Convergence theorems for equilibrium problem and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense

Abstract

In this paper, we introduce an iterative process which converges strongly to a common element of the set of fixed points of an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense and the solution set of generalized equilibrium problem in Banach spaces. Our theorems improve, generalize, and extend several results recently announced.

MSC:47H05, 47H09, 47H10.

1 Introduction

Let E be a real Banach space with the dual space $E^{\ast }$. Let C be a nonempty closed convex subset of E. Let $T:C\to C$ be a nonlinear mapping. We denote by $F(T)$ the set of fixed points of T.

A mapping T is said to be asymptotically nonexpansive if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ such that

$$\parallel T^{n}x-T^{n}y\parallel \le k_n\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,n\ge 1.$$

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. In uniformly convex Banach spaces, they proved that if C is nonempty, bounded, closed, and convex, then every asymptotically nonexpansive self-mapping T on C has a fixed point. Further, the fixed point set of T is closed and convex.

A mapping T is said to be asymptotically nonexpansive in the intermediate sense (see [2]) if it is continuous and the following inequality holds:

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}(\parallel T^{n}x-T^{n}y\parallel -\parallel x-y\parallel )\le 0.$$

(1.1)

If $F(T)\ne \varphi$ and (1.1) holds for all $x\in K$, $y\in F(T)$, then T is called asymptotically quasi-nonexpansive in the intermediate sense. It is well known that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is a self-mapping of C which is asymptotically nonexpansive in the intermediate sense, then T has a fixed point (see [3]). It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.

Iterative approximation of a fixed point for asymptotically nonexpansive mappings in Hilbert or Banach spaces has been studied extensively by many authors (see [4–6] and the references therein).

Let E be a smooth Banach space. The function $\varphi :E\times E\to \mathbb{R}$ defined by

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

is studied by Alber [7]. It follows 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)

Remark 1.1

• (i) If E is a reflexive, strictly convex and smooth Banach space, then for $x,y\in E$, $\varphi (x,y)=0$ if and only if $x=y$.

• (ii) If E is a real Hilbert space, then $\varphi (x,y)={\parallel x-y\parallel }^2$.

Let E be reflexive, strictly convex and smooth Banach space. The generalized projection mapping, introduced by Alber [7], is a mapping ${\mathrm{\Pi }}_C:E\to C$ that assigns to an arbitrary point $x\in E$ the minimum point of the functional $\varphi (y,x)$, that is, ${\mathrm{\Pi }}_{C}x=\overline{x}$, where is $\overline{x}$ is the solution to the minimization problem

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

A point p in C is said to be an asymptotic fixed point of T 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 set of asymptotic fixed points of T will be denoted by $\tilde{F}(T)$. A mapping T is called relatively nonexpansive (see [8]) if $\tilde{F}(T)=F(T)$ and $\varphi (p,Tx)\le \varphi (p,x)$ for all $x\in C$ and $p\in F(T)$.

Recently, Matsushita and Takahashi [9] proved strong convergence theorems for approximation of fixed points of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. More precisely, they proved the following theorem.

Theorem 1.1 Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive mapping from C into itself and let $\{{\alpha }_n\}$ be a sequence of real numbers such that $0\le {\alpha }_n\le 1$ and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\alpha }_n<1$. Suppose that $\{x_n\}$ is given by

$$\{\begin{array}{c}{x}_0=x\in C,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)JT{x}_n),\hfill \\ H_n=\{z\in C:\varphi (z,y_n)\le \varphi (z,x_n)\},\hfill \\ W_n=\{z\in C:〈x_n-z,Jx-J{x}_n〉\ge 0\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{H_n\cap W_n}{x}_0,n=0,1,2,\dots ,\hfill \end{array}$$

(1.3)

where J is the normalized duality mapping on E. If $F(T)$ is nonempty, then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{F(T)}{x}_0$.

In [10], Hao introduced the following iterative scheme for approximating a fixed point of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense in a reflexive, strictly convex and smooth Banach space E: $x_0\in E$, $C_1=C$, $x_1={\mathrm{\Pi }}_{C_1}{x}_0$,

$$\{\begin{array}{c}{y}_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{T}^n{x}_n),\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,y_n)\le \varphi (z,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,n=1,2,\dots ,\hfill \end{array}$$

where ${\xi }_n=max\{0,{sup}_{p\in F(T),x\in C}(\varphi (p,T^{n}x)-\varphi (p,x))\}$.

