Existence Results for a Coupled System of Perturbed Functional Differential Inclusions in Banach Algebras

Abstract

In this paper, we establish some new fixed point theorems for a $2\times 2$ block operator matrix defined on nonempty, bounded, closed, and convex subsets of Banach algebras, where the inputs are multi-valued mappings. The obtained results are then applied to a coupled system of perturbed functional differential inclusions of initial and boundary value problems in order to prove the existence of solutions under a Carathéodory condition.

1 Introduction

The study of differential and integral inclusions in a Banach space has been achieved in the literature for a long time. This study was performed via fixed point techniques by combining the Banach’s metric fixed point theorem with the well-known Schauder’s fixed point theorem. There have been various extensions of Krasnoselskii’s fixed point theorem over the course of time. See for example [7, 10, 18] and the references therein. In this work, we are concerned with fixed point results on Banach algebras of operators defined by a $2\times 2$ block operator matrix

$$\begin{aligned} \displaystyle \left( \begin{array}{l@{\quad }c} A &{} B\cdot B^{\prime } \\ C &{} D \\ \end{array} \right) , \end{aligned}$$

(1.1)

where the entries of the matrix are generally nonlinear multi-valued operators defined on Banach algebras. Our assumptions are as follows: A maps a nonempty, bounded, closed, and convex subset S of a Banach algebra X into the classes, denoted ${\mathcal {P}}_{cl,cv,bd}(X),$ of all closed, convex, and bounded subsets of $X.\, B,B^{\prime }$ from X into ${\mathcal {P}}_{cl,cv,bd}(X),$ C from S into X and D from X into X. A point $(u,v)\in X\times X$ is called a fixed point of a $2\times 2$ block operator matrix (1.1), if we have

$$\begin{aligned} \left\{ \begin{array}{ll} u\in Au+Bv\cdot B^{\prime }v\\ v=Cu+Dv. \end{array} \right. \end{aligned}$$

In this direction, A. Jeribi et al. have established in [17, 20] some fixed point results for the block operator matrix (1.1), where the inputs are nonlinear single-valued mappings based on the convexity of the bounded domain, on the well-known Schauder’s fixed point theorem, and also on the properties of the inputs (cf. completely continuous, weakly sequentially continuous, etc, $\ldots$).

Recently, the authors Ben Amar et al. have established in [3] some existence theorems for a two-dimensional mixed boundary problem, based on a new generalized Schauder’s fixed point theorem, in terms of completely continuous, and on Krasnoselskii’s fixed point theorem for the block operator matrix (1.1) acting on $L_p\times L_p$ space for $p\in ]1, +\infty [$ in the case where $B^{\prime }=1.$ The equations model the evolution of a cell population.

Later, A. Jeribi et al. have established in [19] new variants of fixed point theorems for the operator (1.1), in the case where $B^{\prime }=1$ and applied their results to the existence of solutions for a transport problem occuring in biology. Due to the lack of compactness of the operator $C(I-A)^{-1}$ in $L_1$ space for the transport equations (see [16]), the analysis was carried out via topological arguments and used the weak non-compactness measure results established in [4].

In a series of papers [6, 8], Dhage has focused on the existence of a solution for the following inclusion

$$\begin{aligned} \displaystyle x\in Ax\cdot Bx+Cx \end{aligned}$$

(1.2)

and obtained several valuable results in Banach algebras. They were mainly based on the convexity of the bounded domain and the properties of the operators A, B, and C (cf. lower semi-continuous, upper semi-continuous, completely continuous, ...). For some topological properties of the set of solutions of equation (1.2), see [21].

Our main results are applied to investigate the existence of solutions for the following two-dimensional quadratic initial value problems for differential inclusion

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle {\frac{\partial }{\partial t}}\left( \displaystyle {\frac{x(t) - k(t, x(t))}{f(t, y(t))}}\right) \in G(t,y(t))\\ y(t)=\displaystyle {\frac{1}{1+ b(t)|x(\theta (t))|}}- p\left( t, \frac{1}{1+ b(t)|x(\theta (t))|}\right) + p(t, y(t))\\ (x(0), y(0))=(x_0,y_0)\in {\mathbb {R}}^2, \end{array} \right. \end{aligned}$$

where $t\in J, b,\theta , k, f$ are given functions, $G: J\times {\mathbb {R}} \longrightarrow {\mathcal {P}}_{cp, cv}({\mathbb {R}})$ is a multifunction, whereas $x=x(t)$ and $y=y(t)$ are unknown functions. Notice that this system can be written as a fixed point problem

$$\begin{aligned} \displaystyle \left\{ \begin{array}{lll} x(t)\in k(t, x(t))+ f(t,y(t))\cdot \left[ \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}+\displaystyle \int _0^tG (t,y(s))\mathrm{d}s\right] \\ y(t)=\displaystyle {\frac{1}{1+ b(t)|x(\theta (t))|}}- p\left( t, \frac{1}{1+ b(t)|x(\theta (t))|}\right) + p(t, y(t))\\ (x(0), y(0))=(x_0,y_0)\in {\mathbb {R}}^2. \end{array} \right. \end{aligned}$$

(1.3)

The initial value problem IVP (1.3) is new in the theory of differential inclusions and some special cases belonging to it have been extensively discussed in the literature. The special case when $x=y, p(t,y)=y, f(t,x)=1,$ and k is constant in t,  and IVP (1.3) reduced to IVP

$$\begin{aligned} \displaystyle \left\{ \begin{array}{lll} \displaystyle {\frac{\partial }{\partial t}}\left( x(t)\right) \in G(t,x(t))\\ x(0)=x_0\in {\mathbb {R}}, \end{array} \right. \end{aligned}$$

(1.4)

for all $t\in J,$ where $G: J\times {\mathbb {R}} \longrightarrow {\mathcal {P}}_{cp, cv}({\mathbb {R}})$ has been discussed in the literature for various aspects of the solutions for the IVP (1.4). See Aubin [1] and Papageorgiou [15], and the references therein. Similarly, if $x=y,$ $p(t,x)=x$ and $G(t, x) =\{g(t, x)\},$ we obtain the differential equation

$$\begin{aligned} \displaystyle \left\{ \begin{array}{lll} \displaystyle {\frac{\partial }{\partial t}}\left( \displaystyle {\frac{x(t) - k(t, x(t))}{f(t, x(t))}}\right) = g(t,x(t))\\ x(0)=x_0\in {\mathbb {R}}, \end{array} \right. \end{aligned}$$

(1.5)

for all $t\in J,$ where $g:J\times {\mathbb {R}}\longrightarrow {\mathbb {R}}$ is a nonlinear function. The IVP (1.5) has been discussed in [10, 11] for different aspects of the solutions under suitable conditions.

Our basic strategy is as follows. In Sect. 2 we will recall some definitions and we will give some basic results for a future use. In Sect. 3 we will deal with some fixed point results for $2\times 2$ block operator matrices, which consist of operators acting on nonempty, bounded, closed, and convex subsets in Banach algebras. In Sect. 4, we will use the results of Sect. 3 in order to discuss the existence of solutions for the system (1.3).

2 Auxiliary Facts and Results

Throughout this paper, unless otherwise mentioned, let X be a Banach algebra and let ${\mathcal {P}}_{p}(X)$ be the class of all nonempty subsets of X with property p. Thus ${\mathcal {P}}_{cl}(X), {\mathcal {P}}_{bd}(X), {\mathcal {P}}_{cp}(X)$, and ${\mathcal {P}}_{cv}(X)$ denote, respectively, the classes of all closed, bounded, compact, and convex subsets of X. Similarly, let ${\mathcal {P}}_{cl, bd}(X)$ and ${\mathcal {P}}_{cp,cv}(X)$ be, respectively, the classes of all closed-bounded and compact-convex subsets of X.

A correspondence $Q : X\longrightarrow {\mathcal {P}}_{p}(X)$ is called a multi-valued operator or a multi-valued mapping on X into X. A point $x \in X$ is called a fixed point of Q if $x \in Qx$ and the set of all fixed points of Q in X is denoted by ${\mathcal {F}}_Q.$ For the sake of convenience, we denote $Q(A) = \displaystyle {\cup _{x\in A}}Qx$ for all subsets A of X.

For $x\in X$ and $A,B \in {\mathcal {P}}_{cl}(X)$, we denote by $d(x, A)=\inf \{\Vert x-y\Vert ~;~ y\in A\}.$ Let us define a function $d_H: {\mathcal {P}}_{cl}(X)\times {\mathcal {P}}_{cl}(X) \longrightarrow {\mathbb {R}}_+$ by

$$\begin{aligned} d_H(A,B)=\max \{\displaystyle {\sup _{x\in A}} ~d(x,B),\displaystyle {\sup _{y\in B}}~ d(y,A)\}. \end{aligned}$$

The function $d_H$ is called a Hausdorff metric on X. Note that $\Vert A\Vert _{{\mathcal {P}}}=d_H(A,\{0\}).$ The concept of a Hausdorff metric was used by many authors in order to prove the fixed point and coincidence point results in the setting of metric spaces.

Let $Q: X\longrightarrow {\mathcal {P}}_{p}(X)$ be a multi-valued map. For any subset A of X,  we define

$$\begin{aligned} Q^-(A)=\{x\in X ~;~ Q(x)\cap A\not =\emptyset \} ~~\text { and }~~ Q^+(A)=\{x\in X ~;~ Q(x)\subset A\}. \end{aligned}$$

Definition 2.1

A multi-valued map Q is called lower semi-continuous (l.s.c.) if for every open subset V of X,  the set $Q^-(V)$ is open in X. $\diamondsuit$

Definition 2.2

A multi-valued map Q is called upper semi-continuous (u.s.c.) if for every open subset V of X,  the set $Q^+(V)$ is open in X. $\diamondsuit$

Definition 2.3

A multi-valued map $Q: X\longrightarrow {\mathcal {P}}_{p}(X)$ is called totally bounded if Q(S) is a totally bounded subset of X for all bounded sets S in X. Again, Q is called completely continuous on X,  if it is u.s.c. and totally bounded on X. $\diamondsuit$

We restrict our considerations to multi-valued maps with compact values.

Lemma 2.1

(See  [14], Proposition 6.13) Let $Q: X\longrightarrow {\mathcal {P}}_{cp}(X)$ be an u.s.c. map, and let S be a compact subset of X. Then, Q(S) is compact. $\diamondsuit$

The classical Banach contraction principle states that every self contraction map on a complete metric space has a unique fixed point. In 1969, using the Hausdorff’s metric notion, Covitz and Nadler have proved in [5] a set valued version of the Banach contraction principle. Now, we consider some known results which will be used in the following sections.

