambda$ and $\mathcal{Q}_\lambda$ belong to $\mathcal{L}(X, X^\sigma)$ and satisfy

$$
\| \mathcal{P}_{\lambda} \|_{\mathcal{L}(X, X^{\sigma})} \le \frac{1}{1 - (\mathbf{m}(\text{Re}\lambda, \sigma_0, 1))^2} \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)}{\text{Re}\lambda + \sigma(\xi)},
$$

$$
\| \mathcal{S}_{\lambda} \|_{\mathcal{L}(X, X^{\sigma})} \le \frac{1}{1 - (\mathbf{m}(\text{Re}\lambda, \sigma_0, 1))^2} \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)}{\text{Re}\lambda + \sigma(\xi)},
$$

and

$$
\| \mathcal{Q}_{\lambda} \|_{\mathcal{L}(X, X^{\sigma})} \leq \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)}{\text{Re}\lambda + \sigma(\xi)}.
$$

As an immediate consequence of this lemma we have

**Corollary 2.1.** Assume that the hypotheses of Lemma 2.1 hold true. Then $D(T_H) \subseteq X^\sigma$ with continuous embedding. Further, for $\lambda \in \mathbb{C}$ such that $\text{Re}\lambda > -\sigma_0$, the inverse of the operator $(\lambda - T_H)$ is given by

$$
(\lambda - T_H)^{-1} = \mathcal{P}_\lambda + \mathcal{S}_\lambda + \mathcal{Q}_\lambda \quad (8)
$$

and satisfies

$$
\|(\lambda - T_H)^{-1}\|_{\mathcal{L}(X, X^{\sigma})} \le \frac{3}{1 - (\mathbf{m}(\text{Re}\lambda, \sigma_0, 1))^2}. \quad (9)
$$

The estimate (9) follows from the fact that $\sigma(\cdot)$ is assumed to be bounded because in this case we have

$$
\sup_{\xi \in (-1,1)} \frac{\sigma(\xi)}{\text{Re}\lambda + \sigma(\xi)} \le 1 \quad \text{regardless of } \lambda.
$$

**Remark 2.2.** Note that, for any real number $\lambda$ such that $\lambda > -\sigma_0$, the operators $\mathcal{P}_\lambda$, $\mathcal{S}_\lambda$ and $\mathcal{Q}_\lambda$ are positive (in the lattice sense) on $X$, that is, they map the positive cone of $X$ into the positive cone of $X^\sigma$. Therefore, for such reals, say $\lambda$, the operator $(\lambda - T_H)^{-1}$ is also positive on $X$ (use (8)).
□

<figure><img src="image_3.png" /></figure>
We recall that (this comment is adapted to the one-dimensional setting), if $[-a, a]$ denotes the set of particle positions, the physical collision operators used in linear kinetic theory are bounded operators on $L^1(\Omega, dxd\xi)$ and are in the form

$$ K\psi(x, \xi) = \int_{-1}^{1} \kappa(x, \xi, \xi')\psi(x, \xi')d\xi'. \quad (10) $$

where the kernel $\kappa(\cdot, \cdot, \cdot)$ is assumed to be a measurable nonnegative real valued function from $\Omega \times [-1, 1]$ into $\mathbb{R}$. These operators possess some compactness properties with respect to the velocities (in slab geometry, the variable $\xi$ plays the role of the velocity variable). To highlight these compactness properties, M. Mokhtar-Kharroubi [19] introduced the class of regular collision operators in a general $L^p$-spaces setting ($1 \le p < +\infty$). Basically, a linear operator $K \in \mathcal{L}(L^p(\Omega; dxd\xi))$ is regular if it is local with respect to $x \in [-a, a]$ and compact with respect to $\xi \in [-1, 1]$. However, as it was already noticed in [22] and [23], a linear operator $K \in \mathcal{L}(L^1(\Omega; dxd\xi))$ is regular if it is local with respect to $x \in [-a, a]$ and is weakly compact with respect to $\xi \in [-1, 1]$.

We shall now recall some fact about Nemytskii (or superposition) operators on $L^p$-spaces.

**Definition 2.1.** A function $f: \Omega \times \mathbb{C} \to \mathbb{C}$ is said to satisfy the Carathéodory conditions on $\Omega \times \mathbb{C}$ if

(a) the function $(x, \xi) \mapsto f(x, \xi, u)$ is measurable on $\Omega$ for all $u \in \mathbb{C}$,

(b) the function $u \mapsto f(x, \xi, u)$ is continuous on $\mathbb{C}$ for almost all $(x, \xi) \in \Omega$.

□

If $f$ satisfies the Carathéodory conditions, we can define the operator $N_f$ on the set of functions $\psi: \Omega \to \mathbb{C}$ by

$$ (N_f\psi)(x, \xi) := f(x, \xi, \psi(x, \xi)) $$

for every $(x, \xi) \in \Omega$. The operator $N_f$ is called the Nemytskii operator generated by $f$. In $L^p$-spaces the Nemytskii operator has been extensively investigated (see, for example, [1, 2] and the references therein). The first results on continuity and boundedness of this operator in Lebesgue spaces over bounded domains $\Omega$ in Euclidean space are due to M.A. Krasnosel'skii (see, for example, [2, Theorem 3.1]).

The following result (established in a more general setting in [11, Lemma 2.3] (see also [3])) provides necessary and sufficient conditions under which the Nemytskii operator maps continuously a weighted $L^1$-space into itself. (The statement of this result is adapted to our context.)

**Lemma 2.2.** Assume that $f$ satisfies the Carathéodory conditions. Then $N_f$ maps continuously $X^\sigma$ into itself, if and only if, the function $f$ satisfies the following growth condition

$$ |f(x, \xi, u)| \le \frac{\alpha(x, \xi)}{\sigma(\xi)} + \beta|u|, \quad (11) $$

for a.e. $(x, \xi) \in \Omega$, and every $u \in \mathbb{C}$ with a fixed nonnegative function $\alpha \in X$ and a fixed nonnegative constant $\beta$. Further, it maps bounded sets of $X^\sigma$ into bounded sets of $X^\sigma$.
The continuity of $N_f$ was established in [3, 11]. The fact that it maps bounded sets of $X^\sigma$ into bounded sets of $X^\sigma$ is an immediate consequence of the estimate (11).

