On Some Properties of Multi-Valued Feng–Liu-Type Operators in Metric Spaces

Abstract

In the context of a complete metric space, the most important fixed-point result for multi-valued operators was given in 1969. Many extensions of this fixed-point principle for multi-valued operators were proved by different authors. Based on some of the above-mentioned results, we introduce the notion of the multi-valued Feng–Liu-type operator and we construct an abstract fixed-point theory for this general class of multi-valued operators. Our results extend and complement some theorems in metric fixed-point theory for multi-valued operators. An application to a Cauchy problem related to a first-order differential inclusion is also given. In this case, our theorem improves several previous theorems on this subject by relaxing the contraction-type condition (with respect to its second argument) on the multi-valued right-hand side.

1. Introduction and Preliminaries

Let $(X,d)$ be a metric space and $P\left(X\right)$ be the set of all nonempty subsets of X. We follow the notations from [1]. For the convenience of the reader, we recall some of them.

We define the following families of subsets of X:

$$P_b\left(X\right):=\{Y\in P\left(X\right)|Y \mathrm{is} \mathrm{bounded}\},P_{cl}\left(X\right):=\{Y\in P\left(X\right)|Y \mathrm{is} \mathrm{closed}\}.$$

In the context of a normed space, we denote:

$$P_{cv}\left(X\right):=\{Y\in P\left(X\right)|Y \mathrm{is} \mathrm{convex}\},P_{cl,cv}\left(X\right):=\{Y\in P\left(X\right)|Y \mathrm{is} \mathrm{closed} \mathrm{and} \mathrm{convex}\}.$$

We also introduce in $(X,d)$ the following notations:

$$D_d(a,B):=\inf \left\{d(a,b)\right|b\in B\},e_d(A,B):=sup\{D_d(a,B),a\in A\}.$$

$$H_d(A,B):=\max \{e_d(A,B),e_d(B,A)\}.$$

Let $(X,d)$ be a metric space and $F:X\to P\left(X\right)$ be a multi-valued operator. We denote by $Fix\left(F\right):=\{x\in X:x\in F(x\left)\right\}$ the fixed-point set of F and by $Graph\left(F\right):=\left\{\right(x,y):y\in F(x\left)\right\}$ the graph of the multi-valued operator F.

In the same framework, a sequence of Picard iterates for F starting from arbitrary $x_0\in X$ means a sequence with the property that $x_{n+1}\in F\left(x_n\right)$, for all $n\in \mathbb{N}.$ Also, by a selection of F, we understand a single-valued operator $f:X\to X$ such that $f\left(x\right)\in F\left(x\right)$, for each $x\in X$.

In the context of a complete metric space, the most important fixed-point result for multi-valued operators was given by S.B. Nadler Jr. in 1969. Starting from this result, we introduce the notion of the multi-valued Feng–Liu type operator and we construct a fixed-point theory for this type of multi-valued operator. Our results extend and complement some theorems in the recent literature, see [2,3,4,5,6,7,8,9]. An application to a Cauchy problem related to a first-order differential inclusion is also given. In this case, our theorem improves several previous theorems on this subject by relaxing the contraction-type condition (with respect to its second argument) on the multi-valued right-hand side.

2. Main Results

Let us recall first the main metric fixed-point theorem for multi-valued operators given by S.B. Nadler Jr. in 1969 [6] and slightly extended by H. Covitz and S.B. Nadler Jr. in 1970 [3].

By definition, a multi-valued operator $F:X\to P\left(X\right)$ is called a multi-valued $\alpha$-contraction if there exists $\alpha \in (0,1)$ such that

$$H_d(F\left(x\right),F\left(y\right))\le \alpha d(x,y), \mathrm{for} \mathrm{all} (x,y)\in X\times X.$$

The notion was introduced by S.B. Nadler Jr. in 1969 in [6], where the main metric fixed-point result for multi-valued operators was also proved. We call the following result the Multi-valued Contraction Principle (MCP), and it comes from H. Covitz and S.B. Nadler Jr.’s paper in 1970 [3].

Theorem 1.

Let $(X,d)$ be a complete metric space and $F:X\to P_{cl}\left(X\right)$ be a multi-valued α-contraction. Then:

(i) 

$Fix\left(F\right)\ne \varnothing$;

(ii) 

For each $x_0\in X$, there exists a sequence of Picard iterates for F starting from $x_0$, which converges to a fixed point of F.

If the operator $F:X\to P\left(X\right)$ satisfies the above condition for every $(x,y)\in Graph\left(F\right)$, then F is called a multi-valued graph $\alpha$-contraction.

