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# The modified Mann type iterative algorithm for a countable family of totally quasi-*ϕ*-asymptotically nonexpansive mappings by the hybrid generalized *f*-projection method

- Siwaporn Saewan
^{1}Email author, - Preedaporn Kanjanasamranwong
^{1}Email author, - Poom Kumam
^{2}Email author and - Yeol Je Cho
^{3}Email author

**2013**:63

https://doi.org/10.1186/1687-1812-2013-63

© Saewan et al.; licensee Springer 2013

**Received:**27 July 2012**Accepted:**21 February 2013**Published:**18 March 2013

## Abstract

The purpose of this article is to introduce the modified Mann type iterative sequence, using a new technique, by the hybrid generalized *f*-projection operator for a countable family of totally quasi-*ϕ*-asymptotically nonexpansive mappings in a uniform smooth and strictly convex Banach space with the Kadec-Klee property. Then we prove that the modified Mann type iterative scheme converges strongly to a common element of the sets of fixed points of the given mappings. Our result extends and improves the results of Li *et al.* (Comput. Math. Appl. 60:1322-1331, 2010), Takahashi *et al.* (J. Math. Anal. Appl. 341:276-286, 2008) and many other authors.

**MSC:**47H05, 47H09, 47H10.

## Keywords

- generalized
*f*-projection operator - modified Mann type iterative sequence
- totally quasi-
*ϕ*-asymptotically nonexpansive mapping

## 1 Introduction

*E*be a real Banach space and

*C*be a nonempty closed and convex subset of

*E*. A mapping $T:C\to C$ is said to be

*totally asymptotically nonexpansive*[1] if there exist nonnegative real sequences ${\nu}_{n}$, ${\mu}_{n}$ with ${\nu}_{n}\to 0$, ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$ and a strictly increasing continuous function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\phi (0)=0$ such that

A point $x\in C$ is a *fixed point* of *T* provided $Tx=x$. Denote by $F(T)$ the fixed point set of *T*, that is, $F(T)=\{x\in C:Tx=x\}$. A point $p\in C$ is called an *asymptotic fixed point* of *T* [2] if *C* contains a sequence $\{{x}_{n}\}$ which converges weakly to *p* such that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}-T{x}_{n}\parallel =0$. The asymptotic fixed point set of *T* is denoted by $\stackrel{\u02c6}{F}(T)$.

*E*. We recall that for all $x\in E$ and $x\in {E}^{\ast}$, we denote the value of ${x}^{\ast}$ at

*x*by $\u3008x,{x}^{\ast}\u3009$. Then the

*normalized duality mapping*$J:E\to {2}^{{E}^{\ast}}$ is defined by

*E*is a Hilbert space, then $J=I$, where

*I*is the identity mapping. Next, consider the functional $\varphi :E\times E\to {\mathbb{R}}^{+}\cup \{0\}$ defined by

where *J* is the normalized duality mapping and $\u3008\cdot ,\cdot \u3009$ denotes the duality pairing of *E* and ${E}^{\ast}$.

*E*is a Hilbert space, then $\varphi (x,y)={\parallel x-y\parallel}^{2}$. It is obvious from the definition of

*ϕ*that

*T*is said to be

*relatively nonexpansive*[3, 4] if $\stackrel{\u02c6}{F}(T)=F(T)$ and

*T*is said to be

*relatively asymptotically nonexpansive*[5, 6] if $\stackrel{\u02c6}{F}(T)=F(T)\ne \mathrm{\varnothing}$ and there exists a sequence $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ with ${k}_{n}\to 1$ as $n\to \mathrm{\infty}$ such that

*T*is said to be

*ϕ*-

*nonexpansive*[7, 8] if

*T*is said to be

*quasi-ϕ-nonexpansive*[7, 8] if $F(T)\ne \mathrm{\varnothing}$ and

*T*is said to be

*asymptotically*

*ϕ-nonexpansive*[8] if there exists a sequence $\{{k}_{n}\}\subset [0,\mathrm{\infty})$ with ${k}_{n}\to 1$ as $n\to \mathrm{\infty}$ such that

*T*is said to be

*quasi-ϕ-asymptotically nonexpansive*[8] if $F(T)\ne \mathrm{\varnothing}$ and there exists a sequence $\{{k}_{n}\}\subset [1,\mathrm{\infty})$ with ${k}_{n}\to 1$ as $n\to \mathrm{\infty}$ such that

