Stability Properties for Multi-Valued Contractions in Complete Vector-Valued B-Metric Spaces

Abstract

In this paper, we present some existence and stability results for the fixed point inclusion in the case of multi-valued self contractions, on a complete vector-valued B-metric space. Our main existence result for the fixed point problem extends to the multi-valued setting with a recent result obtained for the single-valued case. Moreover, data dependence on the operator perturbation of the fixed point set and some stability theorems (Ulam–Hyers stability, well-posedness and Ostrowski stability) are proved, in order to have a complete study of the fixed point inclusion.

1. Introduction

There are many extensions of the famous Banach–Caccioppoli Contraction Principle. One of the most interesting extensions was provided by Perov in [1]. Recently, an extension of this result was provided for the case of B-metric spaces (where B is a square matrix), by Precup and Stan [2]. A B-metric is a generalization of the classical vector-valued metric introduced by Perov [1], i.e., the particular case when B is the identity matrix. Meanwhile, the B-metric space is an extension of the well-known b-metric spaces (where $b\ge 1$ is a real number), introduced by Bakhtin and resumed by Czerwik in [3]; see also [4] for the whole history of such spaces.

The main purpose of this paper is to provide a generalization of Precup and Stan existence results (see Theorem 1 in [2]) for the case of multi-valued contractions in complete B-metric spaces. A second goal is to discuss some stability results (data dependence on the operator perturbation, Ulam–Hyers stability, well-posedness and Ostrowski stability) for the fixed point set of a multi-valued self contraction in complete B-metric spaces. Some open questions related to this topic will be addressed. Our results extend some recent theorems provided in [5] for multi-valued contractions of Feng–Liu type in vector-valued metric spaces and those given in [2] for single-valued contractions in vector-valued $\mathbb{B}$-metric spaces.

2. Preliminaries

In this section, we recall some useful notions and results. For the suggested terminology in fixed point theory, see [6].

Throughout this paper, $\mathbb{N}$ denotes the set of all natural numbers including 0, ${\mathbb{N}}^{*}=\mathbb{N}\setminus \left\{0\right\}$, while the symbol ${\mathbb{R}}_{+}$ denotes the set of all nonnegative real numbers. Furthermore, ${\mathcal{M}}_{m,m}\left(\mathbb{R}\right)$ represents the set of all $m\times m$ matrices with real elements, $I_{m,m}$ the identity matrix $m\times m$ and $O_{m,m}$ the null matrix $m\times m$. In addition, $O_m$ represents the null vector in ${\mathbb{R}}^m$.

Definition 1. 

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

(1) 

The distance between a point $x\in X$ and a set $A\in P\left(X\right)$:

$$D(x,A):=\mathit{inf}\left\{d\right(x,a\left)\right|a\in A\}.$$

(2) 

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

$$e(A,B):=sup\left\{D\right(a,B\left)\right|a\in A\}.$$

(3) 

The Hausdorff–Pompeiu distance between the sets $A,B\in P\left(X\right)$:

$$H(A,B)=\mathit{max}\left\{e\right(A,B),e(B,A\left)\right\}.$$

If $u,v\in {\mathbb{R}}^m$, $u=(u_1,\dots ,u_m)$ and $v=(v_1,\dots ,v_m)$, then, by definition,

$$u⪯v \mathrm{i}\mathrm{f} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y} \mathrm{i}\mathrm{f} u_i\le v_i, \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h} i\in \{1,2,\dots ,m\}.$$

Definition 2. 

We say that $(X,d)$ is a vector-valued B-metric space if X is a nonempty set and there exists $B\in {\mathcal{M}}_{m,m}\left(\mathbb{R}\right)$ such that the mapping $d:X\times X\to {\mathbb{R}}_{+}^m$ satisfies

(i) 

  $d(u,v)=O_m$ if and only if $u=v$;

(ii) 

 $d(u,v)=d(v,u)$, for all $u,v\in X$;

(iii) 

$d(u,v)⪯B\left(d\right(u,w)+d(w,v\left)\right),$ for all $u,v,w\in X$.

The mapping d is called a vector-valued B-metric on X. If $B=I_{m,m}$, then we get the classical notion of a vector-valued metric on X; see [1].

For relevant examples of vector-valued B-metrics and other considerations, see [2]. For example, the convergence in a vector-valued B-metric space is defined as follows: a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is convergent to $x^{*}\in X$ if $d(x_n,x^{*})$ converges (componentwise) to $O_m$.

