Fixed Point Problems on Generalized Metric Spaces in Perov’s Sense

Abstract

The aim of this paper is to give some fixed point results in generalized metric spaces in Perov’s sense. The generalized metric considered here is the w-distance with a symmetry condition. The operators satisfy a contractive weakly condition of Hardy–Rogers type. The second part of the paper is devoted to the study of the data dependence, the well-posedness, and the Ulam–Hyers stability of the fixed point problem. An example is also given to sustain the presented results.

1. Introduction and Preliminaries

The well-known Banach contraction principle was extended by Perov in 1964 to the case of spaces endowed with vector-valued metrics. In [1], Perov introduced the concept of vector-valued metric as follows.

Let X be a nonempty set. A mapping $\tilde{d}:X\times X\to {\mathbb{R}}^m$ where $\tilde{d}=\left(\begin{array}{c}{d}_1(x,y)\\ \cdots \\ d_m(x,y)\end{array}\right)$ for every $m\in \mathbb{N}$ is called vector-valued metric on X if the following properties are satisfied.

(1)

$\tilde{d}(x,y)\ge 0$ for all $x,y\in X$, and $\tilde{d}(x,y)=0$ implies $x=y$;

(2)

$\tilde{d}(x,y)=\tilde{d}(y,x)$;

(3)

$\tilde{d}(x,y)\le \tilde{d}(x,z)+\tilde{d}(z,y)$ for all $x,y,z\in X$.

In this case, the pair $(X,\tilde{d})$ is called a generalized metric space in Perov’s sense. Some examples of fixed points on the sense of vector-valued metric are given in [2,3,4,5,6]. Throughout this paper ${\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ will denote the set of all $m\times m$ matrices with positive elements. We also denote by Θ the zero $m\times m$ matrix and $0_{1\times m}=\left(\begin{array}{c}0\\ \cdots \\ 0\end{array}\right)$, by I the identity $m\times m$ matrix and $I_{1\times m}=\left(\begin{array}{c}1\\ \cdots \\ 0\end{array}\right)$ and by U the unity $m\times m$ matrix and $U_{1\times m}=\left(\begin{array}{c}1\\ \cdots \\ 1\end{array}\right)$. If $A\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$, then the symbol $A^{\tau }$ stands for the transpose matrix of A.

Recall that a matrix A is said to be convergent to zero if and only if $A^n\to \Theta$ as $n\to \infty$.

Let us recall the following theorem, which is useful for the proof of the main result, see [7].

Theorem 1.

Let $A\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$. The following assertions are equivalent.

(i) 

A is a matrix convergent to zero;

(ii) 

$A^n\to \Theta$ as $n\to \infty$;

(iii) 

The eigenvalues of A are in the open unit disc, i.e., $\left|\lambda \right|<1$, for each $\lambda \in \mathbb{C}$ with $det(A-\lambda I)=0$;

(iv) 

The matrix $I-A$ is non-singular and

$${(I-A)}^{-1}=I+A+\dots +A^n+\dots ;$$

(v) 

The matrix $I-A$ is non-singular and the matrix ${(I-A)}^{-1}$ has nonnenegative elements.

In [8], one can find that the notion of K-metric, which is an extension of the Perov’s metric. Huang and Zhang reconsidered in [9] the notion of K-metric under the name cone metric.

Hardy and Rogers [10] proved in 1973 a generalization of Reich fixed point theorem. Having this as a starting point, many authors obtained fixed point results for Hardy–Rogers type operators.

Let $(X,d)$ be a metric space. Throughout this paper we use the following notations.

$P(X)$: the set of all nonempty subsets of X;

$P_{cl}(X)$: the set of all nonempty closed subsets of X;

$P_{cp}(X)$: the set of all nonempty compact subsets of X;

$Fix(F):=\{x\in X\mid x\in F(x)\}$: the set of the fixed points of F;

$SFix(F):=\{x\in X\mid \{x\}=F(x)\}$: the set of the strict fixed points of F.

We denote by $\mathbb{N}$ the set of all natural numbers. We also denote by ${\mathbb{N}}^{*}:=\mathbb{N}-\left\{0\right\}$ the set of all natural numbers without 0.

Let $(X,\tilde{d})$ be a generalized metric space in the sense of Perov. Here, if $v,r\in {\mathbb{R}}^m$ have the form $v:=(v_1,v_2,\cdots ,v_m)$ and $r:=(r_1,r_2,\cdots ,r_m)$, then by the inequality $v\le r$ we mean $v_i\le r_i$, for each $i\in \{1,2,\cdots ,m\}$, whereas by the inequality $v<r$, we mean $v_i<r_i$, for each $i\in \{1,2,\cdots ,m\}$. Moreover, $\left|v\right|:=(|v_1|,|v_2|,\cdots ,|v_m|)$ and, if $c\in \mathbb{R}$ then $v\le c$ means $v_i\le c$, for each $i\in \{1,2,\cdots ,m\}$.

We can notice that, in a generalized metric space, some concepts are similar to those given for metric space. Some of these concepts are Cauchy sequence, convergent sequence, completeness, and open and closed subsets.

In [11], Kada et al. introduced the concept of w-distance and improved several results replacing the involved metric by a generalized distance. On the other hand, the notions of single-valued and multivalued weakly contractive maps with respect to w-distance was introduced by Suzuki and Takahashi in [12]. Some recent fixed point results involving the w-distance can be found in [12,13,14,15,16,17,18,19].

Definition 1.

A mapping $w:X\times X\to [0,\infty )$ is a w-distance on X if it satisfies the following conditions for any $x,y,z\in X$.

(1) 

$w(x,z)\le w(x,y)+w(y,z);$

(2) 

the function $w(x,.):X\to [0,\infty )$ is lower semicontinuous;

(3) 

for any $\epsilon >0,$ there exists $\delta >0$ such that $w(z,x)\le \delta$ and $w(z,y)\le \delta$ imply $d(x,y)\le \epsilon .$

In [20], we find the definition of $w_0$-distance as follows.

Definition 2.

Let $(X,d)$ be a metric space. A mapping $w:X\times X\to [0,\infty )$ is called $w_0$-distance if it is w-distance on X with $w(x,x)=0$ for every $x\in \mathbb{R}$.

Remark 1.

Each metric is a $\widetilde{w_0}$-distance, but the reverse is not true.

For the following notations see I.A. Rus [21,22], I.A. Rus, A. Petruşel, A. Sîntămărian [23], and A. Petruşel [24].

