Fixed Point Theorems for the Sum of Two Multivalued Mappings and an Application to an Integral Inclusion

Abstract

In this paper we establish some fixed point results for the sum of two multivalued mappings with weakly sequentially closed graph under weak topology features in a Banach space. An application to illustrate our theory is included.

1 Introduction

In this paper we discuss the nonlinear operator inclusion

$$\begin{aligned} x\in A(x)+B(x),\qquad \qquad x\in \Omega , \end{aligned}$$

(1.1)

where $\Omega$ is a nonempty closed convex subset of a Banach space $E$. We obtain new multivalued analogs of Leray–Schauder alternatives (or Krasnoselskii fixed point theorems) for the sum of two mappings $A+B$ where $A$ is weakly compact with weakly sequentially closed graph and $B$ is $\Phi$-condensing (or hemi-weakly compact) with weakly sequentially closed graph. These results complement the recent literature [2, 4, 12]. As an application we discuss the existence of solutions of the nonlinear integral inclusion

$$\begin{aligned} x(t)\in q(t)+\int ^{\mu (t)}_{0} k_{1}(t,s)F(s,x(\theta (s)))ds+\int ^{\sigma (t)}_{0} k_{2}(t,s)G(s,x(\eta (s)))\mathrm{d}s, \end{aligned}$$

(1.2)

where $q$: $J\rightarrow X, k_{1},k_{2}:J\times J\rightarrow {\mathbb {R}}, F, G:J\times X \rightarrow \mathcal{P}(X)$, $\mu , \theta , \sigma , \eta :J\rightarrow J$, $J=[0,1]$. Note $(1.2)$ was discussed in [7, 8].

2 Preliminaries

We present some definitions and results which we will need.

Let $E$ be a Hausdorff linear topological space. Now

$$\begin{aligned} \mathcal{P}(E)&= \biggl \{D\subset E{:}~D\, \text{ is } \text{ nonempty }\biggr \},\\ \mathcal{P}_\mathrm{{bd}}(E)&= \biggl \{D\subset E{:}~D\, \text{ is } \text{ nonempty } \text{ and } \text{ bounded } \biggr \},\\ \mathcal{P}_\mathrm{{cv}}(E)&= \biggl \{D\subset E{:}~D\, \text{ is } \text{ nonempty } \text{ and } \text{ convex } \biggr \},\\ \mathcal{P}_\mathrm{{cl},\mathrm {bd}}(E)&= \biggl \{D\subset E{:}~D\, \text{ is } \text{ nonempty } \text{ closed } \text{ and } \text{ bounded } \biggr \},\\ \mathcal{P}_\mathrm{{cl},\mathrm {bd},\mathrm {cv}}(E)&= \biggl \{D\subset E{:}~D\, \text{ is } \text{ nonempty } \text{ closed, } \text{ bounded } \text{ and } \text{ convex } \biggr \}. \end{aligned}$$

Let $Z$ be a nonempty subset of a Banach space $Y$ and $F:Z\longrightarrow \mathcal{P}(E)$ be a multivalued mapping. We let

$$\begin{aligned} R(F)=\bigcup _{y \in Z}F(y)\,\, \text{ and } GrF=\{(z,x)\in Z\times X:x\in F(z)\} \end{aligned}$$

be the range and the graph of $F$, respectively. Moreover, for every subset $D$ of $E$, we put $F^{-1}(D)=\{z\in Z:F(z)\cap D \ne \emptyset \}$.

Now we suppose that $E$ is a Banach space with zero $\theta$ and $Z$ is weakly closed in $Y$.

Now $F$ is said to have weakly sequentially closed graph if for every sequence $\{x_{n}\}\subset Z$ with $x_n \rightharpoonup x$ in $Z$ and for every sequence $\{y_{n}\}$ with $y_{n}\in F(x_n), \forall n \in \mathbb {N}, y_n \rightharpoonup y$ in $X$ implies $y \in F(x)$; here $\rightharpoonup$ denotes weak convergence. Now $F$ is called weakly compact, if $F(A)$ is a relatively weakly compact subset of $E$ for all $A \in \mathcal{P}_\mathrm{{bd}}(Z)$. If $F$ is a single-valued mapping, then $F$ is said to be weakly sequentially continuous if for every sequence $\{x_{n}\} \subset Z$ with $x_{n} \rightharpoonup x \in Z$, we have $F(x_{n}) \rightharpoonup F(x)$.

Remark 2.1

If $Z$ is weakly compact, then every sequentially weakly continuous map $F:Z \longrightarrow E$ is weakly continuous. This is an immediately consequence of the Eberlein–Šmulian theorem [10, Theorem 8.12.4, p. 549].

Definition 2.1

Let $E$ be a Banach space and $C$ a lattice with a least element, which is denoted by $0$. By a measure of weak noncompactness $(MWNC)$ on $E$, we mean a function $\Phi$ defined on the set of all bounded subsets of $X$ with values in $C$, such that for any $\Omega _{1}, \Omega _{2} \in \mathcal{P}_\mathrm{{bd}}(E)$:

• (1)  $\Phi (\overline{\text{ conv }}(\Omega _{1}))=\Phi (\Omega _{1})$, where $\overline{\text{ conv }}$ denotes the closed convex hull of $\Omega$,

• (2)  $\Omega _{1} \subseteq \Omega _{2} \Longrightarrow \Phi (\Omega _{1})\le \Phi (\Omega _{2}),$

• (3)   $\Phi (\Omega _{1}\cup \{a\})=\Phi (\Omega _{1})$ for all $a \in E$,

• (4)   $\Phi (\Omega _{1})=0$ if and only if $\Omega _{1}$ is relatively weakly compact in $E$.

If the lattice C is a cone then the $MWNC$ $\Phi$ is said to be positive homogenous if $\Phi (\lambda \Omega )=\lambda \Phi (\Omega )$ for all $\lambda > 0$ and $\Omega \in \mathcal{P}_\mathrm{{bd}}(E)$ and it is called semi-additive if $\Phi (\Omega _{1}+\Omega _{2})\le \Phi (\Omega _{1}) + \Phi (\Omega _{2})$ for all $\Omega _{1}, \Omega _{2} \in \mathcal{P}_\mathrm{{bd}}(E)$.

