New Contributions to Fixed Point Theory for Multi-Valued Feng–Liu Contractions

Abstract

In this paper, we will prove several new results related to the concept of the multi-valued Feng–Liu contraction. An existence, approximation and localization fixed point theorem for a generalized multi-valued nonself Feng–Liu contraction and a new fixed point theorem for multi-valued Feng–Liu contractions in vector-valued metric spaces are proved. Stability results and an application to a system of operatorial inclusions are also given.

1. Introduction and Preliminary Notions and Results

Let $(X,d)$ be a metric space and $P\left(X\right)$ be the set of all nonempty subsets of X. We denote

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

We recall first the following notions:

(1)

The distance from a point $x\in X$ to a set $Y\in P\left(X\right)$:

$$D(x,Y):=\inf \left\{d\right(x,y\left)\right|y\in Y\};$$

(2)

The excess of Y over Z (where $Y,Z\in P\left(X\right)$):

$$e(Y,Z):=sup\left\{D\right(y,Z),y\in Y\};$$

(3)

The Hausdorff–Pompeiu distance between two sets $Y,Z\in P\left(X\right)$:

$$H(Y,Z)=\max \left\{e\right(Y,Z),e(Z,Y\left)\right\}.$$

Notice that H is a generalized metric (in the sense that $H(Y,Z)\in {\mathbb{R}}_{+}\cup \{\infty \}$) on $P_{cl}\left(X\right)$, and it is a classical metric on $P_{b,cl}\left(X\right)$.

For $x\in X$ and $r>0$, we denote by

$$B(x,r):=\{z\in X:d(z,x)<r\}\mathrm{respectively} \mathrm{by}\tilde{B}(x;r):=\{z\in X:d(z,x)\le r\}$$

the open ball (respectively, the closed ball) centered in x with radius r.

If X is a nonempty set and $T:X\to P\left(X\right)$ is a multi-valued operator, then $x\in X$ is called a fixed point for T if $x\in T\left(x\right)$. The set

$$Fix\left(T\right):=\{x\in X|x\in T\left(x\right)\}$$

is the fixed point set of T, while

$$SFix\left(T\right):=\{x\in X|T\left(x\right)=\left\{x\right\}\}$$

is the strict fixed point set of T. We also denote by

$$Graph\left(T\right):=\left\{\right(u,v)\in X\times X|v\in T\left(u\right)\}$$

the graph of the multi-valued operator T.

Remark 1.

If X is a nonempty set and $T:X\to P\left(X\right)$, then the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ satisfying

$$x_0\in X,x_{n+1}\in T\left(x_n\right),for each n\in \mathbb{N}$$

is called an iterative sequence of the Picard type for T starting from $x_0$.

The multi-valued contraction principle was proved in 1969 by Nadler (see [1]), while a slight extension of it was presented by Covitz and Nadler in 1970 (see [2]).

There are several generalizations of the above multi-valued contraction principle of Nadler/Covitz–Nadler. A consistent extension of it appeared in a paper of Feng and Liu (see [3]), as follows.

Definition 1.

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

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

Then, T is called a multi-valued α-contraction of the Feng–Liu type if there exists $\alpha \in ]0,\beta [$, such that for each $x\in X$ there is $y\in I_{\beta }^x$, for which the following assumption holds

$$D(y,T(y\left)\right)\le \alpha d(x,y).$$

Theorem 1.

([3]) Let $(X,d)$ be a complete metric space and $T:X\to P_{cl}\left(X\right)$ be a multi-valued α-contraction of the Feng–Liu type. Suppose that the mapping $h:X\to {\mathbb{R}}_{+}$ defined by $h\left(x\right)=D(x,T(x\left)\right)$ is lower and semi-continuous. Then, $Fix\left(T\right)\ne \varnothing$.

Another generalization of the multi-valued contraction principle involves the notion of the multi-valued graph contraction.

Definition 2.

Ref. [4] Let $(X,d)$ be a metric space and $T:X\to P\left(X\right)$ be a multi-valued operator. Then, T is called a multi-valued graph α-contraction if $\alpha \in ]0,1[$ and

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

(1)

The following theorem, proved in [4], is the main result for multi-valued graph contractions.

Theorem 2.

([4]) Let $(X,d)$ be a complete metric space and $T:X\to P\left(X\right)$ be a multi-valued graph α-contraction with a closed graph. Then, T is a $\frac{1}{1-\alpha }$-multi-valued weak Picard operator; i.e., for each $(x,y)\in Graph\left(T\right)$, there exists in X a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$, such that:

(i) 

$x_0=x$, $x_1=y$;

(ii) 

$x_{n+1}\in F\left(x_n\right)$, for each $n\in \mathbb{N}$;

(iii) 

${\left(x_n\right)}_{n\in \mathbb{N}}$ is convergent to a fixed point $x^{*}(x,y)$ of T;

(iv) 

$d(x,x^{*}(x,y))\le \frac{1}{1-\alpha }d(x,y),for all (x,y)\in Graph\left(T\right)$.

Notice that any multi-valued $\alpha$-contraction is a multi-valued graph $\alpha$-contraction and any multi-valued graph $\alpha$-contraction is a multi-valued $\alpha$-contraction of the Feng–Liu type. For examples and applications of the fixed point theory for the multi-valued graph contraction, see [4].

In this paper, we will prove several new results related to the concept of the multi-valued Feng–Liu contraction. An existence, approximation and localization fixed point theorem for a generalized multi-valued nonself Feng–Liu contraction and a new fixed point theorem for multi-valued Feng–Liu contractions in vector-valued metric spaces are proved. Stability results and an application to a system of operatorial inclusions are also given. For the role and the importance of vector-valued metrics in a nonlinear analysis, see [5,6,7,8].

