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

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.

1 Introduction

Let 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

$$\parallel T^{n}x-T^{n}y\parallel \le \parallel x-y\parallel +{\mu }_n\psi (\parallel x-y\parallel )+{\nu }_n,\mathrm{\forall }x,y\in C,n\ge 1.$$

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 $\hat{F}(T)$.

Let $E^{\ast }$ be a dual space of the Banach space E. We recall that for all $x\in E$ and $x\in E^{\ast }$, we denote the value of $x^{\ast }$ at x by $〈x,x^{\ast }〉$. Then the normalized duality mapping $J:E\to 2^{E^{\ast }}$ is defined by

$$J(x)=\{x^{\ast }\in E^{\ast }:〈x,x^{\ast }〉={\parallel x\parallel }^2,\parallel x^{\ast }\parallel =\parallel x\parallel \},\mathrm{\forall }x\in E.$$

If 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

$$\varphi (x,y)={\parallel x\parallel }^2-2〈x,Jy〉+{\parallel y\parallel }^2,\mathrm{\forall }x,y\in E,$$

(1.1)

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

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

$${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^2,\mathrm{\forall }x,y\in E.$$

(1.2)

T is said to be relatively nonexpansive [3, 4] if $\hat{F}(T)=F(T)$ and

$$\varphi (p,Tx)⩽\varphi (p,x),\mathrm{\forall }x\in C,p\in F(T).$$

T is said to be relatively asymptotically nonexpansive [5, 6] if $\hat{F}(T)=F(T)\ne \mathrm{\varnothing }$ and there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ such that

$$\varphi (p,T^{n}x)⩽k_n\varphi (p,x),\mathrm{\forall }x\in C,p\in F(T),n\ge 1.$$

T is said to be ϕ-nonexpansive [7, 8] if

$$\varphi (Tx,Ty)\le \varphi (x,y),\mathrm{\forall }x,y\in C.$$

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

$$\varphi (p,Tx)\le \varphi (p,x),\mathrm{\forall }x\in C,p\in F(T).$$

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

$$\varphi (T^{n}x,T^{n}y)\le k_n\varphi (x,y),\mathrm{\forall }x,y\in C.$$

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

$$\varphi (p,T^{n}x)\le k_n\varphi (p,x),\mathrm{\forall }x\in C,p\in F(T),n\ge 1.$$

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

$$\varphi (p,T^{n}x)\le \varphi (p,x)+{\nu }_n\phi (\varphi (p,x))+{\mu }_n,\mathrm{\forall }x\in C,p\in F(T),n\ge 1.$$

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 $F(T)=\hat{F}(T)$.

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

On the other hand, Alber [14] introduced that the 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

$$\varphi (\overline{x},x)=\underset{y\in C}{inf}\varphi (y,x).$$

(1.3)

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.

Now, we recall the concept of the generalized f-projection operator. Let $G:C\times E^{\ast }\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ be a functional defined by

$$G(y,\varpi )={\parallel y\parallel }^2-2〈y,\varpi 〉+{\parallel \varpi \parallel }^2+2\rho f(y),$$

(1.4)

where $y\in C$, $\varpi \in E^{\ast }$, 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

$${\pi }_C^f\varpi =\{u\in C:G(u,\varpi )=\underset{y\in C}{inf}G(y,\varpi ),\mathrm{\forall }\varpi \in E^{\ast }\}.$$

In 1953, Mann [17] introduced the following iteration process, which is now well known as Mann’s iteration:

$$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,\mathrm{\forall }n\ge 1,$$

(1.5)

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.

Recently, Takahashi 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

$$\{\begin{array}{c}{y}_n={\alpha }_n{x}_n+(1-{\alpha }_n)T_n{x}_n,\hfill \\ C_{n+1}=\{z\in C:\parallel y_n-z\parallel \le \parallel x_n-z\parallel \},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(1.6)

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 $P_{{\bigcap }_{n=1}^{\mathrm{\infty }}F(T_n)}{x}_0$.

The ideas to generalize the process (1.5) from Hilbert spaces to Banach spaces have recently been made. Especially, Matsushita and Takahashi [11] proposed the following hybrid iteration method with the generalized projection for a relatively nonexpansive mapping T in a Banach space E:

$$\{\begin{array}{c}{x}_0\in C\text{chosen arbitrarily},\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)JT{x}_n),\hfill \\ C_n=\{z\in C:\varphi (z,y_n)\le \varphi (z,x_n)\},\hfill \\ Q_n=\{z\in C:〈x_n-z,J{x}_0-J{x}_n〉\ge 0\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_n\cap Q_n}{x}_0,\mathrm{\forall }n\ge 0.\hfill \end{array}$$

(1.7)

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]).