Theorem 2.1

(See  [5]) Let (X, d) be a complete metric space and let $Q : X\longrightarrow {\mathcal {P}}_{cl}(X)$ be a multi-valued contraction. Then, the fixed point set ${\mathcal {F}}_Q$ of Q is a nonempty and closed subset of X. $\diamondsuit$

Lemma 2.2

(See [23]) Let (X, d) be a Banach space and let $Q_1,Q_2: X\longrightarrow {\mathcal {P}}_{cl,cv}(X)$ be two multi-valued contractions with the same contraction constant k. Then

$$\begin{aligned} d_H({\mathcal {F}}_{Q_1},{\mathcal {F}}_{Q_2})\le \displaystyle \frac{1}{1-k}\sup \left\{ d_H(Q_1(x),Q_2(x)) ~;~ x\in X\right\} . \end{aligned}$$

$\diamondsuit$

The following result is due to Rybinski in [25] and will be useful in the sequel.

Theorem 2.2

Let S be a nonempty and closed subset of a Banach space X and let Y be a metric space. Assume that the multi-valued operator $Q: S\times Y\longrightarrow {\mathcal {P}}_{cl, cv}(S)$ satisfies

• (a) $d_H(Q(x_1,y),Q(x_2,y))\le q\Vert x_1-x_2\Vert$ for all $(x_1,y), (x_2,y)\in S\times Y,$ with $q<1,$

• (b) for every $x\in S,~Q(x,\cdot )$ is lower semi-continuous on Y.

Then, there exists a continuous mapping $f:S\times Y\longrightarrow S$ such that $f(x,y)\in Q(f(x,y), y)$ for each $(x,y)\in S\times Y.$ $\diamondsuit$

The Hausdorff’s measure $\beta$ of non-compactness in a Banach space X is a nonnegative real number $\beta (S)$ defined by

$$\begin{aligned} \beta (S)=\inf \{r>0;~\text { there exists a finite set }\,F\, \text {with}\,S \subseteq F+{\mathcal {B}}(0,r)\}, \end{aligned}$$

for all bounded subset S of X,  where ${\mathcal {B}}(0,r)=\{x\in X ~;~ \Vert x\Vert < r\}.$

Hausdorff’s measure of non-compactness is discussed in Papageorgiou et al. [15], Deimling [13], and the references therein.

Definition 2.4

A multi-valued map $Q : X\longrightarrow {\mathcal {P}}_{cl,bd}(X)$ is called ${\mathcal {D}}$-set-Lipschitz if there is a continuous nondecreasing function $\psi : {\mathbb {R}}^+\longrightarrow {\mathbb {R}}^+$ such that $\beta (Q(S)) \le \psi (\beta (S))$ for all $S \in {\mathcal {P}}_{ cl,bd}(X)$ with $Q(S)\in {\mathcal {P}}_{ cl,bd}(X)$ where $\psi (0)=0.$ Sometimes, the function $\psi$ is called a ${\mathcal {D}}$-function of Q on X. In the special case, when $\psi (r) = kr, k > 0,$ the map Q is called k-set-Lipschitz mapping, and if $k < 1,$ then Q is called a k-set-contraction mapping on X. Mereover, if $\psi (r) < r$ for $r >0,$ then Q is called a nonlinear ${\mathcal {D}}$-set-contraction mapping on X.

We need the following result in the sequel.

Theorem 2.3

(See [12], Theorem 2.1) Let S be a nonempty, closed, convex, and bounded subset of a Banach space X and let $Q: S\longrightarrow {\mathcal {P}}_{cl, cv}(S)$ be a closed and nonlinear ${\mathcal {D}}$-set-contraction mapping. Then, Q has at least one fixed point. $\diamondsuit$

Before reaching the main fixed point results, the useful lemmas for the sequel are stated.

Lemma 2.3

(See [9]) Let X be a Banach space and $S_1, S_2, S_3\in {\mathcal {P}}_{bd,cl}(X).$ Then

$$\begin{aligned} d_H(S_1\cdot S_3,~S_2\cdot S_3)\le d_H(0,S_3)d_H(S_1,S_2). \end{aligned}$$

Lemma 2.4

(See [6]) Let $Q: X\rightarrow {\mathcal {P}}_{cl}(X)$ be a Lipschitz multi-valued mapping. Then, for any bounded subset S in X,  the set Q(S) is bounded. $\diamondsuit$

Lemma 2.5

(See [2]) Assume that X is a Banach space and $S_1, S_2\in {\mathcal {P}}_{bd}(X).$ Then

$$\begin{aligned} \beta (S_1\cdot S_2)\le \beta (S_1)\Vert S_2\Vert _{{\mathcal {P}}}+ \beta (S_2)\Vert S_1\Vert _{{\mathcal {P}}}. \end{aligned}$$

The following result plays the role of a prototype in our discussion of this paper

Theorem 2.4

(See  [26]) Let S be a closed and convex subset of a Banach space X,  and let $Q : S\longrightarrow {\mathcal {P}}_{cl,cv}(S)$ be a multi-valued mapping. Suppose that

• (i) Q is a compact multi-valued mapping, and

• (ii) Q is upper semi-continuous on S.

Then, there exists $x\in S$ such that $x\in Q(x).$

We state a key result due to B. C. Dhage and a proof can be found in [6].

Theorem 2.5

(See [6]) Let S be a nonempty, closed, convex, and bounded subset of the Banach algebra X,  and let $A,C:X \longrightarrow {\mathcal {P}}_{cl,cv,bd}(X)$ and $B:S\longrightarrow {\mathcal {P}}_{cp,cv}(X)$ be three multi-valued operators, such that

• (i) A and C are multi-valued Lipschitz operators with the Lipschitz constants $q_1$ and $q_2$, respectively,

• (ii) B is lower semi-continuous and compact,

• (iii) $Ax\cdot By+Cx\in {\mathcal {P}}_{cl,cv}(X),$ for all $x,y\in S,$ and

• (iv) $q_1M+q_2<1,$ where $M=\Vert B(S)\Vert _{{\mathcal {P}}}=\sup \{\Vert Bx\Vert _{{\mathcal {P}}} ~;~ x\in S\}.$

Then, the equation (1.2) has a solution. $\diamondsuit$

3 Multi-valued Fixed Point Theory

In what follows, we present the fixed point theorem for multi-valued mappings in a Banach algebra.

Theorem 3.1

Let S be a nonempty, bounded, closed, and convex subset of a Banach algebra X,  and let $A: S\longrightarrow {\mathcal {P}}_{cl,cv, bd}(X),$ $B: X\longrightarrow {\mathcal {P}}_{cl,cv, bd}(X), B^{\prime }: X \longrightarrow {\mathcal {P}}_{cp, cv}(X),$ $C: S\longrightarrow X$ and $D: X \rightarrow X$ be five multi-valued operators satisfying:

• (i) A, B, and C are Lipschitzians with the Lipschitz constants $q_1, q_2$, and $q_3$ respectively,

• (ii) D is a contraction mapping with a constant k and $C(S)\subseteq (I-D)(S),$

• (iii) $B^{\prime }$ is lower semi-continuous and totally bounded, and

• (iv) $Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cy \in {\mathcal {P}}_{cl, cv}(S),$ for all $x,y\in S.$

Then, the block operator matrix (1.1) has a fixed point whenever $\frac{q_2 q_3}{1-k}M+ q_1< 1,$ where $M=\Vert B^{\prime }(I-D)^{-1}C(S)\Vert _{{\mathcal {P}}}.$ $\diamondsuit$

Proof of Theorem 3.1

Since D is contraction, it follows that $(I-D)$ is injective and then the operator inverse $(I-D)^{-1}$ exists on $(I-D)(X).$

Let $y\in S$ be fixed and let us define the mapping $T_y: S \longrightarrow \mathrm {P}_{cl,cv}(S)$ by

$$\begin{aligned} T_y(x)=Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cy. \end{aligned}$$

Notice that $T_y$ is a multi-valued contraction for each fixed $y\in S,$ since we have

$$\begin{aligned} \begin{array}{lll} d_H(T_y(x_1),T_y(x_2))&{}\le &{} d_H(Ax_1, Ax_2)~~+\\ &{}&{}\left\| B^{\prime }(I-D)^{-1}C(S) \right\| _{{\mathcal {P}}}d_H(B(I-D)^{-1}Cx_1,B(I-D)^{-1}Cx_2)\\ &{}\le &{} \left( q_1+M\displaystyle {\frac{q_2q_3}{1-k}}\right) \Vert x_1-x_2\Vert \end{array} \end{aligned}$$

whenever $x_1,x_2\in S.$ It follows from Theorem 2.1 that, the fixed point set ${\mathcal {F}}_{T_y}$ given by

$$\begin{aligned} {\mathcal {F}}_{T_y} =\left\{ x\in S\, \text {such that } x\in Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cy\right\} \end{aligned}$$

is a nonempty and closed subset of S for each $y\in S.$ Moreover, by using Theorem 2.2 together with assumption (iii),  we can show that the existence of a continuous mapping $f:S\times S \longrightarrow S$ such that

$$\begin{aligned} f(x,y)\in A(f(x,y))+ B(I-D)^{-1}C(f(x,y))\cdot B^{\prime }(I-D)^{-1}Cy. \end{aligned}$$

Let us define

$$\begin{aligned} \left\{ \begin{array}{ll} G: S \longrightarrow \mathrm {P}_{cl}(S)\\ y \longrightarrow {\mathcal {F}}_{T_y} \end{array} \right. \quad \text { and }\quad \left\{ \begin{array}{ll} g: S\longrightarrow S\\ x \longrightarrow f(x,x). \end{array} \right. \end{aligned}$$

It follows that g is a continuous mapping having the property that

$$\begin{aligned} g(x)=f(x,x)\in A(f(x,x))+ B(I-D)^{-1}C(f(x,x))\cdot B^{\prime }(I-D)^{-1}Cx \end{aligned}$$

for each $x\in S.$ In order to prove that g is completely continuous on S,  it is sufficient to demonstrate that G is continuous and totally bounded on S. In fact, for any $z\in S,$ we have

$$\begin{aligned} \begin{array}{lll} \Vert B(I-D)^{-1}C(z)\Vert _{{\mathcal {P}}}&{}\le &{} \Vert B(I-D)^{-1}C(a)\Vert _{{\mathcal {P}}}+\displaystyle \frac{q_2q_3}{1-k}\Vert z-a\Vert \\ &{}\le &{}\Vert B(I-D)^{-1}C(a)\Vert _{{\mathcal {P}}}+\frac{\Vert z -a\Vert }{M}\\ &{}\le &{} \eta , \end{array} \end{aligned}$$

where

$$\begin{aligned} \displaystyle \eta =\Vert B(I-D)^{-1}C(a)\Vert _{{\mathcal {P}}}+\frac{diam(S)}{M}, \end{aligned}$$

(3.1)

for a fixed point a in S.

