Enriched Multivalued Contractions with Applications to Differential Inclusions and Dynamic Programming

In 1922, Banach [1] demonstrated a significant conclusion for fixed point theory on the metric spaces, known as the Banach contraction principle. Because of its applications in various fields of nonlinear analysis and applied mathematical analysis, this principle has been generalized and extended in different ways.

One of the interesting and famous generalizations was proved by Nadler [2] by applying the concept of the Pompeiu–Hausdorff metric defined on a family of closed and bounded subsets of a complete metric space (see [3]). He established the concept of multi-valued contraction mappings.

Throughout this paper, the standard notations and terminologies in nonlinear analysis (see [4,5]) are used. For the convenience of the reader, we recall some of them.

Let $(X,d)$ be a metric space. Let $CB\left(X\right)$ and $\left(C\right(X\left)\right)$ be the class of all nonempty closed and bounded (compact) subsets of X, respectively.

The symmetric functional $H:CB\left(X\right)\times CB\left(X\right)\to [0,\infty )$ defined by where $D(A,B)={sup}_{x\in A}{inf}_{y\in B}d(x,y)$, for all $A,B\in CB\left(X\right),$ is a metric called the Pompeiu–Hausdorff metric.

Nadler had proved the following fixed point theorem for multivalued mappings satisfying a symmetric contraction condition.

Later, several interesting fixed point theorem for multi-valued mappings were obtained (see [4,5,6,7,8,9,10,11,12,13] and especially the monographs of Rus [14,15,16]).

Let $T:X\to CB\left(X\right)$ be a given mapping. For any fixed $x_0$ in $X,$ a sequence $\left\{x_n\right\}$ in X such that $x_{n+1}\in T{x}_n$ is called a T-orbital sequence around $x_0.$ The set $O(T,x_0)$ denotes the collection of all such sequences. Further, an element $x\in X$ is called a fixed point of $T,$ if $x\in T\left(x\right).$ The set of all fixed points of a multi-valued T is denoted by $Fix\left(T\right),$ that is, $Fix\left(T\right)=\{x\in X:x\in T(x\left)\right\}.$ A multi-valued mapping $T:X\to CB\left(X\right)$ is called a Lipschitzian mapping if, for all $x,y\in X,$ there exists some $L\ge 0$ such that $H(Tx,Ty)\le Ld(x,y)$ holds. If we take $L=1$ in the previous inequality, then the Lipschitzian mapping T is called nonexpansive.

The following data dependence problem is well known.

Let $T_1,T_2:X\to P\left(X\right)$ be two multi-valued mappings such that $Fix\left(T_1\right)$ and $Fix\left(T_2\right)$ are nonempty and there exists $\delta >0$ with the property that $H(T_{1}x,T_{2}x)\le \delta$, for all $x\in X.$

Under these conditions, estimate $H(Fix(T_1,Fix\left(T_2\right)),$ where $P\left(X\right)$ is the power set of $X.$ Several partial answers to this problem are given in [17,18,19].

The stability problems of functional equations originated from a question of Ulam [20] concerning the stability of group homomorphisms. Hyers [21] gave a partial answer to the question of Ulam [20] for Banach spaces.

Let $(X,d)$ be a metric space and $T:X\to 2^X$ a multi-valued mapping. Consider the fixed point problem

Let $ϵ>0.$ An element $w^{*}\in X$ is called an $ϵ$-solution of the fixed point problem (2) if there exists $y^{*}\in X$ such that

The fixed point problem (2) is called Ulam–Hyers stable if there exists a finite constant $c>0$ such that for each $ϵ>0$ and for each $ϵ$-solution $x^{*}\in X$ of the fixed point problem (2) there exist solutions $w^{*}\in X$ of problem (2) such that

Following the authors of [22,23], a single-valued mapping $T:X\to X$ is called an enriched contraction or $(b,\theta )$-enriched contraction [22] if there exist two constants, $b\in [0,\infty )$ and $\theta \in [0,b+1),$ such that, for all $x,y\in X$, and enriched nonexpansive or b-enriched nonexpansive [23] if respectively.

Note that the above two contractive conditions are both symmetric. We state Theorem 2.4 of [22] for convenience and in view of extending to the case of multi-valued mappings.

As shown in [22], a lot of well-known contractive conditions from the literature imply the $(b,\theta )$-enriched contraction. Therefore, Theorem 2 includes as particular cases the Banach contraction principle and several important fixed point theorems in the literature.