Definition 3.

Let (X,d) be a metric space and $f:X\to X$ be a single-valued operator. f is a weakly Picard operator (briefly WPO) if the sequence of successive approximations for f starting from $x\in X$, ${(f^n(x))}_{n\in \mathbb{N}}$, converges, for all $x\in X$ and its limit is a fixed point for f.

If f is a WPO, then we consider the operator

$${\displaystyle f^{\infty }:X\to X\mathrm{defined}\mathrm{by}{f}^{\infty }(x):=\underset{n\to \infty }{lim}{f}^n(x).}$$

Notice that $f^{\infty }(X)=Fix(f)$.

Definition 4.

Let (X,d) be a metric space, $f:X\to X$ be a WPO and $c>0$ be a real number. By definition, the single-valued operator f is c-weakly Picard operator (briefly c-WPO) if and only if the following inequality holds,

$$d(x,f^{\infty }(x))\le cd(x,f(x)),for\hspace{0.17em}all\hspace{0.17em}x\in X.$$

For the theory of weakly Picard operators, for single-valued operators, see [21].

I.A. Rus gave in [22] the definition of Ulam–Hyers stability as follows.

Definition 5.

Let (X,d) be a metric space and $f:X\to X$ be a single-valued operator. By definition, the fixed point equation

$$x=f(x)$$

(1)

is Ulam–Hyers stable if there exists a real number $c_f>0$ such that for each $\epsilon >0$ and each solution $y^{*}$ of the inequation

$$d(y,f(y))\le \epsilon$$

(2)

there exists a solution $x^{*}$ of Equation (1) such that

$$d(y^{*},x^{*})\le c_f\epsilon .$$

Remark 2.

If f is a c-weakly Picard operator, then the fixed point Equation (1) is Ulam–Hyers stable.

The Ulam stability of different functional type equations have been investigated by many authors (see [25,26,27,28,29,30,31,32,33,34,35]).

We present in the first part of this paper some fixed point results in generalized metric spaces in Perov’s sense. The operator satisfies a contractive condition of Hardy–Rogers type. In the second part of the paper, we study the data dependence of the fixed point set. The well-posedness of the fixed point problem and the Ulam–Hyers stability are also studied.

2. Fixed Point Results

First, let us we recall the notion of generalized w-distance defined in [36] by L. Guran.

Definition 6.

Let $(X,\tilde{d})$ be a generalized metric space. The mapping $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ is called generalized w-distance on X if it satisfies the following conditions.

(1) 

$\tilde{w}(x,y)\le \tilde{w}(x,z)+\tilde{w}(z,y)$, for every $x,y,z\in X$;

(2) 

$\tilde{w}$ is lower semicontinuous with respect to the second variable.;

(3) 

For any $\epsilon :=\left(\begin{array}{c}{\epsilon }_1\\ \cdots \\ {\epsilon }_m\end{array}\right)>0$, there exists $\delta :=\left(\begin{array}{c}{\delta }_1\\ \cdots \\ {\delta }_m\end{array}\right)>0$, such that $\tilde{w}(z,x)\le \delta$ and $\tilde{w}(z,y)\le \delta$ implies $\tilde{d}(x,y)\le \epsilon$.

Examples of generalized w-distance and some of its useful properties are also given in [36] and [37]. In the same framework, let us give the definition of generalized $w_0$-distance.

Definition 7.

Let $(X,\tilde{d})$ be a generalized metric space. A mapping $\tilde{w}:X\times X\to [0,\infty )$ is called generalized $\widetilde{w_0}$-distance if it is generalized w-distance on X with $\tilde{w}(x,x)=0_{1\times m}$ for every $x\in \mathbb{R}$.

Let us recall the following useful result.

Lemma 1.

Let $(X,\tilde{d})$ be a generalized metric space, and let $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized w-distance on X. Let $(x_n)$ and $(y_n)$ be two sequences in X, let ${\alpha }_n:=\left(\begin{array}{c}{{\alpha }_n}_1\\ \cdots \\ {{\alpha }_n}_m\end{array}\right)\in {\mathbb{R}}_{+}^m$ and ${\beta }_n=\left(\begin{array}{c}{{\beta }_n}_1\\ \cdots \\ {{\beta }_n}_m\end{array}\right)\in {\mathbb{R}}_{+}^m$ be two sequences such that ${{\alpha }_n}_{(i)}$ and ${{\beta }_n}_{(i)}$ converge to zero for each $i\in \{1,2,\cdots ,m\}$. Let $x,y,z\in X.$ Then, the following assertions hold, for every $x,y,z\in X$.

(1) 

If $\tilde{w}(x_n,y)\le {\alpha }_n$ and $\tilde{w}(x_n,z)\le {\beta }_n$ for any $n\in \mathbb{N},$ then $y=z.$

(2) 

If $\tilde{w}(x_n,y_n)\le {\alpha }_n$ and $\tilde{w}(x_n,z)\le {\beta }_n$ for any $n\in \mathbb{N},$ then $(y_n)$ converges to z.

(3) 

If $\tilde{w}(x_n,x_m)\le {\alpha }_n$ for any $n,m\in \mathbb{N}$ with $m>n,$ then $(x_n)$ is a Cauchy sequence.

(4) 

If $\tilde{w}(y,x_n)\le {\alpha }_n$ for any $n\in \mathbb{N},$ then $(x_n)$ is a Cauchy sequence.

Next, let us give the definition of single-valued weakly Hardy–Rogers type operator on generalized metric space in Perov’s sense.

Definition 8.

Let $(X,\tilde{d})$ be a generalized metric space in Perov’s sense, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized w-distance, and $f:X\to X$ be a single-valued operator. We say that f is a weakly Hardy–Rogers type operator if the following inequality is satisfied,

$$\tilde{w}(f(x),f(y))\le A\tilde{w}(x,y)+B[\tilde{w}(x,f(x))+\tilde{w}(y,f(y))]+C[\tilde{w}(x,f(y))+\tilde{w}(y,f(x))],$$

for all $x,y\in \mathbb{R}$ and $A,B,C\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$.

The first fixed point result of this paper is the following.

Theorem 2.

Let $(X,\tilde{d})$ be a complete generalized metric space in Perov’s sense, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized $w_0$-distance. Let $f:X\to X$ be a single-valued weakly Hardy–Rogers type operator such that

(a) 

f is continuous;

(b) 

there exist matrices $A,B,C\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ such that

(i) 

$M={(I-(B+C))}^{-1}(A+B+C)$ converges to Θ;

(ii) 

$I-(B+C)$ is nonsingular and ${(I-(B+C))}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$;

(iii) 

$I-(A+2B+2C)$ is nonsingular and ${[I-(A+2B+2C)]}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$.

