Krasnoselskii type theorems in product Banach spaces and applications to systems of nonlinear transport equations and mixed fractional differential equations

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(original), Fixed point theory, ISI/JCR, paper

Abstract

In this paper, we use a new technique for the treatment of systems based on the advantage of vector-valued norms and of the weak topology. We first present vector versions of the Leray-Schauder alternative and then some Krasnoselskii type fixed point theorems for a sum of two mappings. Applications are given to a system of nonlinear transport equations, and systems of mixed fractional differential equations.

Authors

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Sana Hadj Amor
Department of Mathematics, LR 11 ES 35, Higher School of Science and Technology, University of Sousse, Tunisia

Abdelhak Traiki
Department of Mathematics, LR 11 ES 35, Higher School of Science and Technology, University of Sousse, Tunisia

Keywords

Krasnoselskii fixed point theorem for a sum of operators; weak topology; generalized contraction; product Banach space; vector-valued norm; system of nonlinear transport equations; convergent to zero matrix; fractional integral.

Paper coordinates

Sana Hadj Amor, Radu Precup and Abdelhak Traiki, Krasnoselskii type theorems in product Banach spaces and applications to systems of nonlinear transport equations and mixed fractional differential equations, Fixed Point Theory, vol 23 (2022) no. 1, 105-126, https://doi.org/10.24193/fpt-ro.2022.1.07

PDF

https://doi.org/10.24193/fpt-ro.2022.1.07

About this paper

Journal

Fixed Point Theory

Publisher Name

Department of Mathematics, ”Babeș-Bolyai” Cluj-Napoca, Romania

DOI

https://doi.org/10.24193/fpt-ro.2022.1.07

Print ISSN

15835022

Online ISSN

20669208

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Krasnoselskii type theorems in product Banach spaces and applications to systems of nonlinear transport equations and mixed fractional differential equations

Sana HADJ AMOR Sana Hadj Amor, Department of Mathematics , Radu PRECUP Radu Precup, Department of Mathematics, Babeş-Bolyai University
Cluj-Napoca, Romania r.precup@math.ubbcluj.ro and Abdelhak TRAIKI Abdelhak Traiki, Department of Mathematics

Abstract.

Mathematics Subject Classification: 47B38, 47H09, 47H08, 47H10

Keywords: Krasnoselskii fixed point theorem for a sum of operators, weak topology, generalized contraction, product Banach space, vector-valued norm, system of nonlinear transport equations, convergent to zero matrix, fractional integral.

1. Introduction

The classical Banach contraction principle is a very useful tool in nonlinear analysis with many applications to integral and differential equations, optimization theory, and other topics. There are many generalizations of this result, one of them is due to A.I. Perov [15] and consists in replacing usual metric spaces by spaces endowed with vector-valued metrics. According to this result, if a space $X$ is a Cartesian product $X=X_{1}\times\cdots X_{n}$ and each component $X_{i}$ is a complete metric space with the metric $d_{i},$ then instead of endowing $X$ with some metric $\delta$ generated by $d_{1},\cdots,d_{n},$ for instance any one of the metrics

	$\displaystyle\delta^{p}(x,y)$	$\displaystyle=$	$\displaystyle(\sum_{i=1}^{n}d_{i}(x_{i},y_{i})^{p})^{\frac{1}{p}}~{}~{}~{}~{}(% 1\leq p<\infty),$
	$\displaystyle\delta^{\infty}(x,y)$	$\displaystyle=$	$\displaystyle\max\{d_{1}(x_{1},y_{1}),\cdots,d_{n}(x_{n},y_{n})\},$

and applying Banach’s contraction principle in the complete metric space $(X,\delta),$ better results are obtained if one considers the vector-valued metric

d(x,y)=(d_{1}(x_{1},y_{1}),\cdots,d_{n}(x_{n},y_{n}))^{T}

and one requires a generalized contraction (in Perov’s sense) condition in the vector-matrix form

d(F(x),F(y))\leq Ad(x,y),\ \ \ x,y\in X,

where $A$ is a square matrix of type $n\times n$ with nonnegative elements having the spectral radius $\rho(A)<1.$ This approach is very fruitful for the treatment of systems of equations arising from various fields of applied mathematics. The advantage of using vector-valued metrics and norms instead of usual scalar ones, in connexion with several techniques of nonlinear analysis, has been pointed out in [20]. Roughly speaking, by a vector approach it is allowed that the component equations of a system behave differently, and thus more general results can be obtained.

In his Ph.D. thesis [22], A. Viorel used generalized contractions in Perov’s sense and gave a vector version of Krasnoselskii’s fixed point theorem [12] for a some of two operators $A$ and $B,$ where $A$ is a compact map and $B$ is a generalized contraction. Applications were given to systems of semi-linear evolution equations. Viorel’s result was extended for multi-valued mappings in [16]. The proofs of these results combine a vector version of the contraction principle (Perov and Perov-Nadler theorems, respectively) with Schauder’s fixed point theorem for maps that are compact with respect to the strong topology.

Alternatively, instead of the strong topology of a Banach space, one may think to use the weak topology. Fixed point results involving the weak topology have been obtained by many authors in the last decades (see, e.g., [2, 4, 5, 6, 8, 19]). The purpose of this paper is to extend the Leray-Schauder and Krasnoselskii’s fixed point theorems to sums of generalized contractions and compact maps with respect to the weak topology. Note that our technique can also be used to give vector versions of the results in [3]. Next, motivated by the papers [6], [13] and [11], we give applications of the theoretical results to a system of transport equations, and a system of mixed fractional differential equations.

The paper is organized as follows: In Section 2, we present some notations and preliminary facts that we will need in what follows. In Section 3, we first give a vector version of the Leray-Schauder fixed point theorem for weakly sequentially continuous mappings and then we extend Viorel’s result by using the weak topology. In Sections 3 and 4, we apply these results to a system of transport equations and a system of mixed fractional differential equations.

2. Preliminaries

In this section, we recall from the literature some notations, definitions, and auxiliary results which will be used throughout this paper.

By a generalized metric space we mean a set $X$ endowed with a vector-valued metric $d,$ that is a mapping $d:X\times X\rightarrow\mathbb{R}_{+}^{n}$ which satisfies all the axioms of a usual metric, with the inequality $\leq$ understood to act componentwise. In such a space, the notions of a Cauchy sequence, convergent sequence, completeness, open and closed set, are defined in a similar way to that of the corresponding notions in a usual metric space.

A mapping $F:X\longrightarrow X,$ where $X$ is a generalized metric space with the vector-valued metric $d$ is said to be a generalized contraction, or a Perov contraction, if there exists a matrix (called Lipschitz matrix) $M\in\mathcal{M}_{n}(\mathbb{R}_{+})$ such that $M^{k}$ tends to the zero matrix as $k\rightarrow\infty$ and

d(F(x),F(y))\leq Md(x,y)\quad\text{ for all }\quad x,y\in X.

Here the vector $d\left(x,y\right)$ and $d\left(F\left(x\right),F\left(y\right)\right)$ are seen like all the vectors in $\mathbb{R}^{n}$ as column matrices. Notice that a matrix $M$ as above is called to be convergent to zero, and that this property is equivalent (see [18]) to each one of the following three properties:

(a)

$I-M$ is non-singular and $(I-M)^{-1}=I+M+M^{2}+\cdots.$ (Here $I$ is the unit matrix of size $n$ ).
(b)

$|\lambda|<1$ for every $\lambda\in C$ with $det(M-\lambda I)=0$ .
(c)

$I-M$ is non-singular and $(I-M)^{-1}$ has nonnegative elements.

Notice that in view of $(c)$ , a vector-matrix inequality like $x\leq Mx$ for a nonnegative vector-column $x=(x_{1},\dots,x_{n})^{T}\in\mathbb{R}_{+}^{n}$ first yields $(I-M)x\leq 0,$ and then $x\leq(I-M)^{-1}0,$ whence $x=0_{\mathbb{R}^{n}}.$

Recall Perov’s fixed point theorem which states that any generalized contraction $F$ on a complete generalized metric space $\left(X,d\right)$ has a unique fixed point $x^{\ast},$ and for each $x\in X$ one has

d(F^{k}(x),x^{\ast})\leq M^{k}(I-M)^{-1}d(x,F(x))~{}~{}~{}\text{ for all }~{}k% \in\mathbb{N}.

Notice that, under the assumptions of Perov’s theorem, and if $J$ is the identity mapping of $X,$ the mapping $J-F$ is bijective and $(J-F)^{-1}$ is continuous.

By a vector-valued norm on a linear space $X$ we mean a mapping $\left\|\cdot\right\|:X\rightarrow\mathbb{R}_{+}^{n}$ which satisfies the usual axioms of a norm, with the inequality $\leq$ understood to act componentwise. Any linear space $X$ endowed with a vector-valued norm $\left\|\cdot\right\|$ is a generalized metric space with respect to the vector-valued metric $\ d\left(x,y\right)=\left\|x-y\right\|.$ In case that $\left(X,d\right)$ is complete, we say that $X$ is a generalized Banach space.

In particular, if $X=X_{1}\times\cdots\times X_{n},$ where $(X_{i},\|.\|_{i})$ is a Banach space for $i=1,\cdots,n,$ then $X$ is a Banach space with respect to the norm

\left|x\right|=\left\|x_{1}\right\|_{1}+\cdot\cdot\cdot+\left\|x_{n}\right\|_{% n},

and a generalized Banch space with respect to the vector-valued norm

\|x\|=\left(\left\|x_{1}\right\|_{1},\cdot\cdot\cdot,\left\|x_{n}\right\|_{n}% \right)^{T},

where $x=\left(x_{1},\cdot\cdot\cdot,x_{n}\right).$ On such a space one can define a vector measure of weak noncompactness by

\omega\left(V\right)=\left(\omega_{1}\left(V_{1}\right),\cdot\cdot\cdot,\omega% _{n}\left(V_{n}\right)\right)^{T}\text{ }

for $V=V_{1}\times\cdot\cdot\cdot\times V_{n}$ and any bounded sets $V_{i}\subset X_{i},$ $i=1,\cdot\cdot\cdot,n,$ where $\omega_{i}$ is the De Blasi measure of weak noncompactness on $X_{i}$ (see [8]). Recall that, if $\left(Y,\left\|.\right\|_{Y}\right)$ is any Banach space, the De Blasi weak measure of noncompactness $\omega_{Y}\left(C\right)$ of any bounded set $C\subset Y$ is given by

\omega_{Y}(C)=\inf\left\{r>0:\text{ there is a weakly compact set }K\subset Y% \text{ such that }C\subset K+\overline{B}_{Y}(0,r)\right\},

where $\overline{B}_{Y}(0,r)=\left\{y\in Y:\left\|y\right\|_{Y}\leq r\right\}.$ For completeness we recall some properties of $\omega_{Y}$ needed below (for the proofs we refer to [1]). Let $C_{1},C_{2}\subset Y$ be bounded. Then

(i)

Monotonicity : If $C_{1}\subset C_{2},$ then $\omega_{Y}(C_{1})\leq\omega_{Y}(C_{2}).$
(ii)

Regularity: $\omega_{Y}(C_{1})=0$ if and only if $C_{1}$ is relatively weakly compact.
(iii)

Invariance under closure: $\omega_{Y}(\overline{C_{1}^{\omega}})=\omega_{Y}(C_{1}),$ where $\overline{C_{1}^{\omega}}$ is the weak closure of $C_{1}.$
(iv)

Semi-homogeneity : $\omega_{Y}(\lambda C_{1})=|\lambda|\omega_{Y}(C_{1})$ for all $\lambda\in\mathbb{R}.$
(v)

Invariance under passage to the convex hull : $\omega_{Y}(conv(C_{1}))=\omega_{Y}(C_{1}).$
(vi)

Semi-additivity : $\omega_{Y}(C_{1}+C_{2})\leq\omega_{Y}(C_{1})+\omega_{Y}(C_{2}).$
(vii)

Cantor’s intersection property: If $(C_{k})_{k\geqslant 1}$ is a decreasing sequence of nonempty, bounded and weakly closed subsets of $Y$ with $\lim\limits_{k\rightarrow+\infty}\omega_{Y}(C_{k})=0,$ then $\bigcap\limits_{k=1}^{\infty}C_{k}\neq\emptyset$ and $\omega_{Y}\left(\bigcap\limits_{k=1}^{\infty}C_{k}\right)=0,$ i.e. $\bigcap\limits_{k=1}^{\infty}C_{k}$ is relatively weakly compact.