If d is a vector-valued B-metric, then we denote by $D_d,e_d,H_d$ the distance functional, the excess functional and the Hausdorff–Pompeiu functional generated by d in the sense of Definition 1. More precisely, if

$$d(w,z):=\left(\begin{array}{c}{d}_1\left(w,z\right)\\ \dots \\ d_m\left(w,z\right)\end{array}\right), \mathrm{f}\mathrm{o}\mathrm{r} w,z\in X,$$

then,

• For $w\in X$ and $A,B\in P\left(X\right)$, we denote

  $$D_d(w,A):=\left(\begin{array}{c}{D}_{d_1}(w,A)\\ ...\\ D_{d_m}(w,A)\end{array}\right) \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} w\in X \mathrm{t}\mathrm{o} A\in P\left(X\right);$$
• For $A,B\in P\left(X\right)$, we denote

  $$e_d(A,B):=\left(\begin{array}{c}{e}_{d_1}(A,B)\\ ...\\ e_{d_m}(A,B)\end{array}\right) \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l};$$
• For $A,B\in P\left(X\right)$, we denote

  $$H_d(A,B):=\left(\begin{array}{c}{H}_{d_1}(A,B)\\ ...\\ H_{d_m}(A,B)\end{array}\right) \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{H}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{d}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{f}–\mathrm{P}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{i}\mathrm{u} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}.$$

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

By definition, a matrix $K\in M_{m,m}\left({\mathbb{R}}_{+}\right)$ is called convergent to zero if $K^n\to O_{m,m}$ as $n\to \infty$. The property of a matrix K to converge to zero is equivalent to the fact that the spectral radius of K is strictly less than 1; i.e., the maximum of the absolute values of its eigenvalues is strictly less than 1. In this case, the matrix $(I_{m,m}-K)$ is nonsingular and ${\left(I_{m,m}-K\right)}^{-1}=I_{m,m}+K+\dots +K^n+\dots$. For other details related to matrix analysis, see [7,8].

3. A Fixed Point Result

In this section, some new fixed point theorems for multi-valued contractions in vector-valued B-metric spaces are provided.

Definition 3. 

Let $(X,d)$ be a vector-valued B-metric space. A multi-valued operator $T:X\to P\left(X\right)$ for which there exists a matrix $K\in M_{m,m}\left({\mathbb{R}}_{+}\right)$ convergent to zero such that

$$H_d(T\left(x\right),T\left(y\right))⪯Kd(x,y),forall(x,y)\in X\times X$$

is said to be a multi-valued vectorial K-contraction.

If $T:X\to P\left(X\right)$ is a multi-valued operator and $(x,y)\in X\times X$, then an iterative sequence of Picard type for T starting from $(x,y)$ is a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ with $x_0=x,x_1=y$ and $x_{n+1}\in T\left(x_n\right)$, for every $n\in \mathbb{N}$.

The following multi-valued version of the well-known Perov’s fixed point theorem in a complete vector-valued metric space is well known; see, for example, [9]. The result is also an extension, from the case of classical metric spaces to vector-valued metric spaces, of the well-known Multi-valued Contraction Principle of Covitz and Nadler [10].

Theorem 1. 

Let $(X,d)$ be a complete vector-valued metric space and $T:X\to P\left(X\right)$ be a multi-valued vectorial K-contraction with closed values. Then, for each $(x_0,x_1)\in Graph\left(T\right)$, there exists an iterative sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ of Picard type for T starting from $(x_0,x_1)$ such that the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ converges to $x^{*}:=x^{*}(x_0;x_1)\in Fix\left(T\right)$.

Example 1. 

If the set $X=[0,1]\times [0,1]$ is endowed with the vector-valued metric

$$d((x,y),(u,v)):=\left(\begin{array}{c}|x-u|\\ |y-v|\end{array}\right),for(x,y),(u,v)\in X$$

and $T:X\to P_{b,cl}\left(X\right)$ is defined by $T(x,y):=[0,{\displaystyle \frac{x}{2}}]\times [0,{\displaystyle \frac{y}{2}}]$, then it is easy to check that T is a multi-valued vectorial K-contraction, with ${\displaystyle K:=\left(\begin{array}{cc}\frac{1}{2}& 0\\ 0& \frac{1}{2}\end{array}\right)}$.