*T*is said to be

*totally quasi-ϕ-asymptotically nonexpansive*if $F(T)\ne \mathrm{\varnothing}$ and there exist nonnegative real sequences ${\nu}_{n}$, ${\mu}_{n}$ with ${\nu}_{n}\to 0$, ${\mu}_{n}\to 0$ as $n\to \mathrm{\infty}$ and a strictly increasing continuous function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\phi (0)=0$ such that

**Remark 1.1** (1) Every relatively nonexpansive mapping implies a relatively quasi-nonexpansive mapping, a quasi-*ϕ*-nonexpansive mapping implies a quasi-*ϕ*-asymptotically nonexpansive mapping and a quasi-*ϕ*-asymptotically nonexpansive mapping implies a totally quasi-*ϕ*-asymptotically nonexpansive mapping, but the converses are not true.

(2) A relatively quasi-nonexpansive mapping is sometimes called hemi-relatively nonexpansive mapping. The class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings (see [4, 9–13]), which requires the strong restriction $F(T)=\stackrel{\u02c6}{F}(T)$.

(3) For other examples of relatively quasi-nonexpansive mappings such as the generalized projections and others, see [[7], Examples 2.3 and 2.4].

*generalized projection*${\mathrm{\Pi}}_{C}:E\to C$ is a mapping that assigns to an arbitrary point $x\in E$ the minimum point of the functional $\varphi (x,y)$, that is, ${\mathrm{\Pi}}_{C}x=\overline{x}$, where $\overline{x}$ is a solution of the minimization problem

In 2006, Wu and Huang [15] introduced a new generalized *f*-projection operator in Banach spaces. They extended the definition of generalized projection operators introduced by Abler [16] and proved the properties of the generalized *f*-projection operator.

*f*-projection operator. Let $G:C\times {E}^{\ast}\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ be a functional defined by

*ρ*is a positive number and $f:C\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ is proper, convex and lower semicontinuous. From the definition of

*G*, Wu and Huang [15] studied the following properties:

- (1)
$G(y,\varpi )$ is convex and continuous with respect to

*ϖ*when*y*is fixed; - (2)
$G(y,\varpi )$ is convex and lower semicontinuous with respect to

*y*when*ϖ*is fixed.

**Definition 1.2**Let

*E*be a real Banach space with the dual space ${E}^{\ast}$ and

*C*be a nonempty closed and convex subset of

*E*. We say that ${\pi}_{C}^{f}:{E}^{\ast}\to {2}^{C}$ is a

*generalized*

*f-projection operator*if

where the initial guess element ${x}_{1}\in C$ is arbitrary and $\{{\alpha}_{n}\}$ is a sequence in $[0,1]$. Mann’s iteration has been extensively investigated for nonexpansive mappings and some mappings. In an infinite-dimensional Hilbert space, Mann’s iteration can conclude *only weak convergence* (see [18, 19]). Bauschke and Combettes [20] introduced a modified Mann iteration method (1.5) in a Hilbert space and proved, under appropriate conditions, some strong convergence.

*et al.*[21] studied the strong convergence theorem by the new hybrid method $\{{x}_{n}\}$ for a family of nonexpansive mappings in Hilbert spaces: ${x}_{0}\in H$, ${C}_{1}=C$, ${x}_{1}={P}_{{C}_{1}}{x}_{0}$ and

where $0\le {\alpha}_{n}\le a<1$ for all $n\ge 1$ and $\{{T}_{n}\}$ is a sequence of nonexpansive mappings of *C* into itself such that ${\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})\ne \mathrm{\varnothing}$. They proved that if $\{{T}_{n}\}$ satisfies some appropriate conditions, then $\{{x}_{n}\}$ converges strongly to ${P}_{{\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})}{x}_{0}$.

*T*in a Banach space

*E*:

They proved that $\{{x}_{n}\}$ converges strongly to a point ${\mathrm{\Pi}}_{F(T)}{x}_{0}$. Many authors studied methods for approximating fixed points of a countable family of (relatively quasi-) nonexpansive mappings (see [22–26]).

*et al.*[27] proved a new strong convergence result of the regularized successive approximation method for a total asymptotically nonexpansive mapping in a Hilbert spaces. In 2010, Li

*et al.*[28] introduced the following hybrid iterative scheme $\{{x}_{n}\}$ for approximation fixed points of a relatively nonexpansive mapping using the generalized

*f*-projection operator in a uniformly smooth real Banach space which is also uniformly convex: ${x}_{0}\in C$ and

They proved strong convergence theorems for finding an element in the fixed point set of *T*.