Throughout this paper, for a mapping $F:D\rightarrow X,$ where $X$ is the Cartesian product $X_{1}\times\cdot\cdot\cdot\times X_{n}$ of $n$ Banach spaces and $D=D_{1}\times\cdot\cdot\cdot\times D_{n},$ for $D_{i}\subset X_{i}$ a weakly closed subset of $X_{i}$ $\left(i=1,\cdot\cdot\cdot,n\right),$ we shall say that $F$ is sequentially weakly continuous if for any sequence $\left(x^{k}\right)\subset D$ such that $x_{i}^{k}\rightarrow x_{i}$ weakly in $X_{i},$ $i=1,\cdot\cdot\cdot,n,$ one has $F_{i}\left(x^{k}\right)\rightarrow F_{i}\left(x\right)$ weakly in $X_{i}$ for $i=1,\cdot\cdot\cdot,n.$

3. Fixed point results

We first state a useful result in terms of the vector measure of weak noncompactness.

Proposition 3.1.

Let $\left(X_{i},\left\|\cdot\right\|_{i}\right),$ $i=1,\cdot\cdot\cdot,n$ be Banach spaces, and let $X=X_{1}\times\cdot\cdot\cdot\times X_{n}.$ If $F:X\rightarrow X$ is weakly sequentially continuous and there is a matrix $M\in\mathcal{M}_{n}(\mathbb{R}_{+})$ such that

(3.1)

\left\|F\left(x\right)-F\left(y\right)\right\|\leq M\left\|x-y\right\|\ \ \ % \text{for all \ }x,y\in X,

then for any bounded sets $V_{i}\subset X_{i},$ $i=1,\cdot\cdot\cdot,n$ and $V=V_{1}\times\cdot\cdot\cdot\times V_{n},$ one has

(3.2)

\omega(F(V))\leq M\omega(V).

Proof.

For each $i\in\{1,\dots,n\}$ , denote $\alpha_{i}=\omega_{i}(V_{i}).$ Then for any $\varepsilon_{i}>0,$ there exists a weakly compact subset $K_{i}$ of $X_{i}$ such that $V_{i}\subset K_{i}+\overline{B}_{X_{i}}(0,\alpha_{i}+\varepsilon_{i}).$ Hence, for every $x=(x_{1},\cdots,x_{n})\in V,$ there is an $y=(y_{1},\cdots,y_{n})\in K=K_{1}\times\cdot\cdot\cdot\times K_{n}$ such that $\|x_{i}-y_{i}\|_{i}\leqslant\alpha_{i}+\varepsilon_{i}$ for $i=1,\dots,n.$ Let $F=\left(F_{1},\cdot\cdot\cdot,F_{n}\right),$ where $F_{i}:X\rightarrow X_{i}$ and let $M=(m_{ij})_{1\leq i,j\leq n}.$ Then using (3.1) gives

(3.3)

\|F_{i}(x)-F_{i}(y)\|_{i}\leq\sum\limits_{j=1}^{n}m_{ij}\|x_{j}-y_{j}\|_{j}% \leq\sum\limits_{j=1}^{n}m_{ij}(\alpha_{j}+\varepsilon_{j}).

As a result, $F_{i}(x)-F_{i}(y)\in\overline{B}_{X_{i}}(0,\sum\limits_{j=1}^{n}m_{ij}(\alpha_% {j}+\varepsilon_{j}))$ for $i=1,\dots,n.$ Hence,

F_{i}(x)\in F_{i}(K)+\overline{B}_{X_{i}}\left(0,\sum\limits_{j=1}^{n}m_{ij}(% \alpha_{j}+\varepsilon_{j})\right),\text{\ \ }i=1,\dots,n.

Consequently,

(3.4)

F_{i}(V)\subset F_{i}(K)+\overline{B}_{X_{i}}\left(0,\sum\limits_{j=1}^{n}m_{% ij}(\alpha_{j}+\varepsilon_{j})\right),\text{ \ }i=1,\dots,n.

Since $F_{i}$ is weakly sequentially continuous and $K$ is weakly compact, we have $F_{i}:K\longrightarrow X_{i}$ is weakly continuous. Thus, $F_{i}(K)$ is weakly compact. As a result

(3.5)

\omega_{i}\left(F_{i}(V)\right)\leq\sum\limits_{j=1}^{n}m_{ij}(\alpha_{j}+% \varepsilon_{j}),\ \ \text{ }i=1,\dots,n.

Letting $\varepsilon_{i}\rightarrow 0$ for all $i$ yields

(3.6)

\omega_{i}\left(F_{i}(V)\right)\leq\sum\limits_{j=1}^{n}m_{ij}\alpha_{j}=\sum% \limits_{j=1}^{n}m_{ij}\omega_{j}(V_{j}),\ \ \ i=1,\cdot\cdot\cdot,n,

or equivalently, in the vector form, (3.2).

We now give some vector versions of the Leray-Schauder fixed point theorem for weakly sequentially continuous mappings.

Theorem 3.1.

Let $\left(X_{i},\left\|\cdot\right\|_{i}\right),$ $i=1,\cdot\cdot\cdot,n$ be Banach spaces. For each $i\in\left\{1,\cdot\cdot\cdot,n\right\},$ consider a nonempty closed and convex set $\Omega_{i}\subset X_{i}$ and a weakly open subset $U_{i}$ of $\Omega_{i}$ with $0\in U_{i}$ such that $\overline{U_{i}^{\omega}}$ is a weakly compact subset of $\Omega_{i}.$ Let $\Omega=\Omega_{1}\times\cdot\cdot\cdot\times\Omega_{n},$ $D=\overline{U_{1}^{\omega}}\times\cdot\cdot\cdot\times\overline{U_{n}^{\omega}},$ and $F:D\rightarrow\Omega$ a weakly sequentially continuous mapping . Then, either

(i):: $F$ has a fixed point, or
(ii):: there exist $i\in\left\{1,\cdot\cdot\cdot,n\right\},$ a point $x=\left(x_{1},\cdot\cdot\cdot,x_{n}\right)\in D$ with $x_{i}\in\partial_{\Omega_{i}}U_{i}=\overline{U_{i}^{\omega}}\setminus U_{i},$ and a number $\lambda\in(0,1)$ with $x=\lambda F(x).$

Proof.

Suppose (ii) does not hold. Let $\Sigma$ be the set defined by

\Sigma=\{x\in D:\ x=\lambda F(x)\ ~{}\text{for some}~{}\lambda\in[0,1]\}.

The set $\Sigma$ is non-empty because $0\in D$ . We will show that $\Sigma$ is weakly compact. First we show that $\Sigma$ is weakly sequentially closed. For that, let $(x_{n})_{n}$ be a sequence of $\Sigma$ such that $x_{n}\rightarrow x$ weakly. Clearly $x\in D.$ For all $n\in\mathbb{N},$ there exists a $\lambda_{n}\in[0,1]$ such that $x_{n}=\lambda_{n}F(x_{n}).$ Since $\lambda_{n}\in[0,1],$ we can extract a subsequence $(\lambda_{n_{j}})_{j}$ such that $\lambda_{n_{j}}\rightarrow\lambda\in[0,1].$ Then since $F$ is weakly sequentally continuous, one has $\lambda_{n_{j}}F(x_{n_{j}})\rightarrow\lambda F(x)$ weakly. Hence $x=\lambda F(x),$ that is $x\in\Sigma.$ Next we prove that $\Sigma=\overline{\Sigma^{\omega}}.$ Obviously $\Sigma\subset\overline{\Sigma^{\omega}}.$ To show the converse inclusion, take any $x\in\overline{\Sigma^{\omega}}.$ Since $\overline{\Sigma^{\omega}}$ is weakly compact, by the Eberlein-Smulian theorem ([10], Theorem 8.12.4, p. 549), there exists a sequence $(x_{n})_{n}\subset\Sigma$ such that $x_{n}\rightarrow x$ weakly, so $x\in\Sigma.$ Hence $\overline{\Sigma^{\omega}}=\Sigma,$ and consequently, $\Sigma$ is a weakly closed subset of the weakly compact set $D .$ Therefore, $\Sigma$ is weakly compact. Because $X$ endowed with its weak topology is a Hausdorff locally convex space, we have that $X$ is completely regular ([21], p. 16). From the assumption that (ii) does not hold, one has $\Sigma\cap(\Omega\setminus U_{1}\times\dots\times U_{n})=\emptyset,$ where both sets $\Sigma$ and $\Omega\setminus U_{1}\times\dots\times U_{n}$ are weakly closed. Then (see [James], p. 146), there is a weakly continuous function $\varphi:\Omega\rightarrow[0,1],$ with $\varphi(x)=1$ for all $x\in\Sigma$ and $\varphi(x)=0$ for all $x\in\Omega\setminus U_{1}\times\dots\times U_{n}.$ Let $F^{\ast}:\Omega\rightarrow\Omega$ be the mapping defined by

F^{\ast}(x)=\varphi(x)F(x).

Since $\varphi$ is weakly continuous and $F$ is weakly sequentially continuous, we have that $F^{\ast}$ is weakly sequentially continuous. In addition, for each $i,$

F_{i}^{\ast}(\Omega)\subset\overline{conv}(F_{i}(D)\cup\{0\})=:D_{i}^{\ast}.

Let $D^{\ast}=D_{1}^{\ast}\times\cdot\cdot\cdot\times D_{n}^{\ast}.$ From the Krein-Smulian theorem ( [9], p. 434) using the weak sequential continuity of $F,$ we have that $D^{\ast}$ is a weakly compact convex set. Moreover $F^{\ast}(D^{\ast})\subset D^{\ast}.$ Now the Arino-Gautier-Penot theorem [2] guarantees that $F^{\ast}$ has a fixed point $x_{0}\in D^{\ast}.$ If $x_{0}\notin U_{1}\times\dots\times U_{n},$ then $\varphi(x_{0})=0,$ whence $x_{0}=0$ , which contradicts our hypothesis $0\in U_{1}\times\dots\times U_{n}.$ Therefore $x_{0}\in U_{1}\times\dots\times U_{n}\subset D$ and $x_{0}=\varphi(x_{0})F(x_{0}),$ which shows that $x_{0}\in\Sigma.$ This implies that $\varphi(x_{0})=1,$ and thus the proof is complete.

In the next result, the weak compactness of the sets $\overline{U_{i}^{\omega}}$ is removed and replaced by a stronger condition on $F .$ The proof is standard and we omit it.

Theorem 3.2.

Let $\left(X_{i},\left\|\cdot\right\|_{i}\right),$ $i=1,\cdot\cdot\cdot,n$ be Banach spaces. For each $i\in\left\{1,\cdot\cdot\cdot,n\right\},$ consider a nonempty closed and convex set $\Omega_{i}\subset X_{i}$ and a weakly open subset $U_{i}$ of $\Omega_{i}$ with $0\in U_{i}.$ Let $\Omega=\Omega_{1}\times\cdot\cdot\cdot\times\Omega_{n},$ $D=\overline{U_{1}^{\omega}}\times\cdot\cdot\cdot\times\overline{U_{n}^{\omega}},$ and $F:D\rightarrow\Omega$ a weakly sequentially continuous mapping such that $F(D)$ is relatively weakly compact. Then the alternative result given by Theorem 3.1 holds.

Theorem 3.2 will now be exploited to derive a Krasnoselskii type fixed point theorem which is the analogue for the weak topology of Viorel’s theorem [22], and a vector version of Theorem $3.4$ in [5].

Theorem 3.3.

Let $X_{i},$ $\Omega_{i},$ $U_{i}$ $(i=1,\cdot\cdot\cdot,n),$ $\Omega$ and $D$ be as in Theorem 3.1, and $X=X_{1}\times\cdot\cdot\cdot\times X_{n}.$ Let $A:D\longrightarrow X$ and $B:X\longrightarrow X$ be two weakly sequentially continuous mappings such that:

(a)

$A(D)$ is relatively weakly compact;
(b)

$B$ is a Perov contraction;
(c)

$\left(J-B\right)^{-1}A\left(D\right)\subset\Omega.$

Then, either

(i):: $A+B$ has a fixed point, or
(ii):: there exist $i\in\left\{1,\cdot\cdot\cdot,n\right\},$ a point $x=\left(x_{1},\cdot\cdot\cdot,x_{n}\right)\in D$ with $x_{i}\in\partial_{\Omega_{i}}U_{i}=\overline{U_{i}^{\omega}}\setminus U_{i},$ and a number $\lambda\in(0,1)$ such as $x=\lambda A(x)+\lambda B(\frac{x}{\lambda}).$

Proof.