Then, $Fix(f)\ne \varnothing$. Moreover, if $x^{*}=f(x^{*})$, then $w(x^{*},x^{*})=0$.

Proof. 

Fix $x_0\in X$. Let $x_1=f(x_0)$ and $x_2=f(x_1)$. Then, we have

$$\begin{gathered} \tilde{w}(x_1,x_2)=\tilde{w}(f(x_0),f(x_1))A\tilde{w}(x_0,x_1)+B[\tilde{w}(x_0,f(x_0))+\tilde{w}(x_1,f(x_1))]+C[\tilde{w}(x_0,f(x_1)) \\ +\tilde{w}(x_1,f(x_0))]=A\tilde{w}(x_0,x_1)+B[\tilde{w}(x_0,x_1)+\tilde{w}(x_1,x_2)]+C[\tilde{w}(x_0,x_2)+\tilde{w}(x_1,x_1)] \\ =(A+B)\tilde{w}(x_0,x_1)+B(\tilde{w}(x_1,x_2))+C[\tilde{w}(x_0,x_1)+\tilde{w}(x_1,x_2)] \\ =(A+B+C)\tilde{w}(x_0,x_1)+(B+C)\tilde{w}(x_1,x_2). \end{gathered}$$

Then, we have $[I-(B+C)]\tilde{w}(x_1,x_2)\le (A+B+C)\tilde{w}(x_0,x_1)$.

We get the inequality

$$\tilde{w}(x_1,x_2)\le {[I-(B+C)]}^{-1}(A+B+C)\tilde{w}(x_0,x_1)=M\tilde{w}(x_0,x_1).$$

(3)

For the next step, we have

$$\begin{gathered} \tilde{w}(x_2,x_3)=\tilde{w}(f(x_1),f(x_2))A\tilde{w}(x_1,x_2)+B[\tilde{w}(x_1,f(x_1))+\tilde{w}(x_2,f(x_2))]+C[\tilde{w}(x_1,f(x_2)) \\ +\tilde{w}(x_2,f(x_1))]=A\tilde{w}(x_1,x_2)+B[\tilde{w}(x_1,x_2)+\tilde{w}(x_2,x_3)]+C[\tilde{w}(x_1,x_3)+\tilde{w}(x_2,x_2)] \\ =(A+B)\tilde{w}(x_1,x_2)+B(\tilde{w}(x_2,x_3))+C[\tilde{w}(x_1,x_2)+\tilde{w}(x_2,x_3)] \\ =(A+B+C)\tilde{w}(x_1,x_2)+(B+C)\tilde{w}(x_2,x_3). \end{gathered}$$

Then, we have $[I-(B+C)]\tilde{w}(x_2,x_3)\le (A+B+C)\tilde{w}(x_1,x_2)$.

Using (3) we obtain the inequality

$$\tilde{w}(x_2,x_3)\le {[I-(B+C)]}^{-1}(A+B+C)\tilde{w}(x_1,x_2)=M\tilde{w}(x_1,x_2)\le M^2\tilde{w}(x_0,x_1).$$

(4)

By induction we obtain a sequence ${(x)}_{n\in \mathbb{N}}\in X$, with $x_n=f(x_{n-1})$ such that

$$\tilde{w}(x_n,x_{n+1})\le M^n\tilde{w}(x_0,x_1),$$

(5)

with $M\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ and $n\in \mathbb{N}$.

We will prove next that ${(x_n)}_{n\in \mathbb{N}}$ is a Cauchy sequence, by estimating $\tilde{w}(x_n,x_m)$, for every $m,n\in \mathbb{N}$ with $m>n$.

$$\begin{gathered} \tilde{w}(x_n,x_m)\le \tilde{w}(x_n,x_{n+1})+\tilde{w}(x_{n+1},x_{n+2})+\dots +\tilde{w}(x_{m-1},x_m) \\ \le M^n(\tilde{w}(x_0,x_1))+M^{n+1}(\tilde{w}(x_0,x_1))+\dots +M^{m-1}(\tilde{w}(x_0,x_1)) \\ \le M^n(I+M+M^2+\dots +M^{m-n-1})(\tilde{w}(x_0,x_1))\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)). \end{gathered}$$

Note that $(I-M)$ is nonsingular since M is convergent to zero. This implies

$${\displaystyle \underset{n\to \infty }{lim}w(x_n,x_m)\le \underset{n\to \infty }{lim}{M}^n{(I-M)}^{-1}\tilde{w}(x_0,x_1))\stackrel{d}{\to }{0}_{1\times m}.}$$

By Lemma 1 (3) the sequence ${(x_n)}_{n\in \mathbb{N}}$ is a Cauchy sequence.

By $(a)$ we have $\tilde{w}(f(x_{n-1}),f(x^{*}))\stackrel{d}{\to }{0}_{1\times m}$, as $n\to \infty$. As $(X,d)$ is complete, there exists $x^{*}\in X$ such that ${\displaystyle \underset{n\to \infty }{lim}{x}_n\stackrel{d}{\to }{x}^{*}}$ as $n\to \infty$. From the continuity of f, it follows that $x_{n+1}=f(x_n)\stackrel{d}{\to }f(x^{*})$ as $n\to \infty$. By the uniqueness of the limit, we get $x^{*}=f(x^{*})$, that is, $x^{*}$ is a fixed point of f. Then $Fix(f)\ne \varnothing$.

Let $x^{*}\in X$ such that $x^{*}=f(x^{*})$. Then, we have

$$\begin{gathered} \tilde{w}(x^{*},x^{*})=\tilde{w}(f(x^{*}),f(x^{*}))\le A\tilde{w}(x^{*},x^{*}) \\ +B[\tilde{w}(x^{*},f(x^{*}))+\tilde{w}(x^{*},f(x^{*}))]+C[\tilde{d}(x^{*},f(x^{*}))+\tilde{d}(x^{*},f(x^{*}))] \\ =A\tilde{w}(x^{*},x^{*})+2B\tilde{w}(x^{*},x^{*})+2C\tilde{w}(x^{*},x^{*}). \end{gathered}$$

(6)

This implies $[I-(A+2B+2C)]\tilde{w}(x^{*},x^{*})\le 0_{1\times m}$. By hypothesis $(iii)$ we get $\tilde{w}(x^{*},x^{*})=0_{1\times m}$. □

We can replace the continuity condition on the operator f and we obtain the following fixed point theorem.