One question is raised naturally as follows:

*Are the results of Alber et al.* [27]*, Li et al.* [28]*and Takahashi et al.* [21]*true in the framework of strictly convex Banach spaces for totally quasi-ϕ-asymptotically nonexpansive mappings?*

Motivated and inspired by the works mentioned above, in this article we aim to introduce a new hybrid projection algorithm of the generalized *f*-projection operator for a countable family of totally quasi-*ϕ*-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Our result extends and improves the results of Li *et al.* [28], Takahashi *et al.* [21] and many other authors.

## 2 Preliminaries

*E*with the norm $\parallel \cdot \parallel $ is called

*strictly convex*if $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. Let $U=\{x\in E:\parallel x\parallel =1\}$ be a unit sphere of

*E*. A Banach space

*E*is called

*smooth*if the limit

*uniformly smooth*if the limit exists uniformly for all $x,y\in U$. The

*modulus of smoothness*of

*E*is the function ${\rho}_{E}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by

*modulus of convexity*of

*E*(see [29]) is the function ${\delta}_{E}:[0,2]\to [0,1]$ defined by

In this paper, we denote the strong convergence and weak convergence of a sequence $\{{x}_{n}\}$ by ${x}_{n}\to x$ and ${x}_{n}\rightharpoonup x$, respectively.

**Remark 2.1**The basic properties of

*E*, ${E}^{\ast}$,

*J*and ${J}^{-1}$ are as follows (see [30]):

- (1)
If

*E*is an arbitrary Banach space, then*J*is monotone and bounded; - (2)
If

*E*is strictly convex, then*J*is strictly monotone; - (3)
If

*E*is smooth, then*J*is single-valued and semi-continuous; - (4)
If

*E*is uniformly smooth, then*J*is uniformly norm-to-norm continuous on each bounded subset of*E*; - (5)
If

*E*is reflexive smooth and strictly convex, then the normalized duality mapping*J*is single-valued, one-to-one and onto; - (6)
If

*E*is a reflexive strictly convex and smooth Banach space and*J*is the duality mapping from*E*into ${E}^{\ast}$, then ${J}^{-1}$ is also single-valued, bijective and is also the duality mapping from ${E}^{\ast}$ into*E*and thus $J{J}^{-1}={I}_{{E}^{\ast}}$ and ${J}^{-1}J={I}_{E}$; - (7)
If

*E*is uniformly smooth, then*E*is smooth and reflexive; - (8)
*E*is uniformly smooth if and only if ${E}^{\ast}$ is uniformly convex; - (9)
If

*E*is a reflexive and strictly convex Banach space, then ${J}^{-1}$ is norm-weak^{∗}-continuous.

**Remark 2.2** If *E* is a reflexive, strictly convex and smooth Banach space, then $\varphi (x,y)=0$ if and only if $x=y$. It is sufficient to show that if $\varphi (x,y)=0$ then $x=y$. From (1.1) we have $\parallel x\parallel =\parallel y\parallel $. This implies that $\u3008x,Jy\u3009={\parallel x\parallel}^{2}={\parallel Jy\parallel}^{2}$. From the definition of *J*, one has $Jx=Jy$. Therefore, we have $x=y$ (see [30–32] for more details).

Recall that a Banach space *E* has the *Kadec-Klee property* [30, 31, 33] if, for any sequence $\{{x}_{n}\}\subset E$ and $x\in E$ with ${x}_{n}\rightharpoonup x$ and $\parallel {x}_{n}\parallel \to \parallel x\parallel $, $\parallel {x}_{n}-x\parallel \to 0$ as $n\to \mathrm{\infty}$. It is well known that if *E* is a uniformly convex Banach space, then *E* has the Kadec-Klee property.

*generalized projection*[14] from

*E*into

*C*is defined by

The existence and uniqueness of the operator ${\mathrm{\Pi}}_{C}$ follows from the properties of the functional $\varphi (y,x)$ and the strict monotonicity of the mapping *J* (see, for example, [14, 30, 31, 34, 35]). If *E* is a Hilbert space, then $\varphi (x,y)={\parallel x-y\parallel}^{2}$ and ${\mathrm{\Pi}}_{C}$ becomes the metric projection ${P}_{C}:H\to C$. If *C* is a nonempty closed and convex subset of a Hilbert space *H*, then ${P}_{C}$ is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces.