**Definition 2.2.** Let $X$ be a Banach space and let $A: D(A) \subseteq X \to X$ be a nonlinear mapping. We say that $A$ is ws-compact if $A$ is continuous and if $(x_n)_{n \in \mathbb{N}} \subseteq D(A)$ is a weakly convergent sequence in $X$, then $(A(x_n))_{n \in \mathbb{N}}$ has a strongly convergent subsequence in $X$. $\square$

**Remark 2.3.** (1) It is clear that each compact operator is ws-compact and when $X$ is reflexive, these two notions are equivalent. However, the fact that $A$ is ws-compact does not necessarily imply the compactness of $A$ even if $A$ is linear. It is well known that a compact linear operator from a Banach space $X$ into a Banach space $Y$ transforms weakly convergent sequences into norm convergent sequences. If $X$ is not reflexive, the converse of this assertion is, in general, not true even if the space $Y$ is reflexive.

(2) Note that linear weakly compact operators acting on Banach spaces with the Dunford-Pettis property are ws-compact. (A Banach space $X$ has the Dunford-Pettis property if every weakly compact operator defined on $X$ takes weakly compact sets into norm compact sets.) Indeed, if $X$ is a Banach space with the Dunford-Pettis property, then every weakly compact linear operator from $X$ into an arbitrary Banach space $Y$ maps weakly convergent sequences in $X$ onto norm convergent sequences in $Y$.

(3) ws-compact operators are not necessarily weakly continuous (see, for example, Theorem 2.6 in [1]). $\square$

We close this section by recalling a fixed point theorem of Schauder type for weak topology (see [17, Theorem 2.1] or [15, Theorem 4.3.1, p. 222]).

**Theorem 2.1.** Let $\Delta$ be a nonempty closed convex subset of a Banach space $X$ and assume that $A: \Delta \to \Delta$ is a ws-compact operator. If $A(\Delta)$ is relatively weakly compact, then $A$ has a fixed point in $\Delta$.

Further fixed point results for the class of ws-compact operators can be found in [13, 14, 15] or [16].

## 3 Existence results

Let us recall some facts concerning weakly compact operators on $L^1$-spaces. Let $(\Sigma_1, A_1, \mu)$ and $(\Sigma_2, A_2, \nu)$ be $\sigma$-finite positive measure spaces with $\Sigma_i \subset \mathbb{R}^N$ ($i=1,2$). An operator from $L^1(\Sigma_1, d\mu)$ into $L^1(\Sigma_2, d\nu)$ is said to be weakly compact if it transforms bounded subsets of $L^1(\Sigma_1, d\mu)$ into relatively weakly compact subsets of $L^1(\Sigma_2, d\nu)$. The following result shows that weakly compact operators from $L^1(\Sigma_1, d\mu)$ into $L^1(\Sigma_2, d\nu)$ are kernel operators (see [12, p. 508]).

**Proposition 3.1.** Let $(\Sigma_1, A_1, \mu)$ and $(\Sigma_2, A_2, \nu)$ be $\sigma$-finite positive measure spaces with $\Sigma_i \subset \mathbb{R}^N$ ($i=1,2$). Let $B \in \mathcal{L}(L^1(\Sigma_1, d\mu), L^1(\Sigma_2, d\nu))$ be a weakly compact operator. Then there exists a $\mu \otimes \nu$-essentially unique, $\mu \otimes \nu$-measurable function $\zeta(\cdot, \cdot)$ defined on $\Sigma_1 \times \Sigma_2$ such that

$$ \mu - \text{ess sup}_{y \in \Sigma_1} \int_{\Sigma_2} \zeta(x, y)d\nu(x) < +\infty $$
and

$$B\psi(x) = \int_{\Sigma_1} \zeta(x, y)\psi(y)d\mu(y), \quad \psi \in L^1(\Sigma_1, d\mu)$$

Moreover, we have

$$\|B\| = \mu - \underset{y \in \Sigma_1}{\operatorname{ess sup}} \int_{\Sigma_2} |\zeta(x, y)| d\nu(x)$$

We assume that the kernel of the collision operator (10) is non-negative and satisfies the conditions

$$\underset{(x, \xi') \in [-a, a] \times [-1, 1]}{\operatorname{ess\,sup}} \frac{1}{\sigma(\xi')} \int_{-1}^{1} \kappa(x, \xi, \xi') d\xi' < +\infty. \quad (12)$$

It is clear that under the assumption (12), we have

$$K \in \mathcal{L}(X^{\sigma}, X). \quad (13)$$

**Definition 3.1.** Let K be the operator

$$\left\{ \begin{array}{l} K : X^{\sigma} \to X \\ \psi \mapsto K\psi(x, \xi) = \int_{-1}^{1} \kappa(x, \xi, \xi')\psi(x, \xi')d\xi'. \end{array} \right.$$

We say that $K \in \mathcal{L}(X^{\sigma}, X)$ is a regular operator if $\{\kappa(x, \cdot, \xi') : (x, \xi') \in \Omega\}$ is relatively weakly compact in L. $\square$

Using Proposition 3.1 and some facts from [22], the following result was established in [18] (the statement below is adapted to our context).

**Proposition 3.2.** Let $K \in \mathcal{L}(X^{\sigma}, X)$ be a non-negative regular collision operator. Then, there exists a sequence $(K_n)_n$ of $\mathcal{L}(X^{\sigma}, X)$ such that

(a) $0 \le K_n \le K$ for any $n \in \mathbb{N}$;

(b) for any $n \in \mathbb{N}$, $K_n$ is dominated by a rank-one operator in $\mathcal{L}(L^{\sigma}, L)$;

(c) $\lim_{n \to \infty} \|K_n - K\|_{\mathcal{L}(X^{\sigma}, X)} = 0$.