It is also known that, in a complete metric space $(X,d)$, any multi-valued graph $\alpha$-contraction $F:X\to P\left(X\right)$ which has a closed graph (i.e., the set $Graph\left(f\right)$ is closed) has at least one fixed point, and for each $x_0\in X$, there exists a sequence of Picard iterates for F starting from $x_0$, which converges to a fixed point of F, see [8]. This result is called the Multi-valued Graph Contraction Principle (MGCP).

We recall now the definition of a multi-valued contraction of the Feng–Liu type, see [4].

Definition 1.

Let $(X,d)$ be a metric space, $F:X\to P\left(X\right)$ be a multi-valued operator, $\beta \in (0,1)$ and $x\in X$. Consider the set

$$I_{\beta }^x:=\{y\in F\left(x\right):\beta d(x,y)\le D_d(x,F\left(x\right))\}.$$

Then, by definition, F is called a multi-valued α-contraction of the Feng–Liu type if $\alpha \in (0,\beta )$ and for each $x\in X$ there is $y\in I_{\beta }^x$ such that

$$D_d(y,F\left(y\right))\le \alpha d(x,y).$$

It is easy to see that if $\beta \in (0,1)$, then the set $I_{\beta }^x$ is not empty for every $x\in X$.

Remark 1.

Notice any multi-valued α-contraction $F:X\to P\left(X\right)$ is a multi-valued α-contraction of the Feng–Liu type, but the reverse implication does not hold. For such examples, see the paper [4]. Also, any multi-valued graph α-contraction is a multi-valued α-contraction of the Feng–Liu type.

The following result was basically given by Feng and Liu in [4]. The version presented here contains, as a second conclusion, the convergence of the sequence of Picard iterates and the strong retraction displacement condition. For the retraction-displacement condition see also [10].

Theorem 2.

Let $(X,d)$ be a complete metric space and $F:X\to P\left(X\right)$ be a multi-valued α-contraction of the Feng–Liu type. Suppose that either the mapping $g:X\to {\mathbb{R}}_{+}$, $g\left(x\right)=D_d(x,F\left(x\right))$ is lower semi-continuous and F has closed values or F has a closed graph. Then, the following conclusions hold:

(i) 

$Fix\left(F\right)\ne \varnothing$;

(ii) 

For every $x_0\in X$, there exists an iterative sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ of the Picard type for F starting from $x_0$ which converges to $x^{*}\left(x_0\right)\in Fix\left(F\right)$, and the following relation holds:

$$d(x_0,x^{*}\left(x_0\right))\le \frac{1}{1-\frac{\alpha }{\beta }}{D}_d(x_0,F\left(x_0\right)).$$

In particular, the following fixed-point theorem also holds.

Theorem 3.

Let $(X,d)$ be a complete metric space and $F:X\to P\left(X\right)$ be a multi-valued operator with a closed graph. Suppose there exists $\alpha \in \left(0.1\right)$ such that

$$D_d(y,F\left(y\right))\le \alpha d(x,y),for all(x,y)\in Graph\left(F\right).$$

Then, F has at least one fixed point, and for every $x_0\in X$ there exists an iterative sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ of the Picard type for F starting from $x_0$, which converges to $x^{*}\left(x_0\right)\in Fix\left(F\right)$ such that the following relation holds:

$$d(x_0,x^{*}\left(x_0\right))\le \frac{1}{1-\alpha }{D}_d(x_0,F\left(x_0\right)).$$

The conclusion of the above fixed-point theorems generates the following important notion.

Definition 2.

Let $(X,d)$ be a metric space and $F:X\to P\left(X\right)$ be a multi-valued operator. Then, by the definition, F is called a multi-valued Feng–Liu-type operator (briefly MF-LT operator) if for each $x_0\in X$ there exists a sequence of Picard iterates for F starting from $x_0$, which converges to a fixed point of F.

From Theorem 1, it is obvious that in a complete metric space, any multi-valued $\alpha$-contraction with closed values, as well as any multi-valued graph contraction with a closed graph, is an MF-LT operator.

The above definition generates a multi-valued operator defined as follows: $\tilde{F}:X\to P\left(Fix\left(F\right)\right)$, given by $\tilde{F}\left(x\right):=\{x^{*}\in Fix\left(F\right)|$ there exists a sequence of Picard iterates for F starting from x that converges to $x^{*}\}$.

Definition 3.