We also need the following lemmas for the proof of our main results.

*T*be a nonlinear mapping,

*T*is said to be

*uniformly asymptotically regular*on

*C*if

A mapping *T* from *C* into itself is said to be *closed* if, for any sequence $\{{x}_{n}\}\subset C$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}={x}_{0}$ and ${lim}_{n\to \mathrm{\infty}}T{x}_{n}={y}_{0}$, we have $T{x}_{0}={y}_{0}$.

**Lemma 2.3** (Chang *et al.* [36])

*Let* *C* *be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space* *E* *with the Kadec*-*Klee property*. *Let* $T:C\to C$ *be a closed and total quasi*-*ϕ*-*asymptotically nonexpansive mapping with the sequences* ${\nu}_{n}$ *and* ${\mu}_{n}$ *of nonnegative real numbers with* ${\nu}_{n}\to 0$, ${\mu}_{n}\to 0$ *as* $n\to \mathrm{\infty}$ *and a strictly increasing continuous function* $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ *with* $\psi (0)=0$. *If* ${\mu}_{1}=0$, *then the fixed point set* $F(T)$ *is a closed convex subset of* *C*.

**Lemma 2.4** (Wu and Hung [15])

*Let*

*E*

*be a real reflexive Banach space with the dual space*${E}^{\ast}$

*and*

*C*

*be a nonempty closed and convex subset of*

*E*.

*The following statements hold*:

- (1)
${\pi}_{C}^{f}\varpi $

*is a nonempty*,*closed and convex subset of**C**for all*$\varpi \in {E}^{\ast}$; - (2)
*If**E**is smooth*,*then for all*$\varpi \in {E}^{\ast}$, $x\in {\pi}_{C}^{f}\varpi $*if and only if*$\u3008x-y,\varpi -Jx\u3009+\rho f(y)-\rho f(x)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C;$ - (3)
*If**E**is strictly convex and*$f:C\to \mathbb{R}\cup \{+\mathrm{\infty}\}$*is positive homogeneous*(*i*.*e*., $f(tx)=tf(x)$*for all*$t>0$*such that*$tx\in C$,*where*$x\in C$),*then*${\pi}_{C}^{f}\varpi $*is a single*-*valued mapping*.

In the following lemma, Fan *et al.*[37] showed that Lemma 2.1(iii) in [37] can be removed.

**Lemma 2.5** (Fan *et al.* [37])

*Let* *E* *be a real reflexive Banach space with its dual space* ${E}^{\ast}$ *and* *C* *be a nonempty closed and convex subset of* *E*. *If* *E* *is strictly convex*, *then* ${\pi}_{C}^{f}\varpi $ *is single*-*valued*.

*J*is a single-valued mapping when

*E*is a smooth Banach space. There exists a unique element $\varpi \in {E}^{\ast}$ such that $\varpi =Jx$, where $x\in E$. This substitution in (1.4) gives the following:

Now, we consider the second generalized *f*-projection operator in Banach spaces (see [28]).

**Definition 2.6**Let

*E*be a real smooth Banach space and

*C*be a nonempty, closed and convex subset of

*E*. We say that ${\mathrm{\Pi}}_{C}^{f}:E\to {2}^{C}$ is the

*generalized*

*f-projection operator*if

**Lemma 2.7** (Deimling [38])

*Let*

*E*

*be a Banach space and*$f:E\to \mathbb{R}\cup \{+\mathrm{\infty}\}$

*be a lower semicontinuous convex function*.

*Then there exist*${x}^{\ast}\in {E}^{\ast}$

*and*$\alpha \in \mathbb{R}$

*such that*

**Lemma 2.8** (Li *et al.* [28])

*Let*

*E*

*be a reflexive smooth Banach space and*

*C*

*be a nonempty*,

*closed and convex subset of*

*E*.

*The following statements hold*:

- (1)
${\mathrm{\Pi}}_{C}^{f}x$

*is nonempty*,*closed and convex subset of**C**for all*$x\in E$; - (2)
*For all*$x\in E$, $\stackrel{\u02c6}{x}\in {\mathrm{\Pi}}_{C}^{f}x$*if and only if*$\u3008\stackrel{\u02c6}{x}-y,Jx-J\stackrel{\u02c6}{x}\u3009+\rho f(y)-\rho f(\stackrel{\u02c6}{x})\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C;$ - (3)
*If**E**is strictly convex*,*then*${\mathrm{\Pi}}_{C}^{f}$*is a single*-*valued mapping*.