**Remark 3.1.** Item (b) of the proposition above means that each operator $K_n$ is dominated by an operator belonging to $\mathcal{L}(X^{\sigma}, X)$ which acts as follows

$$X^{\sigma} \ni \psi \mapsto \int_{-1}^{1} f_n(\xi)\psi(x, \xi')\sigma(\xi')d\xi'$$

where $f_n$ is a nonnegative function belonging to L. $\square$

We shall introduce the following assumptions

(A2) $f(\cdot, \cdot, \cdot)$ is a Carathéodory function satisfying the growth condition (11),
$$ (A3) \quad \beta \|K\|_{\mathcal{L}(X^{\sigma}, X)} \le \frac{1}{12}, $$

where $\beta$ is the constant appearing in the growth condition (11).

For any $r > 0$, we denote by $\mathbb{B}_r^{\sigma}$ the set

$$ \mathbb{B}_r^{\sigma} = \{ \psi \in \mathbf{X}^{\sigma} : \|\psi\|_{\mathbf{X}^{\sigma}} \le r \}. $$

**Theorem 3.1.** Let $K \in \mathcal{L}(\mathbf{X}^{\sigma}, \mathbf{X})$ be a non-negative regular collision operator and assume that the conditions (A1)-(A3) be satisfied. Then, for all $\lambda \in \mathbb{C}$ and $r > 0$ satisfying

$$ \mathrm{Re}\lambda \ge \frac{\ln(2)}{4a} - \sigma_0 \quad \text{and} \quad r \ge 12 \|K\|_{\mathcal{L}(\mathbf{X}^{\sigma}, \mathbf{X})} \|\alpha\|_{\mathbf{X}}, \quad (14) $$

the problem

$$ \begin{cases} \xi \frac{\partial \psi}{\partial x}(x, \xi) + \sigma(\xi)\psi(x, \xi) + \lambda \psi(x, \xi) = \int_{-1}^{1} \kappa(x, \xi, \xi') f(x, \xi', \psi(x, \xi')) d\xi' \\ \psi^i = H(\psi^o), \end{cases} \quad (15) $$

has at least one solution on $\mathbb{B}_r^{\sigma}$.

The following two results are devoted to the existence of positive solutions to the boundary value problem (15). Note that $\mathbf{X}$ and $\mathbf{X}^{\sigma}$ are Banach lattice spaces; their positive cones will be denoted by $\mathbf{X}^{+}$ and $\mathbf{X}^{\sigma+}$ respectively.

Let $r > 0$. We define the set $\mathbb{B}_r^{\sigma,+}$ by

$$ \mathbb{B}_r^{\sigma,+} = \mathbb{B}_r^{\sigma} \cap \mathbf{X}^{\sigma,+}. $$

It is clear from Remark 2.2, that, for all real $\lambda$ such that $\lambda > -\sigma_0$, then the operator $(\lambda - T_H)^{-1}$ is positive (in the lattice sense, that is $(\lambda - T_H)^{-1}(\mathbf{X}^{\sigma+}) \subset \mathbf{X}^+)$. If further the hypotheses of Theorem 3.1 are satisfied, $\mathcal{N}_f(\mathbf{X}^{\sigma+}) \subset \mathbf{X}^+$ and $K$ is positive, then $(\lambda - T_H)^{-1}KN_f$ leaves the positive cone $\mathbf{X}^{\sigma+}$ invariant. If we replace $\mathbb{B}_r^{\sigma}$ by $\mathbb{B}_r^{\sigma,+}$ and we argue as in the proof of Theorem 3.1, then we obtain the following result.

**Corollary 3.1.** Assume that the hypotheses of Theorem 3.1 hold true and $\mathcal{N}_f(\mathbf{X}^{\sigma+}) \subseteq \mathbf{X}^{\sigma+}$. Then, for all reals $\lambda$ and $r$ satisfying (14), Problem (15) has at least one solution on $\mathbb{B}_r^{\sigma,+}$.

In the following, we will focus our attention on the existence of nontrivial positive solutions.

**Corollary 3.2.** Assume that the hypotheses of Corollary 3.1 are satisfied. If there is $\rho > 0$ and $0 \ne \psi_0 \in \mathbb{B}_r^{\sigma,+}$ such that

(a) $\psi_0 \notin N(K)$ where $N(K)$ denotes the null space of $K$,

(b) $(\mathcal{N}_f\psi)(x, \xi) \ge \rho\psi_0(x, \xi)$ for all $\psi \in \mathbb{B}_r^{\sigma,+}$.

Then, for all reals $\lambda$ and $r$ satisfying (14), there exist $\eta > 0$ such that the problem

$$ \begin{cases} \xi \frac{\partial \psi}{\partial x}(x, \xi) + \sigma(\xi)\psi(x, \xi) + \lambda \psi(x, \xi) = \eta \int_{-1}^{1} k(x, \xi, \xi') f(x, \xi', \psi(x, \xi')) d\xi', \\ \psi^i = H(\psi^o), \end{cases} \quad (16) $$

has at least one solution $\psi^* \in \mathbb{B}_r^{\sigma,+}$ satisfying $\|\psi^*\|_{\mathbf{X}^{\sigma}} = r$.
# 4 Proofs

In this section we shall proceed to the proof of our main result.

**Lemma 4.1.** Let $\lambda$ be a complex number satisfying $\text{Re}\lambda > -\sigma_0$ and let $K$ be a regular collision operator. Then the operators $\mathcal{P}_\lambda K$, $\mathcal{S}_\lambda K$ and $\mathcal{Q}_\lambda K$ are weakly compact on $X^\sigma$.

**Proof.** According to Proposition 3.2, there exists a sequence of rank-one operators of $\mathcal{L}(X^\sigma, X)$ which converges, in the operator norm, to $K$. So, according to Remark 3.1, it suffices to establish the result for an operator $K$ of the form

$$K\psi(x, \xi) = \int_{-1}^{1} \zeta(\xi)\psi(x, \xi')\sigma(\xi')d\xi'$$

where $\zeta(\cdot)$ is a nonnegative function belonging to $L$. To do so, let $\psi \in X^\sigma$. Since