The notion above is a generalization of the well-known De Blasi measure of weak noncompactness $\beta$ (see [6]) defined on each bounded set $\Omega$ of $E$ by

$$\begin{aligned} \beta (\Omega )=\inf \{\varepsilon >0{:}\,{\text { there exists a weakly compact set }} D {\text { such that }} \Omega \subset D+B_{\varepsilon }(0) \}, \end{aligned}$$

where $B_{\varepsilon }(0)$ is the closed ball with radius $\varepsilon$ and center $\theta$.

It is well known that $\beta$ enjoys these properties: for any $\Omega _{1}, \Omega _{2} \in \mathcal{P}_\mathrm{{bd}}(E)$,

• (5)  $\beta (\Omega _{1} \cup \Omega _{2})=\max \{\beta (\Omega _{1}), \beta (\Omega _{2})\}$.

• (6)  $\beta (\lambda \Omega _{1})=\lambda \beta (\Omega _{1})$ for all $\lambda >0$.

• (7)  $\beta (\Omega _{1}+\Omega _{2})\le \beta (\Omega _{1}) + \beta (\Omega _{2})$.

Definition 2.2

Let $\Omega$ be a nonempty subset of a Banach space $E$ and $\Phi$ a MWNC on $E$. Let $F:\Omega \longrightarrow \mathcal{P}(E)$. We say

• (a)  $F$ is $\Phi$-condensing if $F$ is bounded and $\Phi (F(D))< \Phi (D)$ for all $D\in \mathcal{P}_\mathrm{{bd}}(\Omega )$ with $\Phi (D)\ne 0$.

• (b)   $F$ is hemi-weakly compact if each sequence $\{x_{n}\}$ has a weakly convergent subsequence whenever there exist $y_{n} \in F(x_{n})$ such that the sequence $\{x_{n}-y_{n}\}$ is weakly convergent.

Theorem 2.1

([3] Theorem (2.2)) Let $\Omega$ be a nonempty, closed, convex subset of Banach $E$. Suppose that $F:\Omega \rightarrow \mathcal{P}_{cv}(\Omega )$ has weakly sequentially closed graph, is weakly compact and $F(\Omega )$ is bounded. Then $F$ has a fixed point.

3 Fixed Point Theorems for the Sum of Two Multivalued Mappings

In this section we discuss the operator inclusion $x\in A(x)+B(x)$.

First, we prove some new versions of Krasnoselskii type fixed point theorems for the sum of two multivalued mappings.

Theorem 3.1

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$. Assume $A: \Omega \rightarrow \mathcal{P}(E)$ and $B:\Omega \rightarrow \mathcal{P}(E)$ are two multivalued mappings satisfying the following conditions,

• (i) $A$ is weakly compact with weakly sequentially closed graph,

• (ii) $B$ is hemi-weakly compact with weakly sequentially closed graph,

• (iii) For each $x\in \Omega$, $(I-B)^{-1}A(x)\in \mathcal{P}_{cv}(\Omega ),$

• (iv) $(I-B)^{-1}A(\Omega )$ is bounded.

Then, there exists $x\in \Omega$ with $x\in A(x)+B(x).$

Proof

Let $F:=(I-B)^{-1}A$. Note $F:\Omega \rightarrow \mathcal{P}_{cv}(\Omega )$ is well defined. We show that $F$ has weakly sequentially closed graph. Let $x_{n}\rightharpoonup x\in \Omega$ and $y_{n}\in F(\Omega )$, and $y_{n}\rightharpoonup y$ with $y_{n}\in F(x_{n})$. Then $(I - B)(y_{n})\cap A(x_{n})\ne \emptyset$, so there exists $z_{n}\in A(x_{n})$ such that $z_{n}\in (I - B)(y_{n})$. Hence $z_{n}=y_{n}-w_{n}$ where $w_{n}\in B(y_{n})$. Since $A$ is weakly compact and $\{x_{n}\}$ is bounded, by the Eberlein–Šmulian’s theorem [10, Theorem 8.12.4, p. 549], $\{z_{n}\}$ has a subsequence $\{z_{n_{k}}\}$ which weakly converges to some $z\in A(x).$ Also $w_{n_{k}}=y_{n_{k}}-z_{nk}$ and since $B$ has weakly sequentially closed graph, it follows that $y-z\in B(y)$, that is, $z\in (I-B)(y).$ Then $(I-B)(y)\cap A(x) \ne \emptyset$ and consequently, $y\in (I-B)^{-1}A(x)=F(x).$ Hence $F$ has weakly sequentially closed graph. Now, let $D$ be an arbitrary bounded subset of $\Omega$. We claim that $F(D)$ is relatively weakly compact. Let $y_{n}\in F(D)$. Choose $\{x_{n}\}\subset D$ such that $y_{n}\in F(x_{n})$, that is $y_{n}\in B(y_{n})+ A(x_{n})$. Thus there exist $u_{n}\in A(x_{n})$ and $v_{n}\in B(y_{n})$ such that $y_{n}=u_{n}+v_{n}.$ Since $A(D)$ is relatively weakly compact, by the Eberlein–Šmulian’s theorem there is a subsequence $\{u_{n_{k}}\}$ which weakly converges to some $u$ in $E$. It follows that $y_{n_{k}}-v_{n_{_{k}}}=u_{n_{k}}\rightharpoonup u$. Now since $B$ is hemi-weakly compact, we obtain that $\{y_{n_{k}}\}$ has a weakly convergent subsequence, say $\{y_{n_{k_{j}}}\}$. Hence $F(D)$ is relatively weakly compact. Consequently, $F$ is weakly compact. From Theorem 2.1, $F$ has a fixed point. Then there exists $x \in \Omega$ such that $x \in A(x)+B(x)$.$\square$

Theorem 3.2

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$. Assume $A: \Omega \rightarrow \mathcal{P}(E)$ and $B:\Omega \rightarrow \mathcal{P}(E)$ are two multivalued mappings satisfying the following conditions,

• (i) $A$ has weakly sequentially closed graph and $A(\Omega )$ is relatively weakly compact,

• (ii) $B$ is hemi-weakly compact with weakly sequentially closed graph,

• (iii) For each $x\in \Omega$, $(I-B)^{-1}A(x)\in \mathcal{P}_{cv}(\Omega ).$

Then, there exists $x\in \Omega$ with $x\in A(x)+B(x).$

Proof

Let $F:=(I-B)^{-1}A :\Omega \rightarrow \mathcal{P}_{cv}(\Omega )$. The same reasoning as above guarantees that $F$ has weakly sequentially closed graph and $F(\Omega )$ is relatively weakly compact. From Theorem 2.1, $F$ has a fixed point. Thus, there exists $x\in \Omega$ with $x\in A(x)+B(x).$ $\square$

Theorem 3.3

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$ and $\Phi$ is a semi-additive $MWNC$ on $E$. Assume $A:\Omega \rightarrow \mathcal{P}(E)$ and $B:\Omega \rightarrow \mathcal{P}(E)$ satisfy the following conditions,

• (i) $A$ is weakly compact with weakly sequentially closed graph,

• (ii) $B$ is $\Phi$-condensing with weakly sequentially closed graph and $B(\Omega )$ is bounded,

• (iii) For each $x\in \Omega$, $(I-B)^{-1}A(x)\in \mathcal{P}_{cv}(\Omega )$,

• (iv) $(I-B)^{-1}A(\Omega )$ is bounded.