Let $(X,d)$ be a metric space and $F:X\to P\left(X\right)$ be an MF-LT operator. Then F is called a ψ-multi-valued Feng–Liu-type operator (briefly ψ-MF-LT operator) if $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is increasing, continuous in 0 with $\psi \left(0\right)=0$ and there exists a selection $\tilde{f}$ of $\tilde{F}$ such that

$$d(x,\tilde{f}\left(x\right))\le \psi \left(D_d(x,F\left(x\right))\right), for all x\in X.$$

A multi-valued weakly Feng–Liu operator for which there exists $C>0$ such that

$$d(x,\tilde{f}\left(x\right))\le C{D}_d(x,F\left(x\right)), \mathrm{for} \mathrm{all} x\in X.$$

is called a C-multi-valued Feng–Liu-type operator (briefly, C-MF-LT operator).

Remark 2.

It is easy to observe that F satisfying the assumptions of Theorem 2 is a C-MF-LT operator with $C:=\frac{1}{1-\frac{\alpha }{\beta }}$. Also, a multi-valued operator F satisfying the conditions from Theorem 3 is a C-MF-LT operator with $C:=\frac{1}{1-\alpha }$.

We discuss now the concept of multi-valued quasicontraction. Probably the most natural definition of a multi-valued quasicontraction is the following one.

Definition 4.

Let $(X,d)$ be a metric space and $F:X\to P\left(X\right)$ be a multi-valued operator. We say that F is a multi-valued k-quasicontraction if $k\in (0,1)$, $Fix\left(F\right)\ne \varnothing$ and

$$e_d(F\left(x\right),x^{*})\le kd(x,x^{*}), for every x\in X and every x^{*}\in Fix\left(F\right).$$

In this case, we observe that if F is a multi-valued k-quasicontraction in the sense of Definition 4, then $Fix\left(F\right)=\left\{x^{*}\right\}$. Indeed, let $x^{*}\in Fix\left(F\right)$ and $u\in Fix\left(F\right)$ with $u\ne x^{*}$. Then, using the above quasicontraction condition, we have

$$d(u,x^{*})\le e_d(F\left(u\right),x^{*})\le kd(u,x^{*}).$$

This is a contradiction with $k<1$. Thus, $Fix\left(F\right)=\left\{x^{*}\right\}$.

Another possibility to define quasicontractions in the multi-valued case is as follows.

Definition 5.

Let $(X,d)$ be a metric space and $F:X\to P\left(X\right)$ be a multi-valued operator. We say that F is a multi-valued k-quasicontraction if $k\in (0,1)$, $Fix\left(F\right)\ne \varnothing$ and

$$H_d(F\left(x\right),Fix\left(F\right)\le kd(x,x^{*}), for every x\in X and every x^{*}\in Fix\left(F\right).$$

Remark 3.

Suppose that $(X,d)$ is a metric space, and $F:X\to P\left(X\right)$ is a multi-valued k-quasicontraction in the sense of Definition 5. Assume that $SFix\left(F\right)\ne \varnothing .$ Then, by a similar approach to that above, we can prove that $Fix\left(F\right)=SFix\left(F\right)=\left\{x^{*}\right\}$.

We now present some stability properties for the fixed-point inclusion

$$x\in F\left(x\right).$$

(1)

We recall below some concepts related to Ulam–Hyers stability, see [1,7,11,12].

Definition 6.

The fixed-point inclusion (1) is said to be Ulam–Hyers stable if there exists $c>0$ such that for every $ϵ>0$ and any $z\in X$ with $D_d(z,F\left(z\right))\le ϵ$, there exists $x^{*}\in Fix\left(F\right)$ such that $d(z,x^{*})\le c·ϵ$.

If the above relation has the following form

$$d(z,x^{*})\le \psi \left(ϵ\right),$$

with a function $\psi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, increasing, continuous in 0 and with $\psi \left(0\right)=0$, then we say that the fixed-point inclusion is generalized Ulam–Hyers stable.

Now we can present the main Ulam–Hyers stability result for the multi-valued inclusion $x\in F\left(x\right)$ with an MF-LT operator.

Theorem 4.

Let $(X,d)$ be a metric space and $F:X\to P\left(X\right)$ be a ψ-MF-LT operator. Then, the fixed-point inclusion $x\in F\left(x\right)$ is generalized Ulam–Hyers stable.

Proof. 