$$
\begin{aligned}
\mathcal{P}_\lambda K \psi(x, \xi) &= \chi_{(-1,0)}(\xi) \mathcal{P}_\lambda^- K \psi(x, \xi) + \chi_{(0,1)}(\xi) \mathcal{P}_\lambda^+ K \psi(x, \xi) \\
&= \frac{1}{(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)|\xi|} \left[ \chi_{(-1,0)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a+x-y)}{|\xi|}} K\varphi(y, \xi)dy \right. \\
&\qquad \left. + \chi_{(0,1)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a-x+y)}{|\xi|}} K\varphi(y, \xi)dy \right] \\
&= \frac{1}{(1 - \mathbf{m}((\lambda, \sigma(\xi), \xi))^2)|\xi|} \left[ \chi_{(-1,0)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a+x-y)}{|\xi|}} \zeta(\xi) \int_{-1}^{1} \varphi(y, \xi')dy\sigma(\xi')d\xi' \right. \\
&\qquad \left. + \chi_{(0,1)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a-x+y)}{|\xi|}} \zeta(\xi) \int_{-1}^{1} \varphi(y, \xi')dy\sigma(\xi')d\xi' \right],
\end{aligned}
$$

the operator $\mathcal{P}_\lambda K$ may be written in the following form $\mathcal{P}_\lambda K = \Lambda_\lambda R$, where $R$ is the following bounded operator

$$\left\{ \begin{array}{l} R : X^\sigma \to L^1((-a, a); dx) \\ \psi \mapsto (R\psi)(x) := \int_{-1}^{1} \psi(x, \xi)\sigma(\xi)d\xi \end{array} \right.$$

and $\Lambda_\lambda$ is defined by

$$
\left\{
\begin{array}{l}
\Lambda_{\lambda} : L^1((-a, a); dx) \to X^{\sigma} \\
\\
\varphi \mapsto \frac{\zeta(\xi)}{(1 - (\mathbf{m}(\lambda, \sigma(\xi), \xi))^2)|\xi|} \\
\\
\qquad \times \left[ \chi_{(-1,0)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a+x-y)}{|\xi|}} \varphi(y)dy + \chi_{(0,1)}(\xi) \int_{-a}^{a} e^{-\frac{(\lambda+\sigma(\xi))(4a-x+y)}{|\xi|}} \varphi(y)dy \right].
\end{array}
\right.
$$

We shall prove now that the operator $\Lambda_\lambda$ is weakly compact. To do so, let $\mathcal{U}$ be a bounded subset of $L^1((-a, a); dx)$ and let $\mathcal{A}$ be a measurable subset of $(-a, a) \times (-1, 1)$ with respect to the measure $dx\sigma(\xi)d\xi$. For any $f \in \mathcal{U}$, we have
$$
\begin{aligned}
|\langle \Lambda_\lambda f \rangle(x, \xi)| & \le \frac{\zeta(\xi)}{1 - (\mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1))^2 |\xi|} \left[ \chi_{(-1,0)}(\xi) \int_{-a}^a e^{-\frac{(\mathrm{Re}\lambda + \sigma(\xi))(a+x)}{|\xi|}} |f(y)| dy \right. \\
& \qquad \left. + \chi_{(0,1)}(\xi) \int_{-a}^a e^{-\frac{(\mathrm{Re}\lambda + \sigma(\xi))(a-x)}{|\xi|}} |f(y)| dy \right] \\
& \le \frac{\|f\|_{L^1((-a,a);dx)}}{1 - (\mathbf{m}(\mathrm{Re}\lambda, \sigma_0, 1))^2} g(x, \xi)
\end{aligned}
$$

where

$$
g(x, \xi) = \frac{\zeta(\xi)}{|\xi|} \left[ \chi_{(-1,0)}(\xi) e^{-\frac{(\mathrm{Re}\lambda + \sigma(\xi))(a+x)}{|\xi|}} + \chi_{(0,1)}(\xi) e^{-\frac{(\mathrm{Re}\lambda + \sigma(\xi))(a-x)}{|\xi|}} \right].
$$

Since

$$
\begin{aligned}
\int_{-1}^{1} \int_{-a}^{a} |g(x, \xi)| dx \sigma(\xi) d\xi &\le \int_{-1}^{0} \int_{-a}^{a} \frac{\zeta(\xi)}{|\xi|} e^{-\frac{(\mathrm{Re}\lambda + \sigma(\xi))(a+x)}{|\xi|}} dx \sigma(\xi) d\xi \\
&\quad + \int_{0}^{1} \int_{-a}^{a} \frac{\zeta(\xi)}{|\xi|} e^{-\frac{(\mathrm{Re}\lambda + \sigma(\xi))(a-x)}{|\xi|}} dx \sigma(\xi) d\xi \\
&= \int_{-1}^{1} \frac{\zeta(\xi) \sigma(\xi)}{\mathrm{Re}\lambda + \sigma(\xi)} \left(1 - e^{-2a \frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|}}\right) d\xi \\
&\le \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)}{\mathrm{Re}\lambda + \sigma(\xi)} \|\zeta\|_{L},
\end{aligned}
$$

one concludes that, for all $f \in \mathcal{U}$, we have

$$
\lim_{|A| \to 0} \int_A |(\Lambda_\lambda f)(x, \xi)| dx \sigma(\xi) d\xi = 0
$$

where $|A|$ denotes the measure of $A$ with respect to measure $dx\sigma(\xi)d\xi$. Hence, according to Corollary 11 in [12, p. 294], the set $\Lambda_\lambda(\mathcal{U})$ is weakly sequentially compact. Applying Eberlein-Smulian's theorem [12, p. 430], we conclude that $\Lambda_\lambda(\mathcal{U})$ is a weakly compact set. This proves the weak compactness of the operator $\Lambda_\lambda$. Next, using the fact that $R \in \mathcal{L}(X^\sigma; L^1((-a, a); dx))$, we infer that the operator $\mathcal{P}_\lambda K$ is weakly compact.

Note that the operator $S_\lambda K$ has the same structure as $\mathcal{P}_\lambda K$, so the same reasoning as above leads
to the weak compactness of $S_\lambda K$.