Theorem 3.

Let $(X,\tilde{d})$ be a complete generalized metric space in Perov’s sense and $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized $w_0$-distance. Let $f:X\to X$ be a single-valued weakly Hardy–Rogers type operator such that the following conditions are satisfied,

(a) 

$inf\{\tilde{w}(x,y)+\tilde{w}(x,f(x)):x\in X\}>0$;

(b) 

there exist matrices $A,B,C\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ such that:

(i) 

$M={(I-(B+C))}^{-1}(A+B+C)$ converges to Θ;

(ii) 

$I-(B+C)$ is nonsingular and ${(I-(B+C))}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$;

(iii) 

$I-(A+2B+2C)$ is nonsingular and ${[I-(A+2B+2C)]}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$.

Then $Fix(f)\ne \varnothing$. Moreover, if $x^{*}=f(x^{*})$, then $w(x^{*},x^{*})=0$.

Proof. 

Following the same steps as in the previous theorem, Theorem 2, we have the estimation

$$\tilde{w}(x_n,x_m)\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)$$

(7)

with $M\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ and $n\in \mathbb{N}$.

By Lemma 1 (3), the sequence ${(x_n)}_{n\in \mathbb{N}}$ is a Cauchy sequence. As $(X,\tilde{d})$ is complete, there exists $x^{*}\in X$ such that $x_n\stackrel{d}{\to }{x}^{*}$. Let $n\in \mathbb{N}$ be fixed. Then, as ${(x_m)}_{m\in \mathbb{N}}\stackrel{d}{\to }{x}^{*}$ and $\tilde{w}(x_n,·)$ is lower semicontinuous, we have

$${\displaystyle \tilde{w}(x_n,x^{*})\le \underset{m\to \infty }{lim\; inf}\tilde{w}(x_n,x_m)\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1).}$$

(8)

Assume that $x^{*}\ne f(x^{*})$. Then, for every $x\in X$, by hypothesis $(a)$ we have

$$\begin{gathered} 0<inf\{\tilde{w}(x,x^{*})+\tilde{w}(x,f(x)):x\in X\}\le inf\{\tilde{w}(x_n,x^{*})+\tilde{w}(x_n,x_{n+1}):n\in \mathbb{N}\} \\ \le inf\{M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)+M^n\tilde{w}(x_0,x_1)\}=0. \end{gathered}$$

This is a contradiction. Therefore $x^{*}=f(x^{*})$, so $Fix(f)\ne \varnothing$. For the proof of the last part of this theorem we use the same steps as is the previous theorem, Theorem 2. □

Further we give a more general fixed point result concerning this new type of operators.

Theorem 4.

Let $(X,\tilde{d})$ be a complete generalized metric space in Perov’s sense, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized $w_0$-distance, and $f:X\to X$ be a single-valued weakly Hardy–Rogers type operator. There exist matrices $A,B,C\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ such that

(i) 

$M={(I-(B+C))}^{-1}(A+B+C)$ converges to Θ;

(ii) 

$I-(B+C)$ is nonsingular and ${(I-(B+C))}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$;

(iii) 

$I-(A+2B+2C)$ is nonsingular and ${[I-(A+2B+2C)]}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$.

Then $Fix(f)\ne \varnothing$. Moreover, if $x^{*}=f(x^{*})$, then $w(x^{*},x^{*})=0$.

Proof. 

Following the same steps as in Theorem 2, we get the estimation

$$\tilde{w}(x_n,x_m)\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)$$

(9)

with $M\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ and $n\in \mathbb{N}$.

By Lemma 1 (3) the sequence ${(x_n)}_{n\in \mathbb{N}}$ is a Cauchy sequence; since $(X,\tilde{d})$ is complete there exists $x^{*}\in X$ such that $x_n\stackrel{d}{\to }{x}^{*}$.

Let $n\in \mathbb{N}$ be fixed. Then, as ${(x_m)}_{m\in \mathbb{N}}\stackrel{d}{\to }{x}^{*}$, $\tilde{w}(x_n,·)$ is lower semicontinuous and letting $n\to \infty$ we have

$${\displaystyle \tilde{w}(x_n,x^{*})\le \underset{m\to \infty }{lim\; inf}\tilde{w}(x_n,x_m)\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)\stackrel{d}{\to }{0}_{1\times m}.}$$

(10)

Let $f(x^{*})\in X$. By triangle inequality and using (6) we obtain

$$\begin{gathered} \tilde{w}(x_n,f(x^{*}))=\tilde{w}(x_n,x^{*})+\tilde{w}(x^{*},f(x^{*}))\le \tilde{w}(x_n,x^{*})+\tilde{w}(f(x^{*}),f(x^{*})) \\ \le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)+[I-(A+2B+2C)]\tilde{w}(x^{*},x^{*})\stackrel{d}{\to }{0}_{1\times m}. \end{gathered}$$

(11)

Using Lemma 1(1), by Equations (10) and (11), we get $x^{*}=f(x^{*})$. Then, $Fix(f)\ne \varnothing$.

For the last part of the proof we use the same steps as in Theorem 2. □

Another fixed point result concerning the single-valued weakly Hardy–Rogers operators in generalized metric space is the following.

Theorem 5.

Let $(X,\tilde{d})$ be a complete generalized metric space in Perov’ sense, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized $w_0$-distance and $f:X\to X$ be a single-valued Hardy–Rogers type operator. Suppose that all the hypothesis of Theorem 2 hold. Then, we have

(1) 

$Fix(f)\ne \varnothing$.

(2) 

There exists a sequence ${(x_n)}_{n\in \mathbb{N}}\in X$ such that $x_{n+1}=f(x_n)$, for all $n\in \mathbb{N}$ and converge to a fixed point of f.

(3) 

$\tilde{d}(x_n,x^{*})\le M^n\tilde{d}(x_0,x_1)$, where $x^{*}\in Fix(f).$

Example 1.