Indeed, by the $\psi$-MF-LT operator assumption on F, the fixed-point set $Fix\left(F\right)$ is nonempty and there exists a selection $\tilde{f}:X\to Fix\left(F\right)$ of $\tilde{F}$. Now, for $ϵ>0$ and for any $ϵ$-fixed point $z\in X$ of F (i.e., $D_d(z,F\left(z\right))\le ϵ$), using again the fact that F is a $\psi$-MF-LT operator, we have

$$d(z,\tilde{f}\left(z\right))\le \psi \left(D_d(z,F\left(z\right))\right)\le \psi \left(ϵ\right).$$

The proof is complete. □

We recall now the concept of the well-posedness of the fixed-point inclusion (1), see [12,13].

Definition 7.

The fixed-point inclusion (1) is said to be well posed in the sense of Reich and Zaslavski if $Fix\left(F\right)\ne \varnothing$ and there exists a mapping $r:X\to Fix\left(F\right)$ such that, for each $x^{*}\in Fix\left(F\right)$ and for any sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subset r^{-1}\left(x^{*}\right)$ with $D_d(u_n,F\left(u_n\right))\to 0$, we have that $u_n\to x^{*}$ as $n\to \infty$.

An abstract well-posedness result for the multi-valued inclusion $x\in F\left(x\right)$ with a $\psi$-MF-LT operator is the following.

Theorem 5.

Let $(X,d)$ be a metric space and $F:X\to P\left(X\right)$ be a ψ-MF-LT operator. Then, the fixed-point inclusion $x\in F\left(x\right)$ is well posed in the sense of Reich and Zaslavski.

Proof. 

Indeed, by the $\psi$-MF-LT operator assumption, the fixed-point set $Fix\left(F\right)$ is nonempty. For the well-posedness property, we consider a selection $\tilde{f}:X\to Fix\left(F\right)$ of $\tilde{F}$. Let $x^{*}\in Fix\left(F\right)$ and consider any sequence ${\left\{u_n\right\}}_{n\in \mathbb{N}}\subset {\tilde{f}}^{-1}\left(x^{*}\right)$ such that $D_d(u_n,F\left(u_n\right))\to 0$. Then, we have that

$$d(u_n,x^{*})=d(u_n,\tilde{f}\left(u_n\right))\le \psi \left(D_d(u_n,F\left(u_n\right))\right)\to 0,\mathrm{as} n\to \infty .$$

The proof is complete. □

By the above results and by Feng–Liu Theorem 2, it follows that a multi-valued operator satisfying the conditions of the Feng–Liu theorem is a $\psi$-MF-LT operator with $\psi \left(t\right):=\frac{1}{1-\frac{\alpha }{\beta }}t$, $t\in {\mathbb{R}}_{+}$, i.e., $Fix\left(F\right)\ne \varnothing$ and

$$d(x_0,\tilde{f}\left(x_0\right))\le \frac{1}{1-\frac{\alpha }{\beta }}{D}_d(x_0,F\left(x_0\right)),$$

where $\tilde{f}\left(x_0\right)\in \tilde{F}\left(x_0\right)$ is one of the fixed points of F, which is the limit of a sequence of Picard iterates for F starting from $x_0$.

As a consequence, we obtain the following theorem.

Theorem 6.

In the conditions of the Feng–Liu theorem for a multi-valued operator F, the fixed-point inclusion $x\in F\left(x\right)$ is Ulam–Hyers stable and is well posed in the sense of Reich and Zaslavski.

The following lemma (see, e.g., [1]) is important for the proof of our next theorems.

Lemma 1.

(Cauchy-Toeplitz Lemma) Let ${\left(a_n\right)}_{n\in \mathbb{N}}$ be a sequence in ${\mathbb{R}}_{+}$ such that the series ${\displaystyle \sum _{n\ge 0}{a}_n}$ is convergent, and let ${\left(b_n\right)}_{n\in \mathbb{N}}\in \mathbb{R}$ be a sequence with non-negative terms such that ${\displaystyle \underset{n\to \infty }{lim}{b}_n=0}$. Then,

$${\displaystyle \underset{n\to \infty }{lim}(\sum _{k=0}^n{a}_{n-k}{b}_k)=0.}$$

We continue our study by recalling the Ostrowski property of the fixed-point inclusion $x\in F\left(x\right)$, see also [11,12].

Definition 8.

We say that the fixed-point inclusion has the Ostrowski stability property if $Fix\left(F\right)\ne \varnothing$ and there exists a mapping $r:X\to Fix\left(F\right)$ such that for each $x^{*}\in Fix\left(F\right)$ and for any sequence ${\left\{v_n\right\}}_{n\in \mathbb{N}}\subset r^{-1}\left(x^{*}\right)$ with $D_d(v_{n+1},F\left(v_n\right))\to 0$ as $n\to \infty$, we have that $v_n\to x^{*}$ as $n\to \infty$.