It remains to show that $\mathcal{Q}_\lambda K$ is weakly compact. Indeed, for $\psi \in X^\sigma$, we have

$$
\begin{align*}
\|(\mathcal{Q}_\lambda K)\psi\|_{X^\sigma} &= \int_{-a}^{a} \int_{-1}^{1} |(\mathcal{Q}_\lambda K)\psi(x, \xi)| dx \sigma(\xi) d\xi \\
&\le \int_{-a}^{a} \int_{-1}^{0} \frac{\zeta(\xi)\sigma(\xi)}{|\xi|} \int_{x}^{a} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} |x-y|} \int_{-1}^{1} |\psi(x, \xi')| dy \sigma(\xi') d\xi' dx d\xi \\
&\quad + \int_{-a}^{a} \int_{0}^{1} \frac{\zeta(\xi)\sigma(\xi)}{|\xi|} \int_{-a}^{x} e^{-\frac{\mathrm{Re}\lambda + \sigma(\xi)}{|\xi|} |x-y|} \int_{-1}^{1} |\psi(x, \xi')| dy \sigma(\xi') d\xi' dx d\xi
\end{align*}
$$
Using Fubini's theorem, we get

$$ \begin{aligned} \|(Q_\lambda K)\psi\|_{X^\sigma} & \le \int_{-a}^a dy \int_{-1}^0 \frac{\zeta(\xi)\sigma(\xi)}{|\xi|} \int_{-a}^y e^{-\frac{\mathrm{Re}\lambda+\sigma(\xi)}{|\xi|}(y-x)} dx \int_{-1}^1 |\psi(x, \xi')|\sigma(\xi')d\xi' d\xi \\ & \quad + \int_{-a}^a dy \int_0^1 \frac{\zeta(\xi)\sigma(\xi)}{|\xi|} \int_y^a e^{-\frac{\mathrm{Re}\lambda+\sigma(\xi)}{|\xi|}(x-y)} dx \int_{-1}^1 |\psi(x, \xi')|\sigma(\xi')d\xi' d\xi \\ & \le \sup_{\xi \in (-1,1)} \frac{\sigma(\xi)}{\mathrm{Re}\lambda + \sigma(\xi)} \|\psi\|_{X^\sigma} \|\zeta\|_{L^1} \end{aligned} $$

The last inequality shows that $Q_\lambda K$ depends continuously on $\zeta(\cdot) \in L$. Since $Q_\lambda K$ is an integral operator, and the set of bounded functions which vanish in a neighborhood of $\xi = 0$ is dense in $L$, it follows that $Q_\lambda K$ is a limit, in the uniform topology, of integral operators with bounded kernels. Hence $Q_\lambda K$ is weakly compact on $X^\sigma$. Q.E.D.

**Corollary 4.1.** Let $\lambda$ be a complex number satisfying $\mathrm{Re}\lambda > -\sigma_0$ and let $K$ be a regular collision operator. Then the operator $(\lambda - T_H)^{-1}K$ is weakly compact on $X^\sigma$.

**Proof.** Let $\lambda \in \mathbb{C}$ be such that $\mathrm{Re}\lambda > -\sigma_0$. To prove the weak compactness of the operator $(\lambda - T_H)^{-1}K$, it suffices to establish that the operators $\mathcal{P}_\lambda K$, $\mathcal{S}_\lambda K$ and $Q_\lambda K$ are weakly compact. This follows from Lemma 4.1. Q.E.D.

**Proof of Theorem 3.1.** Let $\lambda$ be a complex number such that $\mathrm{Re}\lambda > -\sigma_0$. Then according to Corollary 2.1, Problem (15) may be written in the form

$$ \psi = \mathcal{F}(\lambda)(\psi), \quad \psi^i = H(\psi^o), $$

where $\mathcal{F}(\lambda) = (\lambda - T_H)^{-1}K\mathcal{N}_f$. Hence, in order to prove that Problem (15) has a solution, it suffices to prove that the operator $\mathcal{F}(\lambda)$ has a fixed point in $\mathbb{B}_r^\sigma$.

It follows from Corollary 2.1 and Lemma 2.2 that the operator $\mathcal{F}(\lambda)$ is continuous.

We claim that the operator $\mathcal{F}(\lambda)$ is ws-compact on $X^\sigma$. Indeed, let $(\psi_n)_{n \in \mathbb{N}}$ be a weakly convergent sequence in $X^\sigma$. Hence the set $\Psi = \{\psi_n : n \in \mathbb{N}\}$ is a relatively weakly compact set of $X^\sigma$. Let $A$ be a measurable subset of $(-a, a) \times (-1, 1)$ with respect to the measure $\sigma(\xi)dxd\xi$. It follows from [12, Corollary 11, p. 294] that

$$ \lim_{|A| \to 0} \int_A |\psi(x, \xi)| dx \sigma(\xi) d\xi = 0 $$

uniformly for $\psi \in \Psi$ where $|A|$ denotes the measure of $A$ with respect to $\sigma(\xi)dxd\xi$. Using Eq. (11), we get

$$ |\int_A (\mathcal{N}_f \psi)(x, \xi) dx \sigma(\xi) d\xi| \le \int_A |\alpha(x, \xi)| dx d\xi + \beta \int_A |\psi(x, \xi)| dx \sigma(\xi) d\xi $$

and therefore

$$ \lim_{|A| \to 0} \int_A (\mathcal{N}_f \psi)(x, \xi) dx \sigma(\xi) d\xi = 0, $$

uniformly for $\psi \in \Psi$. Using [12, Corollary 11, p. 294], we conclude that the set $\{\mathcal{N}_f \psi_n : n \in \mathbb{N}\}$ is relatively weakly compact on $X^\sigma$. Now, using Corollary 4.1 and the fact that $X^\sigma$ possesses the
Dunford-Pettis property (cf. Remark 2.3(2)), we infer that the set $\mathcal{F}(\lambda)(\Psi)$ is relatively compact, which proves our claim.