Let $X={\mathbb{R}}^2$ be a normed linear space endowed with the generalized norm $\tilde{d}$ defined by$\tilde{d}(x,y)(=\left(\begin{array}{c}\left|\right|x_1-y_1\left|\right|\\ \left|\right|x_2-y_2\left|\right|\end{array}\right)$and $\tilde{w}$ a generalized $w_0$-distance defined by $\tilde{w}(x,y)(=\left(\begin{array}{c}\left|\right|y_1\left|\right|\\ \left|\right|y_2\left|\right|\end{array}\right)$, for each $x=(x_1,x_2),y=(y_1,y_2)\in {\mathbb{R}}^2$. Let $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be an operator given by

$$f(x,y)=\left\{\begin{array}{ll}\frac{4x}{5}+\frac{6y}{5}-1,\frac{6y}{5}-1,& for(x,y)\in {\mathbb{R}}^2,with\hspace{0.17em}x\le 5;\\ \frac{x}{5}+\frac{y}{3}-1,\frac{y}{5},& for(x,y)\in {\mathbb{R}}^2,with\hspace{0.17em}x>5.\end{array}\right.$$

We take $f(x,y)=(f_1(x,y),f_2(x,y))$ where $f_1(x,y)=\left\{\begin{array}{ll}\frac{4x}{5}+\frac{6y}{5}-1,& for(x,y)\in {\mathbb{R}}^2,with\hspace{0.17em}x\le 5;\\ \frac{x}{5}+\frac{y}{3}-1,& for(x,y)\in {\mathbb{R}}^2,with\hspace{0.17em}x>5.\end{array}\right.$ and $f_2(x,y)=\left\{\begin{array}{ll}\frac{6y}{5}-1,& for(x,y)\in {\mathbb{R}}^2,with\hspace{0.17em}x\le 5;\\ \frac{y}{5},& for(x,y)\in {\mathbb{R}}^2,with\hspace{0.17em}x>5.\end{array}\right.$

Next, we show that weakly Hardy–Rogers type condition takes place.

Let $A=\left(\begin{array}{cc}\frac{4}{5}& \frac{6}{5}\\ 0& \frac{6}{5}\end{array}\right)$.

Case 1. If $1\le x_1,x_2,y_1,y_2\le 5$ we have

$$\begin{gathered} \tilde{w}(f(x),f(y))=\left(\begin{array}{c}\left|\right|f_1(y_1,y_2)\left|\right|\\ \left|\right|f_2(y_1,y_2)\left|\right|\end{array}\right)=\left(\begin{array}{c}\left|\right|\frac{4}{5}{y}_1+\frac{6}{5}{y}_2-1\left|\right|\\ \left|\right|0·y_1+\frac{6}{5}{y}_2-1\left|\right|\end{array}\right)\le \left(\begin{array}{c}\frac{4}{5}\left|\right|y_1\left|\right|+\frac{6}{5}\left|\right|y_2\left|\right|-1\\ 0·\left|\right|y_1\left|\right|+\frac{6}{5}\left|\right|y_2\left|\right|-1\end{array}\right) \\ \le \left(\begin{array}{cc}\frac{4}{5}& \frac{6}{5}\\ 0& \frac{6}{5}\end{array}\right)\left(\begin{array}{c}\left|\right|y_1\left|\right|\\ \left|\right|y_2\left|\right|\end{array}\right)=A\tilde{w}(x,y). \end{gathered}$$

Case 2. If $x_1,x_2,y_1,y_2>5$ we have

$$\begin{gathered} \tilde{w}(f(x),f(y))=\left(\begin{array}{c}\left|\right|f_1(y_1,y_2)\left|\right|\\ \left|\right|f_2(y_1,y_2)\left|\right|\end{array}\right)=\left(\begin{array}{c}\left|\right|\frac{1}{5}{y}_1+\frac{1}{3}{y}_2-1\left|\right|\\ \left|\right|0·y_1+\frac{1}{5}{y}_2\left|\right|\end{array}\right)\le \left(\begin{array}{c}\frac{1}{5}\left|\right|y_1\left|\right|+\frac{1}{3}\left|\right|y_2\left|\right|-1\\ 0·\left|\right|y_1\left|\right|+\frac{1}{5}\left|\right|y_2\left|\right|\end{array}\right) \\ \le \left(\begin{array}{cc}\frac{1}{5}& \frac{1}{3}\\ 0& \frac{1}{5}\end{array}\right)\left(\begin{array}{c}\left|\right|y_1\left|\right|\\ \left|\right|y_2\left|\right|\end{array}\right)<\left(\begin{array}{cc}\frac{4}{5}& \frac{6}{5}\\ 0& \frac{6}{5}\end{array}\right)\left(\begin{array}{c}\left|\right|y_1\left|\right|\\ \left|\right|y_2\left|\right|\end{array}\right)=A\tilde{w}(x,y). \end{gathered}$$

Case 3. For other choices of $x_1,x_2,y_1,y_2$ we have

$\tilde{w}(f(x),f(y))=\left(\begin{array}{c}0\\ 0\end{array}\right)\le \left(\begin{array}{cc}\frac{4}{5}& \frac{6}{5}\\ 0& \frac{6}{5}\end{array}\right)\left(\begin{array}{c}\left|\right|y_1\left|\right|\\ \left|\right|y_2\left|\right|\end{array}\right)=A\tilde{w}(x,y).$

Thus, the weakly Hardy–Rogers type condition is satisfied for $A=\left(\begin{array}{cc}\frac{4}{5}& \frac{6}{5}\\ 0& \frac{6}{5}\end{array}\right)$ and $B=C=\Theta$ or $B+C=\Theta$.

As all the hypothesis of Theorem 3 hold, f has a fixed point and it is easy to check that $x=f(x)=(f_1(x),f_2(x))$, where $x=(1,1)$.

Next, let us give some common fixed point results.

Theorem 6.

Let $(X,\tilde{d})$ be a complete generalized metric space in Perov’s sense, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized w-distance, and $f,g:X\to X$ be two continuous single-valued weakly Hardy–Rogers type operators. There exist matrices $A,B,C\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ such that

(i) 

$I-(B+C)$ is nonsingular and ${(I-(B+C))}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$;

(ii) 

$M={(I-(B+C))}^{-1}(A+B+C)$ converges to Θ.

Then, f and g have a common fixed point $x^{*}\in X$.

Proof. 