For any given $x\in D,$ let $F_{x}:X\longrightarrow X$ be defined by

F_{x}(y)=A\left(x\right)+B(y),\ \ \ y\in X.

Using (b) we have

\|F_{x}(y)-F_{x}(z)\|=\|B(y)-B(z)\|\leqslant M\|y-z\|,\quad\text{ for all }y,z% \in X,

where $M$ is the Lipschitz matrix of $B .$ This shows that $F_{x}$ is a Perov contraction with the same Lipschitz matrix $M .$ Perov’s theorem guarantees the existence of a unique point $y_{x}\in X$ such that $y_{x}=A\left(x\right)+B\left(y_{x}\right).$ Let $F:D\rightarrow X$ be defined as

F\left(x\right)=y_{x},\ \ \ x\in D.

From (c), we have $F(D)\subset\Omega.$ Notice that

F(x)=(J-B)^{-1}A(x),\ \ \ x\in D.

Our next task is to show that the mapping $F:=(J-B)^{-1}A$ fulfills the conditions of Theorem 3.2. Indeed, since from (a), the set $A(D)$ is relatively weakly compact, it is also a bounded set. Next using

\|(J-B)^{-1}(x)-(J-B)^{-1}(y)\|\leq(I-M)^{-1}\|x-y\|\quad\text{ for all }x,y% \in X,

we see that $F(D)=(J-B)^{-1}A(D)$ is also bounded. We now claim that $F(D)$ is relatively weakly compact. Indeed, from

(3.7)

F(D)\subset A(D)+B(F(D)),

we obtain

(3.8)

\omega\left(F(D)\right)\leq\omega\left(A(D)+B(F(D))\right).

Further, taking into account that $A(D)$ is relatively weakly compact and using the property (vi) of $\omega_{i}$ we deduce that

(3.9)

\omega\left(F(D)\right)\leq\omega\left(A(D)\right)+\omega\left(B(F(D))\right)=% \omega\left(B(F(D))\right).

Now, by Proposition 3.1 and inequality (3.9), we get

\omega(F(D))\leq M\omega(F(D)).

So $(I-M)\omega(F(D))\leq 0_{\mathbb{R}^{n}}.$ Since matrix $M$ is convergent to zero, we then have $\omega(F(D))=0_{\mathbb{R}^{n}}$ and so $\omega_{i}(F_{i}(D))=0$ for all $i\in\{1,\cdot\cdot\cdot,n\}$ . Consequently, $F(D)$ is relatively weakly compact as claimed.

Next, we show that $F:D\rightarrow\Omega$ is weakly sequentially continuous. To do so, let $(x^{k})_{k}\subset D$ be such that $x_{i}^{k}\rightarrow x_{i}$ weakly as $k\rightarrow\infty,$ for $i=1,\cdot\cdot\cdot,n.$ Because $F(D)$ is relatively weakly compact, it follows by the Eberlein-Smulian theorem ([9], p. 430) that there exists a subsequence of $(x^{k})$ (still denoted by $\left(x^{k}\right))$ and $y\in\Omega$ such that $F_{i}(x^{k})\rightarrow y_{i}$ weakly, for $i=1,\cdot\cdot\cdot,n.$ Now the weak sequentially continuity of $B$ guarantees that $B\left(F(x^{k})\right)\rightarrow B\left(y\right)$ weakly. Also, from the equality $BF=-A+F,$ it follows that

-A(x^{k})+F(x^{k})\rightarrow-A(x)+y\text{ \ \ weakly.}

So $y=F(x).$ It is now easy to see that the whole sequence $\left(F\left(x^{k}\right)\right)$ weakly converges to $F\left(x\right),$ which proves that $F$ is weakly sequentially continuous. Finally, we note that the fixed points of $F$ are the same as the fixed points of $A+B,$ and that the equation $x=\lambda F\left(x\right),$ where $x\in D,$ is equivalent to the equation

x=\lambda A(x)+\lambda B(\frac{x}{\lambda}).

Now we state a variant of the previous result where the assumptions on mapping $B$ are relaxed.

Theorem 3.4.

Let $X_{i},$ $\Omega_{i},$ $U_{i}$ $(i=1,\cdot\cdot\cdot,n),$ $\Omega,$ $D$ and $X$ be as in Theorem 3.3. Let $A:D\longrightarrow X$ and $B:\Omega\longrightarrow X$ be two weakly sequentially continuous mappings such that:

(a)

$A(D)$ is relatively weakly compact;
(b)

$A(D)\subset\left(J-B\right)(\Omega);$
(c)

If $(J-B)(x_{k})\rightarrow y$ weakly, then $(x_{k})_{k}$ has a weakly convergent subsequence;
(d)

$J-B$ is invertible.

Then the alternative of Theorem 3.3 holds.

Proof.

Define $F:D\rightarrow\Omega$ by $F(y):=(J-B)^{-1}A(y).$ $F$ is well defined by assumptions (b) and (d).

First we show that $F(D)$ is relatively weakly compact. Let $(y_{n})_{n}\subset F(D)$ be any sequence and let $(x_{n})_{n}\subset D$ be such that $y_{n}=F(x_{n}).$ Taking into account assumption (a), and using the Eberlein-Smulian’s theorem (see [9], p. 430), we get a subsequence $(y_{\varphi_{1}(n)})_{n}$ of $(y_{n})_{n}$ such that $(J-B)(y_{\varphi_{1}(n)})\rightarrow z$ weakly, for some $z\in\Omega.$ Then, by assumption (c), the sequence $(y_{\varphi_{1}(n)})_{n}$ has a weakly convergent subsequence. Hence, $F(D)$ is relatively weakly compact.

Next, we show that $F:D\rightarrow\Omega$ is weakly sequentially continuous. To do so, let $(x^{k})_{k}\subset D$ be such that $x_{i}^{k}\rightarrow x_{i}$ weakly as $k\rightarrow\infty,$ for $i=1,\cdot\cdot\cdot,n.$ Because $F(D)$ is relatively weakly compact, it follows by the Eberlein-Smulian theorem that there exists a subsequence of $(x^{k})$ (still denoted by $\left(x^{k}\right))$ and $y\in\Omega$ such that $F_{i}(x^{k})\rightarrow y_{i}$ weakly, for $i=1,\cdot\cdot\cdot,n.$ Now the weak sequentially continuity of $B$ guarantees that $B\left(F(x^{k})\right)\rightarrow B\left(y\right)$ weakly. Also, from the equality $BF=-A+F,$ we have

-A(x^{k})+F(x^{k})\rightarrow-A(x)+y\text{ \ \ weakly.}

It follows that $B\left(y\right)=-A\left(x\right)+y,$ whence $y=F(x).$ It is now easy to see that the whole sequence $\left(F\left(x^{k}\right)\right)_{k}$ weakly converges to $F\left(x\right),$ which proves that $F$ is weakly sequentially continuous.
Therefore Theorem 3.2 applies and gives the conclusion.

Remark 3.1.

Any Perov contraction $B:\Omega\longrightarrow X,$ with $B(\Omega)$ bounded, satisfies condition (c) in Theorem 3.4. To prove this, assume that $(J-F)\left(x_{k}\right)\rightarrow y$ weakly, for some $(x_{k})_{k}\subset\Omega$ and $y\in X.$ Writing $x_{k}$ as $x_{k}=(J-B)\left(x_{k}\right)+B\left(x_{k}\right)$ and using the subadditivity of the De Blasi measure of weak noncompactness, we get

\omega(\{x_{k}\})\leq\omega(\{(J-B)\left(x_{k}\right)\})+\omega(\{B\left(x_{k}% \right)\}).

Since $\omega(\{(J-B)\left(x_{k}\right)\})=0_{\mathbb{R}^{n}},$ we obtain $\omega(\{x_{k}\})\leq\omega(\{B(x_{k})\}).$ On the other hand, if $M$ is the Lipschitz matrix of $B,$ then

\omega(\{B(x_{k})\})\leq M\omega(\{x_{k}\}).

It follows that $(I-M)\omega(\{x_{k}\})\leq 0_{\mathbb{R}^{n}},$ and then $\omega(\{x_{k}\})=0_{\mathbb{R}^{n}}.$ Consequently, $\left\{x_{k}\right\}$ is relatively weakly compact and then by the Eberlein-Smulian’s theorem, it has a weakly convergent subsequence. Hence, condition $($ c $)$ is satisfied.

As a consequence of Theorem 3.4 and Remark 3.1, we have the following result.

Corollary 3.1.

Let $X_{i},$ $\Omega_{i},$ $U_{i}$ $(i=1,\cdot\cdot\cdot,n),$ $\Omega,$ $D$ and $X$ be as in Theorem 3.3. Assume that $A:D\longrightarrow X$ and $B:\Omega\longrightarrow X$ are two weakly sequentially continuous mappings such that:

(1)

$A(D)$ is relatively weakly compact;
(2)

$B$ is a Perov contraction and $B(\Omega)$ is bounded;
(3)

$A(D)+B(\Omega)\subset\Omega.$

Then the alternative of Theorem 3.3 holds.

Notice that the vector versions of the original theorems applied to the product space $X=X_{1}\times\cdots\times X_{n}$ allow to use different measures of noncompactness on the factor spaces $X_{i},$ such is the case in paper [7].

4. Application I: Solutions of a system of nonlinear transport equations

We consider the following system :

(4.1)

\left\{\begin{array}[]{ll}v_{3}\dfrac{\partial\Psi_{1}}{\partial x}(x,v)+% \sigma_{1}(x,v,\Psi_{1}(x,v),\Psi_{2}(x,v))-\lambda_{1}\Psi_{1}(x,v)=\int_{K}r% _{1}(x,v,v^{\prime},\Psi_{1}(x,v^{\prime}),\Psi_{2}(x,v^{\prime}))dv^{\prime}&% \\ v_{3}\dfrac{\partial\Psi_{2}}{\partial x}(x,v)+\sigma_{2}(x,v,\Psi_{1}(x,v),% \Psi_{2}(x,v))-\lambda_{2}\Psi_{2}(x,v)=\int_{K}r_{2}(x,v,v^{\prime},\Psi_{1}(% x,v^{\prime}),\Psi_{2}(x,v^{\prime}))dv^{\prime}&\end{array}\right.

where $(x,v)\in D=(0,1)\times K$ with $K$ the unit sphere of $\mathbb{R}^{3}$ , $x\in(0,1),v=(v_{1},v_{2},v_{3})\in K$ , $r_{j}(.,.,.,.),j=1,2$ is a nonlinear function of $\Psi_{j}$ , $\sigma_{j}(.,.,.,.),j=1,2$ is a function on $[0,1]\times K\times\mathbb{C}^{2}$ and $\lambda_{j},j=1,2$ is a complex number. The boundary conditions are modeled by

(4.2)

{\Psi_{j}}_{|D^{i}}=H^{j}({\Psi_{j}}_{|D^{0}}),\quad\text{ for }j=1,2

where $D^{i}$ (resp. $D^{0}$ ) is the incoming ( resp. outgoing) part of the space boundary and are given by

D^{i}=D^{i}_{1}\cup D^{i}_{2}=\{0\}\times K^{1}\cup\{1\}\times K^{0},

D^{0}=D^{0}_{1}\cup D^{0}_{2}=\{0\}\times K^{0}\cup\{1\}\times K^{1},

for

K^{0}=K\cap\{v_{3}<0\}\quad\text{ and }\quad K^{1}=K\cap\{v_{3}>0\}.