We shall show that for suitable $\lambda \in \mathbb{C}$, the operator $\mathcal{F}(\lambda)$ maps $\mathbb{B}_r^\sigma$ into itself. Note that, according to Lemma 2.2, there exist a constant $\beta > 0$ and a function $\alpha \in \mathcal{X}$ such that

$$f(x, \xi, \psi(x, \xi)) \leq \frac{\alpha(x, \xi)}{\sigma(\xi)} + \beta|\psi(x, \xi)| \quad \text{a.e. } (x, \xi) \in \Omega \text{ and for all } \psi \in \mathcal{X}^{\sigma}. \quad (17)$$

Let first observe that the function $M: (-\sigma_0, +\infty) \to \mathbb{R}$, $x \mapsto \frac{1}{1 - (\mathbf{m}(x, \sigma_0, 1))^2}$ is strictly decreasing, so for all $\lambda \in \mathbb{C}$ such that $\text{Re}\lambda > \frac{\ln 2}{4a} - \sigma_0$, we have

$$M(\text{Re}\lambda) \leq M(\frac{\ln 2}{4a} - \sigma_0) = 2.$$

Clearly, for all $\lambda \in \mathbb{C}$ such that $\text{Re}\lambda > \frac{\ln 2}{4a} - \sigma_0$, the operator $\lambda - T_H$ is invertible, so for such $\lambda$ and all $\psi$ in $\mathbb{B}_r^\sigma$ we have

$$\begin{aligned}
\|\mathcal{F}(\lambda)(\psi)\|_{\mathcal{X}^{\sigma}} &= \|(\lambda - T_H)^{-1} K \mathcal{N}_f(\psi)\|_{\mathcal{X}^{\sigma}} \\
&\leq \|(\lambda - T_H)^{-1}\|_{\mathcal{L}(\mathcal{X}, \mathcal{X}^{\sigma})} \|K\|_{\mathcal{L}(\mathcal{X}^{\sigma}, \mathcal{X})} \|\mathcal{N}_f(\psi)\|_{\mathcal{X}^{\sigma}} \\
&\leq 3(\|\alpha\|_{\mathcal{X}} + \beta r) \|K\|_{\mathcal{L}(\mathcal{X}^{\sigma}, \mathcal{X})} M(\text{Re}\lambda) \\
&\leq 6 \|\alpha\|_{\mathcal{X}} \|K\|_{\mathcal{L}(\mathcal{X}^{\sigma}, \mathcal{X})} + 6\beta \|K\|_{\mathcal{L}(\mathcal{X}^{\sigma}, \mathcal{X})} r \\
&\leq 6 \|\alpha\|_{\mathcal{X}} \|K\|_{\mathcal{L}(\mathcal{X}^{\sigma}, \mathcal{X})} + \frac{r}{2}, \quad (\text{use the hypothesis (A3)}).
\end{aligned}$$

So, in order to show that $\mathcal{F}(\lambda)$ maps $\mathbb{B}_r^\sigma$ into itself, it suffices to take $r \geq 12 \|\alpha\|_{\mathcal{X}} \|K\|_{\mathcal{L}(\mathcal{X}^\sigma, \mathcal{X})}$. Therefore, for all $r \geq 12 \|\alpha\|_{\mathcal{X}} \|K\|_{\mathcal{L}(\mathcal{X}^\sigma, \mathcal{X})}$ and $\lambda \in \mathbb{C}$ such that $\text{Re}\lambda \geq \frac{\ln 2}{4a} - \sigma_0$, we have $\mathcal{F}(\lambda)(\mathbb{B}_r^\sigma) \subset \mathbb{B}_r^\sigma$.

In order to apply Theorem 2.1 to the operator $\mathcal{F}(\lambda)$ on $\mathbb{B}_r^\sigma$, it suffices to show that the set $\mathcal{F}(\lambda)(\mathbb{B}_r^\sigma)$ is weakly compact. This follows from the fact that $\mathcal{N}_f$ maps bounded sets into bounded sets and Corollary 4.1.

Accordingly, by Theorem 2.1, the operator $\mathcal{F}(\lambda)$ has at least a fixed point in $\mathbb{B}_r^\sigma$, which completes the proof. Q.E.D.

**Proof of Corollary 3.2.** Arguing as in the proofs of Theorems 3.1 and Corollary 3.1 we infer that, for all real number $\lambda$ such that $\lambda > \bar{\lambda} := \frac{\ln 2}{4a} - \sigma_0$ and for all $r \geq 12 \|\alpha\|_{\mathcal{X}} \|K\|_{\mathcal{L}(\mathcal{X}^\sigma, \mathcal{X})}$, the operator $\mathcal{F}(\lambda)$ maps $\mathbb{B}_r^{\sigma,+}$ into itself. On the other hand, for $\lambda \geq \bar{\lambda}$, using the positivity of $(\lambda - T_H)^{-1}$ and $K$, one sees that

$$\mathcal{F}(\lambda)(\psi) \geq \rho(\lambda - T_0)^{-1} K(\psi_0) \geq 0 \text{ and } (\lambda - T_0)^{-1} K(\psi_0) \neq 0 \text{ for all } \psi \in \mathbb{B}_r^{\sigma,+},$$

where $(\lambda - T_0)^{-1}$ is nothing else but the operator $Q_\lambda$. Accordingly, for all $\psi \in \mathbb{B}_r^{\sigma,+}$, we have
$$ \|\mathcal{F}(\lambda)(\psi)\|_{X^{\sigma}} \geq \rho \|(\lambda - T_H)^{-1}(\psi_0)\|_{X^{\sigma}} > 0 $$

and consequently

$$ \inf \left\{ \|\mathcal{F}(\lambda)(\psi)\|_{X^{\sigma}} : \psi \in \mathbb{B}_{r}^{\sigma,+} \right\} > 0. $$

Hence, for $\lambda \geq \bar{\lambda}$, one can define the operator

$$ G_{\lambda}(\psi) = r \frac{\mathcal{F}(\lambda)(\psi)}{\|\mathcal{F}(\lambda)(\psi)\|_{X^{\sigma}}} \quad \text{for all} \quad \psi \in \mathbb{B}_{r}^{\sigma,+}. $$

Using the same arguments as in the proof of the Theorem 3.1, we see that $G_{\lambda}$ is continuous, ws-compact and maps $\mathbb{B}_{r}^{\sigma,+}$ into itself. Further, the use of Corollary 4.1 insures that $G_{\lambda}(\mathbb{B}_{r}^{\sigma,+})$ is weakly compact. So, applying Theorem 2.1 we conclude that $G_{\lambda}$ has a fixed point $\psi^* \in \mathbb{B}_{r}^{\sigma,+}$ satisfying $\|\psi^*\| = r$. Putting $\eta = \frac{\|G_{\lambda}(\psi^*)\|_{X^{\sigma}}}{r}$, we get $G_{\lambda}(\psi^*) = \eta^{-1}\psi^*$. Consequently, $\psi^* \in D(T_H) \cap \mathbb{B}_{r}^{\sigma,+}$ and

$$ \xi \frac{\partial \psi^*}{\partial x}(x, \xi) + \sigma(\xi) \psi^*(x, \xi) + \lambda \psi^*(x, \xi) = \eta \int_{-1}^{1} k(x, \xi, \xi') f(x, \xi', \psi^*(x, \xi')) d\xi'. $$

This completes the proof.