(1) Let $x_0\in X$. We consider ${(x_n)}_{n\in \mathbb{N}}$ the sequence of successive approximations for f and g, defined by

$$x_{2n+1}=f(x_{2n}),n=0,1,\dots$$

$$x_{2n+2}=g(x_{2n+1}),n=0,1,\dots$$

Then, we have

$$\begin{gathered} \tilde{w}(x_{2n},x_{2n+1})=\tilde{w}(g(x_{2n-1}),f(x_{2n}))\le A\tilde{w}(x_{2n-1},f(x_{2n}) \\ +B[\tilde{w}(x_{2n},f(x_{2n}))+\tilde{w}(x_{2n-1},g(x_{2n-1}))]+C[\tilde{w}(x_{2n},g(x_{2n-1}))+\tilde{w}(x_{2n-1},f(x_{2n}))] \\ =A\tilde{w}(x_{2n-1},x_{2n})+B[\tilde{w}(x_{2n},x_{2n+1})+\tilde{w}(x_{2n-1},x_{2n})]+C\tilde{w}(x_{2n-1},x_{2n+1}) \\ \le A\tilde{w}(x_{2n-1},x_{2n})+B[\tilde{w}(x_{2n},x_{2n+1})+\tilde{w}(x_{2n-1},x_{2n})]+C[\tilde{w}(x_{2n-1},x_{2n})+\tilde{w}(x_{2n},x_{2n+1})]. \end{gathered}$$

Then, we have $\tilde{w}(x_{2n},x_{2n+1})\le {(I-(B+C))}^{-1}(A+B+C)\tilde{w}(x_{2n-1},x_{2n})=M\tilde{w}(x_{2n-1},x_{2n}).$

By the same argument as above, we get

$$\begin{gathered} \tilde{w}(x_{2n+1},x_{2n+2})=\tilde{w}(f(x_{2n}),g(x_{2n+1}))\le A\tilde{d}(x_{2n},f(x_{2n+1}) \\ +B[\tilde{w}(x_{2n},f(x_{2n}))+\tilde{w}(x_{2n+1},g(x_{2n+1}))]+C[\tilde{w}(x_{2n},g(x_{2n+1}))+\tilde{w}(x_{2n+1},f(x_{2n}))] \\ =A\tilde{w}(x_{2n},x_{2n+1})+B[\tilde{w}(x_{2n},x_{2n+1})+\tilde{w}(x_{2n+1},x_{2n+2})]+C\tilde{w}(x_{2n},x_{2n+2}) \\ \le A\tilde{w}(x_{2n},x_{2n+1})+B[\tilde{w}(x_{2n},x_{2n+1})+\tilde{w}(x_{2n+1},x_{2n+2})]+C[\tilde{w}(x_{2n},x_{2n+1})+\tilde{w}(x_{2n+1},x_{2n+2})]. \end{gathered}$$

Then, we have $\tilde{w}(x_{2n+1},x_{2n+2})\le {(I-(B+C))}^{-1}(A+B+C)\tilde{w}(x_{2n},x_{2n+1})=M\tilde{w}(x_{2n},x_{2n+1}).$

Further, we obtain $\tilde{w}(x_n,x_{n+1})\le M^n\tilde{w}(x_0,x_1)$ for each $n\in \mathbb{N}$.

Following the same steps as in the proof of Theorem 2 we estimate $\tilde{w}(x_n,x_m)$, for every $m,n\in \mathbb{N}$ with m > n.

$$\begin{gathered} \tilde{w}(x_n,x_m)\le \tilde{w}(x_n,x_{n+1})+\tilde{w}(x_{n+1},x_{n+2})+\dots +\tilde{w}(x_{m-1},x_m) \\ \le M^n(\tilde{w}(x_0,x_1))+M^{n+1}(\tilde{w}(x_0,x_1))+\dots +M^{m-1}(\tilde{w}(x_0,x_1)) \\ \le M^n(I+M+M^2+\dots +M^{m-n-1})(\tilde{w}(x_0,x_1))\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)). \end{gathered}$$

Note that $(I-M)$ is nonsingular since M is convergent to Θ. Using Lemma 1 (3) the sequence ${(x_n)}_{n\in \mathbb{N}}$ is a Cauchy sequence.

Using the lower semicontinuity of the generalized w-distance, by relation (8) we have $\tilde{w}(x_n,x^{*})\stackrel{d}{\to }{0}_{1\times m}$ as $n\to \infty$. Then, we have $\tilde{w}(x_{2n},x^{*})\stackrel{d}{\to }{0}_{1\times m}$ as $n\to \infty$. By the continuity of f it follows $x_{2n+1}=f(x_{2n})\stackrel{d}{\to }f(x^{*})$ as $n\to \infty$. By the uniqueness of the limit we get $x^{*}=f(x^{*})$.

By $\tilde{w}(x_n,x^{*})\stackrel{d}{\to }{0}_{1\times m}$ as $n\to \infty$ we have that $\tilde{w}(x_{2n+1},x^{*})\stackrel{d}{\to }{0}_{1\times m}$ as $n\to \infty$. By the continuity of g it follows $x_{2n+2}=g(x_{2n+1})\stackrel{d}{\to }g(x^{*})$ as $n\to \infty$. By the uniqueness of the limit we get $x^{*}=g(x^{*})$.

Then, $x^{*}$ is a common fixed point for f and g. □

By replacing the continuity condition for the mappings f and g, we can state the following result.

Theorem 7.

Let $(X,\tilde{d})$ be a complete generalized metric space in Perov’s sense, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized w-distance, and $f,g:X\to X$ be two single-valued Hardy–Rogers type operators. There exist matrices $A,B,C\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ such that

(i) 

$I-(B+C)$ is nonsingular and ${(I-(B+C))}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$;

(ii) 

$I-(A+2B+2C)$ is nonsingular and ${[I-(A+2B+2C)]}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$;

(iii) 

$M={(I-(B+C))}^{-1}(A+B+C)$ converges to Θ.

Then f and g have a common fixed point $x^{*}\in X$.

Proof. 

(1) As in the proof of the previous theorem, Theorem 6, for $x_0\in X$ we consider ${(x_n)}_{n\in \mathbb{N}}$ the sequence of successive approximations for f and g, defined by

$$x_{2n+1}=f(x_{2n}),n=0,1,\dots$$

$$x_{2n+2}=g(x_{2n+1}),n=0,1,\dots$$

We define the sequence ${(x_n)}_{n\mathbb{N}}\in X$ such that

$$\tilde{w}(x_{2n+1},x_{2n+2})\le {(I-(B+C))}^{-1}(A+B+C)\tilde{w}(x_{2n},x_{2n+1})=M\tilde{w}(x_{2n},x_{2n+1}).$$

Further, we obtain $\tilde{w}(x_n,x_{n+1})\le M^n\tilde{d}(x_0,x_1)$ for each $n\in \mathbb{N}$.