Then, there exists $x\in \Omega$ with $x\in A(x)+B(x).$

Proof

The result follows from Theorem 3.1 if we show $B$ is hemi-weakly compact. Let $\{x_{n}\}$ be a sequence in $\Omega$ and $y_n \in B(x_n)$ such that the sequence $\{x_{n}-y_{n}\}$ is weakly convergent. We have $\{x_{n}\} \subset \{x_{n}-y_{n}\}+ \{B(x_n)\}$. Since $\{x_{n}-y_{n}\}$ is bounded and $\{B(x_n)\}$ is bounded, then $\{x_{n}\}$ is bounded. Thus if $\Phi (\{x_{n}\})\ne 0$ we obtain

$$\begin{aligned}&\Phi (\{x_n\})\le \Phi (\{x_{n}-y_{n}\}+{B(x_n)})\le \Phi (\{x_{n}-y_{n}\})+\Phi ({B(x_n)})\\&\quad \le \Phi ({B(x_n)})<\Phi (\{x_n\}), \end{aligned}$$

a contradiction. Thus $\Phi ({x_n})= 0$. Consequently, the set $\{x_n\}$ is relatively weakly compact. Applying the Eberlein–Šmulian’s theorem, we obtain that $\{x_n\}$ possesses a weakly convergent subsequence $\{x_{n_k}\}$.$\square$

Theorem 3.4

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$ and $\Phi$ is a semi-additive $MWNC$ on $E$. Assume $A:\Omega \rightarrow \mathcal{P}(E)$ and $B:\Omega \rightarrow \mathcal{P}(E)$ satisfy the following conditions,

• (i) $A$ has weakly sequentially closed graph and $A(\Omega )$ is relatively weakly compact,

• (ii) $B$ is $\Phi$-condensing with weakly sequentially closed graph and $B(\Omega )$ is bounded,

• (iii) For each $x\in \Omega$, $(I-B)^{-1}A(x)\in \mathcal{P}_{cv}(\Omega )$.

Then, there exists $x\in \Omega$ with $x\in A(x)+B(x).$

Proof

The result is a consequence of Theorem 3.2, since by assumption $(ii)$, $B$ is hemi-weakly compact.$\square$

Our next result is a Leray–Schauder type principle.

Theorem 3.5

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$ and $U$ be a weakly open subset of $\Omega$ with $\theta \in U$. Assume $F{:}~\overline{U^{w}}\rightarrow \mathcal{P}_{cv} (\Omega )$ has weakly sequentially closed graph. In addition, suppose that $F(\overline{U^{w}})$ is relatively weakly compact. Then, either

$(A_{1})$ :

$F$ has a fixed point, or

$(A_{2})$ :

there is a point $x\in \partial _{\Omega }U$ (the weak boundary of $U$ in $\Omega$ ) and $\lambda \in (0,1)$ with $x\in \lambda F(x)$.

Proof

Suppose $(A_{2})$ does not hold and $F$ does not have a fixed point in $\partial _{\Omega }U$ (otherwise we are finished). Let

$$\begin{aligned} D=\{x\in \overline{U^{w}}{:} ~ x\in \lambda F(x),\quad \text{ for } \text{ some } \lambda \in [0,1]\}. \end{aligned}$$

The set $D$ is nonempty and bounded since $\theta \in D$ and $F(\overline{U^{w}})$ is bounded. We have $D \subset \text {conv}(\{\theta \}\cup F(D))$ so $D$ is relatively weakly compact (note the Krein-Šmulian theorem [1], p.604] and the fact that $F(\overline{U^{w}})$ is relatively weakly compact). Now, we prove that $D$ is weakly sequentially closed. Let $\{x_{n}\}\subset D$ such that $x_{n}\rightharpoonup x$. Note $x\in \overline{U^{w}}$. For all $n$, there exists $\lambda _n \in [0,1]$ such that $x_{n}\in \lambda _{n} F(x_n)$. Now $\lambda _n \in [0,1]$ and we can extract a subsequence $(\lambda _{n_j})_j$ such that $\lambda _{n_j}\longrightarrow \lambda \in [0,1]$. Note $x_{n_j}=\lambda _{n_j}y_{n_j}$, where $y_{n_j}\in F(x_{n_j})$. If $\lambda =0$, then $x_{n_j}\rightharpoonup \theta$ ($F(\overline{U^{w}})$ is bounded) and $x \in \{\theta \} \subseteq \overline{U^{w}}$. If $\lambda \ne 0$, then without loss of generality, we can suppose that $\lambda _{n_j}\ne 0$ for all $j$. Now $\lambda _{n_j}^{-1}x_{n_j}=y_{n_j}$ for all $j$ implies $y_{n_j} \rightharpoonup \lambda ^{-1}x$. Since $F$ has weakly sequentially closed graph, we have $y \in F(x)$, which means that $x \in \lambda F(x)$. Thus $x \in D$, so $D$ is weakly sequentially closed. Next let $z \in \overline{D^{w}}$. Since $\overline{D^{w}}$ is weakly compact, by the Eberlein–Šmulian theorem [10, Theorem 8.12.4, p. 549], there exists a sequence $\{z_n\} \subset D$ such that $z_n \rightharpoonup z$. Now $z \in D$ since $D$ is weakly sequentially closed. Hence $\overline{D^{w}}=D$ and $D$ is weakly compact. Now $E$ endowed with the weak topology is a Hausdorff locally convex space, so $E$ is completely regular. Since $D\cap \partial _{\Omega }U =\emptyset$, there exists a weakly continuous function $\pi :\Omega \rightarrow [0,1]$ such that $\pi (D)=1,$ for $x\in D$ and $\pi (x)=0,$ for $x \in \partial _{\Omega }U$. Define the multivalued map $F^{*}:\Omega \rightarrow \mathcal{P}_{cv}(\Omega )$ by

$$\begin{aligned} F^{*}(x)=\left\{ \begin{array}{l@{\quad }l} \pi (x)F(x),&{} if \quad x\in \overline{U^{w}},\\ \{\theta \},&{} if \quad x\in \Omega \backslash \overline{U^{w}}. \end{array} \right. \end{aligned}$$