2. A Local Fixed Point Theorem for a Generalized Multi-Valued Feng–Liu Operator

We start this section by proving a local fixed point theorem for a generalized multi-valued $\alpha$-contraction of the Feng–Liu type. For this purpose, we adapt the notion of the multi-valued $\alpha$-contraction of the Feng–Liu type to a nonself setting.

Definition 3.

Let $(X,d)$ be a metric space, and $Y\in P\left(X\right)$ and $T:Y\to P\left(X\right)$ be a multi-valued operator. Consider $\beta \in ]0,1[$ and $x\in Y$. Define

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

Then, T is called a generalized multi-valued nonself $(\alpha ,\gamma )$-contraction of the Feng–Liu type if there exist $\gamma \in ]0,1[$ and $\alpha \in ]0,\beta -\gamma (1+\beta \left)\right[$, such that, for each $x\in Y$ there is $y\in I_{\beta }^x$, for which the following implication holds

$$y\in Y\Rightarrow D(y,T(y\left)\right)\le \alpha d(x,y)+\gamma D(x,T(y\left)\right).$$

Notice that for $Y=X$ and $\gamma =0$, we obtain the classical concept of the multi-valued $\alpha$-contraction of the Feng–Liu type.

We present now an existence, approximation and localization fixed point theorem for a generalized multi-valued nonself contraction of the Feng–Liu type on a closed ball.

Theorem 3.

Let $(X,d)$ be a complete metric space, let $x_0\in X$ and $r>0$. We consider $T:\tilde{B}(x_0;r)\to P_{cl}\left(X\right)$ a multi-valued nonself $(\alpha ,\gamma )$-contraction of the Feng–Liu type, such that $D(x_0,T\left(x_0\right))\le \left(\beta -\frac{\alpha +\gamma }{1-\gamma }\right)r$. Suppose that the mapping of $h:\tilde{B}(x_0;r)\to {\mathbb{R}}_{+}$ defined by $h\left(x\right)=D(x,T(x\left)\right)$ is lower and semi-continuous on $\tilde{B}(x_0;r)$, or that T has a closed graph. Then, there exists a sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ of Picard iterates for T, starting from $x_0$, which converges to a fixed point $x^{*}\left(x_0\right)\in \tilde{B}(x_0;r)$ of T. Moreover, if $k:=\frac{\alpha +\gamma }{\beta (1-\gamma )}$, then the following relations hold:

(a) 

$d(x_n,x^{*}\left(x_0\right))\le \frac{k^n}{1-k}d(x_0,x_1)$, for each $n\in \mathbb{N}$;

(b) 

$d(x_0,x^{*}\left(x_0\right))\le \frac{1}{1-k}d(x_0,x_1)$.

Proof. 

We will show that there exists in $\tilde{B}(x_0;r)$ a sequence ${\left\{x_n\right\}}_{\mathbb{N}}$ of Picard iterates for T, starting from $x_0$, which converges to a fixed point of T. For $x_0$ there exists $x_1\in I_{\beta }^{x_0}$, having the properties that $x_1\in T\left(x_0\right)$ and $\beta d(x_0,x_1)\le D(x_0,T\left(x_0\right))$. Then, we observe that

$$\beta d(x_0,x_1)\le D(x_0,T\left(x_0\right))\le \beta \left(1-\frac{\alpha +\gamma }{\beta (1-\gamma )}\right)r,$$

which shows that

$$d(x_0,x_1)\le \left(1-\frac{\alpha +\gamma }{\beta (1-\gamma )}\right)r.$$

Hence, since $x_1\in \tilde{B}(x_0;r)$, by the definition of the multi-valued nonself $(\alpha ,\gamma )$-contraction of the Feng–Liu type, we obtain that

$$D(x_1,T\left(x_1\right))\le \alpha d(x_0,x_1)+\gamma D(x_0,T\left(x_1\right))\le \alpha d(x_0,x_1)+\gamma \left(d(x_0,x_1)+D(x_1,T\left(x_1\right))\right).$$

Thus, we have

$$D(x_1,T\left(x_1\right))\le \frac{\alpha +\gamma }{1-\gamma }d(x_0,x_1).$$

For $x_1\in \tilde{B}(x_0;r)$, there exists $x_2\in I_{\beta }^{x_1},$ such that $x_2\in T\left(x_1\right)$ and $\beta d(x_1,x_2)\le D(x_1,T\left(x_1\right))$. Additionally, $D(x_2,F\left(x_2\right))\le \alpha d(x_1,x_2)+\gamma D(x_1,T\left(x_2\right))$. As a consequence, we obtain

$$d(x_1,x_2)\le \frac{1}{\beta }D(x_1,T\left(x_1\right))\le \frac{\alpha +\gamma }{\beta (1-\gamma )}d(x_0,x_1).$$

By our assumptions, we have that $k:=\frac{\alpha +\gamma }{\beta (1-\gamma )}<1$.

Now, we observe that

$$d(x_0,x_2)\le d(x_0,x_1)+d(x_1,x_2)\le d(x_0,x_1)+\frac{\alpha +\gamma }{\beta (1-\gamma )}d(x_0,x_1)\le$$

$$\left(1+\frac{\alpha +\gamma }{\beta (1-\gamma )}\right)\frac{1}{\beta }D(x_0,T\left(x_0\right))\le \left[1-{\left(\frac{\alpha +\gamma }{\beta (1-\gamma )}\right)}^2\right]r.$$

As a consequence, $x_2\in \tilde{B}(x_0;r)$.

By this procedure, we obtain a sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ of Picard iterates for T, starting from $x_0$ with the following properties:

(1)

$x_{n+1}\in T\left(x_n\right)\cap \tilde{B}(x_0;r),n\in \mathbb{N}$;

(2)

$d(x_n,x_{n+1})\le k^{n}d(x_0,x_1),n\in \mathbb{N}$;

(3)

$D(x_n,T\left(x_n\right))\le k^{n}D(x_0,T\left(x_0\right)),n\in \mathbb{N}$.