In 2008, Alber 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

$$\{\begin{array}{c}{y}_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)JT{x}_n),\hfill \\ C_{n+1}=\{w\in C_n:G(w,J{y}_n)\le G(w,J{x}_n)\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}^f{x}_0,\mathrm{\forall }n\ge 0.\hfill \end{array}$$

(1.8)

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

A Banach space 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

$$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

exists for each $x,y\in U$. It is also called 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

$${\rho }_E(t)=sup\{\frac{\parallel x+y\parallel +\parallel x-y\parallel }{2}-1:\parallel x\parallel =1,\parallel y\parallel \le t\}.$$

The modulus of convexity of E (see [29]) is the function ${\delta }_E:[0,2]\to [0,1]$ defined by

$${\delta }_E(\epsilon )=inf\{1-\parallel \frac{x+y}{2}\parallel :x,y\in E,\parallel x\parallel =\parallel y\parallel =1,\parallel x-y\parallel \ge \epsilon \}.$$

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 $〈x,Jy〉={\parallel x\parallel }^2={\parallel Jy\parallel }^2$. 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.

The generalized projection [14] from E into C is defined by

$${\mathrm{\Pi }}_C(x)=\underset{y\in C}{arg\hspace{0.17em}min}\varphi (y,x).$$

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.

Let T be a nonlinear mapping, T is said to be uniformly asymptotically regular on C if

$$\underset{n\to \mathrm{\infty }}{lim}\left(\underset{x\in C}{sup}\parallel T^{n+1}x-T^{n}x\parallel \right)=0.$$

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

  $$〈x-y,\varpi -Jx〉+\rho f(y)-\rho f(x)\ge 0,\mathrm{\forall }y\in C;$$

• (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.

Note that 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:

$$G(y,Jx)={\parallel y\parallel }^2-2〈y,Jx〉+{\parallel x\parallel }^2+2\rho f(y).$$

(2.1)

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

$${\mathrm{\Pi }}_C^{f}x=\{u\in C:G(u,Jx)=\underset{y\in C}{inf}G(y,Jx),\mathrm{\forall }x\in E\}.$$

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

$$f(x)\ge 〈x,x^{\ast }〉+\alpha ,\mathrm{\forall }x\in E.$$

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 $x\in E$, $\hat{x}\in {\mathrm{\Pi }}_C^{f}x$ if and only if

  $$〈\hat{x}-y,Jx-J\hat{x}〉+\rho f(y)-\rho f(\hat{x})\ge 0,\mathrm{\forall }y\in C;$$

• (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 $\hat{x}\in {\mathrm{\Pi }}_C^{f}x$ for all $x\in E$, then

$$\varphi (y,\hat{x})+G(\hat{x},Jx)\le G(y,Jx),\mathrm{\forall }y\in C.$$

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:

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 $\zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\zeta (0)=0$ such that

$$G(p,J{T}^{n}x)\le G(p,Jx)+{\nu }_n\zeta G(p,Jx)+{\mu }_n,\mathrm{\forall }x\in C,p\in F(T),n\ge 1.$$

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

$$\{\begin{array}{c}{y}_{n,i}=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{T}_i^n{x}_n),\hfill \\ C_{n+1,i}=\{z\in C_n:G(z,J{y}_{n,i})\le G(z,J{x}_n)+{\beta }_n\},\hfill \\ C_{n+1}={\bigcap }_{i=1}^{\mathrm{\infty }}{C}_{n+1,i},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}^f{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.1)

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.

Step 1. We first show that $C_{n+1}$ is closed and convex for all $n\ge 1$. From the definition, $C_1={\bigcap }_{i=1}^{\mathrm{\infty }}{C}_{1,i}=C$ for all $i\ge 1$ is closed and convex. Suppose that $C_{n,i}$ is closed and convex for some $n\ge 1$. For any $z\in C_{n,i}$, we know that $G(z,J{y}_{n,i})\le G(z,J{x}_n)+{\beta }_n$ is equivalent to the following:

$$2〈z,J{x}_n-J{y}_{n,i}〉\le {\parallel x_n\parallel }^2-{\parallel y_{n,i}\parallel }^2+{\beta }_n,\mathrm{\forall }i\ge 1.$$

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 $n\ge 1$.