Motivated and inspired by the works mentioned above, in this paper, we introduce a new iterative scheme of the generalized f-projection operator for finding a common element of the set of fixed points of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense and the solution set of generalized equilibrium problem in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.

2 Preliminaries

Let E be a real Banach space with the norm $\parallel \cdot \parallel$ and let $E^{\ast }$ be the dual space of E. The normalized duality mapping $J:E\to 2^{E^{\ast }}$ is defined by

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

By the Hahn-Banach theorem, $J(x)$ is nonempty.

A Banach space E is called strictly convex if $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in U$ with $x\ne y$, where $U=\{x\in E:\parallel x\parallel =1\}$ is the unit sphere of E. A Banach space E is called smooth if the limit

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

exists for each $x,y\in U$. It is also called uniformly smooth if the limit exists uniformly for all $x,y\in U$. 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 a Banach space E related to the normalized duality mapping J are as follows (see [11]):

• (1) If E is a smooth Banach space, then J is single-valued and semicontinuous;

• (2) If E is a uniformly smooth Banach space, then J is uniformly norm-to-norm continuous on each bounded subset of E;

• (3) If E is a uniformly smooth Banach space, then E is smooth and reflexive;

• (4) If E is a reflexive and strictly convex Banach space, then $J^{-1}$ is norm-weak^∗-continuous;

• (5) E is a uniformly smooth Banach space if and only if $E^{\ast }$ is uniformly convex.

Recall that a Banach space E has the Kadec-Klee property if for any sequence $\{x_n\}\subset E$ and $x\in E$ with $x_n\rightharpoonup x$ and $\parallel x_n\parallel \to \parallel x\parallel$, then $\parallel x_n-x\parallel \to 0$ 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.

Definition 2.1 A mapping $T:C\to C$ is said to be

• (1) quasi-ϕ-nonexpansive if $F(T)\ne \varphi$ and

  $$\varphi (p,Tx)\le \varphi (p,x)$$

  for all $x\in C$ and $p\in F(T)$;

• (2) asymptotically quasi-ϕ-nonexpansive in the intermediate sense if $F(T)\ne \varphi$ and

  $$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{p\in F(T),x\in C}{sup}(\varphi (p,T^{n}x)-\varphi (p,x))\le 0$$

put

$${\xi }_n=max\{0,\underset{p\in F(T),x\in C}{sup}(\varphi (p,T^{n}x)-\varphi (p,x))\}.$$

Remark 2.2 From the definition of asymptotically quasi-ϕ-nonexpansiveness in the intermediate sense, it is obvious that ${\xi }_n\to 0$ as $n\to \mathrm{\infty }$ and

$$\varphi (p,T^{n}x)\le \varphi (p,x)+{\xi }_n,\mathrm{\forall }p\in F(T),x\in C.$$

Recall that T is said to be asymptotically regular on C if for any bounded subset K of C,

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\{\parallel T^{n+1}x-T^{n}x\parallel :x\in K\}=0.$$

Definition 2.2 A mapping $T:C\to C$ is said to be closed if for any sequence $\{x_n\}\subset C$ with $x_n\to x$ and $T{x}_n\to y$, $Tx=y$.

Following Alber [7], the generalized projection ${\mathrm{\Pi }}_C:E\to C$ is defined by

$${\mathrm{\Pi }}_C(x)=\{u\in C:\varphi (u,x)=\underset{y\in C}{min}\varphi (y,x)\},\mathrm{\forall }x\in E.$$

In 2006, Wu and Huang [12] introduced a generalized f-projection operator in a Banach space, which extends the definition of the generalized projection ${\mathrm{\Pi }}_C$. Let $G:C\times E^{\ast }\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ be a functional defined as follows:

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

for all $(y,\overline{w})\in C\times E^{\ast }$, where 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, it is easy to see the following properties:

• (i) $G(y,\overline{w})$ is convex and continuous with respect to $\overline{w}$ when y is fixed;

• (ii) $G(y,\overline{w})$ is convex and lower semicontinuous with respect to y when $\overline{w}$ is fixed.

Definition 2.3 ([13])

Let E be a real smooth Banach space and let C be a nonempty closed and convex subset of E. We say that ${\mathrm{\Pi }}_C^f:E\to 2^C$ is a 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 2.1 ([14])

Let E be a Banach space and $f:E\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ be a lower semicontinuous and convex function. Then there exist $x^{\ast }\in E^{\ast }$ and $\alpha \in \mathbb{R}$ such that

$$f(x)\ge 〈x,x^{\ast }〉+\alpha$$

for all $x\in E$.