The purpose of this paper is to extend the concept of enriched contraction and enriched nonexpansive from the case of single-valued mappings to multi-valued mappings, which not only includes the class of multi-valued contraction mappings but also the class of some Lipschitzian mappings as a subclass. We study a data dependence problem of the fixed point sets and Ulam–Hyers stability of the fixed point problem for enriched multi-valued mappings. We present applications of our results in establishing the existence of a solution of a differential inclusion problem, dynamic programming; introduce a algorithm in real Hilbert space; prove a strong convergence theorem for approximating a common solution of fixed point inclusion for enriched multi-valued nonexpansive mapping and equilibrium problem of a bifunction.

We introduce now the notions of enriched multi-valued contraction and enriched multi-valued nonexpansive mapping by means of the following two symmetric contractive type inequalities.

We need the following Lemma of Nadler [2] (see also [7] ).

We now prove the following fixed point theorem for a $(b,\theta )$-enriched multi-valued contraction in normed space. In the sequel, the letters $\mathbb{R}$ and $\mathbb{N}$ denote the set of all real numbers and the set of all natural numbers, respectively.

We now prove the following fixed point theorem for a $(b,\theta )$-enriched multi-valued contraction in a Banach space.

As a corollary of our result, we can obtain Nadler’s fixed point theorem for multi-valued mappings, in the setting of a Banach space.

If in Corollary 1 for T we take a single valued mapping, then we obtain Theorem $2.4$ of [22].

We now prove the following fixed point theorem for b-enriched multi-valued nonexpansive mappings in a uniformly convex Banach space.

As a corollary of our result, we can obtain Theorem 1 of [24] for multi-valued mappings.

Let H be a real Hilbert space with inner product $〈·,·〉.$ Let D be a nonempty and convex subset of H and let $F:D\times D\to \mathbb{R}$ be a bifunction. The equilibrium problem for F is to find $u\in D$ such that

The solutions set of (26) is denoted by $EP\left(F\right).$ The equilibrium problem (26) includes as special cases numerous problems in physics, optimization, and economics. Some methods have been continuously constructed for solving the equilibrium problem (see, for example, Refs. [25,26] and references therein). Let X be a Banach space and D a subset of $X.$ A multi-valued mapping $T:D\to CB\left(X\right)$ is said to satisfy the condition (A), if $\parallel x-p\parallel =D(x,Tp)$ for all $x\in X$ and $p\in Fix\left(T\right).$

When $\left\{x_n\right\}$ is a sequence in H, $x_n\rightharpoonup x$ implies that $\left\{x_n\right\}$ converges weakly to x and $x_n\to x$ means the strong convergence. In a real Hilbert space H, we have for all $x,y\in H$ and $\lambda \in [0,1].$ Let D be a closed and convex subset of $H.$ For every point $x\in H,$ there exists a unique nearest point in $D,$ denoted by $P_{D}x,$ such that

$P_D$ is called the metric projection of H onto $D.$

For solving the equilibrium problem for a bifunction $F:D\times D\to \mathbb{R},$ let us assume that F satisfies the following conditions:

The following is used for the proof if our main results in the sequel.

Using the above results, we study convergence of the following iteration (27). Let D be a nonempty, closed, and convex subset of a Hilbert space H. Let $T:D\to C\left(H\right)$ be a multi-valued nonself mapping, $f:H\to H$ a contraction, and $F:D\times D\to \mathbb{R}$ a bifunction. Let $\left\{{\alpha }_n\right\}$ be a sequence in $[0,1]$ and $\left\{r_n\right\}$ a sequence in $(0,\infty ).$ For a given $x_0\in H,$ we compute

$u_0\in D$such that$F(u_0,y)+\frac{1}{r_0}〈y-u_0,u_0-x_0〉\ge 0$, $\forall y\in D,$

then we let $z_0\in T_{\lambda }{u}_0$ for some $\lambda \in (0,1)$ and define $x_1\in D$ by

We next compute

$u_1\in D$such that$F(u_1,y)+\frac{1}{r_1}〈y-u_1,u_1-x_1〉\ge 0$, $\forall y\in D.$

From Nadler’s Theorem (see [2]), it follows that there exists $z_1\in T_{\lambda }{u}_1$ such that $\parallel z_1-z_0\parallel \le H(T_{\lambda }{u}_1,T_{\lambda }{u}_0).$ Inductively, we construct the sequence $\left\{x_n\right\}$ as follows:

Here, $z_n\in T_{\lambda }{u}_n$ is such that $\parallel z_{n+1}-z_n\parallel \le H(T_{\lambda }{u}_{n+1},T_{\lambda }{u}_n)$ for $n\ge 1.$

Now, we prove a strong convergence theorem of the iteration (27) to find the common element of the solutions set of an equilibrium problem and the fixed points set of a multi-valued nonself mapping.