Step 2. We show, by induction, that $\mathcal{F}={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\ne \mathrm{\varnothing }\subset C_n$ for all $n\ge 1$. It is obvious that $\mathcal{F}\subset C_1=C$. Suppose that $\mathcal{F}\subset C_n$ for some $n\ge 1$. Let $q\in \mathcal{F}$. Since $\{T_i\}$ is a totally quasi-ϕ asymptotically nonexpansive mapping, for each $i\ge 1$, we have

(3.2)

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 $n\ge 1$.

Step 3. We show that $x_n\to p$ as $n\to \mathrm{\infty }$. Since $f:E\to \mathbb{R}$ is a convex and lower semicontinuous function, from Lemma 2.7, it follows that there exist $x^{\ast }\in E^{\ast }$ and $\alpha \in \mathbb{R}$ such that $f(x)\ge 〈x,x^{\ast }〉+\alpha$ for all $x\in E$. Since $x_n\in E$, it follows that

$$\begin{array}{rcl}G(x_n,J{x}_1)& =& {\parallel x_n\parallel }^2-2〈x_n,J{x}_1〉+{\parallel x_1\parallel }^2+2\rho f(x_n)\\ \ge & {\parallel x_n\parallel }^2-2〈x_n,J{x}_1〉+{\parallel x_1\parallel }^2+2\rho 〈x_n,x^{\ast }〉+2\rho \alpha \\ =& {\parallel x_n\parallel }^2-2〈x_n,J{x}_1-\rho x^{\ast }〉+{\parallel x_1\parallel }^2+2\rho \alpha \\ \ge & {\parallel x_n\parallel }^2-2\parallel x_n\parallel \parallel J{x}_1-\rho x^{\ast }\parallel +{\parallel x_1\parallel }^2+2\rho \alpha \\ =& {(\parallel x_n\parallel -\parallel J{x}_1-\rho x^{\ast }\parallel )}^2+{\parallel x_1\parallel }^2-{\parallel J{x}_1-\rho x^{\ast }\parallel }^2+2\rho \alpha .\end{array}$$

(3.3)

For any $q\in \mathcal{F}$, since $x_n={\mathrm{\Pi }}_{C_n}^f{x}_1$, we have

$$G(q,J{x}_1)\ge G(x_n,J{x}_1)\ge {(\parallel x_n\parallel -\parallel J{x}_1-\rho x^{\ast }\parallel )}^2+{\parallel x_1\parallel }^2-{\parallel J{x}_1-\rho x^{\ast }\parallel }^2+2\rho \alpha .$$

This implies that $\{x_n\}$ is bounded and so are $\{G(x_n,J{x}_1)\}$ and $\{y_{n,i}\}$. From the fact that $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}^f{x}_1\in C_{n+1}\subset C_n$ and $x_n={\mathrm{\Pi }}_{C_n}^f{x}_1$, it follows from Lemma 2.9 that

$$0\le {(\parallel x_{n+1}-x_n\parallel )}^2\le \varphi (x_{n+1},x_n)\le G(x_{n+1},J{x}_1)-G(x_n,J{x}_1).$$

(3.4)

This implies that $\{G(x_n,J{x}_1)\}$ is nondecreasing. Hence we know that ${lim}_{n\to \mathrm{\infty }}G(x_n,J{x}_1)$ exists. Taking $n\to \mathrm{\infty }$, we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},x_n)=0.$$

(3.5)

Since $\{x_n\}$ is bounded, 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

$$G(x_n,J{x}_1)\le G(p,J{x}_1),\mathrm{\forall }n\ge 1.$$

(3.6)

Since f is convex and lower semicontinuous, we have

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}G(x_n,J{x}_1)& =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\{{\parallel x_n\parallel }^2-2〈x_n,J{x}_1〉+{\parallel x_1\parallel }^2+2\rho f(x_n)\}\\ \ge & {\parallel p\parallel }^2-2〈p,J{x}_1〉+{\parallel x_1\parallel }^2+2\rho f(p)\\ =& G(x_n,J{x}_1).\end{array}$$

(3.7)

By (3.6) and (3.7), we get

$$G(p,J{x}_1)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}G(x_n,J{x}_1)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}G(x_n,J{x}_1)\le G(p,J{x}_1).$$

