Strong convergence for total quasi-ϕ-asymptotically nonexpansive semigroups in Banach spaces

is called the normalized duality mapping. Let $T:C\to C$ be a nonlinear mapping; $F(T)$ denotes the set of fixed points of mapping T.

Alber et al.[1] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied the methods of approximation of fixed points. They are defined as follows.

Definition 1.1 Let $T:C\to C$ be a mapping. T is said to be total asymptotically nonexpansive if there exist sequences $\{{\mu }_n\}$, $\{{\nu }_n\}$ with ${\mu }_n,{\nu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\psi :\mathbb{R}\to \mathbb{R}$ with $\psi (0)=0$ such that $\parallel T^{n}x-T^{n}y\parallel \le \parallel x-y\parallel +{\mu }_n\psi (\parallel x-y\parallel )+{\nu }_n$ holds for all $x,y\in C$ and all $n\in \mathbb{N}$.

T is said to be total asymptotically quasi-nonexpansive if $F(T)\ne \mathrm{\varnothing }$, there exist sequences $\{{\mu }_n\}$, $\{{\nu }_n\}$ with ${\mu }_n,{\nu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\psi :\mathbb{R}\to \mathbb{R}$ with $\psi (0)=0$ such that $\parallel T^{n}x-p\parallel \le \parallel x-p\parallel +{\mu }_n\psi (\parallel x-p\parallel )+{\nu }_n$ holds for all $x\in C$, $p\in F(T)$ and all $n\in \mathbb{N}$.

Chidume and Ofoedu [2] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings and total asymptotically quasi-nonexpansive mappings in Banach spaces. Chidume et al.[3] gave a new iterative sequence and necessary and sufficient conditions for this sequence to converge to a common fixed point of finite total asymptotically nonexpansive mappings. Chang [4] established some new approximation theorems of common fixed points for a countable family of total asymptotically nonexpansive mappings in Banach spaces.

Recently, many researchers have focused on studying the convergence of iterative algorithms for quasi-ϕ-asymptotically nonexpansive (see [5–9]) and total quasi-ϕ-asymptotically nonexpansive (see [10–12]) mappings. Ye et al.[13] used a new hybrid projection algorithm to obtain strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. Kim [14] used hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings to prove the strong convergence theorems. Saewan [15] used the shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings.

A Banach space E is said to be strictly convex if $\frac{\parallel x+y\parallel }{2}<1$ for $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$; it is also said to be uniformly convex if ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$ for any two sequences $\{x_n\}$, $\{y_n\}$ in E such that $\parallel x_n\parallel =\parallel y_n\parallel =1$ and ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_n+y_n\parallel }{2}=1$. Let $U=\{x\in E:\parallel x\parallel =1\}$ be the unit sphere of E, then the Banach space E is said to be smooth provided ${lim}_{t\to 0}\frac{\parallel x+ty\parallel -\parallel y\parallel }{t}$ exists for each $x,y\in U$. It is also said to be uniformly smooth if the limit is attained uniformly for each $x,y\in U$. It is well known that if E is reflexive and smooth, then the duality mapping J is single valued. A Banach space E is said to have the Kadec-Klee property if a sequence $\{x_n\}$ of E satisfies that $x_n\rightharpoonup x\in E$ and $\parallel x_n\parallel \to x$, then $x_n\to x$. It is known that if E is uniformly convex, then E has the Kadec-Klee property.

The quasi-ϕ-asymptotically nonexpansive and total quasi-ϕ-asymptotically nonexpansive mappings are defined as follows.

holds for all $x\in C$, $p\in F(T)$ and all $n\in \mathbb{N}$.

holds for all $x\in C$, $p\in F(T)$ and all $n\in \mathbb{N}$.

In recent years, many researchers have considered the convergence of asymptotically nonexpansive semigroups [17, 18]. The asymptotically nonexpansive semigroups are defined as follows.

Definition 1.3[17]

holds for all $x,y\in C$, $n\in \mathbb{N}$.

We use $F(T)$ to denote the common fixed point set of the semigroup T, i.e., $F(T)={\bigcap }_{t\ge 0}F(T(t))$.

Chang [19] used the modified Halpern-Mann type iteration algorithm for quasi-ϕ-asymptotically nonexpansive semigroups to prove the strong convergence in the Banach space. The quasi-ϕ-asymptotically nonexpansive semigroups are defined as follows.

Definition 1.4[19]

holds for all $n\in \mathbb{N}$.

This section contains some definitions and lemmas which will be used in the proofs of our main results in the following section.

holds for all $x\in C$, $p\in F(T)$ and all $n\in \mathbb{N}$.

The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-nonexpansive asymptotically semigroups to prove the strong convergence in Banach spaces. The results presented in the article improve and extend the corresponding results of [5, 6, 9–12, 14, 15, 19] and many others.

In order to prove the results of this paper, we shall need the following lemmas:

Lemma 2.1 (See [16])

Lemma 2.2[19]

Let E be a uniformly convex and smooth Banach space and let$\{x_n\}$and$\{y_n\}$be two sequences of E. If$\varphi (x_n,y_n)\to 0$and either$\{x_n\}$or$\{y_n\}$is bounded, then$\parallel x_n-y_n\parallel \to 0$.

Lemma 2.3[10]

Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and C be a nonempty closed convex subset of E. Let$T:C\to C$be a closed and total quasi-ϕ-asymptotically nonexpansive mapping defined by Definition  1.2. If${\nu }_1=0$, then the fixed point set$F(T)$of T is a closed and convex subset of C.

where${\xi }_n={\mu }_n{M}^{\ast }{sup}_{p\in F(T)}\varphi (p,x_n)$. If${\nu }_1=0$and$F(T)$is bounded in C, then the iterative sequence$\{x_n\}$converges strongly to a common fixed point$x^{\ast }\in F(T)$in C.

Proof (I) We prove $F(T)$ and $C_n(n\in \mathbb{N})$ all are closed and convex subsets in C.

Since J is uniformly continuous, this shows that ${lim}_{n\to \mathrm{\infty }}{T}^n(t)x_n=p^{\ast }=0$ uniformly for $t\ge 0$. Again by the assumptions that the semigroup $\mathbf{T}:=\{T(t):t\ge 0\}$ is closed and uniformly L-Lipschitzian, we have

(7)

In view of the definition of ${\mathrm{\Pi }}_{F(T)}{x}_1$, from (8), we have $p^{\ast }=\omega$. Therefore, $x_n\to p^{\ast }={\mathrm{\Pi }}_{F(T)}{x}_1$. This completes the proof of Theorem 3.1. □