Now $F^{*}$ has weakly sequentially closed graph (note $F$ has weakly sequentially closed graph). Since $F^{*}(\Omega )\subseteq \overline{co}\{\{\theta \}\cup F(\overline{U^{w}})\},$ and since $F(\overline{U^{w}})$ is relatively weakly compact then the Krein-Šmulian theorem [1], p.604] implies $\overline{co}\{\{\theta \}\cup F(\overline{U^{w}})\}$ is weakly compact. As a result $F^{*}(\Omega )$ is relatively weakly compact. From Theorem 2.1 there exists $u\in \Omega$ such that $u\in F^{*}(u)$. If $u\notin U$, we have $F^{*}(u)=\{\theta \}$ and hence $u=\theta$ which contradicts $\theta \in U.$ Then $u\in U$ and $u\in F^{*}(u)=\pi (u)F(u),$ that is $u\in D$ and hence $\pi (u)=1.$ Therefore, $(A_{1})$ holds.$\square$

Theorem 3.6

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$ and $U$ be a weakly open subset of $\Omega$ with $\theta \in U$. Assume $A:\overline{U^{w}}\rightarrow \mathcal{P}(E)$ and $B:E\rightarrow \mathcal{P}(E)$ satisfy the following conditions,

• (i) $A$ has weakly sequentially closed graph and $A(\overline{U^{w}})$ is relatively weakly compact,

• (ii) $B$ is hemi-weakly compact and has weakly sequentially closed graph on $\Omega$,

• (iii) For each $x\in \overline{U^{w}}$, $(I-B)^{-1}A(x)$ is convex,

• (iv) $x\in B(x)+ A(y), y\in \overline{U^{w}}\Rightarrow x\in \Omega$.

Then, either

$(A_{1})$ :

$A+B$ has a fixed point, or

$(A_{2})$ :

there is a point $x\in \partial _{\Omega }U$ and $\lambda \in (0,1)$ with $x\in \lambda A(x)+ \lambda B \left( \frac{x}{\lambda } \right) .$

Proof

First we show $F:=(I-B)^{-1}A:\overline{U^{w}}\longrightarrow \mathcal{P}_{cv}(\Omega )$ is well defined. Let $y\in (I-B)^{-1}A (x)$ for $x\in \overline{U^{w}}$. Then $y\in B(y)+A(x),$ so by $(iv)$ we get $y\in \Omega$ and so $(I-B)^{-1}A (x)\subseteq \Omega$ for $x\in \overline{U^{w}}.$ This together with $(iii)$ guarantees that $F:\overline{U^{w}}\longrightarrow P_{cv}(\Omega )$. Next, we show that $F$ has weakly sequentially closed graph. Let $x_{n}\rightharpoonup x\in \overline{U^{w}}$ and $y_{n}\in F(\overline{U^{w}})$, and $y_{n}\rightharpoonup y$ with $y_{n}\in (I-B)^{-1}A(x_{n})$. Then $(I - B)(y_{n})\cap A(x_{n})\ne \emptyset$, so there exists $z_{n}\in A(x_{n})$ such that $z_{n}\in (I - B)(y_{n})$. Hence $z_{n}=y_{n}-w_{n}$ where $w_{n}\in B(y_{n})$. Since $A(\{x_{n}\})$ is relatively weakly compact, by the Eberlein–Šmulian’s theorem [10, Theorem 8.12.4, p. 549], $\{z_{n}\}$ has a subsequence $\{z_{n_{k}}\}$ which weakly converges to some $z\in A(x).$ Also $w_{n_{k}}=y_{n_{k}}-z_{n_{k}}$ and since $B$ has weakly sequentially closed graph, it follows that $y-z\in B(y)$, that is, $z\in (I-B)(y).$ Then $(I-B)(y)\cap A(x) \ne \emptyset$ and consequently, $y\in (I-B)^{-1}A(x).$ Hence $F:=(I - B)^{-1}A$ has weakly sequentially closed graph. Finally we claim that $F(\overline{U^{w}})$ is relatively weakly compact. Let $y_{n}\in F(\overline{U^{w}})$. Choose $\{x_{n}\}\subset \overline{U^{w}}$ such that $y_{n}\in F(x_{n})$, that is $y_{n}\in B(y_{n})+ A(x_{n})$. Thus there exist $u_{n}\in A(x_{n})$ and $v_{n}\in B(y_{n})$ such that $y_{n}=u_{n}+v_{n}.$ Since $A(\overline{U^{w}})$ is relatively weakly compact, by the Eberlein–Šmulian’s theorem there is a subsequence $\{u_{n_{k}}\}$ which weakly converges to some $u$ in $E$. It follows that $y_{n_{k}}-v_{n_{_{k}}}=u_{n_{k}}\rightharpoonup u$. Now since $B$ is hemi-weakly compact, we obtain that $\{y_{n_{k}}\}$ has a weakly convergent subsequence, say $\{y_{n_{k_{j}}}\}$. Hence $F(\overline{U^{w}})$ is relatively weakly compact. From Theorem 3.5, either $F$ has a fixed point or there exist $x\in \partial _{\Omega }U$ and $\lambda \in (0,1)$ with $x\in \lambda F(x).$ The proof is complete. $\square$

Corollary 3.1

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$, $U$ be a weakly open subset of $\Omega$ with $\theta \in U$, and $\Phi$ is a semi-additive $MWNC$ on $E$. Assume $A:\overline{U^{w}}\rightarrow \mathcal{P}(E)$ and $B:E\rightarrow \mathcal{P}(E)$ satisfy the following conditions,

• (i) $A$ has weakly sequentially closed graph and $A(\overline{U^{w}})$ is relatively weakly compact,

• (ii) $B$ is $\Phi$-condensing with weakly sequentially closed graph and $B(\Omega )$ is bounded,

• (iii) For each $x\in \overline{U^{w}}$, $(I-B)^{-1}A(x)$ is convex,

• (iv) $x\in B(x)+ A(y), y\in \overline{U^{w}}\Rightarrow x\in \Omega$.