Let $\varepsilon >0$ be given. Since $B^{\prime }(I-D)^{-1}C$ is totally bounded on S,  then there exists a subset $Y=\{y_1,\ldots ,y_n\}$ as elements of S such that

$$\begin{aligned} B^{\prime }(I-D)^{-1}C(S)\subset \{w_1, \ldots , w_n\} + {\mathcal {B}}\left( 0, \displaystyle {\frac{(1-q_1)\varepsilon }{\eta }} -\displaystyle {\frac{Mq_2q_3}{\eta (1-k)}\varepsilon }\right) , \end{aligned}$$

where $w_i\in B^{\prime }(I-D)^{-1}C(y_i)$ and ${\mathcal {B}}(0,r)$ is the open ball in X centered at 0 of radius r. Then, for each $y\in S,$ there is an element $y_k\in Y,$ such that

$$\begin{aligned} d_H\left( B^{\prime }(I-D)^{-1}Cy, B^{\prime }(I-D)^{-1}Cy_k\right) <\displaystyle {\frac{(1-q_1) \varepsilon }{\eta }}-\displaystyle {\frac{Mq_2q_3}{\eta (1-k)}\varepsilon }. \end{aligned}$$

This implies that

$$\begin{aligned} \begin{array}{rcl} d_H(G(y),G(y_k))&{}=&{}d_H({\mathcal {F}}_{T_y}, {\mathcal {F}}_{T_{y_k}})\\ &{}\le &{} \displaystyle \frac{1}{1-\left( q_1+\displaystyle {\frac{Mq_2q_3}{1-k}}\right) }\displaystyle {\sup _{x\in S}}\left\{ d_H\left( T_y(x),T_{y_k}(x)\right) \right\} \\ &{}{<}&{} \displaystyle {\frac{\eta }{1-\left( q_1+\displaystyle {\frac{Mq_2q_3}{1-k}}\right) }} \left( \displaystyle {\frac{(1-q_1)\varepsilon }{\eta }} -\displaystyle {\frac{Mq_2q_3}{\eta (1-k)}\varepsilon }\right) \\ &{}\le &{} \varepsilon . \end{array} \end{aligned}$$

For each $u\in G(y),$ there exists $u_k\in G(y_k)$ such that $\Vert u-u_k\Vert <\varepsilon ,$ and thereby, for each $y\in Y,$ one has $G(y) \subseteq \displaystyle {\bigcup _{i=1}^n} {\mathcal {B}}(u_k, \varepsilon ),$ where $u_i\in G(y_i), i=1,\ldots ,n.$ Then,

$$\begin{aligned} g(S)\subset G(S) \subseteq \displaystyle {\bigcup _{i=1}^n} {\mathcal {B}}(u_k, \varepsilon ) \end{aligned}$$

and so, g is a completely continuous operator on S. Now, all the assumptions of Schauder’s fixed point theorem are satisfied by the mapping g. Consequently, there exists $u\in S$ such that $u=g(u)$. So, the vector $v=(I-D)^{-1}Cu$ completes this proof. $\square$

An improved version of Theorem 3.1 under a weaker assumption (v) therefore is provided in the following multi-valued fixed point theorem.

Theorem 3.2

Let S be a nonempty, bounded, closed, and convex subset of a Banach algebra X,  and let $A,B: X\longrightarrow {\mathcal {P}}_{cl,cv, bd}(X), B^{\prime }: X \longrightarrow {\mathcal {P}}_{cp, cv}(X)$ and $C, D: X \rightarrow X$ be five multi-valued operators satisfying:

• (i) A, B, and C are Lipschitzians with the Lipschitz constants $q_1, q_2$, and $q_3$, respectively,

• (ii) D is a contraction with a constant k and $C(S) \subseteq (I-D)(X),$

• (iii) $B^{\prime }$ is lower semi-continuous and compact,

• (iv) $Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cy \in {\mathcal {P}}_{cl, cv}(X),$ for all $x\in X$ and $y\in S,$ and

• (v) $x\in Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cy ~;~ y\in S~\Rightarrow x\in S.$

Then, the block operator matrix (1.1) has a fixed point whenever $q_1+\frac{q_2q_3}{1-k}M<1,$ where $M=\Vert B^{\prime }(S)\Vert _{{\mathcal {P}}}.$ $\diamondsuit$

Proof of Theorem 3.2

Let us define a multi-valued operator $T: X\times S \longrightarrow \mathrm{P}_{cl,cv}(X)$ by

$$\begin{aligned} T(x,y)=Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cy. \end{aligned}$$

Following the same procedures as in the proof of Theorem 3.1, it can be proved that the fixed point set

$$\begin{aligned} {\mathcal {F}}_{T_y} =\left\{ x\in X \text { such that } x\in Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cy\right\} \end{aligned}$$

is a nonempty, and closed subset of X for each $y\in S.$ As shown in assumption (v),  we notice that ${\mathcal {F}}_{T_y}\subset S$ for each $y\in S.$ Now, let us define

$$\begin{aligned} \left\{ \begin{array}{ll} G: S \longrightarrow \mathrm {P}_{cl}(S)\\ y \longrightarrow {{\mathcal {F}}}_{T_y} \end{array} \right. \quad \text { and }\quad \left\{ \begin{array}{ll} g: S\longrightarrow S\\ y \longrightarrow f(y,y). \end{array} \right. \end{aligned}$$

It follows that g is a continuous mapping having the property that

$$\begin{aligned} g(y)=f(y,y)\in A(f(y,y))+ B(I-D)^{-1}C(f(y,y))\cdot B^{\prime }(I-D)^{-1}Cy,~\text { for each } y\in S. \end{aligned}$$

Once again, we proceed with the same arguments as in the proof of Theorem 3.1 and we can show that there exists $u\in S,$ such that

$$\begin{aligned} u\in Au+ B(I-D)^{-1}Cu\cdot B^{\prime }(I-D)^{-1}Cu. \end{aligned}$$

The vector $v=(I-D)^{-1}Cu$ completes this proof. $\square$

Now, we can use Theorem 2.3, in conjunction with Lemma 2.5, in order to reach the following fixed point theorem.

Theorem 3.3

Let S be a nonempty, bounded, closed, and convex subset of a Banach algebra X,  and let $A, B, B^{\prime }: X\longrightarrow {\mathcal {P}}_{bd,cv}(X),$ and $C, D: S\longrightarrow X$ be five multi-valued operators satisfying:

• (i) A and B are ${\mathcal {D}}$-Lipschitzians with the ${\mathcal {D}}$-functions $\psi _1$ and $\psi _2$, respectively,

• (ii) C and D are Lipschitzian with the Lipschitz constants q and k, respectively, such that $q+k<1,$

• (iii) $C(S)\subseteq (I-D)(S),$

• (iv) $B^{\prime }$ is upper semi-continuous and compact, and

• (v) $Ax+ B(I-D)^{-1}Cx \cdot B^{\prime }(I-D)^{-1}Cx \in {\mathcal {P}}_{cl, cv}(S),$ for all $x,y\in S.$

Then, the block operator matrix (1.1) has a fixed point whenever $\psi _1(r)+M\psi _2(r)\le r,$ for each $r>0$ where $M=\Vert B^{\prime }(S)\Vert _{{\mathcal {P}}}.$ $\diamondsuit$

Proof of Theorem 3.3

Let us define the mapping

$$\begin{aligned} \left\{ \begin{array}{ll} T: S\longrightarrow {\mathcal {P}}_{{\mathcal {P}}}(S)\\ x\longrightarrow Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx. \end{array} \right. \end{aligned}$$

Obviously, Tx is a convex subset of S for each $x\in S,$ according to assumption (v). The use of Lemma 2.5, as well as assumption (i) leads to

$$\begin{aligned} \begin{array}{rcl} \beta (Tx)&{}=&{} \beta \left( Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx\right) \\ {} &{}\le &{} \beta (Ax)+\beta \left( B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx\right) \\ {} &{}\le &{} \psi _1\left( \beta (\{x\})\right) +\psi _2\left( \beta (\{\frac{q}{1-k} x\})\right) \Vert B^{\prime }(I-D)^{-1}Cx\Vert _{{\mathcal {P}}} , \end{array} \end{aligned}$$

whenever $x\in S.$ Then, T defines a multi-valued map $T: S\longrightarrow {\mathcal {P}}_{cv,cp}(S).$

Our next task is to show that T is closed. To do so, let $\{x_n\}_{n=0}^{\infty }$ be a sequence in S which converges to $x\in S.$ Let $\{y_n\}_{n=0}^{\infty }$ be a convergent sequence to the point y such that $y_n\in T(x_n)$ for each $n\in \mathbb {N}.$ Then,

$$\begin{aligned} \displaystyle d(y,Tx)\le \Vert y-y_n\Vert +d(y_n, Tx_n)+d_H(Tx_n,Tx). \end{aligned}$$

(3.2)

Now, for any $x,z\in S,$ we have

$$\begin{aligned} \begin{array}{lll} d_H\left( Tx, Tz\right) &{}\le \psi _1(\Vert x-z\Vert )+ M\psi _2\left( \frac{q}{1-k}\Vert x-z\Vert \right) \\ &{}\quad +\,\eta ~ d_H(B^{\prime }(I-D)^{-1}Cx, ~ B^{\prime }(I-D)^{-1}Cz)\\ &{}\le (\psi _1+M\psi _2)\left( \Vert x\!-\!z\Vert \right) +\eta ~ d_H(B^{\prime }(I-D)^{-1}Cx,~ B^{\prime }(I-D)^{-1}Cz), \end{array} \end{aligned}$$

where $\eta$ is defined in (3.1). Since $B^{\prime }$ is upper semi-continuous, then

$$\begin{aligned} d_H\left( B^{\prime }(I-D)^{-1}Cx_n , B^{\prime }(I-D)^{-1}Cx\right) \rightarrow 0, ~\text { whenever } x_n \rightarrow x. \end{aligned}$$

Hence, $d_H\left( Tx_n, Tx\right) \rightarrow 0 \, \text {as}\, n\rightarrow \infty .$ Since Tx is closed for all $x\in S,$ then $y\in Tx$ in view of inequality (3.2) and consequently, the multi-valued $T: S\rightarrow {\mathcal {P}}_{cp,cv}(S)$ is closed.