We shall treat the problem (4.1)-(4.2) in the following functional setting : let

X:=L^{1}(D;dxdv),

and

X^{i}:=L^{1}\Big{(}D^{i},|v_{3}|dv\Big{)}:=L^{1}\Big{(}D_{1}^{i},|v_{3}|dv\Big% {)}\oplus L^{1}\Big{(}D_{2}^{i},|v_{3}|dv\Big{)}:=X_{1}^{i}\oplus X_{2}^{i},

endowed with the norm

\|\Psi\|_{X^{i}}=\|\Psi_{1}^{i}\|_{X_{1}^{i}}+\|\Psi_{2}^{i}\|_{X_{2}^{i}}=% \int_{K^{1}}|\Psi(0,v)||v_{3}|dv+\int_{K^{0}}|\Psi(1,v)||v_{3}|dv,

and

X^{0}:=L^{1}\Big{(}D^{0},|v_{3}|dv\Big{)}:=L^{1}\Big{(}D_{1}^{0},|v_{3}|dv\Big% {)}\oplus L^{1}\Big{(}D_{2}^{0},|v_{3}|dv\Big{)}:=X_{1}^{0}\oplus X_{2}^{0},

endowed with the norm

\|\Psi\|_{X^{0}}=\|\Psi_{1}^{0}\|_{X_{1}^{0}}+\|\Psi_{2}^{0}\|_{X_{2}^{0}}=% \int_{K^{0}}|\Psi(0,v)||v_{3}|dv+\int_{K^{1}}|\Psi(1,v)||v_{3}|dv.

For each $j\in\{1,2\},$ let $H^{j}$ be the following linear bounded boundary operator defined by:

\begin{cases}H^{j}:X_{1}^{0}\oplus X_{2}^{0}\longrightarrow X_{1}^{i}\oplus X_% {2}^{i}\\ H^{j}\begin{pmatrix}u_{1}\\ u_{2}\end{pmatrix}=\begin{pmatrix}H^{j}_{11}&H^{j}_{12}\\ H^{j}_{21}&H^{j}_{22}\end{pmatrix}\begin{pmatrix}u_{1}\\ u_{2}\end{pmatrix}\end{cases}

where $H_{l,k}^{j}\in\mathcal{L}(X_{l}^{0},X_{k}^{i}),$ for $l,k,j=1,2.$ The boundary condition can be written as $\Psi^{i}=H^{j}(\Psi^{0})$ for $j=1,2.$ Now for each $j\in\{1,2\}$ we define the streaming operator $T_{H^{j}}$ with domain including the boundary conditions

\begin{cases}T_{H^{j}}:D(T_{H^{j}})\subseteq X\longrightarrow X,\\ \ \hskip 71.13188pt\Psi\longmapsto T_{H^{j}}\Psi(x,v)=v_{3}\dfrac{\partial\Psi% }{\partial x}(x,v)\\ D(T_{H^{j}})=\big{\{}\Psi\in X\text{ such that }\Psi^{i}=H^{j}(\Psi^{0})\big{% \}},\end{cases}

where $\Psi^{0}=(\Psi_{1}^{0},\Psi_{2}^{0})^{T}$ and $\Psi^{i}=(\Psi_{1}^{i},\Psi_{2}^{i})^{T}$ where $\Psi_{1}^{0},\Psi_{2}^{0},\Psi_{1}^{i}$ and $\Psi_{2}^{i}$ are given by

\begin{cases}\Psi_{1}^{i}(v)=\Psi(0,v),\quad\text{ for }v\in K^{1},\\ \Psi_{2}^{i}(v)=\Psi(1,v),\quad\text{ for }v\in K^{0},\\ \Psi_{1}^{0}(v)=\Psi(0,v),\quad\text{ for }v\in K^{0},\\ \Psi_{2}^{0}(v)=\Psi(1,v),\quad\text{ for }v\in K^{1}.\end{cases}

Remark 4.1.

For each $j\in\{1,2\}$ , the derivative of $\Psi$ in the definition of $T_{H^{j}}$ is meant in distributional sense.

For each $j\in\{1,2\}$ , let $\lambda_{0}^{j}$ be the real defined by

\lambda_{0}^{j}:=\begin{cases}0&\text{ if }\|H^{j}\|\leq 1,\\ -\log(\|H^{j}\|)&\text{ if }\|H^{j}\|>1.\end{cases}

Proposition 4.1.

For each $j\in\{1,2\}$ , we have

\Big{\{}\lambda\in\mathbb{C}\text{ such that }\ Re(\lambda)<\lambda_{0}=\inf(% \lambda_{0}^{1},\lambda_{0}^{2})\Big{\}}\subset\rho(T_{H^{j}}).

Proof.

See reference ([4] Proposition 3.1)

For our subsequent analysis, we need this hypothesis: For each $j\in\{1,2\},$

(\mathcal{A}_{1})~{}~{}~{}~{}\quad r_{j}\big{(}x,v,v^{\prime},\Psi_{1}(x,v^{% \prime}),\Psi_{2}(x,v^{\prime})\big{)}=\kappa_{j}(x,v,v^{\prime})f_{j}\big{(}x% ,v^{\prime},L_{j}(\Psi_{1},\Psi_{2})(x,v^{\prime})\big{)},

with $L_{j}:=(L^{1}([0,1]\times K))^{2}\longrightarrow L^{\infty}([0,1]\times K)$ is a continuous linear map and

\begin{cases}f_{j}:[0,1]\times K\times\mathbb{C}^{2}\longrightarrow\mathbb{C}% \\ \qquad(x,v,u_{1},u_{2})\longmapsto f_{j}(x,v,u_{1},u_{2})).\end{cases}

is a mesurable function. The function $\kappa_{j}(.,.,.),j=1,2$ is a measurable function from $[0,1]\times K\times K$ into $\mathbb{R}.$ It defines a continuous linear operator $F_{j},j=1,2$ by

(4.3)		$\displaystyle F_{j}:$	$\displaystyle X$	$\displaystyle\longrightarrow X$
(4.3)			$\displaystyle\Psi$	$\displaystyle\longmapsto F_{j}(\Psi)(x,v)=\int_{K}\kappa_{j}(x,v,v^{\prime})% \Psi(x,v^{\prime})dv^{\prime}$

Note that $dx\otimes dv-ess-\sup\limits_{(x,v)\in[0,1]\times K}\int_{K}|\kappa_{j}(x,v,v^% {\prime})|dv^{\prime}=\|F_{j}\|<\infty.$

Definition 1.

A collision operator $F_{j},j=1,2$ in form (4.3) is said to be regular if the set

\Big{\{}\kappa_{j}(x,.,v^{\prime})\text{ such that }(x,v^{\prime})\in[0,1]% \times K\Big{\}}

is a relatively weakly compact subset of $L^{1}(K,dx).$

We need also the following result which is an immediate consequence of Lemme $4.1$ in [6] for $\sigma\equiv 0.$

Lemma 1.

If the collision operator $F_{j},j=1,2$ is regular on $X$ , then $(T_{H^{j}}-\lambda I)^{-1}F_{j}$ is weakly compact on $X,$ for $Re(\lambda)<\lambda_{0}.$

Definition 2.

A function $f:[0,1]\times K\times\mathbb{C}^{2}\rightarrow\mathbb{C}$ is a CarathÃ©odory map if the following conditions are satisfied

\begin{cases}(x,v)\longmapsto f(x,v,u_{1},u_{2})\text{ is measurable on }[0,1]% \times K,\text{ for all }(u_{1},u_{2})\in\mathbb{C}^{2}.\\ u\longmapsto f(x,v,u_{1},u_{2})\text{ is continuous on }\mathbb{C}^{2},\text{ for almost all }(x,v)\in[0,1]\times K.\end{cases}

If $f$ satisfies the CarathÃ©odory conditions, we can define the operator $\mathcal{N}_{f}$ on the set of functions $(\Psi_{1},\Psi_{2}):[0,1]\times K\longrightarrow\mathbb{C}^{2}$ by

\mathcal{N}_{f}(\Psi_{1},\Psi_{2})(x,v)=f(x,v,\Psi_{1}(x,v),\Psi_{2}(x,v)),% \text{ for every }\ (x,v)\in[0,1]\times K.

The operator $\mathcal{N}_{f}$ is called the Nemytskii operator generated by $f .$ We assume that

(\mathcal{A}_{2})~{}~{}\quad\begin{cases}\text{ For each }j\in\{1,2\},f_{j}% \text{ is a Carath\~{A}\copyright odory map satisfying }\\ |f_{j}(x,v,u_{1},u_{2})|\leqslant a_{j}(x,v)h_{j}(\|(u_{1},u_{2})\|_{L^{1}% \times L^{1}}),\\ \text{ where }a_{j}\in L^{1}([0,1]\times K,dxdv)\text{ and }h_{j}\in L_{loc}^{% \infty}(\mathbb{R}^{+})\text{ a non-decreasing function. }\end{cases}

The interest that an operator satisfies the property $(\mathcal{A}_{2})$ lies in the following lemma:

Lemma 2.

For each $j\in\{1,2\}$ , let $L_{j}:(L^{1}([0,1]\times K,dxdv))^{2}\longrightarrow L^{\infty}([0,1]\times K,dxdv)$ be a continuous linear map and let $f_{j}:[0,1]\times K\times\mathbb{C}^{2}\longrightarrow\mathbb{C}$ be a map satisfying the hypothesis $(\mathcal{A}_{2}).$ Then the map

\displaystyle\Phi_{j}:=\mathcal{N}_{f_{j}}\circ L_{j}:(L^{1}([0,1]\times K,% dxdv))^{2}\longrightarrow L^{1}([0,1]\times K,dxdv)

is weakly sequentially continuous.

Proof.

Let $(u_{n},v_{n})\rightharpoonup(u,v)$ in $(L^{1}([0,1]\times K,dxdv))^{2}.$ By the Eberlein-Smulian Theorem, the set $G=\{(u_{n},v_{n}),(u,v)\}_{n=1}^{\infty}$ is weakly compact. Let us show that $\Phi_{j}(G),j=1,2$ is relatively weakly compact in $L^{1}([0,1]\times K,dxdv).$ Clearly $\Phi_{j}(G)$ is bounded, once

\|\Phi_{j}(u_{1},u_{2})\|_{L^{1}}\leqslant\|a_{j}\|_{L^{1}}h_{j}(\|L_{j}\|\|(u% _{1},u_{2})\|_{L^{1}\times L^{1}}).

Which also shows that $\Phi_{j}(G)$ is uniformly integrable. Since $\C^{2}$ is reflexive, we get, according to Dunford’s Theorem ([3] Theorem 7.10), that $\Phi_{j}(G)$ is relatively weakly compact in $L^{1}([0,1]\times K,dxdv).$ Up to a subsequence, $\Phi_{j}(u_{n},v_{n})\rightharpoonup w_{j}\in L^{1}([0,1]\times K,dxdv).$ The idea is to show that actually $w_{j}=\Phi_{j}(u,v).$ We know $L_{j}(u_{n},v_{n})(x,\xi)\rightharpoonup L_{j}(u,v)(x,\xi)$ in $\C$ for a.e. $(x,\xi)\in[0,1]\times K.$ Since $f$ is a Caratheodory map, then $\Phi_{j}(u_{n},v_{n})(x,\xi)\rightharpoonup\Phi_{j}(u,v)(x,\xi)$ in $\C$ for almost every $(x,\xi)\in[0,1]\times K.$ Now we shall conclude that $w_{j}=\Phi_{j}(u,v)$ a.e. To this end, we start by throwing away a set $A_{0}$ of measure zero such that, for each $j\in\{1,2\}$ the space

F_{j}:=\overline{span}\Big{(}w_{j}(([0,1]\times K)\setminus A_{0})\cup\Phi_{j}% (u,v)(([0,1]\times K)\setminus A_{0})\Big{)}

is a separable and reflexive Banach space. The existence of such a $A_{0}$ is due to Pettis’ Theorem. Let now $\{\varphi_{k}\}$ be a dense sequence of continuous linear functionals in $F_{j}.$ By Ergorov’s Theorem, for each $\varphi_{k}$ fixed, there exists a negligible set $A_{k},$ such that $\varphi_{k}(w_{j})=\varphi_{k}(\Phi_{j}(u,v))$ in $([0,1]\times K)\setminus A_{k}.$ Finally we define $A=\cup_{k=0}^{\infty}A_{k}.$ In this way $\lambda(A)=0$ and by the Hahn-Banach Theorem, $w_{j}(x,\xi)=\Phi_{j}(u,v)(x,\xi)$ for all $(x,\xi)\in([0,1]\times K)\setminus A.$

The following hypothesis will play a crucial role :