Then, either

$(A_{1})$ :

$A+B$ has a fixed point, or

$(A_{2})$ :

there is a point $x\in \partial _{\Omega }U$ and $\lambda \in (0,1)$ with $x\in \lambda A(x)+ \lambda B \left( \frac{x}{\lambda } \right) .$

Proof

The result follows from Theorem 3.6 since $B$ is hemi-weakly compact.$\square$

Theorem 3.7

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$ and $U$ be a weakly open subset of $\Omega$ with $\theta \in U$. Assume that $\Phi$ is a semi-additive $MWNC$ on $E$, and suppose $A:\overline{U^{w}}\rightarrow \mathcal{P}(E)$ and $B:E\rightarrow \mathcal{P}(E)$ satisfy the following conditions,

• (i) $A$ has weakly sequentially closed graph and $A(\overline{U^{w}})$ is relatively weakly compact,

• (ii) $B$ is $\Phi$-condensing and has weakly sequentially closed graph on $\Omega$,

• (iii) For each $x\in \overline{U^{w}}$, $(I-B)^{-1}A(x)$ is convex,

• (iv) $x\in B(x)+ A(y), y\in \overline{U^{w}}\Rightarrow x\in \Omega$,

• (v) $(I-B)^{-1}A(\overline{U^{w}})$ is bounded.

Then, either

$(A_{1})$ :

$A+B$ has a fixed point, or

$(A_{2})$ :

there is a point $u\in \partial _{\Omega }U$ and $\lambda \in (0,1)$ with $u\in \lambda A(u)+ \lambda B \left( \frac{u}{\lambda } \right) .$

Proof

As in Theorem 3.6 we see that $F:=(I-B)^{-1}A:\overline{U^{w}}\longrightarrow \mathcal{P}_{cv}(\Omega )$ is well defined and $F$ has weakly sequentially closed graph. Now we show that $F(\overline{U^{w}})$ is relatively weakly compact. Suppose $\Phi (F(\overline{U^{w}}))>0$. Since

$$\begin{aligned} F(\overline{U^{w}})\subseteq A(\overline{U^{w}})+ B(F(\overline{U^{w}})), \end{aligned}$$

we have

$$\begin{aligned} \Phi (F(\overline{U^{w}}))\le \Phi ( A(\overline{U^{w}}))+ \Phi (B(F(\overline{U^{w}}))). \end{aligned}$$

Since $A(\overline{U^{w}})$ is relatively weakly compact and $B$ is $\Phi$-condensing, it follows that (note $(v)$)

$$\begin{aligned} \Phi (F(\overline{U^{w}}))\le \Phi (B(F(\overline{U^{w}})) < \Phi (F(\overline{U^{w}})), \end{aligned}$$

a contradiction. Thus $\Phi (F(\overline{U^{w}}))=0$ and therefore $F(\overline{U^{w}})$ is relatively weakly compact. Now apply Theorem 3.5 to obtain the desired result.$\square$

Corollary 3.2

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$ and $U$ be a weakly open subset of $\Omega$ with $\theta \in U$. Assume $A:\overline{U^{w}}\rightarrow \mathcal{P}(E)$ and $B:E\rightarrow E$ satisfy the following conditions,

• (i) $A$ has weakly sequentially closed graph and $A(\overline{U^{w}})$ is relatively weakly compact,

• (ii) $B$ is a $\phi$-nonlinear contraction (i.e., there exists a continuous nondecreasing function $\phi :{\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}$ such that $\Vert Bx-By\Vert \le \phi (\Vert x-y\Vert )$ for all $x,y\in E,$ where $\phi (r)< r$ for $r > 0,$) and is weakly sequentially continuous,

• (iii) For each $x\in \overline{U^{w}}$, $(I-B)^{-1}A(x)$ is convex,

• (iv) $x\in B(x)+ A(y), y\in \overline{U^{w}}\Rightarrow x\in \Omega$,

• (v) $(I-B)^{-1}A(\overline{U^{w}})$ is bounded.

Then, either

$(A_{1})$ :

$A+B$ has a fixed point, or

$(A_{2})$ :

there is a point $u\in \partial _{\Omega }U$ and $\lambda \in (0,1)$ with $u\in \lambda A(u)+ \lambda B \left( \frac{u}{\lambda } \right) .$

Proof

The result follows from Theorem 3.7 once one notices that $B$ is $\beta$-condensing (see [12] for a proof).$\square$

Corollary 3.3

Let $\Omega$ be a nonempty closed convex subset of a Banach space $E$ and $U$ be a weakly open subset of $\Omega$ with $\theta \in U$. Assume that $\Phi$ is a semi-additive $MWNC$ on $E$, and suppose $A:\overline{U^{w}}\rightarrow \mathcal{P}(E)$ and $B:\Omega \rightarrow E$ satisfy the following conditions,

• (i) $A$ has weakly sequentially closed graph and $A(\overline{U^{w}})$ is relatively weakly compact,

• (ii) $B$ is $\Phi$-condensing and weakly sequentially continuous,

• (iii) For every $y$ in $\overline{U^{w}}, D_{y}=\{x\in \Omega$ such that $(I-B)(x) \in A(y) \}$ is a convex set,

• (iv) $A(\overline{U^{w}})\subseteq (I-B)(\Omega )$,

• (v) $(I-B)^{-1}A(\overline{U^{w}})$ is bounded.

Then, either

$(A_{1})$ :

$A+B$ has a fixed point, or

$(A_{2})$ :

there is a point $u\in \partial _{\Omega }U$ and $\lambda \in (0,1)$ with $u\in \lambda A(u)+ \lambda B \left( \frac{u}{\lambda } \right) .$

Proof

Define $F:\overline{U^{w}}\rightarrow \mathcal{P}(\Omega )$ by $F(x):=(I-B)^{-1}A(x)$. From conditions $(iii)$ and $(iv)$, $F$ is well defined and takes convex values. Since weakly sequentially continuous mappings have weakly sequentially closed graph, the result follows from Theorem 3.7.$\square$

4 Functional Integral Inclusions

Let $J=[0,1] \subseteq {\mathbb {R}}$, $L^{1}(J,X)$ the space of all integrable $X$-valued functions with norm $\Vert x\Vert _{L^{1}}=\displaystyle \int ^{1}_{0}|x(t)|dt$ and let $C(J,X)$ be the Banach space of all continuous $X$-valued functions defined on $J$ endowed with the norm $\Vert x\Vert =\sup _{t\in J}|x(t)|$.