In what follows, a generalization of the above result to a vector-valued B-metric space is presented. The next result is also an extension of the multi-valued case of the main result of the paper [2]. For related results and developments, see [11,12,13,14].

Theorem 2. 

Let $(X,d)$ be a complete vector-valued B-metric space and $T:X\to P\left(X\right)$ be a multi-valued vectorial K-contraction with closed values.

Then, for each $(x_0,x_1)\in Graph\left(T\right)$, there exists an iterative sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ of Picard type for T starting from $(x_0,x_1)$ such that

(a) 

${\left(x_n\right)}_{n\in \mathbb{N}}$ is convergent to $x^{*}:=x^{*}(x_0;x_1)\in Fix\left(T\right)$;

(b) 

If, additionally, the mapping $d(x,·):X\to {\mathbb{R}}_{+}^n$ is continuous for each $x\in X$ and the matrix $qBK$ is convergent to zero for some $q>1$, then the following estimation holds:

$$d(x_n,x^{*})⪯{\left(qK\right)}^n{(I_{m,m}-qBK)}^{-1}Bd(x_0,x_1),n\in \mathbb{N};$$

(c) 

Under the conditions stated in (b), the following retraction-displacement condition holds:

$$d(x_0,x^{*})⪯{(I_{m,m}-qBK)}^{-1}Bd(x_0,x_1).$$

Proof. 

(a) Let $(x_0,x_1)\in Graph\left(T\right)$ be arbitrary and $q>1$ such that the matrix $qK$ remains convergent to zero. Since the matrix $qK$ is convergent to zero, for arbitrary $a>0$, let $\tilde{n}:=\tilde{n}\left(a\right)$ such that ${\left(qK\right)}^{\tilde{n}}⪯\tilde{K}$, where $\tilde{K}$ is the square $(m,m)$-matrix with all the elements equal to a. For $x_1\in T\left(x_0\right)$ and above $q>1$, there exists $x_2\in T\left(x_1\right)$ such that

$$d(x_1,x_2)⪯q{H}_d(T\left(x_0\right),T\left(x_1\right))⪯qKd(x_0,x_1).$$

Inductively, we obtain a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ of Picard type for T starting from $(x_0,x_1)$ such that

$$d(x_n,x_{n+1})⪯qKd(x_{n-1},x_n), \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} n\in \mathbb{N}.$$

By the above relation, for $n,p\in \mathbb{N},p\ge 1$, we inductively get

$$d(x_n,x_{n+p})⪯\left(qK\right)d(x_{n-1},x_{n-1+p})⪯{\left(qK\right)}^{2}d(x_{n-2},x_{n-2+p})⪯\dots ⪯{\left(qK\right)}^{n}d(x_0,x_p).$$

Thus, we have that

$$d(x_n,x_{n+p})⪯{\left(qK\right)}^{n}d(x_0,x_p), \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y} n\in \mathbb{N} \mathrm{a}\mathrm{n}\mathrm{d} p\in {\mathbb{N}}^{*}.$$

(1)

Then, we obtain

$$d(x_n,x_p)⪯Bd(x_n,x_{n+\tilde{n}})+Bd(x_p,x_{n+\tilde{n}})⪯$$

$$B{\left(qK\right)}^{n}d(x_0,x_{\tilde{n}})+B^{2}d(x_p,x_{p+\tilde{n}})+B^{2}d(x_{p+\tilde{n}},x_{n+\tilde{n}})⪯$$

$$B{\left(qK\right)}^{n}d(x_0,x_{\tilde{n}})+B^2{\left(qK\right)}^{p}d(x_0,x_{\tilde{n}})+B^2{\left(qK\right)}^{\tilde{n}}d(x_p,x_n)⪯$$

$$B{\left(qK\right)}^{n}d(x_0,x_{\tilde{n}})+B^2{\left(qK\right)}^{p}d(x_0,x_{\tilde{n}})+B^2\tilde{K}d(x_p,x_n).$$

Hence, we get that

$$(I_{m,m}-B^2\tilde{K})d(x_p,x_n)⪯B{\left(qK\right)}^{n}d(x_0,x_{\tilde{n}})+B^2{\left(qK\right)}^{p}d(x_0,x_{\tilde{n}}).$$

Notice that ${\tilde{K}}^n={\left(ma\right)}^{n-1}\tilde{K}$. Thus, choosing $a>0$ smaller than 1 divided by the greatest element of $B^2$ multiplied with m, we observe that the matrix $B^2\tilde{K}$ is convergent to zero. Hence, it is invertible and we get

$$d(x_p,x_n)⪯{(I_{m,m}-B^2\tilde{K})}^{-1}\left[B{\left(qK\right)}^{n}d(x_0,x_{\tilde{n}})+B^2{\left(qK\right)}^{p}d(x_0,x_{\tilde{n}})\right]\to 0,$$

(2)

as $n,p\to \infty$. Hence, the sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is Cauchy and it converges to an element $x^{*}\in X$ (depending on $x_0$ and $x_1$).