Finally, we demonstrate that T is a nonlinear ${\mathcal {D}}$-set-contraction on S. Let $S_1\subset S$ be arbitrary. Then, $S_1$ is bounded and also by using Lemma 2.4 to obtain that $A(S_1)$ and $(I-D)^{-1}C(S_1)$ are bounded. Since $B^{\prime }$ is compact, the set $B^{\prime }(I-D)^{-1}C(S_1)$ is relatively compact and hence, bounded in X. Since

$$\begin{aligned} T(S_1)\subset A(S_1)+B(I-D)^{-1}C(S_1)\cdot B^{\prime }(I-D)^{-1}C(S_1), \end{aligned}$$

then $T(S_1)$ is also a bounded subset of X. By using sub-linearity of $\beta ,$ we obtain

$$\begin{aligned} \begin{array}{lll}\beta (T(S_1))&{}\le &{} \beta (A(S_1))+\beta \left( B(I-D)^{-1}C(S_1)\cdot B^{\prime }(I-D)^{-1}C(S_1)\right) \\ {} &{}\le &{}\beta (A(S_1))+\beta \left( B(I-D)^{-1}C(S_1) \right) \Vert B^{\prime }(I-D)^{-1}C(S_1)\Vert _{{\mathcal {P}}}\\ &{}&{}+\,\beta \left( B^{\prime }(I-D)^{-1}C(S_1) \right) \Vert B(I-D)^{-1}C(S_1)\Vert _{{\mathcal {P}}}\\ {} &{}\le &{} \left( \psi _1+M\psi _2\right) (\beta (S_1)). \end{array} \end{aligned}$$

This demonstrates that T is a nonlinear ${\mathcal {D}}$-set-contraction multi-valued mapping on S into itself, and when we apply Theorem 2.3, we notice that T has a fixed point. So, the vector $y=(I-D)^{-1}Cx$ completes the proof. $\square$

A special case of Theorem 3.3, which is useful in applications to differential and integral inclusions, is introduced in the following theorem.

Theorem 3.4

Let S be a nonempty, bounded, closed, and convex subset of a Banach algebra X,  and let $S^{\prime }$ be a nonempty, bounded, closed, and convex subset of a Banach space Y.

Let $A: S\longrightarrow X, B: S^{\prime }\longrightarrow X, C: S \longrightarrow Y, D: S^{\prime } \longrightarrow Y$ and $B^{\prime }: S^{\prime }\longrightarrow {\mathcal {P}}_{cp,cv}(X)$ be five multi-valued operators satisfying:

• (i) A, B, and C are Lipschitzians with the Lipschitz constants $q_1, q_2$, and $q_3$, respectively,

• (ii) D is a contraction with a constant k and $C(S)\subseteq (I-D)(S^{\prime }),$

• (iii) $B^{\prime }$ is upper semi-continuous and compact, and

• (iv) $Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx \subseteq S,$ for all $x\in S.$

Then, the block operator matrix (1.1) has a fixed point whenever $q_1+\frac{q_2 q_3}{1-k}M< \frac{1}{2},$ where $M=\Vert B^{\prime }(I-D)^{-1}C(S)\Vert _{{\mathcal {P}}}.$ $\diamondsuit$

Proof of Theorem 3.4

Clearly, every single-valued Lipschitz mapping with a Lipschitz constant k is a multi-valued Lipschitz mapping with the Lipschitz constant 2k (see Papageorgiou [15]). Once again, $Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx$ is a convex and closed subset of X for each $x\in S,$ and when we apply Theorem 3.3, we reach the desired conclusion. $\square$

Now, we are in a position to combine Theorem 3.3 and Theorem 2.4 in order to deduce the following fixed point theorem.

Theorem 3.5

Let S be a nonempty, closed, and convex subset of a Banach algebra X,  and let $A, B, B^{\prime }: X\longrightarrow {\mathcal {P}}_{cp}(X),$ $C: S\longrightarrow X$ and $D: X\longrightarrow X$ be five multi-valued operators satisfying:

• (i) A, B, and C are Lipschitzians with the Lipschitz constants $q_1, q_2$, and $q_3$, respectively,

• (ii) A and C are compact,

• (iii) D is a contraction with a constant $k\in [0,1],$

• (iv) B and $B^{\prime }$ are upper semi-continuous, and

• (v) $Ax+ B(I-D)^{-1}Cx \cdot B^{\prime }(I-D)^{-1}Cx \in {\mathcal {P}}_{cp, cv}(S),$ for all $x\in S.$

Then, the block operator matrix (1.1) has a fixed point whenever $q_1+\frac{q_2 q_3}{1-k}M< 1,$ where $M=\Vert B^{\prime }(I-D)^{-1}C(S)\Vert _{{\mathcal {P}}}.$ $\diamondsuit$

Proof of Theorem 3.5

From our assumptions, it follows that the inverse operator $(I-D)^{-1}$ exists on X.

Now, we claim that $(I-D)^{-1}C(S_1)$ is a relatively compact subset of X,  for any bounded subset $S_1\subset S.$ If this is not the case, then $d=\beta ((I-D)^{-1}C(S_1))>0.$ Using the following equality:

$$\begin{aligned} (I-D)^{-1}C=C+D(I-D)^{-1}C \end{aligned}$$

and taking into account the subadditivity of $\beta$ and Remark 3.7 in [6], we get

$$\begin{aligned} \begin{array}{lll} \beta \left( (I-D)^{-1}C(S_1)\right) &{}\le &{} \beta (\overline{C(S_1)})+\beta \left( D(I-D)^{-1}C(S_1)\right) \\ &{}\le &{} k\beta \left( (I-D)^{-1}C(S_1)\right) .\end{array} \end{aligned}$$

This is not possible, and the claim is approved. Let us define

$$\begin{aligned} \left\{ \begin{array}{ll} T: S\longrightarrow {\mathcal {P}}_{cl,cv}(S)\\ x\longrightarrow Ax+ B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx. \end{array} \right. \end{aligned}$$

Obviously, T is well defined. The use of both assumption (ii) and Lemma 2.1 implies that T is compact on S. An argument similar to that in the proof of Theorem 3.3 leads to the upper semi-continuity of T. By applying Theorem 2.4, we deduce that T has a fixed point in S. $\square$

4 Differential Inclusions

4.1 Multi-valued Initial Value Problems

The quadratic initial value problems are considered as for first- order ordinary differential inclusions. Given a closed and bounded interval $J = [0, a]$ in ${\mathbb {R}}$ for some $a\in {\mathbb {R}}^{*}_+,$ let us consider the following system

$$\begin{aligned} \displaystyle \left\{ \begin{array}{lll} \left( \displaystyle {\frac{x(t) - k(t, x(t))}{f(t, y(t))}}\right) ^{\prime } \in G(t,y(t))\\ y(t)=\displaystyle {\frac{1}{1+ b(t)|x(\theta (t))|}}- p\left( t, \frac{1}{1+ b(t)|x(\theta (t))|}\right) + p(t, y(t))\\ (x(0), y(0))=(x_0,y_0)\in {\mathbb {R}}^2, \end{array} \right. \end{aligned}$$

(4.1)

where $t\in J,$ and the functions $b,\theta , k, f$ and $G: J\times {\mathbb {R}} \longrightarrow {\mathcal {P}}_{cp, cv}({\mathbb {R}})$ are given, whereas $x=x(t)$ and $y=y(t)$ are unknown functions that satisfy

• (i) The function $t\longrightarrow \displaystyle \frac{x(t) - k(t, x(t))}{f(t, y(t))}$ is differentiable, and

• (ii) $\left( \displaystyle {\frac{x(t) - k(t, x(t))}{f(t, y(t))}}\right) ^{\prime }=v(t),~ t\in J$ for some $v\in L^1(J, {\mathbb {R}})$ such that $v(t)\in G(t, x(t))$ a.e. for $t\in J$ satisfying $(x(0),y(0))=(x_0,y_0).$

We will verify the existence of solutions for system IVP (4.1) in the space $C(J,{\mathbb {R}})$ of continuous and real-valued functions on J. Let us define a norm $\Vert \cdot \Vert$ and a multiplication “$\cdot$” in the Banach algebra $C(J,{\mathbb {R}})$ of continuous and real-valued functions on J by $\Vert x\Vert =\displaystyle {\sup _{t\in J}}|x(t)|$ and

$$\begin{aligned} (x\cdot y)(t)=x(t) y(t); \quad t\in J, \end{aligned}$$

for all $x, y \in C(J,{\mathbb {R}}).$

We need the following definitions in the sequel.

Definition 4.1

A multi-valued map $Q : J \longrightarrow {\mathcal {P}}_{cp}({\mathbb {R}})$ is said to be measurable if for any $y\in X,$ the function $t \mapsto d(y, Q(t)) = \inf \{|y-x| ~;~ x \in Q(t)\}$ is measurable. $\diamondsuit$

Definition 4.2

A multi-valued map $Q : J\times {\mathbb {R}} \longrightarrow {\mathcal {P}}_{cp}({\mathbb {R}})$ is called Carathéodory if

• (i) $t\mapsto Q(t,x)$ is measurable for each $x \in {\mathbb {R}},$ and

• (ii) $x\mapsto Q(t,x)$ is an upper semi-continuous almost everywhere for $t\in J.$

Again, a Carathéodory multi-valued function Q is called $L^1$-Carathéodory, if

• (iii) for each real number $r > 0,$ there exists a function $h_r\in L^1(J,{\mathbb {R}})$ such that

  $$\begin{aligned} \Vert Q(t,x)\Vert _{{\mathcal {P}}}\le h_r(t)~~a.e. \quad \text { for } t\in J, \end{aligned}$$

for all $x\in {\mathbb {R}}$ with $|x|\le r.$

Further, a Carathéodory multi-valued function Q is called $L^1_{{\mathbb {R}}}$-Carathéodory, if

• (iv) there exists a function $h\in L^1(J,{\mathbb {R}}),$ such that

  $$\begin{aligned} \Vert Q(t,x)\Vert _{{\mathcal {P}}}\le h(t)~~a.e. \quad \text { for } t\in J, \end{aligned}$$

for all $x\in {\mathbb {R}},$ and the function h is called a growth function of Q on $J \times {\mathbb {R}}.$ $\diamondsuit$

For any multi-valued function $Q : J \times {\mathbb {R}} \longrightarrow {\mathcal {P}}_{cp}({\mathbb {R}}),$ we denote

$$\begin{aligned} S_Q^1(x)=\{v\in L^1(J, {\mathbb {R}});~ v(t)\in Q(t,x(t)) \text { for all } t\in J\}, \end{aligned}$$

for some $x\in C(J, {\mathbb {R}}).$ The integral of the multi-valued function Q is defined as

$$\begin{aligned} \displaystyle {\int _0^t}Q(s,x(s))\mathrm{d}s=\left\{ \displaystyle {\int _0^t}v(s)\mathrm{d}s \quad v\in S_Q^1(x)\right\} ;\quad \text { for ~each } x\in C(J, {\mathbb {R}}). \end{aligned}$$

Then, we have the following lemmas due to Lasota [22].