Assume that $(X,\Sigma ,\mu )$ is a measurable space. A multivalued map $F:X\rightarrow \mathcal{P}(Y)$ (where $(Y,d)$ is a separable metric space) with closed values is called measurable if $F^{-1}(V)\in \Sigma$ for each open subset $V\subseteq Y,$ and is called weakly measurable if $F^{-1}(U)\in \Sigma$ for each closed subset $U\subseteq Y$. Furthermore, $F$ is weakly measurable if and only if the distance functions $f_{y}:X\rightarrow {\mathbb {R}}$, $f_{y}=dist(y,F(x))= \inf \{d(y,z) : z\in F(x)\}$ is measurable for all $y\in Y.$

Let $D$ be a nonempty subset of $X$ and let $F: D\rightarrow \mathcal{P}(Y)$, where $X$ and $Y$ are any topological vector spaces. Then the graph of $F$ is convex if and only if the set $D$ is convex and $\lambda F(x)+(1-\lambda )F(y)\subseteq F(\lambda x+(1-\lambda )y)$ for all $x,y \in D$ and $\lambda \in (0,1)$ see ([5]).

Throughout this section $X$ will be a finite-dimensional Banach space.

Definition 4.1

A multivalued map $F:J\times X\rightarrow \mathcal{P}(X)$ satisfies the $L^{1}_{X}$-Carathéodory condition if $(1).$ For each $x\in X,$ $F_{x}=F(.,x)$ is weakly measurable, $(2).$ For each $t\in J$, $F_{t}=F(t,.)$ is weakly upper semi-continuous, $(3).$ There exists a function $\alpha \in L^{1}(J,X)$ such that

$$\begin{aligned} \Vert F(t,x)\Vert =\sup \bigr \{ |u| \,: u \in F(t,x) \bigr \}\le \alpha (t),\quad a.e. \,\,\, t\in J \end{aligned}$$

for all $x\in X,$ where $\alpha$ is the growth function of $F$ on $J\times X.$

For a function $x$ defined on $J$ we define the set $S_{F}(x)=\{u\in L^{1}(J,X): u(t)\in F(t,x(t))$ a.e. $t\in J \}$ which is known as the set of selection functions. Also let $\Vert F(t,x(t))\Vert = \sup \bigl \{ |u(t)|: u(t)\in F(t,x(t))\bigr \}.$

Lemma 1

( [7] Lemma (4.1)) Let $X$ be a Banach space with $\dim (X)<\infty$ and let $F:J\times X\rightarrow \mathcal{P}_{cl,bd}(X)$ be $L^{1}_{X}$-Carathéodory. Then $S_{F}(x)\ne \emptyset$ for all $x\in X.$

Now to discuss the functional integral inclusion $(1.2)$. We list the following assumptions:

$\mathbf{(H_{1})}$ :

The functions $\mu , \theta , \sigma , \eta :J\rightarrow J$ are continuous. The function $q:J\rightarrow X$ is weakly continuous and $|q(t)|\le M$ for all $t\in J.$

$\mathbf{(H_{2})}$ :

The functions $k_{1},k_{2}$ are continuous on $J\times J$ with $K_{1}=\max _{t,s\in J}|k_{1}(t,s)|$ and $K_{2}=\max _{t,s\in J}|k_{2}(t,s)|.$

$\mathbf{(H_{3})}$ :

The multivalued mapping $F: J\times X\rightarrow \mathcal{P}_{cl,bd,cv}(X)$ is $L^{1}_{X}$-Carathéodory.

$\mathbf{(H_{4})}$ :

There exists a function $\gamma \in L^{1}(J,X)$ with $\gamma (t)> 0$ a.e. $t\in J$ and a nondecreasing function $\psi :{\mathbb {R}}\rightarrow (0,\infty )$ such that for all $x\in C(J, X),$

$$\begin{aligned} \Vert F(t,x(t))\Vert \le \gamma (t)\psi (|x(t)|)\quad a.e.\quad t\in J. \end{aligned}$$

$\mathbf{(H_{5})}$ :

The multivalued mapping $G: J\times X \rightarrow \mathcal{P}_{cl,bd}(X)$ is $L^{1}_{X}$-Carathéodory, and $G_{t}=G(t,.)$ has convex graph.

$\mathbf{(H_{6})}$ :

There exists a function $\beta \in L^{1}(J,X)$ such that for all $x,y\in C(J,X),$

$$\begin{aligned} \Vert G(t,x(t))-G(t,y(t))\Vert \le \beta (t)|x(t)-y(t)|\quad a.e.\quad t\in J, \end{aligned}$$

where $\Vert G(t,x(t))-G(t,y(t))\Vert =\sup \bigl \{|u(t)-v(t)|: u(t)\in G(t,x(t))$ and $v(t) \in G(t,y(t))\bigr \}.$

Theorem 4.1

Assume $(H_{1})-(H_{6})$ hold. Suppose there is a number $r>0$ such that

$$\begin{aligned} r=\frac{ M +K_{2}L+K_{1}\Vert \gamma \Vert _{L^{1}}\psi (r)}{1-K_{2}\Vert \beta \Vert _{L^{1}}} \end{aligned}$$

where $L = \int ^{1}_{0}\Vert G(s,0)\Vert ds$ and $K_{2}\Vert \beta \Vert _{L^{1}}< 1.$ Then the inclusion (1.2) has a solution on J.

Proof

Let $E=C(J,X).$ Consider $Q=\{u\in E: \Vert u\Vert \le r\}.$ Clearly, $Q$ is a nonempty, closed, bounded, and convex set. Let us consider the two operators $A,B$ defined on $C(J,X)$ by

$$\begin{aligned} A(x)&= \{ u\in E:u(t)=q(t)+\int ^{\mu (t)}_{0}k_{1}(t,s)v(s)ds,\quad v\in S_{F}(x)\},\\ B(x)&= \{ u\in E:u(t)=\int ^{\sigma (t)}_{0}k_{2}(t,s)v(s)ds,\quad v\in S_{G}(x)\}. \end{aligned}$$

Since $S_{F}(x)\ne \emptyset \ne S_{G}(x)$ then $A, B$ are well defined. We will show that $A, B$ satisfy the conditions of Theorem 3.4 on $Q$.