That is, ${lim}_{n\to \mathrm{\infty }}G(x_n,J{x}_1)=G(p,x_1)$, which implies that $\parallel x_n\parallel \to \parallel p\parallel$ and so, by virtue of the Kadec-Klee property of E, it follows that

$$\underset{n\to \mathrm{\infty }}{lim}{x}_n=p.$$

(3.8)

We also have

$$\underset{n\to \mathrm{\infty }}{lim}{x}_{n+1}=p.$$

(3.9)

Since $\{x_n\}$ is bounded (we denote $M={sup}_{n\ge 0}\{\parallel x_n\parallel \}<\mathrm{\infty }$), it follows that

$$\underset{n\to \mathrm{\infty }}{lim}{\beta }_n=\underset{n\to \mathrm{\infty }}{lim}({\nu }_n\underset{q\in \mathcal{F}}{sup}\psi (G(q,x_n))+{\mu }_n)=0.$$

(3.10)

From (3.8) and (3.9), we have ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_{n+1}\parallel =0$. Since J is uniformly norm-to-norm continuous, it follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_n-J{x}_{n+1}\parallel =0.$$

(3.11)

Since $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}^f{x}_1\in C_{n+1}\subset C_n$ and by the definition of $C_{n+1}$, it follows that

that is, we get

$$\varphi (x_{n+1},y_{n,i})\le \varphi (x_{n+1},x_n)+{\beta }_n,\mathrm{\forall }i\ge 1.$$

From (3.5) and (3.10), it follows that for each $i\ge 1$,

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},y_{n,i})=0.$$

(3.12)

Also, from (1.2), it follows that for each $i\ge 1$,

$$\parallel y_{n,i}\parallel \to \parallel p\parallel (n\to \mathrm{\infty }).$$

(3.13)

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

$$\parallel J{y}_{n,i}\parallel \to \parallel Jp\parallel (n\to \mathrm{\infty }).$$

(3.14)

That is, $\{\parallel J{y}_{n,i}\parallel \}$ bounded in $E^{\ast }$ for all $i\ge 1$. Since 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 $i\ge 1$,

$$\varphi (x_{n+1},y_{n,i})={\parallel x_{n+1}\parallel }^2-2〈x_{n+1},J{y}_{n,i}〉+{\parallel y_{n,i}\parallel }^2={\parallel x_{n+1}\parallel }^2-2〈x_{n+1},J{y}_{n,i}〉+{\parallel J{y}_{n,i}\parallel }^2.$$

Taking ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}$ on the both sides of the equality above and the property of weak lower semicontinuity of the norm $\parallel \cdot \parallel$, it follows that

$$\begin{array}{rcl}0& \ge & {\parallel p\parallel }^2-2〈p,y^{\ast }〉+{\parallel y^{\ast }\parallel }^2\\ =& {\parallel p\parallel }^2-2〈p,Jy〉+{\parallel Jy\parallel }^2\\ =& {\parallel p\parallel }^2-2〈p,Jy〉+{\parallel y\parallel }^2\\ =& \varphi (p,y).\end{array}$$

That is, $p=y$, which implies that $y^{\ast }=Jp$. It follows that for each $i\ge 1$, $J{y}_{n,i}\rightharpoonup Jp\in E^{\ast }$. From (3.14) and the Kadec-Klee property of $E^{\ast }$, we have $J{y}_{n,i}\to Jp$ as $n\to \mathrm{\infty }$ for all $i\ge 1$. Since $J^{-1}:E^{\ast }\to E$ is norm-weak^∗-continuous, that is, $y_{n,i}\rightharpoonup p$, it follows from (3.13) and the Kadec-Klee property of E that

$$\underset{n\to \mathrm{\infty }}{lim}{y}_{n,i}=p,\mathrm{\forall }i\ge 1.$$

(3.15)

From (3.9), (3.15) and the triangle inequality, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-y_{n,i}\parallel =0,\mathrm{\forall }i\ge 1.$$

(3.16)

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

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_{n+1}-J{y}_{n,i}\parallel =0,\mathrm{\forall }i\ge 1.$$

(3.17)