Q.E.D.

# 5 Periodic boundary conditions

It should be noticed that, when dealing with perfect periodic boundary conditions, i.e.

$$ \left\{ \begin{array}{l} H : L_1^o \oplus L_2^o \to L_1^i \oplus L_2^i \\ H \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} := \begin{pmatrix} 0 & H_{12} \\ H_{21} & 0 \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \end{array} \right. \qquad (18) $$

where

$$ \left\{ \begin{array}{l} H_{12} : L_2^o \to L_1^i \\ \psi(-a, \xi) \to \psi(a, \xi), \end{array} \right. \qquad \left\{ \begin{array}{l} H_{21} : L_1^o \to L_2^i \\ \psi(-a, \xi) \to \psi(a, \xi) \end{array} \right. $$

slight modifications shows that the results of Section 3 hold also true. Indeed, for $\text{Re}\lambda > -\sigma_0$, similar arguments and calculations as in Section 2 give

$$ (\lambda - T_H)^{-1} = J_{\lambda} + Q_{\lambda}, $$

where $J_{\lambda}$ is the bounded operator defined by

$$ \left\{ \begin{array}{l} J_{\lambda} : X \to X^{\sigma}, \quad J_{\lambda}\varphi := \chi_{(-1,0)}(\xi)J_{\lambda}^{-}\varphi + \chi_{(0,1)}(\xi)J_{\lambda}^{+}\varphi \quad \text{with} \\ (J_{\lambda}^{+}\varphi)(x, \xi) = \frac{1}{(1 - \mathbf{m}(\lambda, \sigma(\xi), \xi))|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a + x - y)}{|\xi|}\right) \varphi(y, \xi)dy, \quad \xi > 0, \\ (J_{\lambda}^{-}\varphi)(x, \xi) = \frac{1}{(1 - \mathbf{m}(\lambda, \sigma(\xi), \xi))|\xi|} \int_{-a}^{a} \exp\left(-\frac{(\lambda + \sigma(\xi))(2a - x + y)}{|\xi|}\right) \varphi(y, \xi)dy, \quad \xi < 0. \end{array} \right. $$
It is clear that $J_λ$ possesses the same structure as the operator $P_λ$. So, we have

**Lemma 5.1.** Let $λ$ be a complex number such that $\text{Re}\lambda > -\sigma_0$ and assume that the condition (A1) holds true. Then the operators $J_λ$, and $Q_λ$ belong to $\mathcal{L}(X, X^σ)$ and satisfy

$$ \|J_λ\|_{\mathcal{L}(X, X^σ)} \le \frac{1}{1 - (\mathbf{m}(\text{Re}\lambda, \sigma_0, 1))} \sup_{\xi \in (-1, 1)} \frac{\sigma(\xi)}{\text{Re}\lambda + \sigma(\xi)}, $$

and

$$ \|Q_λ\|_{\mathcal{L}(X, X^σ)} \le \sup_{\xi \in (-1, 1)} \frac{\sigma(\xi)}{\text{Re}\lambda + \sigma(\xi)}. $$

Moreover, $D(T_H) \subseteq X^σ$ with continuous embedding and, for $λ \in \mathbb{C}$ such that $\text{Re}\lambda > -\sigma_0$, the operator $(λ - T_H)$ is invertible and its inverse is given by

$$ (λ - T_H)^{-1} = J_λ + Q_λ \qquad (19) $$

and satisfies

$$ \|(λ - T_H)^{-1}\|_{\mathcal{L}(X, X^σ)} \le \frac{2}{1 - (\mathbf{m}(\text{Re}\lambda, \sigma_0, 1))}. \qquad (20) $$

The proof is similar to that of Lemma 2.1 and Corollary 2.1 so it is dropped.

Let $λ$ be a complex number such that $\text{Re}\lambda + \sigma_0 > 0$ and assume that the condition (A1) holds true. We know from Lemma 4.1 that, if the collision operator $K$ is regular, then the operator $Q_λ K$ is weakly compact on $X^σ$. Moreover, since $J_λ$ has the same structure as the operator $P_λ$, a similar reasoning as in the proof of Lemma 4.1 ensures that $J_λ K$ is also weakly compact on $X^σ$. Hence, we have the following result

**Lemma 5.2.** Let $λ$ be a complex number satisfying $\text{Re}\lambda > -\sigma_0$ and assume that the condition (A1) holds true. If $K$ is a regular collision operator. Then the operators $(λ - T_H)^{-1} K$ is weakly compact on $X^σ$.

Let us now introduce the following assumption

$$ (A4) \quad \beta \|K\|_{\mathcal{L}(X^σ, X)} \le \frac{1}{8}. $$

**Theorem 5.1.** Assume that the conditions (A1), (A2) and (A4) are satisfied. Then, for all $λ \in \mathbb{C}$ and $r > 0$ satisfying

$$ \text{Re}\lambda \ge \frac{\ln(2)}{2a} - \sigma_0 \quad \text{and} \quad r \ge 8 \|K\|_{\mathcal{L}(X^\sigma, X)} \|\alpha\|_{X}, \qquad (21) $$

the problem

$$ \begin{cases} \xi \frac{\partial \psi}{\partial x}(x, \xi) + \sigma(\xi)\psi(x, \xi) + \lambda \psi(x, \xi) = \int_{-1}^{1} \kappa(x, \xi, \xi') f(x, \xi', \psi(x, \xi')) d\xi' \\ \psi^i = H(\psi^o), \end{cases} \qquad (22) $$

has at least one solution on $\mathbb{B}_r^σ$.
**Proof.** It is clear that, for all $\lambda \in \mathbb{C}$ such that $\text{Re}\lambda \ge \frac{\ln(2)}{2a} - \sigma_0$, Problem (22) may be written as a fixed point problem (use Lemma 5.1),

$$ \psi = \mathcal{P}(\lambda)(\psi), \qquad \psi^i = \mathbf{H}(\psi^o), $$

where $\mathcal{P}(\lambda) = (\lambda - T_H)^{-1}KN_f$. So, in order to prove that Problem (22) has a solution, it suffices to prove that the operator $\mathcal{P}(\lambda)$ has a fixed point in $\mathbb{B}_r^\sigma$.