Lemma 4.1

Let X be a finite-dimensional Banach space. Assume that Q is a $L^1$-Carathéodory multi-valued map with compact values, then $S_Q^1(x)\not =\emptyset$ for $x\in X.$ $\diamondsuit$

Lemma 4.2

Let X be a Banach space and let $Q: J\times X \longrightarrow {\mathcal {P}}_{cp}(X)$ be a Carathéodory multi-valued with $S_Q^1(x)\not =\emptyset ,$ for each $x\in X.$ Assume that $\mathcal {L}: L^1(J, X)\longrightarrow C(J, X)$ is a linear continuous mapping. Then, the operator

$$\begin{aligned} \mathcal {L}\circ S_Q^1: C(J, X) \longrightarrow {\mathcal {P}}_{cp,cv}(C(J,X)) \end{aligned}$$

is a closed graph operator. $\diamondsuit$

Under the following hypotheses, we could reach the solution of (4.1):

$(\mathcal {H}_0)$ The single-valued mapping $k: J\times {\mathbb {R}}\longrightarrow {\mathbb {R}}$ is continuous, and there exists a bounded function $l_1: J \rightarrow {\mathbb {R}}$ with a bound $\Vert l_1\Vert < \frac{1}{4}$ satisfying

$$\begin{aligned} \left| k(t,x)-k(t,y) \right| \le l_1(t)|x-y|;\quad \text { for all } x,y\in {\mathbb {R}}. \end{aligned}$$

$(\mathcal {H}_1)$ The single-valued mapping $f: J\times {\mathbb {R}}\longrightarrow {\mathbb {R}}\backslash \{0\}$ is continuous, and there exists a bounded function $l_2: J \rightarrow {\mathbb {R}}$ with a bound $\Vert l_2\Vert$ satisfying

$$\begin{aligned} \left| f(t,x)-f(t,y) \right| \le l_2(t)|x-y| ;\quad \text { for all } x,y\in {\mathbb {R}}. \end{aligned}$$

$(\mathcal {H}_2)$ The multi-valued mapping $G: J\times {\mathbb {R}} \longrightarrow {\mathcal {P}}_{cp, cv}({\mathbb {R}})$ is $L^1_{{\mathbb {R}}}$-Carathéodory with a growth function h.

$(\mathcal {H}_3)$ The single-valued mapping $p: J\times {\mathbb {R}} \longrightarrow {\mathbb {R}}$ is such that:

• (a) The partial $x\longrightarrow p(t,x)$ is a q-contraction.

• (b) The partial $t\longrightarrow p(t,x)$ is continuous on J,  for all $x\in C(J, {\mathbb {R}}).$

• (c) $|p(t,x)|\le q|x|,$ for all $t\in J$ and $x\in {\mathbb {R}}.$

$(\mathcal {H}_4)$ The functions $b:J \longrightarrow {\mathbb {R}}_+$ and $\theta : J \longrightarrow J$ are continuous.

Theorem 4.1

Assume that the hypotheses $(\mathcal {H}_0)$–$(\mathcal {H}_4)$ hold. If

$$\begin{aligned} \displaystyle \left\{ \begin{array}{ll} \displaystyle {4\Vert b\Vert _{\infty }\Vert l_2\Vert \left( \left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| +\Vert h\Vert _{L^1}\right) < 1}\\ \\ q <~ \displaystyle {\frac{1-4\Vert l_2\Vert \Vert b\Vert _{\infty }\left( \left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| +\Vert h\Vert _{L^1}\right) }{1+4\Vert l_2\Vert \left( \left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| +\Vert h\Vert _{L^1}\right) }}\\ \\ \displaystyle \max \left\{ \Vert l_1\Vert ~;~ \Vert l_2\Vert \left( \left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| +\Vert h\Vert _{L^1}\right) ~;~ K+\left( \left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| +\Vert h\Vert _{L^1}\right) F\right\} < \frac{1}{4},\\ \\ \end{array} \right. \end{aligned}$$

(4.2)

where $\frac{1}{2}<P=\displaystyle {\sup \{|p(t,0)| ~;~ t\in J\}}<1,$ $K=\displaystyle \sup _{t\in J}\{|k(t,0)|\}$ and $F=\displaystyle \sup _{t\in J}\{|f(t,0)|\},$ then the system (4.1) has a solution. $\diamondsuit$

Proof of Theorem 4.1

Consider the mapping A, B, C, D, and $B^{\prime }$ on $X=C(J, {\mathbb {R}})$ by

• $Ax(t)=k(t,x(t)), ~;~ Bx(t)=f(t,x(t))$ and $Dx(t)=p(t,x(t))$

• $Cx(t)=\displaystyle {\frac{1}{1+ b(t)|x(\theta (t))|}}- p\left( t, \frac{1}{1+ b(t)|x(\theta (t))|}\right) ,$ and

• $B^{\prime }x(t)=\left\{ u\in X \text { such that } u(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}+\displaystyle \int _0^tv(s)\mathrm{d}s;~ v\in S_G^1(x)\right\} ,$

for all $t\in J.$ Then, the problem IVP (4.1) may be abstractly written in the form

$$\begin{aligned} \left\{ \begin{array}{ll} x(t)\in Ax(t)+By(t)\cdot B^{\prime }y(t)\\ y(t)=Cx(t)+Dy(t). \end{array} \right. \end{aligned}$$

We will show that A, B, C, D, and $B^{\prime }$ meet all the conditions of Theorem 3.4. Let us define the subsets S and $S^{\prime }$ on $C(J, {\mathbb {R}})$ by

$$\begin{aligned} S=\{y\in C(J, {\mathbb {R}}) ~;~ \Vert y\Vert _{\infty }\le 1+q\}~\text { and }~ S^{\prime }=\left\{ y\in C(J, {\mathbb {R}}) ~;~ \Vert y\Vert _{\infty }\le \displaystyle {\frac{4}{1-q}}\right\} . \end{aligned}$$

It is obvious that S and $S^{\prime }$ are nonempty, bounded, convex, and closed subsets of $C(J, {\mathbb {R}}),$ and it is similarly clear that the operator $B^{\prime }$ is well defined since $S_{G}^1(x)\not =\emptyset ,$ for each $x\in X$. $\square$

• Step 1. We start by showing that the operators A, B, C and D define single-valued operators $A,B,C,D: X\longrightarrow X$ and $B^{\prime }: X\longrightarrow {\mathcal {P}}_{cp, cv}(X).$ The claim regarding A, B, C, and D is clear, since the functions f and k are continuous on $J\times {\mathbb {R}}.$ We only corroborate the claim for the multi-valued operator $B^{\prime }$ on X. First, we show that $B^{\prime }$ has compact values on X. Notice that the operator $B^{\prime }$ is equivalent to the composition ${\mathcal {K}}\circ S_{G}^1$ of two operators on $L^1(J, {\mathbb {R}}),$ where ${\mathcal {K}}:L^1(J, {\mathbb {R}})\longrightarrow X$ is the continuous operator defined by

  $$\begin{aligned} {\mathcal {K}}v(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tv(s)\mathrm{d}s. \end{aligned}$$

  In order to demonstrate that $B^{\prime }$ has compact values, it is sufficient to prove that the composition operator ${\mathcal {K}}\circ S_{G}^1$ has compact values on X. Let $x\in X$ be arbitrary and let $\{v_n\}_{n=0}^{\infty }$ be a sequence in $S_G^1(x).$ Then, by using the definition of $S_G^1(x),$ we get $v_n(t)\in G(t,x(t))$ a.e. for $t\in J$ and for each $n\in \mathbb {N}.$ Since G(t, x(t)) is compact, there is a convergent subsequence of $v_n(t)$ (for simplicity, call it $v_n(t)$ itself) that converges in measure to some $v(t)\in G(t,x(t))$ for $t\in J.$ From the continuity of ${\mathcal {K}},$ it follows that ${\mathcal {K}}v_n(t)\rightarrow {\mathcal {K}}v(t)$ pointwise on J as $n\rightarrow \infty .$ We need to show that $\{{\mathcal {K}}v_n\}_{n=0}^{\infty }$ is an equi-continuous sequence so as to demonstrate the uniformity convergence. Let $t_1,t_2\in J,$ then

  $$\begin{aligned} \displaystyle \left| {\mathcal {K}}v_n(t_1)-{\mathcal {K}}v_n(t_2)\right| \le \left| \displaystyle \int _{t_1}^{t_2}v_n(s)\mathrm{d}s\right| . \end{aligned}$$

  (4.3)

  Since $v_n\in L^1(J, {\mathbb {R}}),$ the right-hand side of (4.3) tends to 0 as $t_1\rightarrow t_2.$ Thereby, the sequence $\{Kv_n\}_{n=0}^{\infty }$ is equi-continuous, and when we apply the Ascoli’s theorem, it is implied that there is a uniformly convergent subsequence. Then, we deduce that $\left( {\mathcal {K}}\circ S_G^1\right) (x)$ is a compact set for all $x\in X.$ Therefore, $B^{\prime }$ is a compact-valued multi-valued operator on X. Again, let $u_1, u_2\in B^{\prime }x.$ Then, there are $v_1,v_2\in S_G^1(x)$ such that

  $$\begin{aligned} u_1(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tv_1(s)\mathrm{d}s;\quad t\in J, \end{aligned}$$

  and

  $$\begin{aligned} u_2(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tv_2(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

  Now, for any $\lambda \in [0,1],$ we have

  $$\begin{aligned} \begin{array}{lll}\lambda u_1(t)+(1-\lambda )u_2(t) &{}=&{} \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}+\displaystyle \int _0^t \left( \lambda v_1(s)+(1-\lambda )v_2(s)\right) \mathrm{d}s\\ &{}=&{}\displaystyle \frac{x_0-k(0, x_0)}{f(0,y_0)}+\displaystyle \int _0^tv(s)\mathrm{d}s,\end{array} \end{aligned}$$

  where $v(s)=\lambda v_1(s)+(1-\lambda )v_2(s)\in G(s, x(s))$ for all $s\in J.$ Hence $\lambda u_1+(1-\lambda )u_2\in B^{\prime }x$ and consequently, $B^{\prime }x$ is convex for each $x\in X.$ As a result, $B^{\prime }$ defines a multi-valued map with convex values.