(\mathcal{A}_{3})~{}\ \begin{cases}\text{For }j=1,2.\ \mathcal{N}_{\sigma_{j}}% \text{ is weakly sequentially continuous and acts from }\overline{B}_{r_{1}}\times\overline{B}_{r_{2}}\text{% into }\overline{B}_{r_{j}}\\ \big{|}\mathcal{N}_{\sigma_{j}}(\Psi_{1},\Psi_{2})(x,v))-\mathcal{N}_{\sigma_{% j}}(\Psi^{\prime}_{1},\Psi^{\prime}_{2})(x,v))\big{|}\leqslant|\rho_{j,1}(x,v)% ||\Psi_{1}-\Psi^{\prime}_{1}|+|\rho_{j,2}(x,v)||\Psi_{2}-\Psi^{\prime}_{2}|\\ \text{ where }\overline{B}_{r}=\{\Psi\in X\text{ such that }\|\Psi\|\leqslant r% \}\text{ and }\rho_{j,1}(.,.),\rho_{j,2}(.,.)\in L^{\infty}(D,dxdv),\end{cases}

Let $\lambda$ be a complex number such that $Re(\lambda)<\lambda_{0}.$ Then due to Proposition 4.1, the mapping $T_{H^{j}}-\lambda I,$ $j=1,2$ is invertible and therefore, the problem (4.1)-(4.2) is equivalent to the following system:

(4.4)

\begin{cases}\Psi_{1}=\mathcal{F}_{1}(\lambda_{1})(\Psi_{1},\Psi_{2})+\mathcal% {H}_{1}(\lambda_{1})(\Psi_{1},\Psi_{2})\\ \Psi_{2}=\mathcal{F}_{2}(\lambda_{2})(\Psi_{1},\Psi_{2})+\mathcal{H}_{2}(% \lambda_{2})(\Psi_{1},\Psi_{2})\\ \Psi_{1}\in D(T_{H^{1}}),\Psi_{2}\in D(T_{H^{2}}),\ Re(\lambda_{j})<\lambda_{0% }\end{cases}

where

\begin{cases}\mathcal{F}_{j}(\lambda_{j}):=(T_{H^{j}}-\lambda_{j}I)^{-1}F_{j}% \mathcal{N}_{f_{j}}L_{j}&\\ \mathcal{H}_{j}(\lambda_{j}):=(T_{H^{j}}-\lambda_{j}I)^{-1}\mathcal{N}_{-% \sigma_{j}}&j=1,2\\ &\end{cases}

Now, the system (4.4) is equivalent to the following fixed point problem :

(4.5)

\begin{cases}(\Psi_{1},\Psi_{2})=\mathcal{F}(\lambda_{1},\lambda_{2})(\Psi_{1}% ,\Psi_{2})+\mathcal{H}(\lambda_{1},\lambda_{2})(\Psi_{1},\Psi_{2})\\ (\Psi_{1},\Psi_{2})\in D(T_{H^{1}})\times D(T_{H^{2}}),\ Re(\lambda_{j})<% \lambda_{0}\text{ for }j=1,2\end{cases}

where

\mathcal{F}(\lambda_{1},\lambda_{2}):=\begin{pmatrix}\mathcal{F}_{1}(\lambda_{% 1})\\ \mathcal{F}_{2}(\lambda_{2})\end{pmatrix}=\begin{pmatrix}(T_{H^{1}}-\lambda_{1% }I)^{-1}F_{1}\mathcal{N}_{f_{1}}L_{1}\\ (T_{H^{2}}-\lambda_{2}I)^{-1}F_{2}\mathcal{N}_{f_{2}}L_{2}\end{pmatrix},

\mathcal{H}(\lambda_{1},\lambda_{2}):=\begin{pmatrix}\mathcal{H}_{1}(\lambda_{% 1})\\ \mathcal{H}_{2}(\lambda_{2})\end{pmatrix}=\begin{pmatrix}(T_{H^{1}}-\lambda_{1% }I)^{-1}\mathcal{N}_{-\sigma_{1}}\\ (T_{H^{2}}-\lambda_{2}I)^{-1}\mathcal{N}_{-\sigma_{2}}\end{pmatrix}

Theorem 4.1.

Assume that $\mathcal{A}_{1}-\mathcal{A}_{3}$ hold and that for $j=1,2,F_{j}$ is a regular operator on $X .$ Let $U_{r_{1}}\times U_{r_{2}}$ be a weakly open subset of $\overline{B}_{r_{1}}\times\overline{B}_{r_{2}}$ with $0\in U_{r_{1}}\times U_{r_{2}}.$ In addition, suppose that

\begin{cases}\text{ for any solution }(\Psi_{1},\Psi_{2})\in X^{2}\text{ to }(% \Psi_{1},\Psi_{2})=\alpha\mathcal{F}(\lambda)(\Psi_{1},\Psi_{2})+\alpha% \mathcal{H}(\lambda)\left(\frac{\Psi_{1}}{\alpha},\frac{\Psi_{2}}{\alpha}% \right)\text{ a.e., }0<\alpha<1,\\ \text{ we have }(\Psi_{1},\Psi_{2})\notin\partial_{\overline{B}_{r_{1}}}U_{r_{% 1}}\times\partial_{\overline{B}_{r_{2}}}U_{r_{2}}\end{cases}

holds. Then, there exists a $\lambda^{\ast}<0,$ such that for $Re(\lambda_{j})<\lambda^{\ast},j=1,2$ enough small, the problem $(\ref{E1})-(\ref{E2})$ has a solution in $\overline{U_{r_{1}}}^{\omega}\times\overline{U_{r_{2}}}^{\omega}$

Proof.

(1)

The proof will be given in several steps:
$\bullet$

Step 1 : The maps $\mathcal{F}(\lambda_{1},\lambda_{2})$ and $\mathcal{H}(\lambda_{1},\lambda_{2})$ are weakly sequentially continuous for suitable $\lambda_{1},\lambda_{2}.$ Indeed, we have for $j=1,2,$ $\mathcal{N}_{\sigma_{j}}$ is weakly sequentially continuous and for $\Re(\lambda_{j})<\lambda_{0},$ the linear operator $(T_{H^{j}}-\lambda_{j})^{-1},j=1,2$ is bounded, so the operator

$\mathcal{H}(\lambda_{1},\lambda_{2}):=((T_{H^{1}}-\lambda_{1}I)^{-1}\mathcal{N% }_{-\sigma_{1}},\ (T_{H^{2}}-\lambda_{2}I)^{-1}\mathcal{N}_{-\sigma_{2}})$

is weakly sequentially continuous, for $\Re(\lambda_{j})<\lambda_{0},j=1,2.$ Moreover, using ([6] page 89), we have

$\mathcal{F}(\lambda_{1},\lambda_{2}):=\big{(}(T_{H^{1}}-\lambda_{1}I)^{-1}F_{1% }\mathcal{N}_{f_{1}}L_{1},\ (T_{H^{2}}-\lambda_{2}I)^{-1}F_{2}\mathcal{N}_{f_{% 2}}L_{2}\big{)}$

is weakly sequentially continuous, for $\Re(\lambda_{j})<\lambda_{0},j=1,2.$
$\bullet$

Step 2 : $\mathcal{F}(\lambda)(\overline{U_{r_{1}}}^{\omega}\times\overline{U_{r_{2}}}^{% \omega})$ is relatively weakly compact in $X\times X.$ Using the hypothesis $(\mathcal{A}_{2})$ , we get $\ \mathcal{N}_{f_{j}}L_{j}(\overline{U_{r_{1}}}^{\omega}\times\overline{U_{r_{% 2}}}^{\omega})$ is a bounded subset of $X .$ So from Lemma 1 we have $\mathcal{F}(\lambda)(\overline{U_{r_{1}}}^{\omega}\times\overline{U_{r_{2}}}^{% \omega})$ is relatively weakly compact in $X\times X.$

\bullet

Step 3 : $\mathcal{H}(\lambda_{1},\lambda_{2})$ is a contraction mapping on $\overline{B}_{r_{1}}\times\overline{B}_{r_{2}}.$ Indeed, let $(\Psi_{1},\Psi_{2}),(\Psi^{\prime}_{1},\Psi^{\prime}_{2})\in B_{r_{1}}\times B% _{r_{2}}.$ We have

	$\displaystyle\\|\mathcal{H}(\lambda)(\Psi_{1},\Psi_{2})-\mathcal{H}(\lambda)(% \Psi^{\prime}_{1},\Psi^{\prime}_{2})\\|$	$\displaystyle=\left(\begin{matrix}\\|(T_{H^{1}}-\lambda_{1}I)^{-1}(\mathcal{N}_% {-\sigma_{1}}(\Psi_{1},\Psi_{2})-\mathcal{N}_{-\sigma_{1}}(\Psi^{\prime}_{1},% \Psi^{\prime}_{2}))\\|\\ \\|(T_{H^{2}}-\lambda_{2}I)^{-1}(\mathcal{N}_{-\sigma_{2}}(\Psi_{1},\Psi_{2})-% \mathcal{N}_{-\sigma_{2}}(\Psi^{\prime}_{1},\Psi^{\prime}_{2}))\\|\end{matrix}\right)$
		$\displaystyle\leqslant\left(\begin{matrix}\\|(T_{H^{1}}-\lambda_{1}I)^{-1}\\|\\|% \mathcal{N}_{-\sigma_{1}}(\Psi_{1},\Psi_{2})-\mathcal{N}_{-\sigma_{1}}(\Psi^{% \prime}_{1},\Psi^{\prime}_{2})\\|\\ \\|(T_{H^{2}}-\lambda_{2}I)^{-1}\\|\\|\mathcal{N}_{-\sigma_{2}}(\Psi_{1},\Psi_{2}% )-\mathcal{N}_{-\sigma_{2}}(\Psi^{\prime}_{1},\Psi^{\prime}_{2})\\|\end{matrix}\right)$
		$\displaystyle\leqslant\left(\begin{matrix}\\|(T_{H^{1}}-\lambda_{1}I)^{-1}\\|(\\|% \rho_{1,1}\\|_{\infty}\\|\Psi_{1}-\Psi^{\prime}_{1}\\|+\\|\rho_{1,2}\\|_{\infty}\\|% \Psi_{2}-\Psi^{\prime}_{2}\\|)\\ \\|(T_{H^{2}}-\lambda_{2}I)^{-1}\\|(\\|\rho_{2,1}\\|_{\infty}\\|\Psi_{1}-\Psi^{% \prime}_{1}\\|+\\|\rho_{2,2}\\|_{\infty}\\|\Psi_{2}-\Psi^{\prime}_{2}\\|)\end{% matrix}\right)$
		$\displaystyle\leqslant\max_{j\in\{1,2\}}(\\|(T_{H^{j}}-\lambda_{j}I)^{-1}\\|)% \left(\begin{matrix}\\|\rho_{1,1}\\|_{\infty}&\\|\rho_{1,2}\\|_{\infty}\\ \\|\rho_{2,1}\\|_{\infty}&\\|\rho_{2,2}\\|_{\infty}\end{matrix}\right)\left(\begin% {matrix}\\|\Psi_{1}-\Psi^{\prime}_{1}\\|\\ \\|\Psi_{2}-\Psi^{\prime}_{2}\\|\end{matrix}\right)$
		$\displaystyle\leqslant M\\|(\Psi_{1},\Psi_{2})-(\Psi^{\prime}_{1}-\Psi^{\prime}% _{2})\\|$

where

M=\max_{j\in\{1,2\}}(\left\|(T_{H^{j}}-\lambda_{j}I)^{-1}\right\|)\begin{% pmatrix}\|\rho_{1,1}\|_{\infty}&\|\rho_{1,2}\|_{\infty}\\ \|\rho_{2,1}\|_{\infty}&\|\rho_{2,2}\|_{\infty}\end{pmatrix}

On the other hand, we have for $Re(\lambda_{j})<\lambda_{0},j=1,2,$

\|(T_{H^{j}}-\lambda_{j})^{-1}\|\leqslant\dfrac{-1}{\Re(\lambda_{j})}\left(1+% \dfrac{\|H^{j}\|}{1-e^{\Re(\lambda_{j})\|H^{j}\|}}\right).