**Lemma 2.9** (Li *et al.* [28])

*Let*

*E*

*be a real reflexive smooth Banach space and*

*C*

*be a nonempty closed and convex subset of*

*E*.

*If*$\stackrel{\u02c6}{x}\in {\mathrm{\Pi}}_{C}^{f}x$

*for all*$x\in E$,

*then*

**Remark 2.10** Let *E* be a uniformly convex and uniformly smooth Banach space and $f(x)=0$ for all $x\in E$. Then Lemma 2.9 reduces to the property of the generalized projection operator considered by Alber [14].

If $f(y)\ge 0$ for all $y\in C$ and $f(0)=0$, then the definition of totally quasi-*ϕ*-asymptotically nonexpansive *T* is equivalent to the following:

## 3 Main results

**Theorem 3.1**

*Let*

*C*

*be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space*

*E*

*with the Kadec*-

*Klee property*.

*Let*${\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}}$

*be a countable family of closed and uniformly totally quasi*-

*ϕ*-

*asymptotically nonexpansive mappings with the sequences*${\nu}_{n}$, ${\mu}_{n}$

*of nonnegative real numbers with*${\nu}_{n}\to 0$, ${\mu}_{n}\to 0$

*as*$n\to \mathrm{\infty}$

*and a strictly increasing continuous function*$\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$

*with*$\psi (0)=0$.

*Let*$f:E\to \mathbb{R}$

*be a convex and lower semicontinuous function with*$C\subset int(D(f))$

*such that*$f(x)\ge 0$

*for all*$x\in C$

*and*$f(0)=0$.

*Assume that*${T}_{i}$

*is uniformly asymptotically regular for all*$i\ge 1$

*and*$\mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing}$.

*For an initial point*${x}_{1}\in E$,

*let*${C}_{1,i}=C$

*for each*$i\ge 1$

*and*${C}_{1}={\bigcap}_{i=1}^{\mathrm{\infty}}{C}_{1,i}$

*and define the sequence*$\{{x}_{n}\}$

*by*

*where* $\{{\alpha}_{n}\}$ *is a sequence in* $(0,1)$ *and* ${\beta}_{n}={\nu}_{n}{sup}_{q\in \mathcal{F}}\psi G(q,J{x}_{n})+{\mu}_{n}$. *If* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1-{\alpha}_{n})>0$, *then* $\{{x}_{n}\}$ *converges strongly to a point* ${\mathrm{\Pi}}_{\mathcal{F}}^{f}{x}_{1}$.

*Proof* We split the proof into four steps.

Therefore, ${C}_{n+1,i}$ is closed and convex. Hence ${C}_{n+1}={\bigcap}_{i=1}^{\mathrm{\infty}}{C}_{n+1,i}$ is closed and convex for all $n\ge 1$.

*ϕ*asymptotically nonexpansive mapping, for each $i\ge 1$, we have

This shows that $q\in {C}_{n+1}$, which implies that $\mathcal{F}\subset {C}_{n+1}$. Hence $\mathcal{F}\subset {C}_{n}$ for all $n\ge 1$.

*E*is reflexive and ${C}_{n}$ is closed and convex for all $n\ge 1$, we can assume that ${x}_{n}\rightharpoonup p\in {C}_{n}$. From the fact that ${x}_{n}={\mathrm{\Pi}}_{{C}_{n}}^{f}{x}_{1}$ and $p\in {C}_{n}$, we get

*f*is convex and lower semicontinuous, we have

*E*, it follows that

*J*is uniformly norm-to-norm continuous, it follows that

*J*is uniformly norm-to-norm continuous, it follows that for each $i\ge 1$,

*E*is reflexive and ${E}^{\ast}$ is also reflexive, we can assume that $J{y}_{n,i}\rightharpoonup {y}^{\ast}\in {E}^{\ast}$ for all $i\ge 1$. Since

*E*is reflexive, we see that $J(E)={E}^{\ast}$. Hence there exists $y\in E$ such that $Jy={y}^{\ast}$. It follows that for each $i\ge 1$,

^{∗}-continuous, that is, ${y}_{n,i}\rightharpoonup p$, it follows from (3.13) and the Kadec-Klee property of

*E*that

*J*is uniformly norm-to-norm continuous, we obtain

That is, ${T}_{i}^{n+1}{x}_{n}={T}_{i}{T}_{i}^{n}{x}_{n}\to p$ as $n\to \mathrm{\infty}$. From ${T}_{i}^{n}{x}_{n}\to p$ as $n\to \mathrm{\infty}$ and the closedness of ${T}_{i}$, we have ${T}_{i}p=p$ for all $i\ge 1$. We see that $p\in F({T}_{i})$ for all $i\ge 1$, which implies that $p\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})$.

