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

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## Abstract

**:**

## 1. Introduction and Preliminaries

- (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 $x,y,z\in X$.

**Theorem**

**1.**

- (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.

**Definition**

**1.**

- (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 $d(x,y)\le \epsilon .$

**Definition**

**2.**

**Remark**

**1.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Remark**

**2.**

## 2. Fixed Point Results

**Definition**

**6.**

- (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 $.

**Definition**

**7.**

**Lemma**

**1.**

- (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.

**Definition**

**8.**

**Theorem**

**2.**

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

**Proof.**

**Theorem**

**3.**

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

**Proof.**

**Theorem**

**4.**

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

**Proof.**

**Theorem**

**5.**

- (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.**

**Theorem**

**6.**

- (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.

**Proof.**

**Theorem**

**7.**

- (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.

**Proof.**

**Remark**

**3.**

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

**Definition**

**9.**

**Theorem**

**8.**

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

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

- (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 $x\in X$.

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Guran, L.; Bota, M.-F.; Naseem, A.
Fixed Point Problems on Generalized Metric Spaces in Perov’s Sense. *Symmetry* **2020**, *12*, 856.
https://doi.org/10.3390/sym12050856

**AMA Style**

Guran L, Bota M-F, Naseem A.
Fixed Point Problems on Generalized Metric Spaces in Perov’s Sense. *Symmetry*. 2020; 12(5):856.
https://doi.org/10.3390/sym12050856

**Chicago/Turabian Style**

Guran, Liliana, Monica-Felicia Bota, and Asim Naseem.
2020. "Fixed Point Problems on Generalized Metric Spaces in Perov’s Sense" *Symmetry* 12, no. 5: 856.
https://doi.org/10.3390/sym12050856