#
On Iteration S_{n} for Operators with Condition (D)

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**A Banach space X is called uniformly convex if, for each $\epsilon \in (0,2]$, there exists $\delta >0$ such that, for $x,y\in X$ with $\left|\right|x\left|\right|\le 1$, $\left|\right|y\left|\right|\le 1$ and $\left|\right|x-y\left|\right|\ge \epsilon $, the following inequality holds true:

**Definition**

**2**

**.**A Banach space X is said to satisfy the Opial property provided that each weakly convergent sequence $\left\{{x}_{n}\right\}$ in X with weak limit x satisfies the following inequality:

**Proposition**

**1**

**.**Let C be a nonempty subset of a Banach space X and $T:C\to C$. If T satisfies condition ($\mathrm{D}$), then:

- (i)
- $\u2225{T}^{2}x-Tx\u2225\le \u2225Tx-x\u2225$, for all$x\in C$;
- (ii)
- For all$x,y\in C$, at least one of the inequalities$\u2225{T}^{2}x-Ty\u2225\le \u2225Tx-y\u2225$or$\u2225{T}^{2}y-Tx\u2225\le \u2225Ty-x\u2225$is satisfied;
- (iii)
- $\left|\right|x-Ty\left|\right|\le 3\left|\right|Tx-x\left|\right|+\left|\right|x-y\left|\right|$whenever$\u2225Tx-x\u2225\le \u2225Ty-y\u2225$;
- (iv)
- $\u2225Tx-Ty\u2225\le 2\mathrm{min}\left\{\u2225Tx-x\u2225,\u2225Ty-y\u2225\right\}+\u2225x-y\u2225$, for all$x,y\in C$.

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3**

**.**Suppose that X is a uniformly convex Banach space and $0<p\le {t}_{n}\phantom{\rule{-0.166667em}{0ex}}\le q\phantom{\rule{-0.166667em}{0ex}}<1$, for all $n\ge 1$. Let $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ be two sequences in X such that the inequalities $\underset{n\to \infty}{\mathrm{lim}\phantom{\rule{4pt}{0ex}}\mathrm{sup}}\u2225{x}_{n}\u2225\le r$, $\underset{n\to \infty}{\mathrm{lim}\phantom{\rule{4pt}{0ex}}\mathrm{sup}}\u2225{y}_{n}\u2225\le r$ and $\underset{n\to \infty}{\mathrm{lim}\phantom{\rule{4pt}{0ex}}\mathrm{sup}}\u2225{t}_{n}{x}_{n}+\left(1-{t}_{n}\right){y}_{n}\u2225=r$ hold for some $r\ge 0$. Then, $\underset{n\to \infty}{\mathrm{lim}}\u2225{x}_{n}-{y}_{n}\u2225=0$.

**Remark**

**1.**

- (i)
- The asymptotic radius of$\left\{{x}_{n}\right\}$relative to C is defined as$$r\left(C,\left\{{x}_{n}\right\}\right)=\mathrm{inf}\left\{r\left(x,\left\{{x}_{n}\right\}\right)\mid x\in C\right\}.$$
- (ii)
- The asymptotic center of$\left\{{x}_{n}\right\}$relative to C is given by$$A\left(C,\left\{{x}_{n}\right\}\right)=\left\{x\in C\mid r\left(x,\left\{{x}_{n}\right\}\right)=r\left(C,\left\{{x}_{n}\right\}\right)\right\}.$$
- (iii)
- Edelstein [26] showed that, for a nonempty closed convex subset C of a uniformly convex Banach space and for each bounded sequence$\left\{{x}_{n}\right\}$, the set$A\left(C,\left\{{x}_{n}\right\}\right)$is a singleton.

## 3. Examples

**Example**

**1.**

**Proof.**

- (1)
- $C(T,x)=\left\{Tp:d(p,Tp)\le d(x,Tx)\right\}\subseteq \mathrm{Im}\mathrm{T}$;
- (2)
- $F\left(T\right)\subseteq C(T,x)$.

**Case I**: Let $x=0$ and $y\in C(T,x)$. Through direct computations, we find

**Case II**: Let $x\in \left({\displaystyle 0,\frac{1}{2}}\right)$ and $y\in C(T,x)$. Again, we compute

**Case III**: For the conditions $x\in \left[{\displaystyle \frac{1}{2},1}\right]$ and $y\in C(T,x)=\left[{\displaystyle \frac{1}{2},1}\right]$, the conclusion comes out immediately.

**Example**

**2.**

**Proof.**

**Case I.**Let us assume that

**Case II.**Suppose

## 4. Fixed Point and Convergence Results

**Lemma**

**4.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. The Stability of the ${S}_{n}$ Iteration Procedure

**Definition**

**3**

**.**Let C be a nonempty subset of E, where $\left(E,\u2225\xb7\u2225\right)$ is a normed space, and let T be a self-mapping on C with a fixed point. Let $\left\{{t}_{n}\right\}$ be an arbitrary sequence in C and

**Lemma**

**5**

**.**Let $\left\{{\psi}_{n}\right\}$ and $\left\{{\phi}_{n}\right\}$ be nonnegative real number sequences satisfying the following inequality:

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 6. Data Dependence Analysis

**Definition**

**4**

**.**Let $T,\tilde{T}:X\to X$ be two mappings. We say that $\tilde{T}$ is an approximate mapping of T if for some $\epsilon >0$, provided that

**Definition**

**5**

**.**Let X be a Banach space and T be a self-mapping with a fixed point p. Let $\tilde{T}$ be an approximate mapping of T with maximum admissible error $\epsilon >0$, admitting a fixed point $\tilde{p}$. Assume that f defines some iteration procedure such that, for ${x}_{1}\in X$, the sequences ${x}_{n+1}=f(T,{x}_{n})$ and ${\tilde{x}}_{n+1}=f(\tilde{T},{\tilde{x}}_{n})$ converge to p and $\tilde{p}$, respectively. We call the iteration procedure for f data independent if

**Lemma**

**6**

**.**Let $\left\{{\psi}_{n}\right\}$ and $\left\{{\phi}_{n}\right\}$ be nonnegative real number sequences for which one assumes there exists ${n}_{0}\in \mathbb{N}$ such that for all $n\ge {n}_{0}$, the following inequality holds:

**Theorem**

**6.**

**Proof.**

**Remark**

**2.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Ciobanescu, C.; Turcanu, T.
On Iteration *S*_{n} for Operators with Condition (D). *Symmetry* **2020**, *12*, 1676.
https://doi.org/10.3390/sym12101676

**AMA Style**

Ciobanescu C, Turcanu T.
On Iteration *S*_{n} for Operators with Condition (D). *Symmetry*. 2020; 12(10):1676.
https://doi.org/10.3390/sym12101676

**Chicago/Turabian Style**

Ciobanescu, Cristian, and Teodor Turcanu.
2020. "On Iteration *S*_{n} for Operators with Condition (D)" *Symmetry* 12, no. 10: 1676.
https://doi.org/10.3390/sym12101676