Following the same steps as in the proof of Theorem 6 we estimate $\tilde{w}(x_n,x_m)$, for every $m,n\in \mathbb{N}$ with $m>n$ and we get $\tilde{w}(x_n,x_m)\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)).$

Note that $(I-M)$ is nonsingular since M is convergent to Θ. By Lemma 1 (3), the sequence ${(x_n)}_{n\in \mathbb{N}}$ is a Cauchy sequence. Using the lower semicontinuity of the generalized w-distance, by relation (8), we have $\tilde{w}(x_n,x^{*})\stackrel{d}{\to }{0}_{1\times m},$ as $n\to \infty$. By (11) we have $\tilde{w}(x_n,f(x^{*}))\stackrel{d}{\to }{0}_{1\times m},$ as $n\to \infty$. Then, using Lemma 1 (2), we get $x^{*}=f(x^{*})$.

Let us show that $g(x^{*})=x^{*}$. Then, by the definition of Hardy–Rogers type operators we have

$$\begin{gathered} \tilde{w}(x^{*},g(x^{*}))=\tilde{d}(f(x^{*}),g(x^{*})) \\ \le A\tilde{w}(x^{*},x^{*})+B[\tilde{w}(x^{*},f(x^{*}))+\tilde{w}(x^{*},g(x^{*})]+C[\tilde{w}(x^{*},g(x^{*}))+\tilde{w}(x^{*},f(x^{*}))]. \end{gathered}$$

Then, we get

$$\tilde{w}(x^{*},g(x^{*}))\le {(I-(B+C))}^{-1}(A+B+C)\tilde{w}(x^{*},x^{*}).$$

(12)

By (6) we get $\tilde{w}(x^{*},g(x^{*}))=0_{1\times m}.$

Let $g(x^{*})\in X$. By triangle inequality and using (12) we obtain

$$\tilde{w}(x_n,g(x^{*}))=\tilde{w}(x_n,x^{*})+\tilde{w}(x^{*},g(x^{*}))\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)+0_{1\times m}\stackrel{d}{\to }{0}_{1\times m}.$$

(13)

Using (8) and (13), by Lemma 1 (2), we obtain $x^{*}=g(x^{*})$. Then $x^{*}$ is a common fixed point for f and g. □

Remark 3.

In the case of common fixed points, the generalized $\tilde{w}$-distance must not necessarily be a generalized $\widetilde{w_0}$-distance.

3. Ulam–Hyers Stability, Well-Posedness, and Data Dependence of Fixed Point Problem

We begin this section with the extension of Ulam–Hyers stability for fixed point equation for the case of single-valued operators on generalized metric space in Perov’s sense. Then, let us recall the definition of weakly Ulam–Hyers stability.

Definition 9.

Let $(X,\tilde{d})$ be a metric space, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized w-distance, and $f:X\to X$ be an operator. By definition, the fixed point equation

$$x=f(x)$$

(14)

is weakly Ulam–Hyers stable if there exists a real positive matrix $N\in {\mathcal{M}}_{m,m}(\mathbb{R}+)$ such that, for each $\epsilon >0$ and each solution $y^{*}$ of the inequation

$$\tilde{w}(y,f(y))\le \epsilon I_{1\times m}$$

(15)

there exists a solution $x^{*}$ of the Equation (14) such that

$$\tilde{d}(y^{*},x^{*})\le N\epsilon I_{1\times m}.$$

Theorem 8.

Let $(X,\tilde{d})$ be a generalized metric space in Perov’s sense, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized $w_0$-distance and $f:X\to X$ be a single-valued Hardy–Rogers type operator defined in (8). There exist matrices $A,B,C\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ such that

(i) 

$N=M^n{(I-M)}^{-1}$ is nonsingular and $N=M^n{(I-M)}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$, where $M={(I(B+C))}^{-1}(A+B+C)$ converges to Θ;

(ii) 

$I-(A+2B+2C)$ is nonsingular and ${[I-(A+2B+2C)]}^{-1}\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$;

(iii) 

$I-P^2$ is nonsingular and $I-P^2\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ where $P={[I-(A+C)]}^{-1}C\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$.

Then, the fixed point Equation (14) is weakly Ulam–Hyers stable.

Proof. 

Let $\delta I_{1\times m}>0_{1\times m}$ such that $\tilde{w}(x_0,x_1)\le \delta I_{1\times m}$, for every $x_0,x_1\in X$ with $x_1=f(x_0)$. Let $Fix(f)=\left\{x^{*}\right\}$ and $u^{*}\in X$ be a solution of Equation (14). Then, $\tilde{w}(u^{*},f(u^{*}))\le \epsilon I_{1\times m}$. By the definition of the weakly Hardy–Rogers type operator we obtain

$$\begin{gathered} \tilde{w}(x^{*},u^{*})\le \tilde{w}(f(x^{*}),f(u^{*}))\le A\tilde{w}(x^{*},u^{*})+B[\tilde{w}(x^{*},f(x^{*}))+\tilde{w}(u^{*},f(u^{*}))]+C[\tilde{w}(x^{*},f(u^{*}) \\ +\tilde{w}(u^{*},f(x^{*}))]=A\tilde{w}(x^{*},u^{*})+B[\tilde{w}(x^{*},x^{*})+\tilde{w}(u^{*},u^{*})]+C[\tilde{w}(x^{*},u^{*})+\tilde{w}(u^{*},x^{*})] \\ =(A+C)\tilde{w}(x^{*},u^{*})+B[\tilde{w}(x^{*},x^{*})+\tilde{w}(u^{*},u^{*})]+C\tilde{w}(u^{*},x^{*}). \end{gathered}$$

(16)

By (6) we get

$$\tilde{w}(x^{*},x^{*})=\tilde{w}(f(x^{*}),f(x^{*}))\le (A+2B+2C)\tilde{w}(x^{*},x^{*})\mathrm{and}$$

(17)

$$\tilde{w}(u^{*},u^{*})=\tilde{w}(f(u^{*}),f(u^{*}))\le (A+2B+2C)\tilde{w}(u^{*},u^{*}).$$

Using hypothesis $(ii)$ we get $\tilde{w}(x^{*},x^{*})=\tilde{w}(u^{*},u^{*})=0_{1\times m}$.