Lemma 2.2 ([13])

Let E be a reflexive smooth Banach space and let C be a nonempty closed convex subset of E. The following statements hold:

• (1) ${\mathrm{\Pi }}_C^{f}x$ is a nonempty closed convex subset of C for all $x\in E$;

• (2) For all $x\in F$, $\overline{x}\in {\mathrm{\Pi }}_C^{f}x$ if and only if

  $$〈\overline{x}-y,Jx-J\overline{x}〉+\rho f(y)-\rho f(\overline{x})\ge 0$$

for all $y\in C$;

• (3) If E is strictly convex, then ${\mathrm{\Pi }}_C^f$ is a single-valued mapping.

Let θ be a bifunction from $C\times C$ to ℝ, where ℝ denotes the set of real numbers. The equilibrium problem is to find $\overline{x}\in C$ such that

$$\theta (\overline{x},y)\ge 0$$

(2.1)

for all $y\in C$. The set of solutions of (2.1) is denoted by $EP(\theta )$.

For solving the equilibrium problem for a bifunction $\theta :C\times C\to \mathbb{R}$, let us assume that θ satisfies the following conditions:

(A1) $\theta (x,x)=0$ for all $x\in C$;

(A2) θ is monotone; i.e., $\theta (x,y)+\theta (y,x)\le 0$ for all $x,y\in C$;

(A3) for all $x,y,z\in C$,

$$\underset{t\downarrow 0}{lim}\theta (tz+(1-t)x,y)\le \theta (x,y);$$

(A4) for all $x\in C$, $y\mapsto \theta (x,y)$ is convex and lower semicontinuous.

Lemma 2.3 ([15])

Let C be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E and let θ be a bifunction from $C\times C$ to ℝ satisfying the conditions (A1)-(A4). For all $r>0$ and $x\in E$, define a mapping $T_r^{\theta }:E\to C$ as follows:

$$T_r^{\theta }x=\{z\in C:\theta (z,y)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\mathrm{\forall }y\in C\}.$$

Then the following conclusions hold:

• (1) $T_r^{\theta }$ is single-valued;

• (2) $T_r^{\theta }$ is a firmly nonexpansive-type mapping, i.e., for all $x,y\in E$,

  $$〈T_r^{\theta }x-T_r^{\theta }y,J{T}_r^{\theta }x-J{T}_r^{\theta }y〉\le 〈T_r^{\theta }x-T_r^{\theta }y,Jx-Jy〉;$$

• (3) $F(T_r^{\theta })=EP(\theta )$ is closed and convex;

• (4) $T_r^{\theta }$ is quasi-ϕ-nonexpansive;

• (5) $\varphi (q,T_r^{\theta }x)+\varphi (T_r^{\theta }x,x)\le \varphi (q,x)$, $\mathrm{\forall }q\in F(T_r^{\theta })$.

Lemma 2.4 ([10])

Let E be a reflexive, strictly convex and smooth Banach space such that both E and $E^{\ast }$ have the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let $T:C\to C$ be a closed and asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Then $F(T)$ is a closed convex subset of C.

Lemma 2.5 ([13])

Let E be a real reflexive smooth Banach space and let C be a nonempty closed and convex subset of E. Then, for any $x\in E$ and $\overline{x}\in {\mathrm{\Pi }}_C^{f}x$,

$$\varphi (y,\overline{x})+G(\overline{x},Jx)\le G(y,Jx)$$

for all $y\in C$.

3 Main results

Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let θ be a bifunction from $C\times C$ to ℝ satisfying the conditions (A1)-(A4). Let $T:C\to C$ be a closed and asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is asymptotically regular on C, $\mathcal{F}=F(T)\cap EP(\theta )$ is nonempty, and $F(T)$ is bounded. Let $f:E\to {\mathbb{R}}^{+}$ be a convex and lower semicontinuous function with $C\subset int(D(f))$ and $f(0)=0$. Let $\{{\alpha }_n\}$ be a sequence in $[0,1]$ and $\{{\beta }_n\}$, $\{{\gamma }_n\}$ be sequences in $(0,1)$ satisfying the following conditions:

• (i) ${\alpha }_n+{\beta }_n+{\gamma }_n=1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$;

• (iii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$.

Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{x}_1\in E\mathit{\text{ chosen arbitrarily}},\hfill \\ C_1=C,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_1+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{x}_n),\hfill \\ u_n\in C\mathit{\text{ such that }}\theta (u_n,y)+\frac{1}{r_n}〈y-u_n,J{u}_n-J{y}_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ C_{n+1}=\{z\in C_n:G(z,J{u}_n)\le {\alpha }_{n}G(z,J{x}_1)+(1-{\alpha }_n)G(z,J{x}_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}^f{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.1)

where ${\xi }_n=max\{0,{sup}_{p\in F(T),x\in C}(\varphi (p,T^{n}x)-\varphi (p,x))\}$, $\{r_n\}$ is a real sequence in $[a,\mathrm{\infty })$ for some $a>0$ and ${\mathrm{\Pi }}_{C_{n+1}}^f$ is the generalized f-projection operator. Then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$.

Proof It follows from Lemma 2.3 and Lemma 2.4 that ℱ is a closed convex subset of C, so that ${\mathrm{\Pi }}_{\mathcal{F}}^{f}x$ is well defined for any $x\in C$.

We split the proof into six steps.

Step 1. We first show that $C_n$ is nonempty, closed, and convex for all $n\ge 1$.

In fact, it is obvious that $C_1=C$ is closed and convex. Suppose that $C_n$ is closed and convex for some $n\ge 2$. For $z_1,z_2\in C_{n+1}$, we see that $z_1,z_2\in C_n$. It follows that $z=t{z}_1+(1-t)z_2\in C_n$, where $t\in (0,1)$. Notice that

$$G(z_1,J{u}_n)\le {\alpha }_{n}G(z_1,J{x}_1)+(1-{\alpha }_n)G(z_1,J{x}_n)+{\xi }_n,$$

and

$$G(z_2,J{u}_n)\le {\alpha }_{n}G(z_2,J{x}_1)+(1-{\alpha }_n)G(z_2,J{x}_n)+{\xi }_n.$$

The above inequalities are equivalent to

$$\begin{array}{r}2{\alpha }_n〈z_1,J{x}_1〉+2(1-{\alpha }_n)〈z_1,J{x}_n〉-2〈z_1,J{u}_n〉\\ \le {\alpha }_n{\parallel x_1\parallel }^2+(1-{\alpha }_n){\parallel x_n\parallel }^2-{\parallel u_n\parallel }^2+{\xi }_n\end{array}$$

(3.2)

and

$$\begin{array}{r}2{\alpha }_n〈z_2,J{x}_1〉+2(1-{\alpha }_n)〈z_2,J{x}_n〉-2〈z_2,J{u}_n〉\\ \le {\alpha }_n{\parallel x_1\parallel }^2+(1-{\alpha }_n){\parallel x_n\parallel }^2-{\parallel u_n\parallel }^2+{\xi }_n.\end{array}$$

(3.3)

Multiplying t and $1-t$ on both sides of (3.2) and (3.3), respectively, we obtain

$$\begin{array}{r}2{\alpha }_n〈z,J{x}_1〉+2(1-{\alpha }_n)〈z,J{x}_n〉-2〈z,J{u}_n〉\\ \le {\alpha }_n{\parallel x_1\parallel }^2+(1-{\alpha }_n){\parallel x_n\parallel }^2-{\parallel u_n\parallel }^2+{\xi }_n.\end{array}$$

Hence we have

$$G(z,J{u}_n)\le {\alpha }_{n}G(z,J{x}_1)+(1-{\alpha }_n)G(z,J{x}_n)+{\xi }_n.$$

This implies that $C_{n+1}$ is closed and convex for all $n\ge 1$. This shows that ${\mathrm{\Pi }}_{C_{n+1}}^f{x}_1$ is well defined.

Step 2. We show that $\mathcal{F}\subset C_n$ for all $n\ge 1$.

For $n=1$, we have $\mathcal{F}\subset C_1=C$. Now, assume that $\mathcal{F}\subset C_n$ for some $n\ge 2$. Let $q\in \mathcal{F}$. Since T is asymptotically quasi-ϕ-nonexpansive with intermediate sense, we have from Remark 2.2 and Lemma 2.3 that