*v*. By the definition ${x}_{n}={\mathrm{\Pi}}_{{C}_{n}}^{f}{x}_{1}$ and $v\in \mathcal{F}\subset {C}_{n}$, we also have

*G*and

*f*, we know that, for any $x\in E$, $G(\xi ,Jx)$ is convex and lower semicontinuous with respect to

*ξ*and so

From the definition of ${\mathrm{\Pi}}_{\mathcal{F}}^{f}{x}_{1}$, since $p\in \mathcal{F}$, we conclude that $v=p={\mathrm{\Pi}}_{\mathcal{F}}^{f}{x}_{1}$ and ${x}_{n}\to p$ as $n\to \mathrm{\infty}$. This completes the proof. □

Setting ${\nu}_{n}\equiv 0$ and ${\mu}_{n}\equiv 0$ in Theorem 3.1, we have the following.

**Corollary 3.2**

*Let*

*C*

*be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space*

*E*

*with the Kadec*-

*Klee property*.

*Let*${\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}}$

*be a countable family of closed and uniformly quasi*-

*ϕ*-

*asymptotically nonexpansive mappings*.

*Let*$f:E\to \mathbb{R}$

*be a convex and lower semicontinuous function with*$C\subset int(D(f))$

*such that*$f(x)\ge 0$

*for all*$x\in C$

*and*$f(0)=0$.

*Assume that*${T}_{i}$

*is uniformly asymptotically regular for all*$i\ge 1$

*and*$\mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing}$.

*For an initial point*${x}_{1}\in E$,

*let*${C}_{1,i}=C$, ${C}_{1}={\bigcap}_{i=1}^{\mathrm{\infty}}{C}_{1,i}$

*and define the sequence*$\{{x}_{n}\}$

*by*

*where* $\{{\alpha}_{n}\}$ *is a sequence in* $(0,1)$. *If* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1-{\alpha}_{n})>0$, *then* $\{{x}_{n}\}$ *converges strongly to a point* ${\mathrm{\Pi}}_{\mathcal{F}}^{f}{x}_{1}$.

Setting $i=1$ and ${T}_{i}=T$ in Theorem 3.1, we have the following.

**Corollary 3.3**

*Let*

*C*

*be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space*

*E*

*with the Kadec*-

*Klee property*.

*Let*

*T*

*be a closed totally quasi*-

*ϕ*-

*asymptotically nonexpansive mapping with the sequences*${\nu}_{n}$, ${\mu}_{n}$

*of nonnegative real numbers with*${\nu}_{n}\to 0$, ${\mu}_{n}\to 0$

*as*$n\to \mathrm{\infty}$

*and a strictly increasing continuous function*$\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$

*with*$\psi (0)=0$.

*Let*$f:E\to \mathbb{R}$

*be a convex and lower semicontinuous function with*$C\subset int(D(f))$

*such that*$f(x)\ge 0$

*for all*$x\in C$

*and*$f(0)=0$.

*Assume that*

*T*

*is a uniformly asymptotically regular and*$\mathcal{F}=F(T)\ne \mathrm{\varnothing}$.

*For an initial point*${x}_{1}\in E$,

*let*${C}_{1}=C$

*and define the sequence*$\{{x}_{n}\}$

*by*

*where* $\{{\alpha}_{n}\}$ *is a sequence in* $(0,1)$ *and* ${\beta}_{n}={\nu}_{n}sup\psi G(p,{x}_{n})+{\mu}_{n}$. *If* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1-{\alpha}_{n})>0$, *then* $\{{x}_{n}\}$ *converges strongly to a point* ${\mathrm{\Pi}}_{\mathcal{F}}^{f}{x}_{1}$.

Taking $f(x)=0$ for all $x\in C$, we have $G(\xi ,Jx)=\varphi (\xi ,x)$ and ${\mathrm{\Pi}}_{C}^{f}x={\mathrm{\Pi}}_{C}x$. Thus, from Theorem 3.1, we obtain the following.