• Step 2. In this step, we will show that A, B and C are single-valued Lipschitz operators on X. Let $x,y\in X.$ Then,

  $$\begin{aligned} \begin{array}{lll} \Vert Ax-Ay\Vert &{}=&{}\displaystyle \sup _{t\in J}\left| k(t,x(t))-k(t,y(t))\right| \\ &{}\le &{}\displaystyle \sup _{t\in J}|l_1(t)||x(t)-y(t)|\le \Vert l_1\Vert \Vert x-y\Vert \end{array} \end{aligned}$$

  which demonstrates that A is a single-valued Lipschitz operator on X with the Lipschitz constant $\Vert l_1\Vert .$ In a similar way, it can be proved that B and C are again two Lipschitz operators on X with the Lipschitz constants $\Vert l_2\Vert$ and $(\Vert b\Vert _{\infty }+q)$, respectively. Now let us prove that $C(S)\subset (I-D)(S^{\prime }).$ To do it, let $x\in S$ be a fixed point. Let us define a mapping

  $$\begin{aligned}\left\{ \begin{array}{ll} \varphi _x: C(J, {\mathbb {R}})\longrightarrow C(J, {\mathbb {R}})\\ y \longrightarrow Cx+Dy, \end{array} \right. \end{aligned}$$

  by $\varphi _x(y)(t)=\varphi _{x(t)}(y(t)).$ According to assumption $((\mathcal {H})_3)(a),$ we reach the result that the operator $\varphi _x$ is a contraction with a lipschitz constant q,  and then it implies that there is a unique point $y\in X$ such that $Cx+Dy=y.$ Hence,

  $$\begin{aligned} C(S)\subset (I-D)(C(J, {\mathbb {R}})). \end{aligned}$$

  Since $y\in C(J, {\mathbb {R}}),$ then there exists $t^{*}\in J$ such that

  $$\begin{aligned} \begin{array}{lll}\Vert y\Vert _{\infty }&{}=&{} |y(t^{*})|= \left| Cx(t^{*})+Dy(t^{*})\right| \\ &{}\le &{} \displaystyle {\frac{1}{1+ b(t^{*})|x(\theta (t^{*}))|}}+ \left| p\left( t^{*}, \frac{1}{1+ b(t^{*})|x(\theta (t^{*}))|}\right) \right| + \left| p(t^{*}, y(t^{*}))\right| \\ &{}\le &{} 1+ q + 2|p(t,0)|+ q|y(t^{*})|. \end{array} \end{aligned}$$

  This implies that

  $$\begin{aligned} |y(t^{*})| \le \displaystyle \frac{1+q +2P}{1-q}. \end{aligned}$$

  We conclude that $C(S)\subset (I-D)(S^{\prime }).$

• Step 3. In this step, we have to show that $B^{\prime }$ is completely continuous on $S^{\prime }.$ First, let us prove that $B^{\prime }(S^{\prime })$ is a uniformly bounded and equi-continuous set. To see this, let $u\in B^{\prime }(S^{\prime })$ be arbitrary. Then, there is a $v\in S_G^1(x)$ such that

  $$\begin{aligned} u(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tv(s)\mathrm{d}s;\quad t\in J \end{aligned}$$

  for some $x\in S^{\prime }.$ By using assumption $(H_2)$ we get

  $$\begin{aligned} \begin{array}{rcl}|u(t)|&{}\le &{}\left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| +\displaystyle \int _0^t\left| v(s)\right| \mathrm{d}s\\ &{}\le &{}\left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| +\displaystyle \int _0^t\left\| G(s,x(s))\right\| \mathrm{d}s\\ &{}\le &{}\left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| +\displaystyle \int _0^th(s)\mathrm{d}s\\ &{}\le &{}\left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} \right| +\Vert h\Vert _{L^1}, \end{array} \end{aligned}$$

  for all $t\in J$ and so, $B^{\prime }(S^{\prime })$ is a uniformly bounded set. Again, when we proceed with the some arguments as in Step 1, we notice that $B^{\prime }(S^{\prime })$ is an equi-continuous set in X. Our next task is to shows that $B^{\prime }$ is an upper semi-continuous multi-valued mapping. Let $\{x_n\}_{n=0}^{\infty }$ be a sequence in X such that $x_n\rightarrow x$. Let $\{y_n\}_{n=0}^{\infty }$ be a sequence in $B^{\prime }x_n$ such that $y_n\rightarrow y.$ We will prove that $y\in B^{\prime }x.$ Since $y_n\in B^{\prime }x_n,$ there exists a $v_n\in S_G^1(x_n)$ such that

  $$\begin{aligned} y_n(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tv_n(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

  We must prove that there is a $v\in S_G^1(x)$ such that

  $$\begin{aligned} y(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tv(s)\mathrm{d}s \quad t\in J. \end{aligned}$$

  Let us consider the continuous linear operator ${\mathcal {K}}: L^1(J, {\mathbb {R}}) \longrightarrow C(J, {\mathbb {R}})$ defined by

  $$\begin{aligned} {\mathcal {K}}v(t)=\displaystyle \int _0^t v(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

  Then, ${\mathcal {K}}\circ S_G^1(x)$ is a closed graph operator, in view of Lemma 4.2 ( see lemma 5.4 in [6]). Also, from the definition of $({\mathcal {K}}),$ we have

  $$\begin{aligned} y_n(t)- \displaystyle {\frac{x_0-k(0,x_0)}{f(0,y_0)}} \in ({\mathcal {K}})\circ S_G^1(x). \end{aligned}$$

  Since $y_n\rightarrow y,$ there is a point $v\in S_G^1(x)$ such that

  $$\begin{aligned} y(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tv(s)\mathrm{d}s; \quad t\in J. \end{aligned}$$

  This demonstrates that $B^{\prime }$ is an u.s.c. mapping on X. Therefore, $B^{\prime }$ is upper semi-continuous and compact. Summing up, $B^{\prime }$ is a completely continuous multi-valued map on X. Now, from assumption $((\mathcal {H})_2),$ we deduce that

  $$\begin{aligned} \begin{array}{lll} M&{}=&{}\Vert B^{\prime }(I-D)^{-1}C(S)\Vert _{{\mathcal {P}}}\\ &{}=&{}\sup \left\{ \Vert B^{\prime }(I-D)^{-1} Cx\Vert _{{\mathcal {P}}}~;~x\in S\right\} \\ &{}\le &{} \sup \left\{ \Vert B^{\prime }x\Vert _{{\mathcal {P}}}~;~ x\in S\right\} \\ &{}\le &{} \left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| +\Vert h\Vert _{L^1}. \end{array} \end{aligned}$$

  In order to simplify the calculation, it is assumed that $x_0=k(0,x_0).$ From the second inequality of (4.2), it follows that

  $$\begin{aligned} q\left( 1+\displaystyle \frac{1}{4\Vert h\Vert _{L^1}\Vert l_2\Vert }\right) +\Vert b\Vert _{\infty }< \displaystyle {\frac{1}{4\Vert h\Vert _{L^1}\Vert l_2\Vert }}. \end{aligned}$$

  Then,

  $$\begin{aligned} \displaystyle \frac{\Vert l_2\Vert \big (\Vert b\Vert _{\infty }+q\big )}{1-q}\Vert h\Vert _{L^1} <\frac{1}{4}. \end{aligned}$$

  Consequently,

  $$\begin{aligned} \Vert l_1\Vert + \displaystyle \frac{\Vert l_2\Vert \big (\Vert b\Vert _{\infty }+q\big )}{1-q}\Vert h\Vert _{L^1}<\displaystyle \frac{1}{2}. \end{aligned}$$

  Finally, it remains to verify the assumption (iv) of Theorem 3.4. Let $x\in S$ and $y\in X$ such that

  $$\begin{aligned} y\in Ax+B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx. \end{aligned}$$

  Then, there is $z\in S^{\prime }$ and $v\in S_G^1(z)$ such that

  $$\begin{aligned} y(t)=k(t,x(t))+f\left( t,z(t)\right) \cdot \displaystyle \int _0^t v(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

  Therefore, we have

  $$\begin{aligned} \begin{array}{lll} |y(t)|&{}\le &{} \left| k(t,x(t))\right| + \left| f\left( t,z(t)\right) \right| \displaystyle \int _0^t|v(s)|\mathrm{d}s\\ &{}\le &{}\left| k(t,x(t))-k(t,0)\right| +|k(t,0)| +\left( \left| f\left( t,z(t)\right) \right. \right. \\ &{}&{}\left. \left. -f(t,0)\right| +|f(t,0)|\right) \Vert h \Vert _{L^1}\\ &{}\le &{}\Vert l_1\Vert |x(t)|+K +(\Vert l_2\Vert |z(t)|+F)\Vert h\Vert _{L^1}\\ &{}\le &{}(1+q)(\Vert l_1\Vert +\Vert l_2\Vert \Vert h\Vert _{L^1})+ K+\Vert l_2\Vert \Vert h\Vert _{L^1}+F\Vert h\Vert _{L^1}\\ &{}\le &{}1+q. \end{array} \end{aligned}$$

  So, $\Vert y\Vert _{\infty }\le 1+q,$ and consequently

  $$\begin{aligned} Ax+B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx \subset S,~ \text { for all }~ x\in S. \end{aligned}$$

We deduce that A, B, C, D, and $B^{\prime }$ meet all the requirements of Theorem 3.4. Now, the result follows from Theorem 3.4.