Step 1 A has weakly sequentially closed graph and $A(Q)$ is relatively weakly compact.

First, we show $A$ has weakly sequentially closed graph. Let $\{x_n\}\subset Q$ and $x_{n}\rightharpoonup x \in Q,$ $y_{n}\in A(x_{n})$ for all $n\in \mathbb {N}$, $y_{n}\rightharpoonup y$. Then by ([9] Theorem 9) we have $y_{n}(t)\rightharpoonup y(t)$ (and similarly, $x_{n}(t)\rightharpoonup x(t)$). Now for $y_{n}\in A(x_{n})$ there exists $v_{n}\in S_{F}(x_{n})$ such that

$$\begin{aligned} y_{n}(t)= q(t)+ \int ^{\mu (t)}_{0} k_{1}(t,s)v_{n}(s)ds. \end{aligned}$$

Fix $t\in J$. Without loss of generality we may assume that $y_{n}(t)\ne 0$. From the Hahn-Banach theorem there exists $f\in X^{*}$ such that $f(y_{n}(t))=|y_{n}(t)|$ and $|f|_{*}=1.$ Since $v_{n}(t)\in F(t,(x_{n}(t)))$ a.e. $t\in J,$ and $F$ has bounded values then from the reflexivity of $X$ there exists a subsequence $\{v_{n_{k}}(t)\}$ converging weakly to some $v(t).$ i.e., $f(v_{n_{k}}(t))\rightarrow f(v(t)).$ An application of the Lebesgue dominated convergence theorem ([13], p. 91) yields

$$\begin{aligned} |y_{n_{_{k}}}(t)|= f\left( q(t)+\int ^{\mu (t)}_{0} k_{1}(t,s)v_{n_{k}}(s)ds\right) \rightarrow f\left( q(t)+\int ^{\mu (t)}_{0} k_{1}(t,s)v(s)ds\right) . \end{aligned}$$

Then

$$\begin{aligned} y(t)= q(t)+\int ^{\mu (t)}_{0} k_{1}(t,s)v(s)ds. \end{aligned}$$

Moreover, by hypotheses $(H_{3})$, $F(t,.)$ is weakly upper semi-continuous, so $F(t,.)$ has weakly closed graph ([11], p. 68). Then $v_{n_{k}}(t)\in F(t,x_{n_{k}}(t))$ implies that $v (t)\in F(t,x(t)).$ Hence $v\in S_{F}(x)$ and therefore $y\in A(x)$. Consequently, $A$ has weakly sequentially closed graph.

Next we show that $A(Q)$ is relatively weakly compact. Let $t\in J$ be fixed, and $\{u_{n}\}$ be a sequence in $A(Q)$. Thus there exists $x_{n}\in Q$ such that $u_{n}\in A(x_{n}),$ and hence there exists $v_{n}\in S_{F}(x_{n})$ such that

$$\begin{aligned} u_{n}(t)=q(t)+\int ^{\mu (t)}_{0}k_{1}(t,s)v_{n}(s)ds. \end{aligned}$$

Thus

$$\begin{aligned} |u_{n}(t)|&= f\left( q(t)+\int ^{\mu (t)}_{0}k_{1}(t,s)v_{n}(s)ds\right) \\&\le |q(t)|+K_{1}\int ^{1}_{0}|v_{n}(s)|ds \le |q(t)|+K_{1}\int ^{1}_{0}\Vert F(s,x(s))\Vert ds\\&\le |q(t)|+K_{1}\int ^{1}_{0}\alpha (s)ds \le M +K_{1}\Vert \alpha \Vert _{L^{1}}. \end{aligned}$$

Therefore, $\{u_{n}(t)\}$ is weakly equi-bounded. For all $t\in J,$ the reflexivity of $X$ implies that the set $\{u_{n}(t): n\in \mathbb {N} \}$ is relatively weakly sequentially compact ([14], p. 782). Now we show $A(Q)$ is weakly equi-continuous. Let $t_{1},t_{2}\in J$ and assume that $u_{n}(t_{1})\ne u_{n}(t_{2})$. Then there exists $f\in X^{*}$ such that $f(u_{n}(t_{1})- u_{n}(t_{2}))= |u_{n}(t_{1})- u_{n}(t_{2})|$ and $|f|_{*}=1.$ Thus

$$\begin{aligned} |u_{n}(t_{1})- u_{n}(t_{2})|&\le |(q(t_{1})-q(t_{2}))|+K_{1}\int ^{\mu (t_{1})}_{0}|v_{n}(s)|ds- K_{1}\int ^{\mu (t_{2})}_{0}|v_{n}(s)|ds\\&\le |(q(t_{1})-q(t_{2}))|+K_{1}\int ^{\mu (t_{1})}_{\mu (t_{2})}|v_{n}(s)|ds\\&\le |(q(t_{1})-q(t_{2}))|+K_{1}\int ^{\mu (t_{1})}_{\mu (t_{2})} \Vert F(s,x(s))\Vert ds\\&\le |(q(t_{1})-q(t_{2}))|+K_{1}\int ^{\mu (t_{1})}_{\mu (t_{2})} \alpha (s) ds. \end{aligned}$$

Since $q$ and $\mu$ are continuous on a compact interval $J$, they are uniformly continuous and so as $t_{1}\rightarrow t_{2}$, we get $|u_{n}(t_{1})- u_{n}(t_{2})|\rightarrow 0.$ Hence $A(Q)$ is weakly equi-continuous. Now (see the Arzela Ascoli theorem ), $u_{n_{j}}\rightharpoonup u\in A(Q)$ and hence by the Eberlein–Šmulian theorem [10, Theorem 8.12.4, p. 549] we conclude that $A(Q)$ is relatively weakly compact.

Step 2 B has weakly sequentially closed graph and is $\Phi$-condensing.

From an argument similar to that in Step 1, $B(Q)$ is relatively weakly compact and $B$ has weakly sequentially closed graph. Then $B(Q)$ is bounded, and $B$ is $\Phi$-condensing for any $(MWNC)$ $\Phi$ on $E$.