By (2), we find that the sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ is Cauchy. Thus, the sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ converges to an element $x^{*}\left(x_0\right)\in \tilde{B}(x_0;r)$. We only need to demonstrate that $x^{*}\left(x_0\right)$ is a fixed point of T. If T has a closed graph, the conclusion follows immediately by the fact that the sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ is of Picard iterates for T. If we suppose that the mapping $h\left(x\right)=D(x,T(x\left)\right)$ is a lower semi-continuous on $\tilde{B}(x_0;r)$, then the conclusion follows by (3), observing that the sequence ${\left\{D(x_n,T\left(x_n\right))\right\}}_{n\in \mathbb{N}}$ is convergent to 0.

The conclusions (a) and (b) follow (2), taking into account that, for $n,p\in \mathbb{N}$ with $p\ge 1$, we have

$${\displaystyle d(x_n,x_{n+p})\le \sum _{i=0}^{n+p-1}{k}^{i}d(x_0,x_1)-\sum _{i=0}^{n-1}{k}^{i}d(x_0,x_1).}$$

Letting $p\to \infty$, we obtain (a). Then, taking $n=1$, we obtain

$${\displaystyle d(x_1,x^{*}\left(x_0\right))\le \sum _{i=0}^{\infty }{k}^{i}d(x_0,x_1)-d(x_0,x_1)}$$

which immediately gives the conclusion (b). □

Remark 2.

For related results, see [9,10,11,12,13]. For complementary results, see also [14,15,16,17,18].

3. A Fixed Point Theorem for Multi-Valued Feng–Liu Contractions in Vector-Valued Metric Spaces

In this section, we will prove a fixed point result for multi-valued Feng–Liu contractions in complete vector-valued metric spaces. For this purpose, we recall some notions and results.

If $x,y\in {\mathbb{R}}^m$, $x=(x_1,\dots ,x_m)$ and $y=(y_1,\dots ,y_m)$, then, by definition

$$x⪯y\mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if}{x}_i\le y_i,\mathrm{for} \mathrm{each} i\in \{1,2,\cdots ,m\}.$$

We will make an identification between row and column vectors in ${\mathbb{R}}^m$.

We now recall the concept of vector-valued metric space in the sense of Perov, see, e.g., [19]. If X is a nonempty set, then a functional $d:X\times X\to \mathbb{R}$${}_{+}^m$ satisfying the usual axioms of a metric with respect to the above mentioned relation ⪯ is called a vector-valued metric in the sense of Perov. In this case, the pair $(X,d)$ is a vector-valued metric space.

We may suppose that

$$d(x,y):=\left(\begin{array}{c}{d}_1\left(x,y\right)\\ \cdots \\ d_m\left(x,y\right)\end{array}\right),\mathrm{for} x,y\in X.$$

We denote by

$$D(x,Y):=\left(\begin{array}{c}{D}_{d_1}(x,Y)\\ \cdots \\ D_{d_m}(x,Y)\end{array}\right),\mathrm{the} \mathrm{vectorial} \mathrm{distance} \mathrm{from} \mathrm{a} \mathrm{point} x\in X\mathrm{to} \mathrm{a} \mathrm{set} Y\in P\left(X\right)$$

and by

$$H(A,B):=\left(\begin{array}{c}{H}_{d_1}(A,B)\\ \cdots \\ H_{d_m}(A,B)\end{array}\right),\mathrm{the} \mathrm{vectorial} \mathrm{Pompeiu}–\mathrm{Hausdorff} \mathrm{distance} \mathrm{on} P\left(X\right).$$

In this section, $M_{m,m}\left({\mathbb{R}}_{+}\right)$ denotes the set of all $m\times m$ matrices with positive elements, $I_m$ is the identity $m\times m$ matrix and $O_m$ denotes the null $m\times m$ matrix.

By definition, a matrix $K\in M_{m,m}\left({\mathbb{R}}_{+}\right)$ is said to be convergent to zero if $K^n\to O_m$ as $n\to \infty$. The following characterization theorem is useful for the proof of our main results, see, e.g., [20].

Theorem 4.

Let $K\in M_{m,m}\left({\mathbb{R}}_{+}\right)$. The following assertions are equivalent:

(i) 

$K^n\to O_m$ as $n\to \infty$;

(ii) 

The spectral radius $\rho \left(K\right)$ of K is strictly less than 1, i.e., the eigenvalues of K are in the open unit disc;

(iii) 

The matrix $(I_m-K)$ is nonsingular and

$${\left(I_m-K\right)}^{-1}=I_m+K+\dots +K^n+\dots ;$$

(2)

(iv) 

The matrix $\left(I_m-K\right)$ is nonsingular and ${\left(I_m-K\right)}^{-1}$ has nonnegative elements.

The following theorem is the main fixed point result for K-contractions in complete vector-valued metric spaces, see [19].

Theorem 5

(Perov). Let $\left(X,d\right)$ be a complete vector-valued metric space and let $f:X\to X$ be an K-contraction; i.e., $K\in M_{m,m}\left({\mathbb{R}}_{+}\right)$ converges to zero and

$$d\left(f\left(x\right),f\left(y\right)\right)⪯Kd\left(x,y\right),for all x,y\in X.$$

Then:

$\left(1\right)$ Fix$\left(f\right)=\left\{x^{*}\right\}$, i.e., there exists a unique solution $x^{*}\in X$ of the fixed point equation $x=f\left(x\right)$;

$\left(2\right)$ the sequence ${\left(x_n\right)}_{n\in \mathbb{N}},x_n:=f^n\left(x_0\right)$ of successive approximations for f starting from any $x_0\in X$ is convergent to $x^{*}$;

$\left(3\right)$ the following estimation holds

$$d\left(x_n,x^{*}\right)⪯K^n{\left(I_m-K\right)}^{-1}d\left(x_0,x_1\right),for every n\in \mathbb{N};$$

(3)

A multi-valued variant of Perov’s theorem was given in [21].