From the definition of $y_{n,i}$, it follows that

$$\begin{array}{rcl}\parallel J{x}_{n+1}-J{y}_{n,i}\parallel & =& \parallel J{x}_{n+1}-{\alpha }_{n}J{x}_n-(1-{\alpha }_n)J{T}_i^n{x}_n\parallel \\ =& \parallel (1-{\alpha }_n)J{x}_{n+1}-(1-{\alpha }_n)J{T}_i^n{x}_n+{\alpha }_{n}J{x}_{n+1}-{\alpha }_{n}J{x}_n\parallel \\ \ge & (1-{\alpha }_n)\parallel J{x}_{n+1}-J{T}_i^n{x}_n\parallel -{\alpha }_n\parallel J{x}_n-J{x}_{n+1}\parallel ,\end{array}$$

(3.18)

and so

$$\parallel J{x}_{n+1}-J{T}_i^n{x}_n\parallel \le \frac{1}{(1-{\alpha }_n)}(\parallel J{x}_{n+1}-J{y}_{n,i}\parallel +{\alpha }_n\parallel J{x}_n-J{x}_{n+1}\parallel ).$$

(3.19)

Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}(1-{\alpha }_n)>0$, it follows from (3.11) and (3.17) that, for each $i\ge 1$,

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_{n+1}-J{T}_i^n{x}_n\parallel =0.$$

(3.20)

Since $J^{-1}$ is uniformly norm-to-norm continuous, for each $i\ge 1$, we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-T_i^n{x}_n\parallel =0.$$

(3.21)

By using the triangle inequality, for each $i\ge 1$, we have

$$\parallel T_i^n{x}_n-p\parallel \le \parallel T_i^n{x}_n-x_{n+1}\parallel +\parallel x_{n+1}-p\parallel .$$

From (3.21) and $x_{n+1}\to p$ as $n\to \mathrm{\infty }$, it follows that for each $i\ge 1$,

$$\underset{n\to \mathrm{\infty }}{lim}\parallel T_i^n{x}_n-p\parallel =0.$$

(3.22)

For each $i\ge 1$, we have

$$\parallel T_i^{n+1}{x}_n-p\parallel \le \parallel T_i^{n+1}{x}_n-T_i^n{x}_n\parallel +\parallel T_i^n{x}_n-p\parallel .$$

Since $T_i$ is uniformly asymptotically regular for all $i\ge 1$, it follows from (3.22) that

$$\parallel T_i^{n+1}{x}_n-p\parallel =0.$$

(3.23)

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 $p\in {\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)$.

Step 4. We show that $p={\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$. Since ℱ is a closed and convex set, it follows from Lemma 2.8 that ${\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$ is single-valued, which is denoted by v. By the definition $x_n={\mathrm{\Pi }}_{C_n}^f{x}_1$ and $v\in \mathcal{F}\subset C_n$, we also have

$$G(x_n,J{x}_1)\le G(v,J{x}_1),\mathrm{\forall }n\ge 1.$$

By the definition of 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

$$G(p,J{x}_1)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}G(x_n,J{x}_1)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}G(x_n,J{x}_1)\le G(v,J{x}_1).$$

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

$$\{\begin{array}{c}{y}_{n,i}=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{T}_i^n{x}_n),\hfill \\ C_{n+1,i}=\{z\in C_n:G(z,J{y}_{n,i})\le G(z,J{x}_n)\},\hfill \\ C_{n+1}={\bigcap }_{i=1}^{\mathrm{\infty }}{C}_{n+1,i},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}^f{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.24)

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

$$\{\begin{array}{c}{y}_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{T}^n{x}_n),\hfill \\ C_{n+1}=\{z\in C_n:G(z,J{y}_{n,i})\le G(z,J{x}_n)+{\beta }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}^f{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.25)

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

$$\{\begin{array}{c}{y}_{n,i}=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{T}_i^n{x}_n),\hfill \\ C_{n+1,i}=\{z\in C_n:\varphi (z,y_{n,i})\le \varphi (z,x_n)+{\beta }_n\},\hfill \\ C_{n+1}={\bigcap }_{i=1}^{\mathrm{\infty }}{C}_{n+1,i},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.26)

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

$$\{\begin{array}{c}{y}_{n,i}=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{T}_i^n{x}_n),\hfill \\ C_{n+1,i}=\{z\in C_n:\varphi (z,y_{n,i})\le \varphi (z,x_n)\},\hfill \\ C_{n+1}={\bigcap }_{i=1}^{\mathrm{\infty }}{C}_{n+1,i},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.27)

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

$$\{\begin{array}{c}{y}_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{T}^n{x}_n),\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,y_n)\le \varphi (z,x_n)+{\beta }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.28)

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]).