**Corollary 3.4**

*Let*

*C*

*be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space*

*E*

*with the Kadec*-

*Klee property*.

*Let*${\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}}$

*be a countable family of closed and uniformly totally quasi*-

*ϕ*-

*asymptotically nonexpansive mappings with the sequences*${\nu}_{n}$, ${\mu}_{n}$

*of nonnegative real numbers with*${\nu}_{n}\to 0$, ${\mu}_{n}\to 0$

*as*$n\to \mathrm{\infty}$

*and a strictly increasing continuous function*$\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$

*with*$\psi (0)=0$.

*Assume that*${T}_{i}$

*is uniformly asymptotically regular for all*$i\ge 1$

*and*$\mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing}$.

*For an initial point*${x}_{1}\in E$,

*let*${C}_{1,i}=C$, ${C}_{1}={\bigcap}_{i=1}^{\mathrm{\infty}}{C}_{1,i}$

*and define the sequence*$\{{x}_{n}\}$

*by*

*where* $\{{\alpha}_{n}\}$ *is a sequence in* $(0,1)$ *and* ${\beta}_{n}={\nu}_{n}sup\psi (\varphi (p,{x}_{n}))+{\mu}_{n}$. *If* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1-{\alpha}_{n})>0$, *then* $\{{x}_{n}\}$ *converges strongly to a point* ${\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$.

**Corollary 3.5**

*Let*

*C*

*be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space*

*E*

*with the Kadec*-

*Klee property*.

*Let*${\{{T}_{i}\}}_{i=1}^{\mathrm{\infty}}$

*be a countable family of closed and uniformly quasi*-

*ϕ*-

*asymptotically nonexpansive mappings*.

*Assume that*${T}_{i}$

*is uniformly asymptotically regular for all*$i\ge 1$

*and*$\mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\varnothing}$.

*For an initial point*${x}_{1}\in E$,

*let*${C}_{1,i}=C$, ${C}_{1}={\bigcap}_{i=1}^{\mathrm{\infty}}{C}_{1,i}$

*and define the sequence*$\{{x}_{n}\}$

*by*

*where* $\{{\alpha}_{n}\}$ *is a sequence in* $(0,1)$. *If* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1-{\alpha}_{n})>0$, *then* $\{{x}_{n}\}$ *converges strongly to a point* ${\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$.

**Corollary 3.6**

*Let*

*C*

*be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space*

*E*

*with the Kadec*-

*Klee property*.

*Let*

*T*

*be a closed totally quasi*-

*ϕ*-

*asymptotically nonexpansive mapping with the sequences*${\nu}_{n}$, ${\mu}_{n}$

*of nonnegative real numbers with*${\nu}_{n}\to 0$, ${\mu}_{n}\to 0$

*as*$n\to \mathrm{\infty}$

*and a strictly increasing continuous function*$\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$

*with*$\psi (0)=0$.

*Assume that*

*T*

*is uniformly asymptotically regular and*$\mathcal{F}=F(T)\ne \mathrm{\varnothing}$.

*For an initial point*${x}_{1}\in E$,

*let*${C}_{1}=C$

*and define the sequence*$\{{x}_{n}\}$

*by*

*where* $\{{\alpha}_{n}\}$ *is a sequence in* $(0,1)$ *and* ${\beta}_{n}={\nu}_{n}sup\psi (\varphi (p,{x}_{n}))+{\mu}_{n}$. *If* ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1-{\alpha}_{n})>0$, *then* $\{{x}_{n}\}$ *converges strongly to a point* ${\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}$.

**Remark 3.7** (1) Corollary 3.3 extends and improves the results of Li *et al.* [28] from a relatively nonexpansive mapping to a totally quasi-*ϕ*-asymptotically nonexpansive mapping.

(2) Corollary 3.6 extends and generalizes the result of Takahashi *et al.* [21] from a Hilbert space to a Banach space and from a nonexpansive mapping to a totally quasi-*ϕ* asymptotically nonexpansive mapping.

(3) In the case of spaces, we extend Banach spaces from a uniformly smooth and uniformly convex Banach to a uniformly smooth and strictly convex Banach with the Kadec-Klee property, which can be found in the literature works by many authors (see [12, 21, 22, 28]).

## Declarations

### Acknowledgements

This research was supported by Thaksin University. Moreover, the forth author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012-0008170).

## Authors’ Affiliations

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