We will prove now that $x^{*}\in Fix\left(T\right)$. We estimate

$$D_d(x^{*},T\left(x^{*}\right))⪯d(x^{*},x_{n+1})+H_d(T\left(x_n\right),T\left(x^{*}\right))⪯d(x^{*},x_{n+1})+Kd(x_n,x^{*})\to 0,$$

as $n\to \infty$. Thus, $x^{*}\in T\left(x^{*}\right)$.

(b) By (1), letting $p\to \infty$, we get

$$d(x_n,x^{*})⪯{\left(qK\right)}^{n}d(x_0,x^{*}).$$

(3)

By (3), taking $n=1$, we get

$$d(x_1,x^{*})⪯\left(qK\right)d(x_0,x^{*}).$$

(4)

Then, we observe that

$$d(x_n,x^{*})⪯{\left(qK\right)}^{n}d(x_0,x^{*})⪯{\left(qK\right)}^{n}B(d(x_0,x_1)+d(x_1,x^{*})).$$

Also, we have

$$d(x_0,x^{*})⪯Bd(x_0,x_1)+Bd(x_1,x^{*})⪯Bd(x_0,x_1)+qBKd(x_0,x^{*}).$$

Thus, we conclude

$$d(x_n,x^{*})⪯{\left(qK\right)}^n{(I_{m,m}-qBK)}^{-1}Bd(x_0,x_1),forn\in \mathbb{N}.$$

(c) Taking $n=0$ in the above relation, we get the last conclusion. □

Remark 1. 

It is an open question to extend the above result for the case of a B-metric space with an inverse positive matrix B; see [2] for the definition. It is also an open question to obtain the above result without the continuity assumption on the B-metric d.

4. Stability Properties

Let $(X,d)$ be a vector-valued B-metric space. We suppose throughout the section that the mapping $d(x,·):X\to {\mathbb{R}}_{+}^n$ is continuous for each $x\in X$.

In what follows, we emphasize the Ulam–Hyers stability, the well-posedness and the data dependence properties for multi-valued operators.

Definition 4. 

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

$$d(\tilde{x},x^{*})⪯A\epsilon .$$

Definition 5. 

Let $(X,d)$ be a vector-valued B-metric space and $T:X\to P\left(X\right)$ be a multi-valued operator with $Fix\left(T\right)\ne \varnothing$. Suppose that there exists $r:X\to Fix\left(T\right)$ a set retraction; i.e., the restriction of r to $Fix\left(T\right)$ is the identity mapping. The fixed point inclusion $x\in T\left(x\right)$ is well-posed in the sense of Reich and Zaslavski (see [15,16]) if, for each $x^{*}\in Fix\left(T\right)$ and any sequence ${\left(z_n\right)}_{n\in \mathbb{N}}\subset r^{-1}\left(x^{*}\right)$ with

$$D_d(z_n,T\left(z_n\right))\underset{n\to \infty }{\to }0,$$

we have that $z_n\underset{n\to \infty }{\to }{x}^{*}$.

Definition 6. 

Let $(X,d)$ be a vector-valued B-metric space, $T:X\to P\left(X\right)$ be a multi-valued operator, and $U:X\to P\left(X\right)$ be a multi-valued operator with $Fix\left(U\right)\ne \varnothing$. Suppose that there exists $\mu =({\mu }_1,\dots ,{\mu }_m)({\mu }_i>0,i\in \{1,2,\dots ,m\}$) with

$$H_d(T\left(x\right),U\left(x\right))⪯\mu ,forallx\in X.$$

Then, the fixed point inclusion $x\in T\left(x\right)$ has the data dependence property of the operator perturbation if, for each $u^{*}\in Fix\left(U\right)$, there is $x^{*}\in Fix\left(T\right)$ with

$$d(u^{*},x^{*})⪯V\mu , for some V\in M_{m,m}\left({\mathbb{R}}_{+}^{*}\right).$$

In order to obtain the first result for the previous stability concepts, we will use the vectorial retraction-displacement criterion. For related results and applications of this concept in the single-valued case, see [17].