Theorem 6.

Let $(X,d)$ be a complete vector-valued metric space and let $F:X\to P_{cl}\left(X\right)$ be a multi-valued K-contraction, i.e., $K\in M_{m,m}\left({\mathbb{R}}_{+}\right)$ is convergent to zero and

$$H\left(F\left(x\right),F\left(y\right)\right)⪯Kd\left(x,y\right),for all x,y\in X.$$

(4)

Then:

(i) 

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

(ii) 

For each $(x,y)\in Graph\left(F\right)$, there exists a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ (with $x_0=x$, $x_1=y$ and $x_{n+1}\in F\left(x_n\right)$, for each $n\in {\mathbb{N}}^{*}$), such that ${\left(x_n\right)}_{n\in \mathbb{N}}$ is convergent to a fixed point $x^{*}:=x^{*}(x,y)$ of F, and the following relations hold:

$$d(x_n,x^{*})⪯K^n{(I-K)}^{-1}d(x_0,x_1),for each n\in {\mathbb{N}}^{*}$$

and

$$d(x,x^{*})⪯{(I-K)}^{-1}d(x,y).$$

Our next result is a generalization of the previous theorem in terms of a multi-valued Feng–Liu contraction.

We introduce first the following notion.

Definition 4.

Let $(X,d)$ be a vector-valued metric space, $F:X\to P\left(X\right)$ be a multi-valued operator, $B\in M_{m,m}\left({\mathbb{R}}_{+}\right)$ be a diagonal matrix with elements $b_1,\cdots ,b_m\in ]0,1[$ and $x\in X$. Consider the set

$$I_B^x:=\{y\in F\left(x\right):Bd(x,y)⪯D(x,F\left(x\right))\}.$$

Then, F is called a multi-valued vectorial contraction of the Feng–Liu type if there exists a matrix $A\in M_{m,m}\left({\mathbb{R}}_{+}\right)$, such that the matrix $B^{-1}A$ is convergent to zero, and for each $x\in X$, there is $y\in I_B^x$, for which the following relation holds

$$D(y,F(y\left)\right)⪯Ad(x,y).$$

Theorem 7.

Let $(X,d)$ be a complete vector-valued metric space and $F:X\to P\left(X\right)$ be a multi-valued vectorial contraction of the Feng–Liu type. Suppose that F has closed graph. Then, for each $x_0\in X$, there exists an iterative sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ of the Picard type for F starting from $x_0$ with the following properties:

(1) 

${\left(x_n\right)}_{n\in \mathbb{N}}$ converges to $x^{*}\left(x_0\right)\in Fix\left(F\right)$;

(2) 

$d(x_n,x^{*}\left(x_0\right))⪯{\left(B^{-1}A\right)}^n{\left(I_m-B^{-1}A\right)}^{-1}d(x_0,x_1),n\in \mathbb{N}$;

(3) 

$d(x_0,x^{*}\left(x_0\right))⪯{\left(I_m-B^{-1}A\right)}^{-1}d(x_0,x_1)⪯{\left(I_m-B^{-1}A\right)}^{-1}{B}^{-1}D(x_0,F\left(x_0\right))$.

Proof. 

Let $x_0\in X$ be arbitrarily chosen. Then, there exists $x_1\in I_B^{x_0}$ such that

$$D(x_1,F\left(x_1\right))⪯Ad(x_0,x_1).$$

For $x_1\in X$, by the multi-valued vectorial contraction condition of the Feng–Liu type, there exists $x_2\in I_B^{x_1}$, such that

$$D(x_2,F\left(x_2\right))⪯Ad(x_1,x_2).$$

Since $Bd(x_1,x_2)⪯D(x_1,F\left(x_1\right))$, we obtain that

$$d(x_1,x_2)⪯B^{-1}D(x_1,F\left(x_1\right))⪯B^{-1}Ad(x_0,x_1).$$

Let us denote $K:=B^{-1}A$. Notice that $B^{-1}A=A{B}^{-1}$. By the above procedure, there exists a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ in X with the following properties:

(a)

$x_{n+1}\in I_B^{x_n}$, for each $n\in \mathbb{N}$;

(b)

$d(x_n,x_{n+1})⪯Kd(x_{n-1},x_n)⪯\cdots ⪯K^{n}d(x_0,x_1)$, for each $n\in {\mathbb{N}}^{*}$;

(c)

$D(x_{n+1},F\left(x_{n+1}\right))⪯KD(x_n,F\left(x_n\right))⪯\cdots ⪯K^{n+1}D(x_0,F\left(x_0\right))$, for each $n\in \mathbb{N}$.

Then, by (b), the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is Cauchy in $(X,d)$. Hence, it is convergent to an element $x^{*}:=x^{*}\left(x_0\right)\in X$. Since ${\left(x_n\right)}_{n\in \mathbb{N}}$ is an iterative sequence of the Picard type for F starting from $x_0$, and F has a closed graph, we obtain that $x^{*}\in Fix\left(F\right)$. Moreover, by the relation

$$d(x_n,x_{n+p})⪯K^n\left(I_m+K+\cdots K^{p-1}\right)d(x_0,x_1)⪯K^n{\left(I_m-K\right)}^{-1}d(x_0,x_1),$$

letting $p\to \infty$, we obtain the following a priori approximation for the fixed point:

$$d(x_n,x^{*}\left(x_0\right))⪯K^n{\left(I_m-K\right)}^{-1}d(x_0,x_1),n\in \mathbb{N}.$$

Taking $n=0$ in the above relation, we obtain the following retraction–displacement condition:

$$d(x_0,x^{*}\left(x_0\right))⪯{\left(I_m-K\right)}^{-1}d(x_0,x_1)⪯{\left(I_m-K\right)}^{-1}{B}^{-1}D(x_0,F\left(x_0\right)).$$

The proof is complete. □

We now present some stability concepts for the fixed point inclusion $x\in F\left(x\right)$ in the setting of a vector-valued metric space.

