# On Some New Multivalued Results in the Metric Spaces of Perov’s Type

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## Abstract

**:**

## 1. Introduction and Preliminaries

- (1)
- $\tilde{d}(x,y)\u2ab00$ and $\tilde{d}(x,y)=0$ if and only if $x=y$,
- (2)
- $\tilde{d}(x,y)=\tilde{d}(y,x),$
- (3)
- $\tilde{d}(x,y)\u2aaf\tilde{d}(x,z)+\tilde{d}(z,y),$

**Theorem**

**1.**

- (i)
- A matrix A converges to Θ as $n\to \infty $;
- (ii)
- If $\lambda \in \mathbb{C}$ such that $det(A-\lambda I)=0$ then $\left|\lambda \right|<1$;
- (iii)
- The matrix $I-A$ is regular and ${(I-A)}^{-1}=I+A+{A}^{2}+\cdots .$

**Definition**

**1.**

- (i)
- ${y}_{0}=y$, ${y}_{1}=z$;
- (ii)
- ${y}_{n+1}\in T({y}_{n}),$ for each $n\in \mathbb{N}$;
- (iii)
- the sequence ${({y}_{n})}_{n\in \mathbb{N}}$ is convergent and its limit is a fixed point of T.

**Remark**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Remark**

**2.**

## 2. Main Results

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Definition**

**4.**

**Theorem**

**2.**

- (i)
- $I-q(B+C)$ is nonsingular and ${(I-q(B+C))}^{-1}\in {\mathbb{M}}_{m,m}({\mathbb{R}}_{+})$, for $q\in (1,Q)$;
- (ii)
- $M={(I-q(B+C))}^{-1}q(A+B+C)$ converges to Θ.

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

- (1)
- $Fix(T)\ne \varnothing $.
- (2)
- There exists a sequence ${({x}_{n})}_{n\in \mathbb{N}}\in X$ such that ${x}_{n+1}\in T({x}_{n})$, for all $n\in \mathbb{N}$ and converge to a fixed point of T.
- (3)
- One has the estimation $\tilde{d}({x}_{n},{x}^{*})\u2aaf{(I-q(B+C))}^{-1}{[q(A+B+C)]}^{n}\tilde{d}({x}_{0},{x}_{1})$, where ${x}^{*}\in Fix(T).$

**Proof.**

**Example**

**1.**

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

**Theorem**

**5.**

- (i)
- $I-q(B+C)$ is nonsingular and ${(I-q(B+C))}^{-1}\in {\mathbb{M}}_{m,m}({\mathbb{R}}_{+})$, for $q\in (1,Q)$;
- (ii)
- $I-q(A+2C)$ is nonsingular and ${[I-q(A+2C)]}^{-1}\in {\mathbb{M}}_{m,m}({\mathbb{R}}_{+})$;
- (iii)
- $M={(I-q(B+C))}^{-1}q(A+B+C)$ converges to Θ.

- (1)
- T and G have a common fixed point ${x}^{*}\in X$;
- (2)
- ${x}^{*}$ is a unique common fixed point of T and G.

**Proof.**

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

**Definition**

**5.**

**Definition**

**6.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

- (i)
- for $A,B,C,M\in {\mathbb{M}}_{m,m}({\mathbb{R}}_{+})$ with $M={[I-q(B+C)]}^{-1}q(A+B+C)$ a matrix convergent to Θ such that, for every $x,y\in X$ with $i\in \{1,2\}$ and $q\in (1,Q)$, we have:$\tilde{H}({T}_{i}(x),{T}_{i}(y))\u2aafqA\tilde{d}(x,y)+qB[\tilde{D}(x,{T}_{i}(x))+\tilde{D}(y,{T}_{i}(y))]+qC[\tilde{d}(x,{T}_{i}(y))+\tilde{d}(y,{T}_{i}(x))];$
- (ii)
- there exists $\eta >0$ such that $\tilde{H}({T}_{1}(x),{T}_{2}(x))\u2aaf{(I-M)}^{-1}\eta {I}_{m\times 1}$, for all $x\in X$.

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Guran, L.; Bota, M.-F.; Naseem, A.; Mitrović, Z.D.; Sen, M.d.l.; Radenović, S.
On Some New Multivalued Results in the Metric Spaces of Perov’s Type. *Mathematics* **2020**, *8*, 438.
https://doi.org/10.3390/math8030438

**AMA Style**

Guran L, Bota M-F, Naseem A, Mitrović ZD, Sen Mdl, Radenović S.
On Some New Multivalued Results in the Metric Spaces of Perov’s Type. *Mathematics*. 2020; 8(3):438.
https://doi.org/10.3390/math8030438

**Chicago/Turabian Style**

Guran, Liliana, Monica-Felicia Bota, Asim Naseem, Zoran D. Mitrović, Manuel de la Sen, and Stojan Radenović.
2020. "On Some New Multivalued Results in the Metric Spaces of Perov’s Type" *Mathematics* 8, no. 3: 438.
https://doi.org/10.3390/math8030438