4.2 Multi-valued Periodic Boundary Value Problems of First Order

Let $J=[0,{T}]$ be the closed and bounded interval in ${\mathbb {R}}.$ Let $C(J, {\mathbb {R}})$ be the Banach algebra of all continuous functions from J to ${\mathbb {R}}$ endowed with the sup-norm $\Vert .\Vert _{\infty }$ defined by $\Vert f\Vert _{\infty }=\displaystyle {\sup _{t\in J}}|f(t)|,$ for each $f\in C(J, {\mathbb {R}}).$ Consider the periodic boundary value problem (in short PBVP) for the first-order ordinary differential inclusion

$$\begin{aligned} \displaystyle \left\{ \begin{array}{lll} \left( \displaystyle {\frac{x(t) - k(t, x(t))}{f(t, y(t))}}\right) ^{\prime }+h(t) \left( \displaystyle {\frac{x(t) - k(t, x(t))}{f(t, y(t))}}\right) \in G_h(t,x(t),y(t))\\ \\ y(t)=\displaystyle {\frac{1}{1+ b(t)|x(\theta (t))|}}- p\left( t, \frac{1}{1+ b(t)|x(\theta (t))|}\right) + p(t, y(t))\\ \\ (x(0), y(0))=(x({T}),y({T}))\in {\mathbb {R}}^2, \end{array} \right. \end{aligned}$$

(4.4)

where $h\in L^1(J,{\mathbb {R}})$ is bounded and the multi-valued function $G_h: J\times {\mathbb {R}}\times {\mathbb {R}} \longrightarrow {\mathcal {P}}_{cp, cv}({\mathbb {R}})$ is defined by

$$\begin{aligned} G_h(t,x,y)=G(t,y)+h(t)\left( \displaystyle {\frac{x - k(t, x)}{f(t, y)}}\right) . \end{aligned}$$

A solution to the system (4.4) stands for two functions $x,y\in AC(J, {\mathbb {R}})$ that satisfy

• (i) The function $t\longrightarrow \displaystyle \frac{x(t) - k(t, x(t))}{f(t, y(t))}$ is absolutely continuous, and

• (ii) There exists a function $v\in L^1(J, {\mathbb {R}})$ such that $v(t)\in G(t, y(t))$ checking the following equality

  $$\begin{aligned} \left( \displaystyle {\frac{x(t) - k(t, x(t))}{f(t, y(t))}}\right) ^{\prime }=v(t)~;~ (x(0),y(0))=(x({T}),y({T})), \end{aligned}$$

where $AC(J,{\mathbb {R}})$ is the space of all absolutely continuous real-valued functions on J.

The following useful lemma appears in Nieto [24].

Lemma 4.3

For any $h\in L^1(J,{\mathbb {R}}^+)$ and $\sigma \in L^1(J, {\mathbb {R}}),$ x is a solution for the differential equation

$$\begin{aligned} \left\{ \begin{array}{ll} x^{\prime }+h(t)x=\sigma (t) \\ x(0)=x({T}) \end{array} \right. \end{aligned}$$

if, and only if, x is a solution for the integral equation

$$\begin{aligned} x(t)=\displaystyle {\int _0^{{T}}}g_h(t,s)\sigma (s)\mathrm{d}s,\text { where} \end{aligned}$$

$$\begin{aligned} \displaystyle g_h(t,s)=\left\{ \begin{array}{ll} \displaystyle {\frac{e^{H(s)-H(t)+H({T})}}{e^{H({T})}-1}}~~{\mathrm{if} }\quad 0\le s\le t\le {T}\\ \displaystyle {\frac{e^{H(s)-H(t)}}{e^{H({T})}-1}}~~\qquad \quad \,{\mathrm{if} }\quad 0\le t<s \le {T} \end{array} \right. \end{aligned}$$

(4.5)

with $H(t)=\displaystyle {\int _0^t}h(s)\mathrm{d}s.$ $\diamondsuit$

We will use the following assumptions in the sequel.

• $(B_0)$ The functions $t\rightarrow f(t, x)$ and $t\rightarrow k(t, x)$ are periodic of period T for all $x\in {\mathbb {R}}.$

• $(B_1)$ The function $(x,y)\rightarrow \displaystyle \frac{x-k(0,x)}{f(0,y)}$ is injective on ${\mathbb {R}}^2.$

Lemma 4.4

Assume that the hypotheses $(B_0)$ and $(B_1)$ hold. Then, for any bounded integrable function h on J,  $(x,y)\in X\times X$ is a solution for the differential inclusion (4.4) if, and only if, it is a solution of the integral equation

$$\begin{aligned} \left\{ \begin{array}{ll} x(t)\in k(t,x(t))+f(t,y(t))\cdot \displaystyle {\int _0^{{T}}}g_h(t,s)G_h(s,x(s),y(s))\mathrm{d}s\\ y(t)=\displaystyle {\frac{1}{1+ b(t)|x(\theta (t))|}} - p\left( t, \frac{1}{1+ b(t)|x(\theta (t))|}\right) + p(t, y(t))\\ (x(0), y(0))=(x({T}),y({T}))\in {\mathbb {R}}^2, \end{array} \right. \end{aligned}$$

where the Green’s function $g_h$ is defined in (4.5) and $X=C(J,{\mathbb {R}}).$ $\diamondsuit$

Let us assume that the functions involved in system (4.4) satisfy the following conditions:

• $(B_2)$ The single-valued mapping $k: J\times {\mathbb {R}}\longrightarrow {\mathbb {R}}$ is continuous, and there is a bounded function $l_1: J \rightarrow {\mathbb {R}}$ with a bound $0<\Vert l_1\Vert \le \frac{1}{6}$ satisfying

  $$\begin{aligned} \left| k(t,x)-k(t,y) \right| \le l_1(t)|x-y|~;\quad \text { for all } x,y\in {\mathbb {R}}. \end{aligned}$$

• $(B_3)$ The single-valued mapping $f: J\times {\mathbb {R}}\longrightarrow {\mathbb {R}}\backslash \{0\}$ is continuous, and there is a bounded function $l_2: J \rightarrow {\mathbb {R}}$ with a bound $\Vert l_2\Vert$ satisfying

  $$\begin{aligned} \left| f(t,x)-f(t,y) \right| \le l_2(t)|x-y|~;\quad \text { for all } x,y\in {\mathbb {R}}. \end{aligned}$$

• $(B_4)$ The single-valued mapping $p: J\times {\mathbb {R}} \longrightarrow {\mathbb {R}}$ is such that:

  • (a) p is Lipschitzian with a constant q in $]0,\frac{1}{4}[$ with respect to the first variable.

  • (b) p is continuous with respect to the second variable.

  • (c) $|p(t,x)|\le q|x|,$ for all $t\in J$ and $x\in {\mathbb {R}}.$

• $(B_5)$ The functions $b:J \longrightarrow {\mathbb {R}}_+$ and $\theta : J\rightarrow J$ are continuous with $4\Vert b\Vert _{\infty }\le 1.$

• $(B_6)$ The multi-valued operator $G: J\times {\mathbb {R}} \longrightarrow {\mathcal {P}}_{cp, cv}({\mathbb {R}})$ is Carathéodory.

• $(B_7)$ There is a function $\varrho \in L^1(J, {\mathbb {R}}_+^{*})$ and a continuous and nondecreasing function $\varphi : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+$ such that

  $$\begin{aligned} \Vert G_h(t,x,y)\Vert _{{\mathcal {P}}}\le \varrho (t)\varphi (|y|) \end{aligned}$$

  for each $(x,y)\in {\mathbb {R}}^2.$

Theorem 4.2

Assume that the hypotheses $(B_0)$-$(B_7)$ hold. Further, if there exists a real number $r_0>0$ such that

$$\begin{aligned} \displaystyle \left\{ \begin{array}{ll} \Vert l_1\Vert +\Vert l_2\Vert M_h\Vert \varrho \Vert _{L^1}\varphi (r_0)<\frac{1}{2}\\ \displaystyle \max \left\{ K+ FM_h\Vert \varrho \Vert _{L^1}\varphi (r_0)~;~\frac{3}{4}+P\right\} \le \frac{r_0}{2}, \end{array} \right. \end{aligned}$$

(4.6)

where $\frac{1}{2}<P=\displaystyle {\sup \{|p(t,0)|~;~ t\in J\}}<1,$ $K=\displaystyle \sup _{t\in J}\{|k(t,0)|\},$ $F=\displaystyle \sup _{t\in J}\{|f(t,0)|\}$, and $M_h=\displaystyle \sup _{t,s\in J}\{|g_h(t,s)|\},$ then the system (4.4) has a solution. $\diamondsuit$

Proof of Theorem 4.2

Let us consider the mapping A, B, C, D, and $B^{\prime }$ on X by

• $Ax(t)=k(t,x(t)), ~;~ Bx(t)=f(t,x(t))$ and $Dx(t)=p(t,x(t))$

• $Cx(t)=\displaystyle {\frac{1}{1+ b(t)|x(\theta (t))|}}- p\left( t, \frac{1}{1+ b(t)|x(\theta (t))|}\right) ,$ and

• $B^{\prime }x(t)=\left\{ u\in X ~~;~~ u(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}+\displaystyle \int _0^tg_h(t,s)v(s)\mathrm{d}s~;~v\in S_{G_h}^1(x)\right\}$

for all $t\in J.$ Then, the system IVP (4.1) is equivalent to the operator inclusion

$$\begin{aligned} \left\{ \begin{array}{ll} x(t)\in Ax(t)+By(t)\cdot B^{\prime }y(t)\\ y(t)=Cx(t)+Dy(t). \end{array} \right. \end{aligned}$$

We will show that A, B, C, D, and $B^{\prime }$ meet all the conditions of Theorem 3.4 on $\mathcal {{B}}(0,{r_0}).$ Since $S_{G_h}^1(x)\not =\emptyset$ for each x in the closed ball $\mathcal {{B}}(0,{r_0}),$ it follows that $B^{\prime }$ is well defined. We start by showing that $B^{\prime }$ define multi-valued operator $B^{\prime }: \mathcal {{B}}(0,{r_0})\longrightarrow {\mathcal {P}}_{cp,cv}(X).$ In fact, let $u_1, u_2\in B^{\prime }x.$ Then there are $v_1,v_2\in S_{G_h}^1(x)$ such that

$$\begin{aligned} u_1(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tg_h(t,s)v_1(s)\mathrm{d}s;\quad t\in J, \end{aligned}$$

and

$$\begin{aligned} u_2(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tg_h(t,s)v_2(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

Now, for any $\lambda \in [0,1],$ we have

$$\begin{aligned} \begin{array}{lll}\lambda u_1(t)+(1-\lambda )u_2(t) &{}=&{} \displaystyle \frac{x_0 -k(0,x_0)}{f(0,y_0)}+\displaystyle \int _0^tg_h(t,s)\left( \lambda v_1(s)+(1-\lambda )v_2(s)\right) \mathrm{d}s\\ &{}=&{}\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}+\displaystyle \int _0^tg_h (t,s)v(s)\mathrm{d}s,\end{array} \end{aligned}$$

where $v(s)=\lambda v_1(s)+(1-\lambda )v_2(s)\in G_h(s,x(s),y(s))$ for all $s\in J.$

Hence, $\lambda u_1+(1-\lambda )u_2\in B^{\prime }x$ and consequently, $B^{\prime }x$ is convex for each $x\in X.$ So, $B^{\prime }$ defines a multi-valued operator with a compact and convex values.

