# New Construction of Strongly Relatively Nonexpansive Sequences by Firmly Nonexpansive-Like Mappings

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## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1**

**.**The metric projection ${P}_{C}$ from X onto C and the generalized projection ${Q}_{C}$ from X onto C are defined by

**Definition**

**2**

**.**A mapping $T:C\u27f6X$ is said to be a firmly nonexpansive-like mapping, if

**Definition**

**3**

**.**Let $T:C\u27f6X$ be a mapping. A point $p\in C$ is said to be an asymptotic fixed point of $T,$ if there exists a sequence $\left\{{x}_{n}\right\}$ of C such that ${x}_{n}\rightharpoonup p$ and ${x}_{n}-T{x}_{n}\u27f60$. The set of all asymptotic fixed points of T is denoted by $\widehat{F}\left(T\right)$.

**Definition**

**4**

**.**The mapping T is said to be of type $\left(r\right),$ if $F\left(T\right)$ is nonempty and $\phi (u,Tx)\u2a7d\phi (u,x)$ for all $u\in F\left(T\right)$ and $x\in C$.

**Definition**

**5**

**.**The mapping T is said to be of type $\left(sr\right),$ if T is of type $\left(r\right)$ and $\phi (T{z}_{n},{z}_{n})\u27f60$, whenever $\left\{{z}_{n}\right\}$ is a bounded sequence of C such that $\phi (u,{z}_{n})-\phi (u,T{z}_{n})\u27f60$ for some $u\in F\left(T\right)$.

**Definition**

**6**

**.**The sequence $\left\{{T}_{n}\right\}$ is said to satisfy the condition $\left(Z\right),$ if every weak subsequential limit of $\left\{{x}_{n}\right\}$ belongs to $F\left(\left\{{T}_{n}\right\}\right)$, whenever $\left\{{x}_{n}\right\}$ is a bounded sequence of C such that ${x}_{n}-{T}_{n}{x}_{n}\u27f60$.

**Theorem**

**1**

**.**The space X is 2-uniformly convex if and only if there exists $\mu \u2a7e0$ such that

**Lemma**

**1**

**.**Suppose that X is 2-uniformly convex. Then

**Lemma**

**2**

**.**If $T:C\u27f6X$ is a firmly nonexpansive-like mapping, then $F\left(T\right)$ is a closed convex subset of X and $\widehat{F}\left(T\right)=F\left(T\right)$.

**Lemma**

**3**

**.**Suppose that X is uniformly convex. If $S:X\u27f6X$ and $T:C\u27f6X$ are mappings of type $\left(r\right)$ such that $F\left(S\right)\cap F\left(T\right)$ is nonempty and S or T is of type $\left(sr\right)$, then $ST:C\u27f6X$ is of type $\left(r\right)$ and $F\left(ST\right)=F\left(S\right)\cap F\left(T\right)$. Further, if both S and T are of type $\left(sr\right)$, then so is $ST$.

**Lemma**

**4**

**.**Suppose that X is uniformly convex. Let $\left\{{S}_{n}\right\}$ be a sequence of mappings of X into itself and $\left\{{T}_{n}\right\}$ a sequence of mappings of C into X such that $F\left(\left\{{S}_{n}\right\}\right)\cap F\left(\left\{{T}_{n}\right\}\right)$ is nonempty, both $\left\{{S}_{n}\right\}$ and $\left\{{T}_{n}\right\}$ are of type $\left(sr\right),$ and ${S}_{n}$ or ${T}_{n}$ is of type $\left(sr\right)$ for all $n\in \mathbb{N}$. Then the following holds:

- (i)
- $\left\{{S}_{n}{T}_{n}\right\}$ is of type $\left(sr\right)$;
- (ii)
- if X is uniformly smooth and both $\left\{{S}_{n}\right\}$ and $\left\{{T}_{n}\right\}$ satisfy the condition $\left(Z\right)$, then so does $\left\{{S}_{n}{T}_{n}\right\}$.