The concept of the Ulam–Hyers stability is now introduced; see also [22].

Definition 5.

Let $(X,d)$ be a vector-valued metric space and $F:X\to P\left(X\right)$ be a multi-valued operator. The fixed point inclusion $x\in F\left(x\right)$ is called Ulam–Hyers stable if there exists a matrix $C\in M_{m,m}\left({\mathbb{R}}_{+}^{*}\right)$, such that for every $\epsilon :=({\epsilon }_1,\cdots ,{\epsilon }_m)$ (with ${\epsilon }_i>0$ for each $i\in \{1,2,\cdots ,m\}$) and for each ε-fixed point $\tilde{x}\in X$ of F (i.e., $D(\tilde{x},F\left(\tilde{x}\right))⪯\epsilon )$, there exists $x^{*}\in Fix\left(F\right)$, such that

$$d(\tilde{x},x^{*})⪯C\epsilon .$$

The well-posedness of the fixed point inclusion $x\in F\left(x\right)$ in a vector-valued metric space is defined, as follows. The concept is inspired by the single-valued case; see the papers of Reich and Zaslavski [23,24].

Definition 6.

Let $(X,d)$ be a vector-valued metric space. Let $F:X\to P\left(X\right)$ be a multi-valued operator such that $Fix\left(F\right)\ne \varnothing$ and let $r:X\to Fix\left(F\right)$ be a set retraction. Then, the fixed point inclusion $x\in F\left(x\right)$ is called well-posed in the sense of Reich and Zaslavski if for each $x^{*}\in Fix\left(F\right)$ and for any sequence ${\left\{y_n\right\}}_{n\in \mathbb{N}}\subset r^{-1}\left(x^{*}\right)$, such that ${\left\{D(y_n,F\left(y_n\right))\right\}}_{n\in \mathbb{N}}$ converges to zero as $n\to \infty$; we have that

$$y_n\to x^{*} as n\to \infty .$$

The data dependence property is given in our next definition.

Definition 7.

Let $(X,d)$ be a vector-valued metric space and $F:X\to P\left(X\right)$ be a multi-valued operator. Let $G:X\to P\left(X\right)$ be a multi-valued operator satisfying the following conditions:

(i) 

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

(ii) 

There exists $\eta :=({\eta }_1,\cdots ,{\eta }_m)$ (with ${\eta }_i>0$ for each $i\in \{1,2,\cdots ,m\}$), such that $H\left(F\right(x),G(x\left)\right)⪯\eta$, for all $x\in X$.

Then, the fixed point inclusion $x\in F\left(x\right),x\in X$ has the data dependence property if for each $g^{*}\in Fix\left(G\right)$ there exists $x^{*}\in Fix\left(F\right)$, such that

$$d(g^{*},x^{*})⪯S\eta ,for some matrix S\in M_{m,m}\left({\mathbb{R}}_{+}^{*}\right).$$

The notion of the Ostrowski stability property for a fixed point inclusion in the vector-valued metric space is now presented; see also [12].

Definition 8.

Let $(X,d)$ be a vector-valued metric space. Let $F:X\to P\left(X\right)$ be a multi-valued operator such that $Fix\left(F\right)\ne \varnothing$ and let $r:X\to Fix\left(F\right)$ be a set retraction. Then, the fixed point inclusion $x\in F\left(x\right)$ is said to have the Ostrowski stability property if for each $x^{*}\in Fix\left(F\right)$ and for any sequence ${\left\{z_n\right\}}_{n\in \mathbb{N}}\subset r^{-1}\left(x^{*}\right)$, such that ${\left\{D(z_{n+1},F\left(z_n\right))\right\}}_{n\in \mathbb{N}}$ converges to zero as $n\to \infty$; we have that

$$z_n\to x^{*} as n\to \infty .$$

The following retraction–displacement condition will be important for our main results.

Definition 9.

Let $(X,d)$ be a vector-valued metric space and let $F:X\to P\left(X\right)$ be a multi-valued operator such that $Fix\left(F\right)\ne \varnothing$. Then, we say that F satisfies the strong vectorial retraction–displacement condition if there exist a matrix $Q\in M_{m,m}\left({\mathbb{R}}_{+}^{*}\right)$ and a set retraction $r:X\to Fix\left(F\right)$, such that

$$d(x,r(x\left)\right)⪯QD(x,F(x\left)\right),for all x\in X.$$

(5)

An abstract result concerning some stability properties of a multi-valued operator is given in our next result.

Theorem 8.

Let $(X,d)$ be a vector-valued metric space and let $F:X\to P\left(X\right)$ be a multi-valued operator satisfying the strong vectorial retraction–displacement condition, such that $Fix\left(F\right)\ne \varnothing$. Then, the fixed point inclusion $x\in F\left(x\right)$ has the Ulam–Hyers stability property; it is well-posed and satisfies the data dependence property.

Proof. 

Suppose that there exists a matrix $Q\in M_{m,m}\left({\mathbb{R}}_{+}^{*}\right)$ and a set retraction $r:X\to Fix\left(F\right)$, such that

$$d(x,r(x\left)\right)⪯QD(x,F(x\left)\right),\mathrm{for} \mathrm{all} x\in X.$$

In order to prove the Ulam–Hyers stability property, let us consider $\epsilon :=({\epsilon }_1,\cdots ,{\epsilon }_m)$ (with ${\epsilon }_i>0$ for each $i\in \{1,2,\cdots ,m\}$) and $\tilde{x}\in X$ such that $D(\tilde{x},F\left(\tilde{x}\right))⪯\epsilon$. Then, by the strong vectorial retraction–displacement condition, we have

$$d(\tilde{x},r\left(\tilde{x}\right))⪯QD(\tilde{x},F\left(\tilde{x}\right))⪯Q\epsilon .$$

Thus, the fixed point inclusion $x\in F\left(x\right)$ is Ulam–Hyers stable.