Remark 4.

The above notion is related to the so-called limit shadowing property of a multi-valued operator $F:X\to P\left(X\right)$. We say that F has the limit shadowing property if for any sequence ${\left\{y_n\right\}}_{n\in \mathbb{N}}$ in X with $D_d(y_{n+1},F\left(y_n\right))\to 0$, there exists $x_0\in X$ and a sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ of Picard iterates for F starting from $x_0$ such that $d(y_n,x_n)\to 0$ as $n\to \infty$.

Definition 9.

Let $(X,d)$ be a metric space, $F:X\to P\left(X\right)$ be a multi-valued operator such that $Fix\left(F\right)\ne \varnothing$, and there exists a mapping $r:X\to Fix\left(F\right)$. If we denote

$$X_{x^{*}}:=r^{-1}\left(x^{*}\right),for x^{*}\in Fix\left(F\right),$$

then we have

$${\displaystyle X=\bigcup _{x^{*}\in Fix\left(F\right)}{X}_{x^{*}}.}$$

This partition is called the fixed-point partition of X corresponding to r.

For example, if $(X,d)$ is a complete metric space and $F:X\to P\left(X\right)$ is a multi-valued $\alpha$-contraction of the Feng–Liu type with a closed graph, then F is a $\frac{1}{1-\frac{\alpha }{\beta }}$-MF-LT operator and for any selection $\tilde{f}:X\to Fix\left(F\right)$ of the multi-valued operator $\tilde{F}:X\to P\left(Fix\left(F\right)\right)$, defined by $\tilde{F}\left(x\right):=\{x^{*}\in Fix\left(F\right)|$ there exists a sequence of Picard iterates for F starting from x that converges to $x^{*}\}$, we have

$$d(x_0,\tilde{f}\left(x_0\right))\le \frac{1}{1-\frac{\alpha }{\beta }}{D}_d(x_0,F\left(x_0\right)), \mathrm{for} \mathrm{every}{x}_0\in X.$$

Thus, ${\displaystyle X=\bigcup _{x^{*}\in Fix\left(F\right)}{\tilde{f}}^{-1}\left(x^{*}\right)}$ is a fixed-point partition corresponding to $\tilde{f}$.

In terms of a fixed-point partition generated by a mapping $r:X\to Fix\left(F\right)$, we introduce the following notion.

Definition 10.

Let $(X,d)$ be a metric space, $F:X\to P\left(X\right)$ be a multi-valued operator with $Fix\left(F\right)\ne \varnothing$, and $r:X\to Fix\left(F\right)$ be a given mapping. We say that F is a multi-valued k-quasicontraction with respect to the fixed-point partition corresponding to r if $k\in (0,1)$ and

$$e_d(F\left(x\right),r\left(x\right))\le kd(x,r\left(x\right)), for every x\in X.$$

An abstract result concerning the Ostrowski stability property for the multi-valued inclusion $x\in F\left(x\right)$ with an MF-LT operator is the following.

Theorem 7.

Let $(X,d)$ be a metric space and $F:X\to P\left(X\right)$ be an MF-LT operator. If, additionally, F is a multi-valued quasicontraction, then the fixed-point inclusion $x\in F\left(x\right)$ has the Ostrowski stability property.

Proof. 

Indeed, by the MF-LT operator assumption on F, the fixed-point set $Fix\left(F\right)$ is nonempty. Let $x^{*}\in Fix\left(F\right)$, and consider any sequence ${\left\{v_n\right\}}_{n\in \mathbb{N}}\subset r^{-1}\left(x^{*}\right)$ with $D_d(v_{n+1},F\left(v_n\right))\to 0$ as $n\to \infty$. Then, we have

$$\begin{gathered} d(v_{n+1},x^{*})=d(v_{n+1},r\left(v_n\right))\le D_d(v_{n+1},F\left(v_n\right))+e_d(F\left(v_n\right),r\left(v_n\right))\le \\ {D}_d(v_{n+1},F\left(v_n\right))+kd(v_n,r\left(v_n\right))=D_d(v_{n+1},F\left(v_n\right))+kd(v_n,x^{*})\le \\ {D}_d(v_{n+1},F\left(v_n\right))+k[D_d(v_n,F\left(v_{n-1}\right))+kd(v_{n-1},x^{*})]\le \cdots \le \\ {D}_d(v_{n+1},F\left(v_n\right))+k{D}_d(v_n,F\left(v_{n-1}\right))+k^2{D}_d(v_{n-1},F\left(v_{n-2}\right))+\cdots +k^n{D}_d(v_1,F\left(v_0\right))+k^{n+1}d(v_0,x^{*}). \end{gathered}$$

By the Cauchy–Toeplitz lemma, we obtain the desired conclusion. □

Finally, let us discuss the data dependence (on the operator perturbation) of the fixed-point set for MF-LT operators. More precisely, we are interested in the following problem.