(See [6], page 89 ). So, $\|(T_{H^{j}}-\lambda_{j})^{-1}\|\leqslant\Upsilon(\Re(\lambda_{j}))$ where

\Upsilon(t)=\frac{-1}{t}\left(1+\frac{\|H^{j}\|}{1-e^{t\|H^{j}\|}}\right)

Clearly, $\Upsilon$ is continuous and satisfies $\lim\limits_{t\to-\infty}\Upsilon(t)=0.$ Hence there exists $\lambda^{\prime}<0$ such that for $\Re(\lambda_{j})<\min(\lambda_{0},\lambda^{\prime}),$ we have

(\max_{j\in\{1,2\}}\|(T_{H^{j}}-\lambda_{j})^{-1}\|\|\rho_{k,l}\|_{\infty})_{1% \leq k,l\leq 2}

are small enough and so, $M$ is a matrix convergent to zero. In conclusion, the operator $\mathcal{H}(\lambda_{1},\lambda_{2})$ is a contraction mapping on $\overline{B}_{r_{1}}\times\overline{B}_{r_{2}}.$

\bullet

Step 4: Condition (3) of theorem 3.1 holds for suitable $\lambda_{1},\lambda_{2}.$ We will show that for suitable $\lambda=(\lambda_{1},\lambda_{2}),$ we have $\mathcal{F}(\lambda)(\overline{U_{r_{1}}}^{\omega}\times\overline{U_{r_{2}}}^{% \omega})+\mathcal{H}(\lambda)(\overline{B}_{r_{1}}\times\overline{B}_{r_{2}})% \subset\overline{B}_{r_{1}}\times\overline{B}_{r_{2}}.$ To do so, let $(\Psi_{1},\Psi_{2})\in\overline{U_{r_{1}}}^{\omega}\times\overline{U_{r_{2}}}^% {\omega}$ and $(\varphi_{1},\varphi_{2})\in\overline{B}_{r_{1}}\times\overline{B}_{r_{2}}.$ Then we have

	$\displaystyle\\|\mathcal{H}(\lambda)(\varphi_{1},\varphi_{2})+\mathcal{F}(% \lambda)(\Psi_{1},\Psi_{2})\\|$	$\displaystyle=\left\\|\left(\begin{matrix}(T_{H^{1}}-\lambda_{1}I)^{-1}\Big{(}% \mathcal{N}_{-\sigma_{1}}(\varphi_{1},\varphi_{2})+F_{1}\mathcal{N}_{f_{1}}L_{% 1}(\Psi_{1},\Psi_{2})\Big{)}\\ (T_{H^{2}}-\lambda_{2}I)^{-1}\Big{(}\mathcal{N}_{-\sigma_{2}}(\varphi_{1},% \varphi_{2})+F_{2}\mathcal{N}_{f_{2}}L_{2}(\Psi_{1},\Psi_{2})\Big{)}\end{% matrix}\right)\right\\|$
		$\displaystyle\leqslant\left(\begin{matrix}\left\\|(T_{H^{1}}-\lambda_{1}I)^{-1}% \right\\|\left(\left\\|\mathcal{N}_{-\sigma_{1}}(\varphi_{1},\varphi_{2})\right% \\|+\\|F_{1}\\|\\|(\mathcal{N}_{f_{1}}L_{1}(\Psi_{1},\Psi_{2}))\\|\right)\\ \left\\|(T_{H^{2}}-\lambda_{2}I)^{-1}\right\\|\left(\left\\|\mathcal{N}_{-\sigma_% {2}}(\varphi_{1},\varphi_{2})\right\\|+\\|F_{2}\\|\\|(\mathcal{N}_{f_{2}}L_{2}(% \Psi_{1},\Psi_{2}))\\|\right)\end{matrix}\right)$
		$\displaystyle\leqslant\left(\begin{matrix}\left\\|(T_{H^{1}}-\lambda_{1}I)^{-1}% \right\\|\left(M_{1}(r_{1},r_{2})+\\|F_{1}\\|(\\|a_{1}\\|\\|h_{1}\\|_{\infty}\right)% \\ \left\\|(T_{H^{2}}-\lambda_{2}I)^{-1}\right\\|\left(M_{2}(r_{1},r_{2})+\\|F_{2}\\|% (\\|a_{2}\\|\\|h_{2}\\|_{\infty}\right)\end{matrix}\right)$

where for $j=1,2,M_{j}(r_{1},r_{2})$ denotes respectively the upper bounded of $\mathcal{N}_{-\sigma_{j}}$ on $\overline{B}_{r_{1}}\times\overline{B}_{r_{2}}.$ So, for $\Re\lambda<\min(\lambda^{\prime},\lambda_{1}),\lambda_{1}<0,$ we obtain

\|\mathcal{H}(\lambda)(\varphi_{1},\varphi_{2})+\mathcal{F}(\lambda)(\Psi_{1},% \Psi_{2})\|\leq\max_{j\in\{1,2\}}\Upsilon(\Re(\lambda_{j}))\left(\begin{matrix% }M_{1}(r_{1},r_{2})+\|F_{1}\|(\|a_{1}\|\|h_{1}\|_{\infty}\\ M_{2}(r_{1},r_{2})+\|F_{2}\|(\|a_{2}\|\|h_{2}\|_{\infty}\end{matrix}\right),

where $\Upsilon$ is defined in step $3 .$ Thus, there exists $\lambda^{\prime\prime}<0$ such that for $\Re(\lambda_{j})<\min(\lambda_{0},\lambda^{\prime},\lambda^{\prime\prime}),$ we have

\mathcal{H}(\lambda_{1},\lambda_{2})(\varphi_{1},\varphi_{2})+\mathcal{F}(% \lambda_{1},\lambda_{2})(\Psi_{1},\Psi_{2})\in\overline{B}_{r_{1}}\times% \overline{B}_{r_{2}}.

Consequently, there exist $\lambda^{\ast}=\min(\lambda_{0},\lambda^{\prime},\lambda^{\prime\prime})$ such that for $\Re(\lambda)<\lambda^{\ast},$ the mappings $\mathcal{F}(\lambda_{1},\lambda_{2})$ and $\mathcal{H}(\lambda_{1},\lambda_{2})$ satisfy the assumptions of Corollary (3.1) on the nonempty bounded, closed and convex subset $\overline{B}_{r_{1}}\times\overline{B}_{r_{2}}.$ Consequently the problem $(\ref{E1}-\ref{E2})$ has a solution $(\varphi,\psi)$ in $\overline{B}_{r_{1}}\times\overline{B}_{r_{2}}$ for all $\lambda$ such that $\Re\lambda<\lambda^{\ast}.$

5. Application II: Existence of weak solutions

We discuss the existence of weak solutions for a coupled system of mixed fractional differential equations

(5.1)

\left\{\begin{array}[]{ll}D^{\alpha}_{1^{-}}(D^{\beta_{1}}_{0^{+}}u(t))+f_{1}(% t,u(t),v(t))=0,&\\ D^{\alpha}_{1^{-}}(D^{\beta_{2}}_{0^{+}}v(t))+f_{2}(t,u(t),v(t))=0;~{}~{}~{}t% \in I:=[0,1],&\end{array}\right.

with the following initial conditions:

(5.2)

\left\{\begin{array}[]{ll}D^{\beta_{1}}_{0^{+}}u(0)=D^{\beta_{1}}_{0^{+}}u(1)=% D^{\beta_{2}}_{0^{+}}v(0)=D^{\beta_{2}}_{0^{+}}v(1)=0,&\\ u(0)=u^{\prime}(1)=v(0)=v^{\prime}(1)=0;&\end{array}\right.

where $\alpha>1,\beta_{i}<2,$ for $i=\{1,2\},$ $f_{1},f_{2}:I\times E\times E\to E$ are given continuous functions, $E$ is a real (or complex) Banach space with norm $\|.\|_{E}$ and dual $E^{\ast}$ such that $E$ is the dual of a weakly compactly generated Banach space $X .$ Let’s remember that

D^{\alpha}_{a^{+}}f(t)=\frac{1}{\Gamma(n-\alpha)}(\frac{d}{dt})^{n}\int_{a}^{t% }(t-s)^{n-\alpha-1}f(s)ds

and

D^{\alpha}_{b^{-}}f(t)=\frac{1}{\Gamma(n-\alpha)}(-\frac{d}{dt})^{n}\int_{t}^{% b}(s-t)^{n-\alpha-1}f(s)ds

where $n=[\alpha]+1,$ are, respectively, the right and left Riemann-Liouville fractional derivatives of order $\alpha$ and

I^{\alpha}_{a^{+}}f(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}(t-s)^{\alpha-1}f(s% )ds

and

I^{\alpha}_{b^{-}}f(t)=\frac{1}{\Gamma(\alpha)}\int_{t}^{b}(s-t)^{\alpha-1}f(s% )ds

are, respectively, the right and left Riemann-Liouville fractional integrals of order $\alpha.$ Let $C(I,E)$ be the Banach space of all continuous functions $w$ from $I$ into $E$ with the supremum (uniform) norm. As usual, $AC(I)$ denotes the space of absolutely continuous functions from $I$ into $E .$ Also by $C(I,E)^{2}=C^{2},$ we denote the product space of continuous functions with the norm

\|(u,v)\|_{C^{2}}=\left(\begin{matrix}\|u\|_{C}\\ \|v\|_{C}\end{matrix}\right).

Let $(E,w)=(E,\sigma(E,E^{\ast}))$ be the Banach space $E$ with its weak topology.

Definition 3.

A Banach space $X$ is called weakly compactly generated (WCG for short) if it contains a weakly compact set whose linear span is dense in $X$ .

Definition 4.

([17]) The function $u:I\to E$ is said to be Pettis integrable on $I$ if and only if there is an element $u_{J}\in E$ corresponding to each $J\subset I$ such that $\phi(u_{J})=\int_{J}\phi(u(s))ds$ for all $\phi\in E^{*}$ , where the integral on the right-hand side is assumed to exist in the sense of Lebesgue (by definition, $u_{J}=\int_{J}u(s)ds)$

Let $P(I,E)$ be the space of all E-valued Pettis integrable functions on $I$ , and $L^{1}(I,E)$ be the Banach space of Lebesgue integrable functions $u:I\to E$ . Define the class $P_{1}(I,E)$ by

P_{1}(I,E)=\Big{\{}u\in P(I,E):\phi(u)\in L^{1}(I,E)\text{ for every }\phi\in E% ^{*}\Big{\}}.

The space $P_{1}(I,E)$ is normed by

\|u\|_{P_{1}}=\sup\limits_{\phi\in E^{*},\|\phi\|\leqslant 1}\int_{0}^{1}|\phi% (u(x))|d\lambda x,

where $\lambda$ stands for a Lebesgue measure on $I$ . The following result is due to Pettis (see [17], Theorem 3.4 and Corollary 3.41).

Proposition 5.1.

([17]) If $u\in P_{1}(I,E)$ and $h$ is a measurable and essentially bounded E-valued function, then $uh\in P_{1}(J,E)$ .

For all that follows, the symbol $\int$ denotes the Pettis integral.

Proposition 5.2.

Let $E$ be a normed space, and $x_{0}\in E$ with $x_{0}\neq 0$ . Then there exists $\phi\in E^{*}$ with $\|\phi\|=1$ and $\phi(x_{0})=\|x_{0}\|$ .

Let us start by defining what we mean by a weak solution of the coupled system $\ref{appp}-\ref{appp1}.$

Definition 5.

A coupled function $(u,v)\in C^{2}$ is said to be a weak solution of the system $(\ref{appp})-(\ref{appp1})$ if $(u,v)$ satisfies equations $(\ref{appp})$ and conditions $(\ref{appp1})$ on $I .$

The following hypotheses will be used in the sequel:
$(H_{1})$ For a.e. $t\in I,$ the functions $u\mapsto f_{i}(t,u,.)$ and $v\mapsto f_{i}(t,.,v);i=1,2$ are weakly sequentially continuous.
$(H_{2})$ For a.e. $u,v\in C(I,E),$ the functions $t\mapsto f_{i}(t,u,v),i=1,2$ are Pettis integrable a.e. on $I .$
$(H_{3})$ There exist $p_{ij}\in C(I,[0,\infty)),i=1,2,$ such that

\|f_{i}(t,u_{1}(t),u_{2}(t))-f_{i}(t,v_{1}(t),v_{2}(t)\|_{E}\leq p_{i1}(t)\|u_% {1}(t)-v_{1}(t)\|_{E}+p_{i2}(t)\|u_{2}(t)-v_{2}(t)\|_{E}

for a.e. $t\in I$ and each $u_{1},u_{2},v_{1},v_{2}\in C.$
Let

p_{ij}^{\ast}=\sup_{t\in I}p_{i,j}(t),i,j=1,2.