For the well-posedness property of the fixed point inclusion, let us consider the sequence ${\left\{y_n\right\}}_{n\in \mathbb{N}}\subset r^{-1}\left(x^{*}\right)$, such that the sequence $D(y_n,F\left(y_n\right))$ converges to zero as $n\to \infty$. Then, for each $n\in \mathbb{N}$, we have $r\left(y_n\right)=x^{*}$ and, again by the strong vectorial retraction–displacement condition, we conclude that

$$d(y_n,x^{*})=d(y_n,r\left(y_n\right))⪯QD(y_n,F\left(y_n\right))\to 0 \mathrm{as} n\to \infty .$$

Let us now prove the data dependence of the fixed point set. Let us consider a multi-valued operator $G:X\to P\left(X\right)$ to have the properties:

(i) $Fix\left(G\right)\ne \varnothing$;

(ii) There exists $\eta :=({\eta }_1,\cdots ,{\eta }_m)$ (with ${\eta }_i>0$ for each $i\in \{1,2,\cdots ,m\}$), such that

$$H\left(F\right(x),G(x\left)\right)⪯\eta ,\mathrm{for} \mathrm{all} x\in X.$$

Take any $g^{*}\in Fix\left(G\right)$ and denote $x^{*}:=r\left(g^{*}\right)$. Then, by the strong vectorial retraction–displacement condition, we have that

$$d(g^{*},x^{*})=d(g^{*},r\left(g^{*}\right))⪯QD(g^{*},F\left(g^{*}\right))⪯QH(G\left(g^{*}\right),F\left(g^{*}\right))\le Q\eta .$$

The proof is now complete.

□

The following result shows that any multi-valued vectorial contraction of the Feng–Liu type has a strong vectorial retraction–displacement condition.

Theorem 9.

Let $(X,d)$ be a complete vector-valued metric space and $F:X\to P\left(X\right)$ be a multi-valued vectorial contraction of the Feng–Liu type. Suppose that F has a closed graph. Then, F satisfies the strong vectorial retraction–displacement condition.

Proof. 

By Theorem 7, we know that $Fix\left(F\right)\ne \varnothing$ and, for every $x\in X$, there exists an iterative sequence ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ of the Picard type for F starting from the arbitrary $x_0\in X$, which converges to a fixed point $x^{*}\left(x_0\right)$ of F. Moreover, the following relation holds

$$d(x_0,x^{*}\left(x_0\right))⪯{\left(I_m-B^{-1}A\right)}^{-1}{B}^{-1}D(x_0,F\left(x_0\right)).$$

Thus, we can define the set-retraction $r:X\to Fix\left(F\right)$, $x⟼r\left(x\right):=x^{*}\left(x\right)$ with the property

$$d(x,r\left(x\right))⪯{\left(I_m-B^{-1}A\right)}^{-1}{B}^{-1}D(x,F\left(x\right)),x\in X.$$

Hence, the strong vectorial retraction–displacement condition from Definition 9 is satisfied. □

By combining the above two theorems, we obtain the following stability properties for the multi-valued vectorial contraction of the Feng–Liu type.

Theorem 10.

Let $(X,d)$ be a complete vector-valued metric space and $F:X\to P\left(X\right)$ be a multi-valued vectorial contraction of the Feng–Liu type. Suppose that F has a closed graph. Then, the fixed point inclusion $x\in F\left(x\right)$ is well-posed in the sense of Reich and Zaslavsi, has the Ulam–Hyers stability property and satisfies the data dependence property.

Proof. 

By Theorem 7, we have that $Fix\left(F\right)\ne \varnothing$, while Theorem 9 implies that F has the strong vectorial retraction–displacement property. The conclusions follow by Theorem 8.

□

Remark 3.

It is an open question to prove the Ostrowski stability property for a multi-valued vectorial contraction of the Feng–Liu type.

4. An Application to a System of Operatorial Inclusions

In this section, we will present an existence result for a system of operatorial inclusion in complete metric spaces. The approach is based on the vectorial technique for multi-valued Feng–Liu operators.

Let $(X,d_1)$ and $(Y,d_2)$ be two complete metric spaces and let $G_1:X\times Y\to P\left(X\right)$ and $G_2:X\times Y\to P\left(Y\right)$ be two multi-valued operators with a closed graph. We consider the following system of operatorial inclusions

$$\left\{\begin{array}{c}x\in G_1(x,y)\hfill \\ y\in G_2(x,y).\hfill \end{array}\right.$$

(6)

Denote by $Z:=X\times Y$ and define on Z the vectorial metric $\tilde{d}:Z\times Z\to {\mathbb{R}}_{+}^2$ given by

$$\tilde{d}((x,y),(u,v)):=\left(\begin{array}{c}{d}_1\left(x,u\right)\\ d_2\left(y,v\right)\end{array}\right),\mathrm{for} \mathrm{each}(x,y),(u,v)\in Z.$$

Let $b_1,b_2\in ]0,1[$ and define the following nonempty sets:

$$I_{b_1}^{(x,y)}:=\{u\in G_1(x,y):b_1{d}_1(x,u)\le D_1(x,G_1(x,y))\}\subset X$$

and

$$I_{b_2}^{(x,y)}:=\{v\in G_2(x,y):b_2{d}_2(y,v)\le D_2(y,G_2(x,y))\}\subset Y,$$

where $D_1$ and $D_2$ are the distances from a point to a set with respect to $d_1$ and $d_2$, respectively.