Step 3 $(I-B)^{-1}A(x)$ is convex for each $x\in Q$.

Let $u_{1},u_{2}\in (I-B)^{-1}A(x)$. Then $u_{1}\in B(u_{1})+A(x)$ and $u_{2}\in B(u_{2})+A(x)$. Thus there exists $\alpha _{1},\alpha _{2}\in S_{F}(x),$ $\beta \in S_{G}(u_{1})$ and $\gamma \in S_{G}(u_{2})$ such that for all $t\in J$

$$\begin{aligned} u_{1}(t)=q(t)+\int ^{\mu (t)}_{0}k_{1}(t,s) \alpha _{1}(s)ds+\int ^{\sigma (t)}_{0}k_{2}(t,s)\beta (s)ds \end{aligned}$$

and

$$\begin{aligned} u_{2}(t)=q(t)+\int ^{\mu (t)}_{0}k_{1}(t,s) \alpha _{2}(s)ds+\int ^{\sigma (t)}_{0}k_{2}(t,s)\gamma (s)ds . \end{aligned}$$

For all $\lambda \in [0,1],$

$$\begin{aligned} \lambda u_{1}(t)+ (1-\lambda )u_{2}(t)&= q(t)+\int ^{\mu (t)}_{0}k_{1}(t,s)[\lambda \alpha _{1}(s)+(1-\lambda )\alpha _{2}(s)]ds\\&\qquad \quad +\int ^{\sigma (t)}_{0}k_{2}(t,s)[\lambda \beta (s)+(1-\lambda )\gamma (s)]ds\\&= q(t)+\int ^{\mu (t)}_{0}k_{1}(t,s)\alpha (s)ds +\int ^{\sigma (t)}_{0}k_{2}(t,s)\delta (s)ds , \end{aligned}$$

here $\alpha (t)=\lambda \alpha _{1}(t)+(1-\lambda )\alpha _{2}(t)$ and $\delta (t)=\lambda \beta (t)+(1-\lambda )\gamma (t).$ Since $F(t,x(t))$ has convex values then $\alpha (t)\in F(t,x(t)).$ Also, $\alpha \in L^{1}(J,X)$. Thus $\alpha \in S_{F}(x).$ Now we claim that $\delta \in S_{G}(\lambda u_{1}+ (1-\lambda )u_{2}).$ Note

$$\begin{aligned} \lambda \beta (t)+(1-\lambda )\gamma (t)\in \lambda G(t,u_{1}(t))+(1-\lambda )G(t,u_{2}(t)). \end{aligned}$$

From $(H_5)$, $G(t,.)$ has convex graph and so

$$\begin{aligned} \lambda G(t,u_{1}(t))+(1-\lambda )G(t,u_{2}(t)) \subseteq G(t,\lambda u_{1}(t)+ (1-\lambda )u_{2}(t) ), \end{aligned}$$

and hence

$$\begin{aligned} \delta (t)=\lambda \beta (t)+(1-\lambda )\gamma (t)\in G(t,\lambda u_{1}(t)+ (1-\lambda )u_{2}(t) ). \end{aligned}$$

Also $\delta =\lambda \beta +(1-\lambda )\gamma \in L_{1}(J,X).$ Then $\delta \in S_{G}(\lambda u_{1}+ (1-\lambda )u_{2}).$ Hence $\lambda u_{1}+ (1-\lambda )u_{2}\in A(x)+B(\lambda u_{1}+ (1-\lambda )u_{2})$ and therefore $\lambda u_{1}+ (1-\lambda )u_{2}\in (I-B)^{-1}A(x).$

Step 4  If $y\in B(y)+A(x),$ $x\in Q$ then $y\in Q$.

For $x\in Q$ we have $\Vert x\Vert \le r$. Let $y\in B(y)+A(x)$. Then there exists $v\in S_{F}(x)$ and $w\in S_{G}(y)$ such that

$$\begin{aligned} y(t)= q(t)+\int ^{\mu (t)}_{0}k_{1}(t,s)v(s)ds +\int ^{\sigma (t)}_{0}k_{2}(t,s)w(s)ds. \end{aligned}$$

Now

$$\begin{aligned} |y(t)|&\le |q(t)|+K_{1}\int ^{\mu (t)}_{0}|v(s)|ds +K_{2}\int ^{\sigma (t)}_{0}|w(s)|ds\\&\le |q(t)|+K_{1}\int ^{\mu (t)}_{0}\Vert F(s,x(s))\Vert ds +K_{2}\int ^{\sigma (t)}_{0}\Vert G(s,y(s))\Vert ds. \end{aligned}$$

From $(H_{4})$ and $(H_{6})$ we get

$$\begin{aligned} |y(t)|&\le |q(t)|+K_{1}\int ^{\mu (t)}_{0} \gamma (s)\psi (|x(s)|)ds\\&\quad +\,K_{2}\int ^{\sigma (t)}_{0}\Vert G(s,0)\Vert ds+K_{2}\int ^{\sigma (t)}_{0}\beta (s)|y(s)| ds\\&\le |q(t)|+K_{1}\int ^{1}_{0} \gamma (s)\psi (|x(s)|)ds +K_{2}\int ^{1}_{0}\Vert G(s,0)\Vert ds +K_{2}\int ^{1}_{0}\beta (s)|y(s)| ds\\&\le M+ K_{1}\Vert \gamma \Vert _{L^{1}} \psi (r)+K_{2} L\\&\quad +\,K_{2}\Vert \beta \Vert _{L^{1}}\Vert y\Vert , \end{aligned}$$

and we can do this for all $t\in J$. Then

$$\begin{aligned} \Vert y\Vert \le M+ K_{1}\Vert \gamma \Vert _{L^{1}} \psi (r)+K_{2} L+ K_{2}\Vert \beta \Vert _{L^{1}}\Vert y\Vert . \end{aligned}$$

Hence,

$$\begin{aligned} \Vert y\Vert \le \frac{M +K_{2} L+ K_{1}\Vert \gamma \Vert _{L^{1}} \psi (r)}{1- K_{2}\Vert \beta \Vert _{L^{1}}}. \end{aligned}$$

Therefore, $y\in Q$. Thus $(I-B)^{-1}A(Q)\subset Q$.

Applying Theorem 3.4, we obtain that the inclusion (1.2) has a solution on $J.$ The proof is complete. $\square$