**Theorem**

**2**

**.**Let X be a smooth and uniformly convex Banach space, C a nonempty closed convex subset of X, and $\left\{{T}_{n}\right\}$ a sequence of mappings of C into X such that $\left\{{T}_{n}\right\}$ is of type $\left(sr\right)$ and $\left\{{T}_{n}\right\}$ satisfies the condition $\left(Z\right)$. If ${T}_{n}\left(C\right)\subset C$ for all $n\in \mathbb{N}$ and J is weakly sequentially continuous, then the sequence $\left\{{x}_{n}\right\}$ defined by ${x}_{1}\in C$ and ${x}_{n+1}={T}_{n}{x}_{n}$ for all $n\in \mathbb{N}$ converges weakly to the strong limit of $\left\{{Q}_{F}{x}_{n}\right\}$.

## 2. Main Results

**Lemma**

**5.**

**Proof.**

**Theorem**

**3.**

- (i)
- $\left\{{T}_{n}\right\}$, $\left\{{S}_{n}\right\}$ are sequences of firmly nonexpansive-like mappings from C into X such that $F=F\left(\left\{{T}_{n}\right\}\right)\cap F\left(\left\{{S}_{n}\right\}\right)$ is nonempty;
- (ii)
- $\left\{{U}_{n}\right\}$ is a sequence of mappings from C into X defined by$${U}_{n}={J}^{-1}\left(J{T}_{n}-{\beta}_{n}J(I-{S}_{n})\right)$$

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

- (a)
- T is of type $\left(sr\right)$ if and only if $\{T,T,\cdots \}$ is of type $\left(sr\right)$;
- (b)
- $\widehat{F}\left(T\right)=F\left(T\right)$ if and only if $\{T,T,\cdots \}$ satisfies the condition $\left(Z\right)$.

**Corollary**

**1.**

- (i)
- $F\left(U\right)\subset F\left(T\right)\cap F\left(S\right)$ and U is of type $\left(sr\right)$;
- (ii)
- if X is uniformly smooth, then $\widehat{F}\left(U\right)=F\left(U\right)$.

**Theorem**

**4.**

- (i)
- $F\left(\left\{{V}_{n}\right\}\right)\subset F$ and $\left\{{V}_{n}\right\}$ is of type $\left(sr\right)$;
- (ii)
- if X is uniformly smooth and $\left\{{S}_{n}\right\}$ satisfies the condition $\left(Z\right)$, then so does $\left\{{V}_{n}\right\}$.

**Proof.**

**Theorem**

**5.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Aoyama, K.; Kohsaka, F.; Takahashi, W. Three generalizations of firmly nonexpansive mappings: Their relations and continuity properties. J. Nonlinear Convex Anal.
**2009**, 10, 131–147. [Google Scholar] - Aoyama, K.; Kohsaka, F.; Takahashi, W. Strongly relatively nonexpansive sequences in Banach spaces and applications. J. Fixed Point Theory Appl.
**2009**, 5, 201–225. [Google Scholar] [CrossRef] - Aoyama, K.; Kohsaka, F.; Takahashi, W. Strong convergence theorems for a family of mappings of type (P) and applications. Nonlinear Anal. Optim.
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## Share and Cite

**MDPI and ACS Style**

Işık, H.; Haddadi, M.R.; Parvaneh, V.; Park, C.; Kornokar, S. New Construction of Strongly Relatively Nonexpansive Sequences by Firmly Nonexpansive-Like Mappings. *Mathematics* **2020**, *8*, 284.
https://doi.org/10.3390/math8020284

**AMA Style**

Işık H, Haddadi MR, Parvaneh V, Park C, Kornokar S. New Construction of Strongly Relatively Nonexpansive Sequences by Firmly Nonexpansive-Like Mappings. *Mathematics*. 2020; 8(2):284.
https://doi.org/10.3390/math8020284

**Chicago/Turabian Style**

Işık, Hüseyin, Mohammad Reza Haddadi, Vahid Parvaneh, Choonkil Park, and Somayeh Kornokar. 2020. "New Construction of Strongly Relatively Nonexpansive Sequences by Firmly Nonexpansive-Like Mappings" *Mathematics* 8, no. 2: 284.
https://doi.org/10.3390/math8020284