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

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## Abstract

**:**

## 1. Introduction and Preliminary Notions and Results

- (1)
- The distance from a point $x\in X$ to a set $Y\in P\left(X\right)$:$$D(x,Y):=\mathrm{inf}\left\{d\right(x,y\left)\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}y\in Y\};$$
- (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 $Y,Z\in P\left(X\right)$:$$H(Y,Z)=\mathrm{max}\left\{e\right(Y,Z),e(Z,Y\left)\right\}.$$

**Remark**

**1.**

**Definition**

**1.**

**Theorem**

**1.**

**Definition**

**2.**

**Theorem**

**2.**

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

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

**Definition**

**3.**

**Theorem**

**3.**

- (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.**

- (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)
- $D({x}_{n},T\left({x}_{n}\right))\le {k}^{n}D({x}_{0},T\left({x}_{0}\right)),n\in \mathbb{N}$.

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

**Theorem**

**4.**

- (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$${\left({I}_{m}-K\right)}^{-1}={I}_{m}+K+\dots +{K}^{n}+\dots ;$$
- (iv)
- The matrix $\left({I}_{m}-K\right)$ is nonsingular and ${\left({I}_{m}-K\right)}^{-1}$ has nonnegative elements.

**Theorem**

**5**

**.**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

**Theorem**

**6.**

- (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:$$d({x}_{n},{x}^{*})\u2aaf{K}^{n}{(I-K)}^{-1}d({x}_{0},{x}_{1}),foreachn\in {\mathbb{N}}^{*}$$and$$d(x,{x}^{*})\u2aaf{(I-K)}^{-1}d(x,y).$$

**Definition**

**4.**

**Theorem**

**7.**

- (1)
- ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ converges to ${x}^{*}\left({x}_{0}\right)\in Fix\left(F\right)$;
- (2)
- $d({x}_{n},{x}^{*}\left({x}_{0}\right))\u2aaf{\left({B}^{-1}A\right)}^{n}{\left({I}_{m}-{B}^{-1}A\right)}^{-1}d({x}_{0},{x}_{1}),n\in \mathbb{N}$;
- (3)
- $d({x}_{0},{x}^{*}\left({x}_{0}\right))\u2aaf{\left({I}_{m}-{B}^{-1}A\right)}^{-1}d({x}_{0},{x}_{1})\u2aaf{\left({I}_{m}-{B}^{-1}A\right)}^{-1}{B}^{-1}D({x}_{0},F\left({x}_{0}\right))$.

**Proof.**

- (a)
- ${x}_{n+1}\in {I}_{B}^{{x}_{n}}$, for each $n\in \mathbb{N}$;
- (b)
- $d({x}_{n},{x}_{n+1})\u2aafKd({x}_{n-1},{x}_{n})\u2aaf\cdots \u2aaf{K}^{n}d({x}_{0},{x}_{1})$, for each $n\in {\mathbb{N}}^{*}$;
- (c)
- $D({x}_{n+1},F\left({x}_{n+1}\right))\u2aafKD({x}_{n},F\left({x}_{n}\right))\u2aaf\cdots \u2aaf{K}^{n+1}D({x}_{0},F\left({x}_{0}\right))$, for each $n\in \mathbb{N}$.

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

- (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)\u2aaf\eta $, for all $x\in X$.

**Definition**

**8.**

**Definition**

**9.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Remark**

**3.**

## 4. An Application to a System of Operatorial Inclusions

**Theorem**

**11.**

**Proof.**

- (I)
- $\tilde{d}({z}_{n},{z}^{*}\left({z}_{0}\right))\u2aaf{\left({B}^{-1}A\right)}^{n}{\left({I}_{m}-{B}^{-1}A\right)}^{-1}\tilde{d}({z}_{0},{z}_{1}),n\in \mathbb{N}$;
- (II)
- $\tilde{d}({z}_{0},{z}^{*}\left({z}_{0}\right))\u2aaf{\left({I}_{m}-{B}^{-1}A\right)}^{-1}\tilde{d}({z}_{0},{z}_{1})\u2aaf{\left({I}_{m}-{B}^{-1}A\right)}^{-1}{B}^{-1}\tilde{D}({z}_{0},G\left({z}_{0}\right))$.

**Remark**

**4.**

**Theorem**

**12.**

**Example**

**.**

## 5. Conclusions

- (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.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Petruşel, A.; Petruşel, G.; Yao, J.-C.
New Contributions to Fixed Point Theory for Multi-Valued Feng–Liu Contractions. *Axioms* **2023**, *12*, 274.
https://doi.org/10.3390/axioms12030274

**AMA Style**

Petruşel A, Petruşel G, Yao J-C.
New Contributions to Fixed Point Theory for Multi-Valued Feng–Liu Contractions. *Axioms*. 2023; 12(3):274.
https://doi.org/10.3390/axioms12030274

**Chicago/Turabian Style**

Petruşel, Adrian, Gabriela Petruşel, and Jen-Chih Yao.
2023. "New Contributions to Fixed Point Theory for Multi-Valued Feng–Liu Contractions" *Axioms* 12, no. 3: 274.
https://doi.org/10.3390/axioms12030274