By (16) we obtain

$$\tilde{w}(x^{*},u^{*})\le {[I-(A+C)]}^{-1}C\tilde{w}(u^{*},x^{*}).$$

(18)

By the definition of the weakly Hardy–Rogers type operator we get

$$\tilde{w}(u^{*},x^{*})\le {[I-(A+C)]}^{-1}C\tilde{w}(x^{*},u^{*})$$

and using (18) we obtain

$$\tilde{w}(x^{*},u^{*})\le {({[I-(A+C)]}^{-1}C)}^2\tilde{w}(x^{*},u^{*})=P^2\tilde{w}(x^{*},u^{*}).$$

(19)

Then, $(I-P^2)\tilde{w}(x^{*},u^{*})\le 0_{1\times m}$. By hypothesis $(iii)$ we get $\tilde{w}(x^{*},u^{*})=0_{1\times m}$.

Let $x_n\in X$ such that, by Equations (8) and (19) we have

$$\tilde{w}(x_n,x^{*})\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)\le N\delta I_{1\times m}\mathrm{and}$$

(20)

$$\tilde{w}(x_n,u^{*})\le \tilde{w}(x_n,x^{*})+\tilde{w}(x^{*},u^{*})\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)+0_{1\times m}\le N\delta I_{1\times m}.$$

Then, using the definition of generalized w-distance, there exists $\epsilon I_{1\times m}>0_{1\times m}$ such that

$$\tilde{d}(x^{*},u^{*})\le \epsilon I_{1\times m}\le N\epsilon I_{1\times m}.$$

Then, the fixed point Equation (14) is weakly Ulam–Hyers stable. □

The following result assures the well-posedness of the fixed point problem with respect to the generalized $w_0$-distance $\tilde{w}$.

Theorem 9.

Let $(X,\tilde{d})$ be a generalized metric space in Perov’s sense, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized $w_0$-distance, and $f:X\to X$ be a single-valued Hardy–Rogers type operator defined in Equation (8). If all the hypothesis of Theorem 2 (respectively, 3 and 4) are satisfied, the fixed point Equation (14) is well-posed with respect to the generalized $w_0$-distance $\tilde{w}$, i.e., if $Fix(f)=\left\{x^{*}\right\}$ and $x_n\in \mathbb{N}$, with $n\in \mathbb{N}$, such that $\tilde{w}(x_n,f(x_n))\to 0_{1\times m}$ as $n\to \infty$, then $x_n\to x^{*}$ as $n\to \infty$.

Proof. 

Let $x^{*}\in Fix(f)$ and let ${(x)}_{n\in \mathbb{N}}\in X$ such that $\tilde{w}(x_n,f(x_n))\stackrel{d}{\to }{0}_{1\times m}$ as $n\to \infty$. That means $\tilde{w}(x_{n-1},x_n)\stackrel{d}{\to }{0}_{1\times m}$ as $n\to \infty$.

By the lower semicontinuity of the generalized w-distance, using (8) we have

$${\displaystyle \tilde{w}(x_{n-1},x^{*})\le \underset{m\to \infty }{lim\; inf}\tilde{w}(x_n,x_m)\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)\stackrel{d}{\to }{0}_{1\times m}.}$$

Then, using Lemma 1 (3) we get $x_n\stackrel{d}{\to }{x}^{*}$ as $n\to \infty$. □

The next theorem presents a data dependence result.

Theorem 10.

Let $(X,\tilde{d})$ be a generalized metric space in Perov’s sense, $\tilde{w}:X\times X\to {\mathbb{R}}_{+}^m$ be a generalized $w_0$-distance, and $f_1,f_2:X\to X$ be single-valued operators, which satisfy the following conditions,

(i) 

for $A,B,C,M\in {\mathcal{M}}_{m,m}({\mathbb{R}}_{+})$ with $M={[I-(B+C)]}^{-1}(A+B+C)$ a matrix convergent to Θ such that, for every $x,y\in X$ and $i\in \{1,2\}$, we have:$\tilde{w}(f_i(x),f_i(y))\le A\tilde{w}(x,y)+B[\tilde{w}(x,f_i(x))+\tilde{w}(y,f_i(y))]+C[\tilde{w}(x,f_i(y))+\tilde{w}(y,f_i(x))];$

(ii) 

there exists $\eta >0$ such that $\tilde{w}(f_1(x),f_2(x))\le \eta I$, for all $x\in X$.

Then, for $x_1^{*}=f_1(x_1^{*})$ there exists $x_2^{*}=f_2(x_2^{*})$ such that $\tilde{d}(x_1^{*},x_2^{*})\le {(I-M)}^{-1}\eta I_{1\times m}$; (respectively, for $x_2^{*}=f_2(x_2^{*})$ there exists $x_1^{*}=f_1(x_1^{*})$ such that $\tilde{w}(x_2^{*},x_1^{*})\le {(I-M)}^{-1}\eta I_{1\times m}$).

Proof. 

As in the proof of Theorem 2 (respectively, Theorem 3) we construct the sequence of successive approximations ${(x_n)}_{n\in \mathbb{N}}\in X$ of $f_2$ with $x_0:=x_1^{*}$ and $x_1=f_2(x_1^{*})$ having the property $\tilde{w}(x_n,x_{n+1})\le M^n\tilde{w}(x_0,x_1)$, where $M={[I-(B+C)]}^{-1}(A+B+C)$.

If we consider the sequence ${(x_n)}_{n\in \mathbb{N}}\in X$ converges to $x_2^{*}$, we have $x_2^{*}=f(x_2^{*})$. Moreover, for each $n,p\in \mathbb{N}$ we have $\tilde{w}(x_n,x_{n+p})\le M^n{(I-M)}^{-1}\tilde{w}(x_0,x_1)$.

Letting $p\to 0$ we get $\tilde{w}(x_n,x_2^{*})\le I{(I-M)}^{-1}\tilde{w}(x_0,x_1)$.

Choosing $n=0$ we get $\tilde{w}(x_0,x_2^{*})\le I{(I-M)}^{-1}\tilde{w}(x_0,x_1)$ and using above the notations we get our conclusion $\tilde{w}(x_1^{*},x_2^{*})\le {(I-M)}^{-1}\eta I_{1\times m}$. □

4. Conclusions

The purpose of this paper is to establish some fixed point results in generalized metric spaces in Perov’s sense. The generalized metric considered here is the w-distance, for which the symmetry condition is not satisfied. The operators satisfy a contractive weakly condition of Hardy–Rogers type. The second part of the paper is devoted to the study of the data dependence, as well as the well-posedness and the Ulam–Hyers stability of the fixed point problem. In order to prove our main results we had to impose a symmetry condition for the w-distance. The results presented in this paper generalize some recent ones.