Denote also

$$B:=\left(\begin{array}{cc}{b}_1& 0\\ 0& b_2\end{array}\right)$$

(7)

We suppose that for every $(x,y)\in X\times Y$ there exist $u\in I_{b_1}^{(x,y)}$ and $v\in I_{b_2}^{(x,y)}$, such that

$$D_1(u,G_1(u,v))\le a_1{d}_1(x,u)+a_2{d}_2(y,v)$$

(8)

and

$$D_2(v,G_2(u,v))\le a_3{d}_1(x,u)+a_4{d}_2(y,v),$$

(9)

where

$$A:=\left(\begin{array}{cc}{a}_1& a_2\\ a_3& a_4\end{array}\right).$$

(10)

is a matrix with nonnegative elements. We also suppose that the matrix

$$B^{-1}A=\left(\begin{array}{cc}\frac{a_1}{b_1}& \frac{a_2}{b_1}\\ \frac{a_3}{b_2}& \frac{a_4}{b_2}\end{array}\right)$$

(11)

is convergent to zero.

Under the above assumptions, we have the following existence and approximation result.

Theorem 11.

Let us consider the system of operatorial inclusions (6). Under the above assumptions, the system (6) has at least one solution $(x^{*},y^{*})\in X\times Y$. Moreover, for each $(x_0,y_0)\in X\times Y$, there exist two sequences ${\left\{x_n\right\}}_{n\in \mathbb{N}}\subset X$ and ${\left\{y_n\right\}}_{n\in \mathbb{N}}\subset Y$ with the following properties:

(A) $x_{n+1}\in G_1(x_n,y_n)$ and $y_{n+1}\in G_2(x_n,y_n)$, for each $n\in \mathbb{N}$;

(B) ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ converges to $x^{*}$ and ${\left\{y_n\right\}}_{n\in \mathbb{N}}$ converges to $y^{*}$ as $n\to \infty$;

(C) $\left(\begin{array}{c}{d}_1\left(x_n,x^{*}\right)\\ d_2\left(y_n,y^{*}\right)\end{array}\right)⪯{\left(B^{-1}A\right)}^n{\left(I_m-B^{-1}A\right)}^{-1}\left(\begin{array}{c}{d}_1\left(x_0,x_1\right)\\ d_2\left(y_0,y_1\right)\end{array}\right),n\in \mathbb{N};$

(D) $\begin{array}{l}\left(\begin{array}{c}{d}_1\left(x_0,x^{*}\right)\\ d_2\left(y_0,y^{*}\right)\end{array}\right)⪯{\left(I_m-B^{-1}A\right)}^{-1}\left(\begin{array}{c}{d}_1\left(x_0,x_1\right)\\ d_2\left(y_0,y_1\right)\end{array}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}⪯{\left(I_m-B^{-1}A\right)}^{-1}{B}^{-1}\left(\begin{array}{c}{D}_1\left(x_0,G_1(x_0,y_0)\right)\\ D_2\left(y_0,G_2(x_0,y_0)\right)\end{array}\right)\end{array}$.

Proof. 

We denote $Z:=X\times Y$ and, for $z:=(x,y)\in Z$, consider the multi-valued operator $G:Z\to P\left(Z\right)$ given by $G\left(z\right):=G_1\left(z\right)\times G_2\left(z\right)$. Notice that the fixed points $z^{*}=(x^{*},y^{*})$ of G are solutions for the operatorial inclusion (6).

Let $(x,y)\in X\times Y$ and $(u,v)\in I_{b_1}^{(x,y)}\times I_{b_2}^{(x,y)}$, such that

$$D_1(u,G_1(u,v))\le a_1{d}_1(x,u)+a_2{d}_2(y,v)$$

and

$$D_2(v,G_2(u,v))\le a_3{d}_1(x,u)+a_4{d}_2(y,v).$$

We consider the vectorial gap function

$$\tilde{D}(z,Z):=\left(\begin{array}{c}{D}_1(x,X)\\ D_2(y,Y)\end{array}\right).$$

We denote $I_B^{(x,y)}:=\{(u,v)\in G(x,y):B\tilde{d}((x,y),(u,v))⪯\tilde{D}((x,y),G(x,y))\}$. Then, by our assumptions, the set $I_B^{(x,y)}$ is nonempty for each $(x,y)\in Z$. Moreover, by (8) and (9), we obtain that for each $z:=(x,y)\in Z$, there exists $w:=(u,v)\in I_B^{(x,y)}$, such that

$$\tilde{D}\left(w,G\left(w\right))\right)⪯A\tilde{d}(z,w),$$

where

$$A=\left(\begin{array}{cc}{a}_1& a_2\\ a_3& a_4\end{array}\right).$$

Thus, G satisfies all the assumptions of the Theorem 7. As a consequence, for each $z_0:=(x_0,y_0)\in Z$ there exists an iterative sequence of the Picard type ${\{z_n:=(x_n,y_n)\}}_{n\in \mathbb{N}}$, which converges to a fixed point $z^{*}\left(z_0\right)\in Z$ of G, and the following relations hold:

(I)

$\tilde{d}(z_n,z^{*}\left(z_0\right))⪯{\left(B^{-1}A\right)}^n{\left(I_m-B^{-1}A\right)}^{-1}\tilde{d}(z_0,z_1),n\in \mathbb{N}$;

(II)

$\tilde{d}(z_0,z^{*}\left(z_0\right))⪯{\left(I_m-B^{-1}A\right)}^{-1}\tilde{d}(z_0,z_1)⪯{\left(I_m-B^{-1}A\right)}^{-1}{B}^{-1}\tilde{D}(z_0,G\left(z_0\right))$.

Thus, the proof is complete. □

Remark 4.

In the above mentioned conditions, some stability results (well-posedness, Ulam–Hyers stability and data dependence property) for the system of operatorial inclusions (6) can be established by applying the abstract results proved in Section 3.