Definition 11.

Let $(X,d)$ be a metric space and $F,G:X\to P\left(X\right)$ be two MF-LT operators. Suppose that there exists $\eta >0$ such that

$$H_d(F\left(x\right),G\left(x\right))\le \eta , for every x\in X.$$

If there exists a function $\gamma :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, increasing, continuous in 0 and with $\gamma \left(0\right)=0$ such that

$$H_d(Fix\left(F\right),Fix\left(G\right))\le \gamma \left(\eta \right),$$

then we say that the fixed-point inclusion $x\in F\left(x\right)$ satisfies the data dependence property with respect to operator perturbation.

Concerning the above problem, we have the following abstract result.

Theorem 8.

Let $(X,d)$ be a metric space and $F:X\to P\left(X\right)$ be a ψ-MF-LT operator and $G:X\to P\left(X\right)$ be a β-MF-LT operator. Suppose that there exists $\eta >0$ such that

$$H_d(F\left(x\right),G\left(x\right))\le \eta , for every x\in X.$$

Then, the fixed-point inclusion $x\in F\left(x\right)$ has the data dependence property.

Proof. 

Let $x_F^{*}\in Fix\left(F\right)$ be arbitrary. Since G is a $\beta$-MF-LT operator, we have that $Fix\left(G\right)\ne \varnothing$ and there exists a selection $\tilde{g}$ of $\tilde{G}$ such that

$$d(x,\tilde{g}\left(x\right))\le \beta \left(D_d(x,G\left(x\right))\right), \mathrm{for} \mathrm{all} x\in X.$$

For $x:=x_F^{*}$, the above relation becomes

$$d(x_F^{*},\tilde{g}\left(x_F^{*}\right))\le \beta \left(D_d(x_F^{*},G\left(x_F^{*}\right))\right).$$

Thus, we obtain that there exists $\tilde{g}\left(x_F^{*}\right)\in Fix\left(G\right)$ such that

$$d(x_F^{*},\tilde{g}\left(x_F^{*}\right))\le \beta \left(D_d(x_F^{*},G\left(x_F^{*}\right))\right)\le \beta \left(H_d(F\left(x_F^{*}\right),G\left(x_F^{*}\right))\right)\le \beta \left(\eta \right).$$

Similarly, for arbitrary $x_G^{*}\in Fix\left(G\right)$, there exists $\tilde{f}(x_G^{*})\in Fix\left(F\right)$ such that

$$d(x_G^{*},\tilde{f}(x_G^{*}))\le \psi \left(\eta \right).$$

Thus, we obtain that

$$H_d(Fix\left(F\right),Fix\left(G\right))\le max\{\psi \left(\eta \right),\beta \left(\eta \right)),$$

and the proof is complete. □

Remark 5.

It would be of interest to extend the results of this paper to various generalized metrical structures, see, for example, [1,14,15].

3. An Application

We present now an application to differential inclusions. Let us consider the following differential inclusion:

$$x^{\prime }\left(t\right)\in F(t,x\left(t\right)), \text{a.e.} t\in [a,b],$$

(2)

with the initial condition $x\left(a\right)=x^0$, where $F:[a,b]\times {\mathbb{R}}^n\to P_{cl,cv}\left({\mathbb{R}}^n\right)$ satisfies:

(H1)

There exists an integrable function $M:[a,b]\to {\mathbb{R}}_{+}$ such that for each $x\in C([a,b],{\mathbb{R}}^n)$, we have

$$\parallel F(s,x(s\left)\right)\parallel :=max\{\parallel w\parallel :w\in F(s,x\left(s\right)\left)\right\}\le M\left(s\right),a.e.s\in [a,b];$$

(H2)

$F(·,x(·)):[a,b]\to P_{cl,cv}\left({\mathbb{R}}^n\right)$ is measurable for every $x\in C([a,b],{\mathbb{R}}^n)$;

(H3)

For each $s\in [a,b]$, $F(s,·):{\mathbb{R}}^n\to P_{cl,cv}\left({\mathbb{R}}^n\right)$ is lower semi-continuous;

(H4)

There exists a continuous function $p:[a,b]\to {\mathbb{R}}_{+}$ such that for each $s\in [a,b]$ and $u,v\in {\mathbb{R}}^n$, we have

$$D(w,F(s,v\left)\right)\le p\left(s\right)\parallel u-v\parallel , \mathrm{for} \mathrm{all} w\in F(s,u).$$

By a solution of the above initial value problem, we understand an absolutely continuous function $x:[a,b]\to {\mathbb{R}}^n$ which satisfies $x^{\prime }\left(t\right)\in F(t,x\left(t\right))$, a.e. $t\in [a,b]$ and has the property that $x\left(a\right)=x^0.$ For the above notions and related results, see [16,17,18].