We shall transform the system $(\ref{appp})-(\ref{appp1})$ to an equivalent system of integral equations. Consider the corresponding linear system:

D^{\alpha}_{1^{-}}(D^{\beta_{i}}_{0^{+}}u_{i}(t))=-y_{i}(t),~{}0<t<1,

D^{\beta_{i}}_{0^{+}}u_{i}(0)=D^{\beta_{i}}_{0^{+}}u_{i}(1)=0,~{}u_{i}(0)=u_{i% }^{\prime}(1)=0,

here $i\in\{1,2\}.$

Lemma 3.

[1] Assume that $y_{i}\in C(0,1)\cap L_{1}(0,1),$ for $i\in\{1,2\},$ then the boundary value problem $(\ref{appp})-(\ref{appp1}),$ has a unique solution given by

u_{i}(t)=\int_{0}^{1}G_{i}(t,r)y_{i}(r)dr+g_{i}(t)\int_{0}^{1}s^{\alpha-1}y_{i% }(s)ds.

Where

G_{i}(t,r)=\frac{1}{\Gamma(\beta_{i})\Gamma(\alpha)}\left\{\begin{array}[]{ll}% \int_{0}^{r}(t^{\beta_{i}-1}(1-s)^{\beta_{i}-2}-(t-s)^{\beta i-1})(r-s)^{% \alpha-1}ds,0\leq r\leq t\leq 1,&\\ t^{\beta_{i}-1}\int_{0}^{r}(1-s)^{\beta_{i}-2}(r-s)^{\alpha-1}ds-\int_{0}^{t}(% t-s)^{\beta_{i}-1}(r-s)^{\alpha-1}ds,0\leq t\leq r\leq 1.&\end{array}\right.

g_{i}(t)=\frac{1}{\Gamma(\beta_{i})\Gamma(\alpha)}(\int_{0}^{t}(t-s)^{\beta_{i% }-1}(1-s)^{\alpha-1}ds-\frac{t^{\beta_{i}-1}}{\alpha+\beta_{i}-2}).

Lemma 4.

[1] The functions $g_{i}$ and $G_{i},$ for all $i\in\{1,2\}$ are continuous and satisfy the following properties:

0\leq G_{i}(t,r)\leq\frac{1}{(\alpha+\beta_{i}-2)\Gamma(\beta_{i})\Gamma(% \alpha)},0\leq t,r\leq 1

g_{i}(t)\leq 0,~{}|g_{i}(t)|\leq\frac{1}{(\alpha+\beta_{i}-2)\Gamma(\beta_{i})% \Gamma(\alpha)},0\leq t\leq 1.

Define the integral operators $A$ and $B$ on $C^{2}$ by

A(u_{1},u_{2})(t)=\left(\begin{matrix}A_{1}(u_{1},u_{2})(t)\\ A_{2}(u_{1},u_{2})(t)\end{matrix}\right),~{}\text{and}~{}B(u_{1},u_{2})(t)=% \left(\begin{matrix}B_{1}(u_{1},u_{2})(t)\\ B_{2}(u_{1},u_{2})(t)\end{matrix}\right),

where

A_{i}(u_{1},u_{2})(t)=\int_{0}^{1}G_{i}(t,r)f_{i}(r,u_{1}(r),u_{2}(r))dr,

B_{i}(u_{1},u_{2})(t)=g_{i}(t)\int_{0}^{1}s^{\alpha-1}f_{i}(s,u_{1}(s),u_{2}(s% ))ds.

First notice that the hypotheses $(H_{1},H_{2})$ imply that the operators $A$ and $B$ are well defined. By $\cite[cite]{[\@@bibref{}{1}{}{}]},$ The function $u=(u_{1},u_{2})\in C^{2}$ is a solution of the system $(\ref{appp})-(\ref{appp1})$ if, and only if, $Au(t)+Bu(t)=u(t)$ for all $t\in I.$ Let $R>0$ be such that

R>\sup\{\frac{4L}{(\alpha+\beta_{1}-2)\Gamma(\beta_{1})\Gamma(\alpha)},\frac{4% L}{(\alpha+\beta_{2}-2)\Gamma(\beta_{2})\Gamma(\alpha)}\}

where $L=\sup\{|f_{i}(t,0,0)|,0\leq t\leq 1,i=1,2\},$ and consider the closed subset of $(C(I,E))^{2}$ defined by:

\mathcal{B}_{R}=\{(u,v)\in(C(I,E))^{2};\|(u,v)\|_{C^{2}}\leq\left(\begin{% matrix}R\\ R\end{matrix}\right)\}.

Theorem 5.1.

Assume that hypotheses $(H_{1})-(H_{3})$ hold. Let $U$ be a weakly open subset of $\mathcal{B}_{R}.$ If

(5.3)

\frac{p_{i1}^{\ast}+p_{i2}^{\ast}}{(\alpha+\beta_{i}-2)\Gamma(\beta_{i})\Gamma% (\alpha)}<\frac{1}{4}

for $i\in\{1,2\}$ and if for any solution $(u,v)$ of $(u,v)=\lambda A(u,v)+\lambda B(\frac{u}{\lambda},\frac{v}{\lambda})$ with $\lambda\in(0,1),$ we have $(u,v)\not\in\partial_{\mathcal{B}_{R}}U,$ then the coupled system $(\ref{appp})-(\ref{appp1})$ has at least one weak solution defined on $I .$

Proof.

We shall show that the operators $A$ and $B$ satisfies all the assumptions of Corollary $\ref{cor1}.$ The proof will be given in several steps.
Step $1:$ $A$ and $B$ are relatively weakly compact. Let $(u_{n},v_{n})$ be a sequence in $\mathcal{B}_{R}$ and let $(u_{n}(t),v_{n}(t))\rightharpoonup(u(t),v(t))$ in $(E\times E,\omega)$ for each $t\in I.$ Fix $t\in I,$ since the functions $f_{i},i=1,2$ satisfy the assumption $(H_{1}),$ we have $f_{i}(t,u_{n}(t),v_{n}(t))$ converge weakly uniformly to $f_{i}(t,u(t),v(t)).$ Hence the Lebesgue dominated convergence theorem for Pettis integral implies that $A(u_{n},v_{n})(t)$ (respectively $B(u_{n},v_{n})(t)$ ) converges weakly uniformly to $A(u,v)(t)$ (respectively $B(u,v)(t)$ ) in $(E\times E,\omega),$ for each $t\in I.$ Thus, $A(u_{n},v_{n})\rightharpoonup A(u,v)$ and $B(u_{n},v_{n})\rightharpoonup B(u,v).$ Hence, $A$ and $B$ are weakly sequentially continuous.
Step $2:$ The operator $A$ is relatively weakly compact. Let $U$ be a weakly open subset of $B_{R}$ such that $0\in U.$ Let $(u,v)\in\overline{U}^{\omega}$ be an arbitrary point. We shall prove $A(u,v)\in\mathcal{B}_{R}.$ Fix $t\in I$ and consider $A(u,v)(t).$ Without loss of generality, we may assume that $A_{i}(u,v)(t)\neq 0.$ By the Hahn-Banach Theorem there exists $\varphi\in E^{\ast}$ with $\|\varphi\|=1$ such that $\|A_{i}(u,v)(t)\|_{E}=\varphi(A_{i}(u,v)(t)).$ Thus,

	$\displaystyle\\|A_{i}(u,v)(t)\\|_{E}$	$\displaystyle\leq\int_{0}^{1}G_{i}(t,r)\varphi(f_{i}(r,u(r),v(r)))dr$
		$\displaystyle\leq\frac{1}{(\alpha+\beta_{i}-2)\Gamma(\beta_{i})\Gamma(\alpha)}% \int_{0}^{1}\varphi(f_{i}(r,u(r),v(r))-f_{i}(r,0,0))+f_{i}(r,0,0)dr$
		$\displaystyle\leq\frac{1}{(\alpha+\beta_{i}-2)\Gamma(\beta_{i})\Gamma(\alpha)}% (p_{i1}^{\ast}\\|u\\|_{E}+p_{i2}^{\ast}\\|v\\|_{E}+L)$
		$\displaystyle\leq\frac{p_{i1}^{\ast}R+p_{i2}^{\ast}R+L}{(\alpha+\beta_{i}-2)% \Gamma(\beta_{i})\Gamma(\alpha)}$
		$\displaystyle\leq\frac{R}{2}$

Let $(A_{i}(u_{n},v_{n}))$ be any sequence in $A_{i}(\overline{U}^{\omega}).$ Notice that $\overline{U}^{\omega}$ is bounded. By reflexiveness, for each $t\in I$ the set $\{A_{i}(u_{n},v_{n})(t),n\in\mathbb{N}\}$ is relatively weakly compact. Let $(u,v)\in\overline{U}^{\omega},0\leq t\leq s\leq 1,$ we have

	$\displaystyle\\|$	$\displaystyle A_{i}(u,v)(t)-A_{i}(u,v)(s)\\|_{E}\leq\int_{0}^{t}\|G_{i}(t,r)-G_{% i}(s,r)\|\varphi(f_{i}(r,u(r),v(r)))dr$
		$\displaystyle+\int_{t}^{s}\|G_{i}(t,r)-G_{i}(s,r)\|\varphi(f_{i}(r,u(r),v(r)))dr% +\int_{s}^{1}\|G_{i}(t,r)-G_{i}(s,r)\|\varphi(f_{i}(r,u(r),v(r)))dr$
		$\displaystyle\leq\frac{L}{\Gamma(\beta_{i})\Gamma(\alpha)}(\frac{3(s^{\beta_{i% }-1}-t^{\beta_{i}-1})}{\beta_{i}-1}+\frac{2((s^{\beta_{i}}-t^{\beta_{i}})-(s-t% )^{\beta_{i}})}{\beta_{i}}+3(s-t)).$

Consequently, $\|A_{i}(u,v)(t)-A_{i}(u,v)(s)\|_{E}\to 0,$ when $t\mapsto s,$ for all $i\in\{1,2\}.$ One shows that $\{A(u_{n},v_{n});n\in\mathbb{N}\}$ is a weakly equicontinuous subset of $C^{2}.$ It follows now from the Ascoli-Arzela Theorem that $(A(u_{n},v_{n}))$ is relatively weakly compact.
Step $3:$ $B$ is $M-$ contraction and $B(\mathcal{B}_{R})$ is bounded. Indeed, let $(u,v)\in\overline{U}^{\omega},$ then by using hypothesis $(H_{3})$ it yields

	$\displaystyle\\|B_{i}(u_{1},u_{2})(t)$	$\displaystyle-B_{i}(v_{1},v_{2})(t)\\|_{E}$
		$\displaystyle\leq\|g_{i}(t)\|\int_{0}^{1}s^{\alpha-1}\varphi(f_{i}(s,u_{1}(s),u_% {2}(s))-f_{i}(s,v_{1}(s),v_{2}(s)))ds$
		$\displaystyle\leq\frac{p_{i1}^{\ast}\\|u_{1}-v_{1}\\|_{C}+p_{i2}^{\ast}\\|u_{2}-v% _{2}\\|_{C}}{(\alpha+\beta_{i}-2)\Gamma(\beta_{i})\Gamma(\alpha)}$

Then

\|B(u_{1},u_{2})-B(v_{1},v_{2})\|_{C^{2}}\leq M\|u-v\|_{C^{2}}

where

M=\left(\begin{matrix}\frac{p_{11}^{\ast}}{(\alpha+\beta_{1}-2)\Gamma(\beta_{1% })\Gamma(\alpha)}&\frac{p_{12}^{\ast}}{(\alpha+\beta_{1}-2)\Gamma(\beta_{1})% \Gamma(\alpha)}\\ \frac{p_{21}^{\ast}}{(\alpha+\beta_{2}-2)\Gamma(\beta_{2})\Gamma(\alpha)}&% \frac{p_{22}^{\ast}}{(\alpha+\beta_{2}-2)\Gamma(\beta_{2})\Gamma(\alpha)}\end{% matrix}\right)

Also as in step $2,$ we have

\|B_{i}(u,v)(t)\|_{E}\leq\frac{R}{2}

Step $4:$ Let $(u_{1},u_{2})\in\overline{U}^{\omega}$ and $(v_{1},v_{2})\in\mathcal{B}_{R}.$ It follows that $A(u_{1},u_{2})+B(v_{1},v_{2})\in\mathcal{B}_{R}.$ Hence, the result follows.