In particular, an existence and approximation result for the multi-valued altering points problem can be obtained. We notice that, if $(X,d_1)$ and $(Y,d_2)$ are two metric spaces and $G_1:Y\to P\left(X\right)$ and $G_2:X\to P\left(Y\right)$ are two multi-valued operators, then the following system of operatorial inclusions is called an altering points problem for multi-valued operators:

$$\left\{\begin{array}{c}x\in G_1\left(y\right)\hfill \\ y\in G_2\left(x\right).\hfill \end{array}\right.$$

(12)

The above problem has important applications in the theory of generalized/multivalued variational inequalities, see, e.g., [25].

Theorem 12.

Let $(X,d_1)$ and $(Y,d_2)$ be two complete metric spaces and let $G_1:Y\to P\left(X\right)$ and $G_2:X\to P\left(Y\right)$ be two multi-valued operators with a closed graph. Let $b_1,b_2\in ]0,1[$ and the sets

$$J_{b_1}^{(x,y)}:=\{u\in G_1\left(y\right):b_1{d}_1(x,u)\le D_1(x,G_1\left(y\right))\}$$

and

$$J_{b_2}^{(x,y)}:=\{v\in G_2\left(x\right):b_2{d}_2(y,v)\le D_2(y,G_2\left(x\right))\}.$$

We suppose that for every $(x,y)\in X\times Y$, there exists $(u,v)\in J_{b_1}^{(x,y)}\times J_{b_2}^{(x,y)}$, such that

$$D_1(u,G_1\left(v\right))\le a_1{d}_1(x,u)+a_2{d}_2(y,v)$$

and

$$D_2(v,G_2\left(u\right))\le a_3{d}_1(x,u)+a_4{d}_2(y,v),$$

where

$$A:=\left(\begin{array}{cc}{a}_1& a_2\\ a_3& a_4\end{array}\right).$$

is a matrix with nonnegative elements. We also suppose that the matrix

$$B^{-1}A=\left(\begin{array}{cc}\frac{a_1}{b_1}& \frac{a_2}{b_1}\\ \frac{a_3}{b_2}& \frac{a_4}{b_2}\end{array}\right)$$

is convergent to zero, where

$$B:=\left(\begin{array}{cc}{b}_1& 0\\ 0& b_2\end{array}\right).$$

Then, the altering points problem (12) has at least one solution in $X\times Y$. Moreover, for each $(x_0,y_0)\in X\times Y$, there exist two sequences ${\left\{x_n\right\}}_{n\in \mathbb{N}}\subset X$ and ${\left\{y_n\right\}}_{n\in \mathbb{N}}\subset Y$ with the following properties:

(A) $x_{n+1}\in G_1\left(y_n\right)$ and $y_{n+1}\in G_2\left(x_n\right)$, for each $n\in \mathbb{N}$;

(B) ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ converges to $x^{*}$ and ${\left\{y_n\right\}}_{n\in \mathbb{N}}$ converges to $y^{*}$ as $n\to \infty$;

(C) $\left(\begin{array}{c}{d}_1\left(x_n,x^{*}\right)\\ d_2\left(y_n,y^{*}\right)\end{array}\right)⪯{\left(B^{-1}A\right)}^n{\left(I_m-B^{-1}A\right)}^{-1}\left(\begin{array}{c}{d}_1\left(x_0,x_1\right)\\ d_2\left(y_0,y_1\right)\end{array}\right),n\in \mathbb{N};$

(D) $\begin{array}{l}\left(\begin{array}{c}{d}_1\left(x_0,x^{*}\right)\\ d_2\left(y_0,y^{*}\right)\end{array}\right)⪯{\left(I_m-B^{-1}A\right)}^{-1}\left(\begin{array}{c}{d}_1\left(x_0,x_1\right)\\ d_2\left(y_0,y_1\right)\end{array}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}⪯{\left(I_m-B^{-1}A\right)}^{-1}{B}^{-1}\left(\begin{array}{c}{D}_1\left(x_0,G_1\left(y_0\right)\right)\\ D_2\left(y_0,G_2\left(x_0\right)\right)\end{array}\right)\end{array}$.

Example. 

Let $X:=\{0,\frac{1}{2},\cdots ,\frac{1}{2^n},\cdots \}$ and $Y:=\{0,1,\frac{1}{2},\cdots ,\frac{1}{2^n},\cdots \}$, both endowed with the absolute value metric. Let $G_1:Y\to P\left(X\right)$ and $G_2:X\to P\left(Y\right)$ be given by

$$G_1\left(y\right):=\left\{\begin{array}{cc}\left\{\frac{1}{2^{n+1}}\right\},& y\in \{\frac{1}{2^n}:n\in \mathbb{N}\}\\ \{0,\frac{1}{2}\},& y=0\end{array}\right.$$

and

$$G_2\left(x\right):=\left\{\begin{array}{cc}\left\{\frac{1}{2^{n+1}}\right\},& x\in \{\frac{1}{2^n}:n\in \mathbb{N},n\ge 2\}\\ \{0,\frac{1}{2},1\},& x\in \{0,\frac{1}{2}\}\end{array}\right.$$

The above operators satisfy the multi-valued vectorial contraction condition of the Feng–Liu type, having as solutions of the altering point problem the pairs $(0,0)$, $(\frac{1}{2},0)$ and $(\frac{1}{2},1)$.

The above results generalize some altering points theorems, as given for the single-valued case in [26].

5. Conclusions

In this paper, some new contributions to the study of the fixed point inclusion are given. The two main results of this work are:

(1)

An existence, approximation and localization result for the fixed points of a multi-valued Feng–Liu contraction;

(2)

A study of the fixed point inclusion $x\in T\left(x\right)$ with a multi-valued Feng–Liu contraction $T:X\to P\left(X\right)$ in the context of a vector-valued metric space; the study includes existence, approximation and stability results for the fixed point inclusion $x\in T\left(x\right)$; the importance of the fixed point theory in vector-valued metric spaces is illustrated by an application to a system of operatorial inclusions. The particular case of altering points for multi-valued operators is also considered.