Our result improves several previous theorems on this subject by relaxing the contraction-type condition on F, with respect to its second argument, see [16,17,19].

Theorem 9.

Consider the Cauchy problem (2). Suppose that hypotheses (H1)–(H4) hold. Then the following conclusions hold:

(i) 

There exists at least one solution for the initial value problem:

$$\left\{\begin{array}{c}{x}^{\prime }\left(t\right)\in F(t,x\left(t\right)), \text{a.e.} t\in [a,b],\hfill \\ x\left(a\right)=x^0,x^0\in {\mathbb{R}}^n.\hfill \end{array}\right.$$

(3)

(ii) 

The initial value problem (3) is Ulam–Hyers stable, i.e., there exists $C>0$ such that for each $\epsilon >0$ and for each function $y\in C([a,b],{\mathbb{R}}^n)$, a solution of the inequation

$$D_{{\parallel ·\parallel }_{{\mathbb{R}}^n}}(y\left(t\right),F(t,y\left(t\right)))\le \epsilon ,t\in [a,b],$$

there exists a solution x of differential inclusion (2) such that

$$\left|x\right(t)-y(t\left)\right|\le c·\epsilon , for each t\in [a,b].$$

Proof. 

The problem (3) is equivalent to an integral inclusion of the Volterra type:

$$x\left(t\right)\in x^0+{\int }_a^{t}F(s,x\left(s\right))ds,t\in [a,b].$$

(4)

Let us consider the multi-valued operator $W:C([a,b],{\mathbb{R}}^n)\to P\left(C([a,b],{\mathbb{R}}^n)\right)$ defined by

$$W\left(x\right):=\{w\in C([a,b],{\mathbb{R}}^n):w\left(t\right)\in x^0+{\int }_a^{t}F(s,x\left(s\right))ds,t\in [a,b]\}.$$

If for $x\in C([a,b],{\mathbb{R}}^n)$ we denote by $S_{F,x}$ the set of integrable selections of F, i.e.,

$$S_{F,x}:=\{f\in L^1([a,b],{\mathbb{R}}^n):f\left(s\right)\in F(s,x\left(s\right)) \mathrm{for} \text{a.e.} s\in [a,b]\},$$

then, by (H1)–(H3), we obtain that $S_{F,x}$ is nonempty for each $x\in C([a,b],{\mathbb{R}}^n)$. Thus, the set $W\left(x\right)$ is nonempty for each $x\in C([a,b],{\mathbb{R}}^n)$. Moreover, W has closed values (by Theorem 8.6.3 in [16]).

We denote by $\parallel ·\parallel$ the usual supremum (Cebîsev) norm on $C([a,b],{\mathbb{R}}^n)$. For our approach, we consider $C([a,b],{\mathbb{R}}^n)$ with a Bielecki-type norm in $C([a,b],{\mathbb{R}}^n)$, given by

$${\parallel x\parallel }_B:=\underset{t\in [a,b]}{sup}{(\parallel x\left(t\right)\parallel }_{{\mathbb{R}}^n}·e^{-\tau q\left(t\right)}), \mathrm{where} q\left(t\right):={\int }_a^{t}p\left(s\right)ds \mathrm{and} \tau >1.$$

We denote this space by $X:=(C([a,b],{\mathbb{R}}^n),\parallel ·{\parallel }_B)$. For sake of simplicity, we denote by $|·|$ the usual Euclidean norm in ${\mathbb{R}}^n$.

Under the above notations, our problem (4) is equivalent to the following fixed-point inclusion:

$$x\in W\left(x\right),x\in X,$$

(5)

where $W:X\to P_{cl}\left(X\right)$.