Proceeding as in the proof of Theorem 4.1 we deduce that the single-valued operators A, B and C are Lipschitzian with the Lipschitz constants $\Vert l_1\Vert _{\infty },\Vert l_2\Vert _{\infty }$ and $\Vert b\Vert _{\infty }+q$, respectively. Under the assumption $(B_4) (a),$ we get $C(\mathcal {{B}}(0,r_0))\subset (I-D)(X).$ Since $y\in X,$ then there exists $t^{*}\in J$ such that

$$\begin{aligned} \Vert y\Vert _{\infty }=|Cx(t^{*})+Dy(t^{*})|\le \frac{1+q+2P}{1-q}\le r_0. \end{aligned}$$

As a result, $y\in \mathcal {{B}}(0,{r_0})$ and consequently $C(\mathcal {{B}}(0,{r_0}))\subset (I-D)(\mathcal {{B}}(0,r_0)).$

Our next task is to shows that $B^{\prime }$ is completely continuous on $\mathcal {{B}}(0,r_0).$ First, we have to prove that $B^{\prime }(\mathcal {{B}}(0,{r_0}))$ is a totally bounded subset of X. To do this, it is enough to showing that $B^{\prime }(\mathcal {{B}}(0,{r_0}))$ is a uniformly bounded and equi-continuous subset of X. To see this, let $u\in B^{\prime }(\mathcal {{B}}(0,{r_0}))$ be arbitrary. Then, there is a $v\in S^1_{G_h}(x)$ such that

$$\begin{aligned} u(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tg_h (t,s)v(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

Hence,

$$\begin{aligned} \begin{array}{lll} |u(t)|&{}\le &{} \left| \displaystyle \frac{x_0-k(0, x_0)}{f(0,y_0)}\right| +\displaystyle \int _0^tg_h(t,s)\left\| G_h(s, x(s),y(s))\right\| _{{\mathcal {P}}}\mathrm{d}s;\quad t\in J,\\ &{}\le &{}\left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} \right| +\displaystyle \int _0^tg_h(t,s)\varrho (s)\varphi (|y(s)|)\mathrm{d}s;\quad t\in J,\\ &{}\le &{}\left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} \right| +\displaystyle \int _0^tg_h(t,s)\Vert \varrho \Vert _{L^1}\varphi (r)\mathrm{d}s;\quad t\in J,\\ &{}\le &{}\left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} \right| +M_h\Vert \varrho \Vert _{L^1}\varphi (r);\quad t\in J.\end{array} \end{aligned}$$

Consequently, $B^{\prime }(\mathcal {{B}}(0,{r_0}))$ is a uniformly bounded subset of X. Finally, it is sufficient to show that $B^{\prime }(\mathcal {{B}}(0,{r_0}))$ is an equi-continuous subset of X. Indeed, for any $t_1,t_2\in [0,{T}],$ we have

$$\begin{aligned} \begin{array}{rcl} |B^{\prime }x(t_1)-B^{\prime }x(t_2)| &{}\le &{} \left( \displaystyle \int _0^t\frac{\partial }{\partial t}g_h(t,s)\left\| G_h(s,x(s),y(s))\right\| _{{\mathcal {P}}}\mathrm{d}s\right) |t_1-t_2|\\ &{}\le &{}\left( \displaystyle \int _0^t(-h(t))\Vert \varrho \Vert _{L^1}\varphi (r)\mathrm{d}s\right) |t_1-t_2|\\ &{}\le &{}\left( \displaystyle {\max _{t\in J}}(h(t))M_h\Vert \varrho \Vert _{L^1}\varphi (r)\right) |t_1-t_2|. \end{array} \end{aligned}$$

This shows that $B^{\prime }(\mathcal {{B}}(0,{r_0}))$ is totally bounded by using Arzelà-Ascoli theorem.

Now, let us demonstrate that $B^{\prime }$ is an upper semi-continuous multi-valued mapping on X. Let $\{x_n\}_{n=0}^{\infty }$ be a sequence in X such that $x_n\rightarrow x$. Let $\{y_n\}_{n=0}^{\infty }$ be a sequence such that $y_n\in B^{\prime }x_n$ and $y_n\rightarrow y.$ We will prove that $y\in B^{\prime }x.$ Since $y_n\in B^{\prime }x_n,$ there exists a $v_n\in S_G^1(x_n)$ such that

$$\begin{aligned} y_n(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tg_h(t,s)v_n(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

We must prove that there is a $v\in S_{G_h}^1(x)$ such that

$$\begin{aligned} y(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tg_h(t,s)v(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

Let us consider the continuous linear operator $({\mathcal {K}}): L^1(J, {\mathbb {R}}) \longrightarrow C(J, {\mathbb {R}})$ defined by

$$\begin{aligned} ({\mathcal {K}})v(t)=\displaystyle \int _0^t g_h(t,s)v(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

Then, $({\mathcal {K}})\circ S_{G_h}^1(x)$ is a closed graph operator, in view of Lemma 5.4 in [6]. Also, from the definition of $({\mathcal {K}}),$ we deduce that

$$\begin{aligned} y_n(t)- \displaystyle {\frac{x_0-k(0,x_0)}{f(0,y_0)}} \in ({\mathcal {K}})\circ S_{G_h}^1(x). \end{aligned}$$

Since $y_n\rightarrow y,$ there is a point $v\in S_{G_h}^1(x)$ such that

$$\begin{aligned} y(t)=\displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)} +\displaystyle \int _0^tg_h(t,s)v(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

This demonstrates that $B^{\prime }$ is a u.s.c. operator on X. Hence, $B^{\prime }$ is a completely continuous multi-valued operator on X.

Now, it remains to verify the assumption (iv) of Theorem 3.4 on $\mathcal {{B}}(0,{r_0}).$ It follows that

$$\begin{aligned} \begin{array}{lll} M&{}=&{}\sup \left\{ \Vert B^{\prime }(I-D)^{-1}Cx\Vert _{{\mathcal {P}}}~;~x\in S\right\} \\ &{}\le &{}\sup \left\{ \Vert B^{\prime }x\Vert _{{\mathcal {P}}}~;~ x\in S\right\} \\ &{}\le &{}\left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| + \displaystyle {\int _{0}^t}|g_h(t,s)|\Vert G_h(s,x(s),y(s))\Vert _{{\mathcal {P}}}\mathrm{d}s\\ &{}\le &{} \left| \displaystyle \frac{x_0-k(0,x_0)}{f(0,y_0)}\right| + M_h\Vert \varrho \Vert _{L^1}\varphi (r_0). \end{array} \end{aligned}$$

In order to simplify the calculation, it is assumed that $x_0=k(0,x_0).$ From the first inequality of (4.6), it follows that

$$\begin{aligned} \displaystyle \frac{\Vert l_2\Vert \big (\Vert b\Vert _{\infty }+q\big )}{1-q}M+\Vert l_1\Vert <\displaystyle {\frac{\Vert b\Vert _{\infty }+q}{1-q}}\left( \frac{1}{2}-\Vert l_1\Vert \right) +\Vert l_1\Vert <\frac{1}{2}. \end{aligned}$$

Finally, it remains to verify the assumption (iv) of Theorem 3.4. Let $x\in \mathcal {{B}}(0,{r_0})$ and

$$\begin{aligned} y\in Ax+B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx. \end{aligned}$$

Then, there is $z\in \mathcal {{B}}(0,{r_0})$ and $v\in S_{G_h}^1(z)$ such that

$$\begin{aligned} y(t)=k(t,x(t))+f\left( t,z(t)\right) \cdot \displaystyle \int _0^t g(h,s) v(s)\mathrm{d}s;\quad t\in J. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{array}{lll} |y(t)|&{}\le &{} \left| k(t,x(t))\right| + \left| f\left( t,z(t)\right) \right| \displaystyle \int _0^tg(h,s) |v(s)|\mathrm{d}s\\ &{}\le &{}\left| k(t,x(t))-k(t,0)\right| +|k(t,0)| +\,\left( \left| f\left( t,z(t)\right) -f(t,0)\right| \right. \\ &{}&{}\left. +|f(t,0)|\right) M_h\Vert \varrho \Vert _{L^1}\varphi (r_0)\\ &{}\le &{}\Vert l_1\Vert |x(t)|+K +(\Vert l_2\Vert |z(t)|+F) M_h\Vert \varrho \Vert _{L^1}\varphi (r_0)\\ {} &{}\le &{}\left( \Vert l_1\Vert +\Vert l_2\Vert M_h\Vert \varrho \Vert _{L^1}\varphi (r_0)\right) r_0+ K+F M_h\Vert \varrho \Vert _{L^1}\varphi (r_0)\\ {} &{}<&{}r_0. \end{array} \end{aligned}$$

So, $\Vert y\Vert _{\infty }\le r_0$ and consequently,

$$\begin{aligned} Ax+B(I-D)^{-1}Cx\cdot B^{\prime }(I-D)^{-1}Cx \subset \mathcal {{B}}(0,{r_0}),~ \text { for all }~ x\in \mathcal {{B}}(0,{r_0}). \end{aligned}$$

We deduce that A, B, C, D, and $B^{\prime }$ meet all the requirements of Theorem 3.4. Now, the result follows from Theorem 3.4. $\square$