Definition 7. 

Let $(X,d)$ be a vector-valued B-metric space and $T:X\to P\left(X\right)$ be a multi-valued operator with $Fix\left(T\right)\ne \varnothing$. Then, T satisfies the vectorial retraction-displacement criterion if there is a matrix $A\in M_{m,m}\left({\mathbb{R}}_{+}^{*}\right)$ and a set retraction $r:X\to Fix\left(T\right)$ with

$$d(x,r\left(x\right))⪯A{D}_d(x,T\left(x\right)),forallx\in X.$$

By the above definitions, it is easy to check the following general abstract result; see [5,18] for the approach in the case of classical vector-valued metric spaces. For the sake of completeness, we present the proof of the first conclusion.

Theorem 3. 

Let $(X,d)$ be a vector-valued B-metric space and $T:X\to P\left(X\right)$ be a multi-valued operator such that $Fix\left(T\right)\ne \varnothing$. Suppose that T satisfies the vectorial retraction-displacement criterion. Then,

• The fixed point inclusion $x\in T\left(x\right)$ is Ulam–Hyers stable;
• The fixed point inclusion $x\in T\left(x\right)$ is well-posed in the sense of Reich and Zaslavski;
• The fixed point inclusion $x\in T\left(x\right)$ fulfills the data dependence property.

Proof. 

Let $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_m)$ (where ${\epsilon }_i>0,i\in \{1,2,\dots ,m\}$) and take any $\epsilon$-fixed point $\tilde{x}\in X$ of T. Then, $D_d(\tilde{x},T\left(\tilde{x}\right))⪯\epsilon$. Since the vectorial retraction-displacement criterion holds, we have

$$d(x,r\left(x\right))⪯A{D}_d(x,T\left(x\right)), \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} x\in X.$$

Taking $x:=\tilde{x}$, we obtain

$$d(\tilde{x},r\left(\tilde{x}\right))⪯A{D}_d(\tilde{x},T\left(\tilde{x}\right))⪯A\epsilon .$$

Thus, there exists $x^{*}:=r\left(\tilde{x}\right)\in Fix\left(T\right)$ such that

$$d(\tilde{x},x^{*})⪯A\epsilon .$$

The last two conclusions can be obtained in a similar way, using the vectorial retraction-displacement criterion. □

In what follows, we obtain, under certain conditions, that any multi-valued K-contraction satisfies the vectorial retraction-displacement criterion.

Theorem 4. 

Let $(X,d)$ be a complete vector-valued B-metric space. Let $T:X\to P\left(X\right)$ be a vectorial multi-valued K-contraction with closed values. Suppose that the matrix $qBK$ is convergent to zero for some $q>1$. Then, T satisfies the vectorial retraction-displacement criterion.

Proof. 

By Theorem 2 (c), we get that, for every $(x_0,x_1)\in Graph\left(T\right)$, we have

$$d(x_0,x^{*})⪯{(I_{m,m}-qBK)}^{-1}Bd(x_0,x_1).$$

Consider now $l=(l_1,\dots ,l_m)$ with $l_i\in (0,1)$, for every $i\in \{1,2,\dots ,m\}$. Then, there exists $x_1\in T\left(x_0\right)$ such that $d(x_0,x_1)\prec l{D}_d(x_0,T\left(x_0\right))$. Thus, we have

$$d(x_0,x^{*})⪯{(I_{m,m}-qBK)}^{-1}Bl{D}_d(x_0,T\left(x_0\right)).$$

The conclusion follows from Theorem 3. □

Furthermore, we investigate the Ostrowski stability property in vector-valued B-metric spaces. The above definitions appear in [18], for the setting of a vector-valued metric space.

Definition 8. 

Let $(X,d)$ be a vector-valued B-metric space, $T:X\to P\left(X\right)$ be a multi-valued operator with $Fix\left(T\right)\ne \varnothing$, and $r:X\to Fix\left(T\right)$ a set retraction. The fixed point inclusion $x\in T\left(x\right)$ has the Ostrowski stability property if, for each $x^{*}\in Fix\left(T\right)$ and any sequence ${\left(y_n\right)}_{n\in \mathbb{N}}\subset r^{-1}\left(x^{*}\right)$ with

$$D_d(y_{n+1},T\left(y_n\right))\underset{n\to \infty }{\to }0,$$

we have that $y_n\underset{n\to \infty }{\to }{x}^{*}$.