We now show that

$$D_{{\parallel ·\parallel }_B}(y,W\left(y\right))\le \frac{1}{\tau }{\parallel x-y\parallel }_B, \mathrm{for} \mathrm{every} x\in C([a,b],{\mathbb{R}}^n) \mathrm{and} y\in W\left(x\right),$$

where ${\displaystyle D_{{\parallel ·\parallel }_B}(y,W\left(y\right)):=\underset{w\in W\left(y\right)}{inf}{\parallel y-w\parallel }_B}$.

For this purpose, it is enough to show that for $x\in C([a,b],{\mathbb{R}}^n)$ and every $y\in W\left(x\right)$, there exists $w\in W\left(y\right)$ such that ${\parallel y-w\parallel }_B\le \frac{1}{\tau }{\parallel x-y\parallel }_B$.

By the above approach, for $x\in C([a,b],{\mathbb{R}}^n)$ and for $y\in W\left(x\right)$, there exists an integrable selection $\tilde{f}\left(s\right)\in F(s,x\left(s\right))$ such that $y\left(t\right)=x^0+{\int }_a^t\tilde{f}\left(s\right)ds$, $t\in [a,b]$. For the above $\tilde{f}$, we show that there exists an integrable selection $\widehat{f}\left(s\right)\in F(s,y\left(s\right))$ and $w\left(t\right):=x^0+{\int }_a^t\widehat{f}\left(s\right)ds$, $t\in [a,b]$ such that

$$|\tilde{f}\left(s\right)-\widehat{f}\left(s\right)|<p\left(s\right)|x\left(s\right)-y\left(s\right)|.$$

By (H4), we obtain that

$$D(\tilde{f}\left(s\right),F(s,y\left(s\right)))<p\left(s\right)\parallel x\left(s\right)-y\left(s\right)\parallel ,s\in [a,b].$$

Thus, there exists $z\in F(s,y(s\left)\right)$ such that $|\tilde{f}\left(s\right)-z|<p\left(s\right)|x\left(s\right)-y\left(s\right)|$, $s\in [a,b]$. We define now a multi-valued operator G by

$$G\left(s\right):=\{z:|\tilde{f}\left(s\right)-z|<p\left(s\right)|x\left(s\right)-y\left(s\right)\left|\right\}|\cap F(s,y\left(s\right)).$$

Notice first that $G:[a,b]\to P_{cl,cv}\left({\mathbb{R}}^n\right)$ is measurable by Proposition 3.4 (a) from [17]. Let $\widehat{f}$ be a measurable selection of G. Then, by (H1), $\widehat{f}$ is integrable, and for $s\in [a,b]$ we have that $\widehat{f}\left(s\right)\in F(s,y\left(s\right))$ and

$$|\tilde{f}\left(s\right)-\widehat{f}\left(s\right)|<p\left(s\right)|x\left(s\right)-y\left(s\right)|.$$

Hence, we obtain

$$|y\left(t\right)-w\left(t\right)|\le {\int }_a^t|\tilde{f}\left(s\right)-\widehat{f}\left(s\right)|ds\le {\int }_a^{t}p\left(s\right)|x\left(s\right)-y\left(s\right)|ds=$$

$${\int }_a^t{e}^{\tau q\left(s\right)}p\left(s\right)e^{-\tau q\left(s\right)}|x\left(s\right)-y\left(s\right)|ds\le \frac{1}{\tau }{e}^{\tau q\left(t\right)}{\parallel x-y\parallel }_B.$$

Thus, we immediately obtain that

$${\parallel y\left(t\right)-w\left(t\right)\parallel }_B\le \frac{1}{\tau }{\parallel x-y\parallel }_B.$$

The conclusions of the theorem (via the fixed-point inclusion (5)) follow now by Theorem 3. □

4. Conclusions

In this paper, we considered the fixed-point inclusion $x\in F\left(x\right),x\in X$, where $(X,d)$ is a complete metric space and $F:Z\to P\left(X\right)$ is a multi-valued operator. Under a very general and abstract assumption on F (inspired by the conclusions of Nadler’s Multi-valued Contraction Principle and of Feng–Liu fixed-point theorem), we derive some useful stability properties of our fixed-point inclusion. Namely, if F is a $\psi$-multi-valued Feng–Liu-type operator, the fixed-point inclusion $x\in F\left(x\right)$ is Ulam–Hyers stable, satisfies the well-posedness property in the sense of Reich and Zaslavski, and has the data dependence property with respect to operator perturbation. Additionally, if F is a multi-valued Feng–Liu-type operator and satisfies a multi-valued quasicontraction condition with constant $k\in (0,1)$, then the fixed-point inclusion $x\in F\left(x\right)$ is Ostrowski stable. Our abstract results are applied to an initial value problem associated to a differential inclusion.