By Lemmas 2.2 and 5.1, the operator $\mathcal{P}(\lambda)$ is continuous and the use of Lemma 5.2 implies that it is weakly compact. Further, using Lemma 5.2 and the fact that the space $X^\sigma$ possesses the Dunford-Pettis property, we infer that $\mathcal{P}(\lambda)$ is ws-compact.

It is clear that the function $t \mapsto \frac{1}{1 - \mathbf{m}(t, \sigma_0, 1)}$ is strictly decreasing on $(-\sigma_0, +\infty)$ and, for all $\lambda \in \mathbb{C}$ such that $\text{Re}\lambda > \frac{\ln 2}{2a} - \sigma_0$, we have $\frac{1}{1 - \mathbf{m}(\text{Re}\lambda, \sigma_0, 1)} \le \frac{1}{1 - \mathbf{m}(\frac{\ln 2}{2a} - \sigma_0, \sigma_0, 1)} = 2$. Hence, for such $\lambda$, the operator $\lambda - T_H$ is invertible and, for all $\psi$ in $\mathbb{B}_r^\sigma$, we have

$$ \begin{aligned} \|\mathcal{P}(\lambda)(\psi)\|_{X^\sigma} &\le \|(\lambda - T_H)^{-1}\|_{\mathcal{L}(X, X^\sigma)}\|K\|_{\mathcal{L}(X^\sigma, X)}\|\mathcal{N}_f(\psi)\|_{X^\sigma} \\ &\le \frac{2(\|\alpha\|_X + \beta r)\|K\|_{\mathcal{L}(X^\sigma, X)}}{1 - \mathbf{m}(\text{Re}\lambda, \sigma_0, 1)} \quad \text{(use the estimate (20))} \\ &\le 4 \|\alpha\|_X \|K\|_{\mathcal{L}(X^\sigma, X)} + \frac{r}{2}. \quad \text{(use the hypoythesis (A4))}. \end{aligned} $$

So, for all $r \ge 8 \|\alpha\|_X \|K\|_{\mathcal{L}(X^\sigma, X)}$, we have $\|\mathcal{P}(\lambda)(\psi)\|_{X^\sigma} \le r$ and therefore $\mathcal{P}(\lambda)$ maps $\mathbb{B}_r^\sigma$ into itself.

Accordingly, the operator $\mathcal{P}(\lambda)$ satisfies all the hypotheses of Theorem 2.1 on the convex set $\mathbb{B}_r^\sigma$. Accordingly, $\mathcal{P}(\lambda)$ possesses a fixed point in $\mathbb{B}_r^\sigma$, which concludes the proof. Q.E.D.

The existence of positive solutions to the boundary value problem (22) may be discussed in the same way as in Section 3. It is clear that, for any real number $\lambda > -\sigma_0$, the operator $J_\lambda$ is positive in the lattice sense. Hence, as in Section 3 we infer that, for such $\lambda$, $(\lambda - T_H)^{-1}$ is a positive operator. If further the hypotheses of Theorem 5.1 are satisfied, $\mathcal{N}_f(X_+^\sigma) \subseteq X_+^\sigma$ and $K$ is positive, then $(\lambda - T_H)^{-1}KN_f$ leaves the positive cone $X^{\sigma,+}$ invariant. Now replacing $\mathbb{B}_r^\sigma$ by $\mathbb{B}_r^{\sigma,+}$ and arguing as in the proof of Theorem 5.1 we get

**Corollary 5.1.** Let the hypotheses of Theorem 5.1 be satisfied. Further, if $\mathcal{N}_f(X^{\sigma,+}) \subseteq X^{\sigma,+}$. Then, for all reals $\lambda$ and $r$ satisfying (21), the problem (22) has at least one solution on $\mathbb{B}_r^{\sigma,+}$.

For the existence of nontrivial positive solution to Problem (22), we have

**Corollary 5.2.** Assume that the hypotheses of Corollary 5.1 are satisfied. If there is $\rho > 0$ and $0 \ne \psi_0 \in \mathbb{B}_r^{\sigma,+}$ such that

(i) $\psi_0 \notin N(K)$ where $N(K)$ denotes the null space of $K$,

(ii) $(\mathcal{N}_f)(x, \xi) \ge \rho\psi_0(x, \xi)$ for all $\psi \in \mathbb{B}_r^{\sigma,+}$.
Then, for all reals $\lambda$ and $r$ satisfying (21), there exist $\eta > 0$ such that the problem

$$ egin{cases} \xi \frac{\partial \psi}{\partial x}(x, \xi) + \sigma(\xi)\psi(x, \xi) + \lambda\psi(x, \xi) = \eta \int_{-1}^{1} k(x, \xi, \xi')f(x, \xi', \psi(x, \xi'))d\xi', \\ \psi^i = H(\psi^o), \end{cases} $$

has at least one solution $\psi^* \in \mathbb{B}_r^{\sigma,+}$ satisfying $\|\psi^*\|_{X^\sigma} = r$.

**Proof.** The proof may be modeled in the same way as that of Corollary 3.2, so it is omitted. Q.E.D.

**Concluding remark.**

In this paper we deal with both perfect reflecting and perfect periodic boundary conditions, i.e. $\|Hu\|_{L^i} = \|u\|_{L^o}$ and $\|Hu\|_{L^i} = \|u\|_{L^o}$. However, this hypothesis is not necessary; it was used in order to simplify the calculations (which are tedious). Actually, our analysis works also for reflecting and periodic boundary conditions satisfying $\|Hu\|_{L^i} = \alpha \|u\|_{L^o}$ and $\|Hu\|_{L^i} = \alpha \|u\|_{L^o}$ where $\alpha \in (0,1)$ denotes an accommodation coefficient. $\square$

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