Definition 9. 

Let $(X,d)$ be a vector-valued B-metric space, $T:X\to P\left(X\right)$ be a multi-valued operator with $Fix\left(T\right)\ne \varnothing$, and there exist $r:X\to Fix\left(T\right)$ a set retraction. We say that T is a multi-valued L-quasicontraction with respect to r if there exists $L\in M_{m,m}\left({\mathbb{R}}_{+}\right)$ which is convergent to zero and

$$e_d(T\left(x\right),r\left(x\right))⪯Ld(x,r\left(x\right)), for every x\in X.$$

We will use the following auxiliary result, proved in [14].

Lemma 1. 

Let $A_n,B_n\in M_{m,m}\left({\mathbb{R}}_{+}\right)$ such that

(i) 

${\displaystyle \sum _{k\in \mathbb{N}}}{A}_k<\infty$;

(ii) 

$B_n\underset{n\to \infty }{\to }{O}_{m,m}$.

Then, ${\displaystyle \sum _{k=0}^n}{A}_{n-k}{B}_k\underset{n\to \infty }{\to }{O}_{m,m}$.

Concerning the Ostrowski stability concept, we prove the following theorem.

Theorem 5. 

Let $(X,d)$ be a complete vector-valued B-metric space and $T:X\to P\left(X\right)$ be a vectorial multi-valued K-contraction with closed values. Suppose that T is a multi-valued L-quasicontraction, such that $LB=BL$ and ${\displaystyle \sum _{k\in \mathbb{N}}}{\left(LB\right)}^k<\infty$. Then, the fixed point inclusion $x\in T\left(x\right)$ has the Ostrowski stability property.

Proof. 

Since T is a vectorial multi-valued K-contraction, using Theorem 2, we deduce that $Fix\left(T\right)\ne \varnothing$ and there exists $r:X\to Fix\left(T\right)$ a set retraction (see conclusion (c) and the proof of Theorem 4).

We consider $x^{*}\in Fix\left(T\right)$ and ${\left(y_n\right)}_{n\in \mathbb{N}}\subset r^{-1}\left(x^{*}\right)$ with $D(y_{n+1},T\left(y_n\right))\underset{n\to \infty }{\to }0.$

Thus,

$$d(y_{n+1},x^{*})=d(y_{n+1},r\left(y_n\right))⪯B(D(y_{n+1},T\left(y_n\right))+e(T\left(y_n\right),r\left(y_n\right)))⪯$$

$$B(D(y_{n+1},T\left(y_n\right))+Ld(y_n,x^{*}))⪯$$

$$B(D(y_{n+1},T\left(y_n\right))+BL(D(y_n,T\left(y_{n-1}\right))+Ld(y_{n-1},x^{*})))=$$

$$BD(y_{n+1},T\left(y_n\right))+BLBD(y_n,T\left(y_{n-1}\right))+{\left(BL\right)}^{2}d(y_{n-1},x^{*})⪯\dots ⪯$$

$$B\left[D(y_{n+1},T\left(y_n\right))+\left(LB\right)D(y_n,T\left(y_{n-1}\right))+\dots +{\left(LB\right)}^{n}D(y_1,T\left(y_0\right))\right]+{\left(BL\right)}^{n+1}d(y_0,x^{*}).$$

From Lemma 1, it follows that $y_n\underset{n\to \infty }{\to }0$. □

5. Conclusions

The results of this paper are extensions of some important theorems in the fixed point theory literature, in at least two main directions:

1.

Extensions to the case of complete B-metric spaces of some fixed point theorems in complete vector-valued metric spaces, starting with Perov’s Contraction Principle [1];

2.

Extensions to the case of a complete B-metric spaces of some fixed point results in complete b-metric spaces, starting with Bakhtin’s Theorem and Czerwik’s Theorem; see [3,4].

The paper includes not only existence results but also some stability-type results, which are important approaches in the numerical treatment of fixed point problems. We also notice that such fixed point results (including existence, uniqueness and stability results) are very useful for various applications, especially for the study of systems of operatorial inclusions, such as integral inclusions, coupled fixed point inclusions and some others; see [19,20].