Example 1.

Let

E=l^{1}=\{u=(u_{1},u_{2},\cdots,u_{n},\cdots),\sum_{n=1}^{\infty}|u_{n}|<\infty\}

be the Banach space with the norm

\|u\|_{E}=\sum_{n=1}^{\infty}|u_{n}|.

We consider the following coupled fractional order system

(5.4)

\left\{\begin{array}[]{ll}D_{1^{-}}^{1.2}(D_{0^{+}}^{1.9}u_{n}(t))=f_{n}(t,u(t% ),v(t))&\\ D_{1^{-}}^{1.2}(D_{0^{+}}^{1.9}v_{n}(t))=g_{n}(t,u(t),v(t))&\\ D_{0^{+}}^{1.9}u_{n}(0)=D_{0^{+}}^{1.9}u_{n}(1)=0&\\ D_{0^{+}}^{1.9}v_{n}(0)=D_{0^{+}}^{1.9}v_{n}(1)=0&\\ u_{n}^{\prime}(1)=u_{n}(0)=0,~{}~{}v_{n}^{\prime}(1)=v_{n}(0)=0,&\end{array}\right.

$(\alpha=1.2,\beta_{1}=\beta_{2}=1.9),$ where

f_{n}(t,u(t),v(t))=\frac{c}{n^{2}}(te^{-7}u_{n}(t)+\frac{e^{-(t+5)}}{1+v_{n}(t% )}),

and

g_{n}(t,u(t),v(t))=\frac{c}{n^{2}}(\frac{te^{-6}}{1+v_{n}(t)}),~{}~{}t\in I

with

u=(u_{1},u_{2},\cdots,u_{n},\cdots),~{}~{}v=(v_{1},v_{2},\cdots,v_{n},\cdots),% ~{}~{}c:=\frac{0.1e^{4}}{4}\Gamma(1.2)\Gamma(1.9).

Set

f=(f_{1},f_{2},\cdots,f_{n},\cdots)~{}\text{and}~{}g=(g_{1},g_{2},\cdots,g_{n}% ,\cdots).

Clearly the functions $f$ and $g$ are continuous. For each $u,v\in E$ and $t\in I,$ we have

\|f(t,u_{1}(t),u_{2}(t))-f(t,v_{1}(t),v_{2}(t))\|_{E}\leq c(e^{-7}\|u_{1}(t)-v% _{1}(t)\|+e^{-(t+5)}\|u_{2}(t)-v_{2}(t)\|),

\|g(t,u_{1}(t),u_{2}(t))-g(t,v_{1}(t),v_{2}(t))\|_{E}\leq cte^{-6}\|u_{2}(t)-v% _{2}(t)\|

and

L=\frac{c\pi^{2}}{6}e^{-5}.

Hence, the hypothesis $(H_{3})$ is satisfied with $p_{11}^{\ast}=ce^{-7},p_{12}^{\ast}=ce^{-5},p_{21}^{\ast}=0$ and $p_{22}^{\ast}=ce^{-6}.$ We shall show that condition $(\ref{c1})$ holds. Indeed:

\sup_{i=1,2}\{\frac{p_{i1}^{\ast}+p_{i2}^{\ast}}{(\alpha+\beta_{i}-2)\Gamma(% \beta_{i})\Gamma(\alpha)}\}<\frac{1}{8}

So, all conditions of Theorem 5.1 are satisfied. Let now $U$ be a weakly subset of $\mathcal{B}_{R},(R>\frac{\pi^{2}}{6e}).$ If for any solution $(u,v)$ of $(u,v)=\lambda A(u,v)+\lambda B(\frac{u}{\lambda},\frac{v}{\lambda})$ with $\lambda\in(0,1),$ we have $(u,v)\not\in\partial_{\mathcal{B}_{R}}U$ then the coupled system $(\ref{a})$ has at least one solution $(u,v)$ in $\mathcal{B}_{R}.$

References

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	$\displaystyle\\|\mathcal{H}(\lambda)(\Psi_{1},\Psi_{2})-\mathcal{H}(\lambda)(% \Psi^{\prime}_{1},\Psi^{\prime}_{2})\\|$	$\displaystyle=\left(\begin{matrix}\\|(T_{H^{1}}-\lambda_{1}I)^{-1}(\mathcal{N}_% {-\sigma_{1}}(\Psi_{1},\Psi_{2})-\mathcal{N}_{-\sigma_{1}}(\Psi^{\prime}_{1},% \Psi^{\prime}_{2}))\\|\\ \\|(T_{H^{2}}-\lambda_{2}I)^{-1}(\mathcal{N}_{-\sigma_{2}}(\Psi_{1},\Psi_{2})-% \mathcal{N}_{-\sigma_{2}}(\Psi^{\prime}_{1},\Psi^{\prime}_{2}))\\|\end{matrix}\right)$
		$\displaystyle\leqslant\left(\begin{matrix}\\|(T_{H^{1}}-\lambda_{1}I)^{-1}\\|\\|% \mathcal{N}_{-\sigma_{1}}(\Psi_{1},\Psi_{2})-\mathcal{N}_{-\sigma_{1}}(\Psi^{% \prime}_{1},\Psi^{\prime}_{2})\\|\\ \\|(T_{H^{2}}-\lambda_{2}I)^{-1}\\|\\|\mathcal{N}_{-\sigma_{2}}(\Psi_{1},\Psi_{2}% )-\mathcal{N}_{-\sigma_{2}}(\Psi^{\prime}_{1},\Psi^{\prime}_{2})\\|\end{matrix}\right)$
		$\displaystyle\leqslant\left(\begin{matrix}\\|(T_{H^{1}}-\lambda_{1}I)^{-1}\\|(\\|% \rho_{1,1}\\|_{\infty}\\|\Psi_{1}-\Psi^{\prime}_{1}\\|+\\|\rho_{1,2}\\|_{\infty}\\|% \Psi_{2}-\Psi^{\prime}_{2}\\|)\\ \\|(T_{H^{2}}-\lambda_{2}I)^{-1}\\|(\\|\rho_{2,1}\\|_{\infty}\\|\Psi_{1}-\Psi^{% \prime}_{1}\\|+\\|\rho_{2,2}\\|_{\infty}\\|\Psi_{2}-\Psi^{\prime}_{2}\\|)\end{% matrix}\right)$
		$\displaystyle\leqslant\max_{j\in\{1,2\}}(\\|(T_{H^{j}}-\lambda_{j}I)^{-1}\\|)% \left(\begin{matrix}\\|\rho_{1,1}\\|_{\infty}&\\|\rho_{1,2}\\|_{\infty}\\ \\|\rho_{2,1}\\|_{\infty}&\\|\rho_{2,2}\\|_{\infty}\end{matrix}\right)\left(\begin% {matrix}\\|\Psi_{1}-\Psi^{\prime}_{1}\\|\\ \\|\Psi_{2}-\Psi^{\prime}_{2}\\|\end{matrix}\right)$
		$\displaystyle\leqslant M\\|(\Psi_{1},\Psi_{2})-(\Psi^{\prime}_{1}-\Psi^{\prime}% _{2})\\|$

	$\displaystyle\\|\mathcal{H}(\lambda)(\varphi_{1},\varphi_{2})+\mathcal{F}(% \lambda)(\Psi_{1},\Psi_{2})\\|$	$\displaystyle=\left\\|\left(\begin{matrix}(T_{H^{1}}-\lambda_{1}I)^{-1}\Big{(}% \mathcal{N}_{-\sigma_{1}}(\varphi_{1},\varphi_{2})+F_{1}\mathcal{N}_{f_{1}}L_{% 1}(\Psi_{1},\Psi_{2})\Big{)}\\ (T_{H^{2}}-\lambda_{2}I)^{-1}\Big{(}\mathcal{N}_{-\sigma_{2}}(\varphi_{1},% \varphi_{2})+F_{2}\mathcal{N}_{f_{2}}L_{2}(\Psi_{1},\Psi_{2})\Big{)}\end{% matrix}\right)\right\\|$
		$\displaystyle\leqslant\left(\begin{matrix}\left\\|(T_{H^{1}}-\lambda_{1}I)^{-1}% \right\\|\left(\left\\|\mathcal{N}_{-\sigma_{1}}(\varphi_{1},\varphi_{2})\right% \\|+\\|F_{1}\\|\\|(\mathcal{N}_{f_{1}}L_{1}(\Psi_{1},\Psi_{2}))\\|\right)\\ \left\\|(T_{H^{2}}-\lambda_{2}I)^{-1}\right\\|\left(\left\\|\mathcal{N}_{-\sigma_% {2}}(\varphi_{1},\varphi_{2})\right\\|+\\|F_{2}\\|\\|(\mathcal{N}_{f_{2}}L_{2}(% \Psi_{1},\Psi_{2}))\\|\right)\end{matrix}\right)$
		$\displaystyle\leqslant\left(\begin{matrix}\left\\|(T_{H^{1}}-\lambda_{1}I)^{-1}% \right\\|\left(M_{1}(r_{1},r_{2})+\\|F_{1}\\|(\\|a_{1}\\|\\|h_{1}\\|_{\infty}\right)% \\ \left\\|(T_{H^{2}}-\lambda_{2}I)^{-1}\right\\|\left(M_{2}(r_{1},r_{2})+\\|F_{2}\\|% (\\|a_{2}\\|\\|h_{2}\\|_{\infty}\right)\end{matrix}\right)$

	$\displaystyle\\|$	$\displaystyle A_{i}(u,v)(t)-A_{i}(u,v)(s)\\|_{E}\leq\int_{0}^{t}\|G_{i}(t,r)-G_{% i}(s,r)\|\varphi(f_{i}(r,u(r),v(r)))dr$
		$\displaystyle+\int_{t}^{s}\|G_{i}(t,r)-G_{i}(s,r)\|\varphi(f_{i}(r,u(r),v(r)))dr% +\int_{s}^{1}\|G_{i}(t,r)-G_{i}(s,r)\|\varphi(f_{i}(r,u(r),v(r)))dr$
		$\displaystyle\leq\frac{L}{\Gamma(\beta_{i})\Gamma(\alpha)}(\frac{3(s^{\beta_{i% }-1}-t^{\beta_{i}-1})}{\beta_{i}-1}+\frac{2((s^{\beta_{i}}-t^{\beta_{i}})-(s-t% )^{\beta_{i}})}{\beta_{i}}+3(s-t)).$

	$\displaystyle\\|B_{i}(u_{1},u_{2})(t)$	$\displaystyle-B_{i}(v_{1},v_{2})(t)\\|_{E}$
		$\displaystyle\leq\|g_{i}(t)\|\int_{0}^{1}s^{\alpha-1}\varphi(f_{i}(s,u_{1}(s),u_% {2}(s))-f_{i}(s,v_{1}(s),v_{2}(s)))ds$
		$\displaystyle\leq\frac{p_{i1}^{\ast}\\|u_{1}-v_{1}\\|_{C}+p_{i2}^{\ast}\\|u_{2}-v% _{2}\\|_{C}}{(\alpha+\beta_{i}-2)\Gamma(\beta_{i})\Gamma(\alpha)}$

Krasnoselskii type theorems in product Banach spaces and applications to systems of nonlinear transport equations and mixed fractional differential equations

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Krasnoselskii type theorems in product Banach spaces and applications to systems of nonlinear transport equations and mixed fractional differential equations

Abstract.

1. Introduction

2. Preliminaries

3. Fixed point results

Proposition 3.1.

Proof.

Theorem 3.1.

Proof.

Theorem 3.2.

Theorem 3.3.

Proof.

Theorem 3.4.

Proof.

Remark 3.1.

Corollary 3.1.

4. Application I: Solutions of a system of nonlinear transport equations

Remark 4.1.

Proposition 4.1.

Proof.

Definition 1.

Lemma 1.

Definition 2.

Lemma 2.

Proof.

Theorem 4.1.

Proof.

5. Application II: Existence of weak solutions

Definition 3.

Definition 4.

Proposition 5.1.

Proposition 5.2.

Definition 5.

Lemma 3.

Lemma 4.

Theorem 5.1.

Proof.

Example 1.

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