$$\begin{array}{rl}G(q,J{u}_n)=& G(q,J{T}_{r_n}^{\theta }{y}_n)\\ =& \varphi (q,T_{r_n}^{\theta }{y}_n)+2\rho f(q)\\ \le & \varphi (q,y_n)+2\rho f(q)\\ =& G(q,J{y}_n)\\ =& G(q,{\alpha }_{n}J{x}_1+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{x}_n)\\ =& {\parallel q\parallel }^2-2{\alpha }_n〈q,J{x}_1〉-2{\beta }_n〈q,J{T}^n{x}_n〉-2{\gamma }_n〈q,J{x}_n〉\\ +{\parallel {\alpha }_{n}J{x}_1+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{x}_n\parallel }^2+2\rho f(q)\\ \le & {\parallel q\parallel }^2-2{\alpha }_n〈q,J{x}_1〉-2{\beta }_n〈q,J{T}^n{x}_n〉-2{\gamma }_n〈q,J{x}_n〉\\ +{\alpha }_n{\parallel J{x}_1\parallel }^2+{\beta }_n{\parallel J{T}^n{x}_n\parallel }^2+{\gamma }_n{\parallel J{x}_n\parallel }^2+2\rho f(q)\\ =& {\alpha }_{n}G(q,J{x}_1)+{\beta }_{n}G(q,J{T}^n{x}_n)+{\gamma }_{n}G(q,J{x}_n)\\ =& {\alpha }_{n}G(q,J{x}_1)+{\beta }_n\{\varphi (q,T^n{x}_n)+2\rho f(q)\}+{\gamma }_{n}G(q,J{x}_n)\\ \le & {\alpha }_{n}G(q,J{x}_1)+{\beta }_n\{\varphi (q,x_n)+{\xi }_n+2\rho f(q)\}+{\gamma }_{n}G(q,J{x}_n)\\ \le & {\alpha }_{n}G(q,J{x}_1)+{\beta }_{n}G(q,J{x}_n)+{\gamma }_{n}G(q,J{x}_n)+{\xi }_n\\ =& {\alpha }_{n}G(q,J{x}_1)+(1-{\alpha }_n)G(q,J{x}_n)+{\xi }_n,\end{array}$$

which shows that $q\in C_{n+1}$. This implies that $\mathcal{F}\subset C_{n+1}$ and so $\mathcal{F}\subset C_n$ for all $n\ge 1$.

Step 3. We prove that $\{x_n\}$ is bounded and ${lim}_{n\to \mathrm{\infty }}G(x_n,J{x}_1)$ exists.

By Lemma 2.1, we have the result that there exist $x^{\ast }\in E^{\ast }$ and $\alpha \in \mathbb{R}$ such that

$$f(x)\ge 〈x,x^{\ast }〉+\alpha .$$

Since $x_n\in C_n\subset E$, it follows that

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

For all $q\in \mathcal{F}$ and $x_n={\mathrm{\Pi }}_{C_n}^{f_1}{x}_1$, we have

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

This implies that the sequence $\{x_n\}$ is bounded and so is $\{G(x_n,J{x}_1)\}$. From (1.2) and Lemma 2.5, we obtain

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

(3.4)

This shows that $\{G(x_n,J{x}_1)\}$ is nondecreasing. It follows from the boundedness that ${lim}_{n\to \mathrm{\infty }}G(x_n,J{x}_1)$ exists.

Step 4. Next, we prove that $x_n\to \overline{x}$, $y_n\to \overline{x}$, and $u_n\to \overline{x}$ as $n\to \mathrm{\infty }$, where $\overline{x}$ is some point in C.

By (3.4), we obtain

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

(3.5)

Since $\{x_n\}$ is bounded and E is reflexive, we may assume that $x_n\rightharpoonup \overline{x}$ as $n\to \mathrm{\infty }$. Since $C_n$ is closed and convex, we find that $\overline{x}\in C_n$. From the weak lower semicontinuity of the norm and $x_n={\mathrm{\Pi }}_{C_n}^f{x}_1$, we obtain

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

which implies that ${lim}_{n\to \mathrm{\infty }}G(x_n,J{x}_1)=G(\overline{x},J{x}_1)$. From Lemma 2.5, we obtain

$$\begin{array}{rl}0& \le {(\parallel \overline{x}\parallel -\parallel x_n\parallel )}^2\\ \le \varphi (\overline{x},x_n)\\ \le G(\overline{x},J{x}_1)-G(x_n,J{x}_1).\end{array}$$

Hence we have ${lim}_{n\to \mathrm{\infty }}\parallel x_n\parallel =\parallel \overline{x}\parallel$. In view of the Kadec-Klee property of E, we find that

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

(3.6)

And we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x_{n+1}\parallel =0.$$

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

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

From $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}^f{x}_1\in C_{n+1}\subset C_n$ and (3.1), we have

$$G(x_{n+1},J{u}_n)\le {\alpha }_{n}G(x_{n+1},J{x}_1)+(1-{\alpha }_n)G(x_{n+1},J{x}_n)+{\xi }_n.$$

This is equivalent to the following:

$$\varphi (x_{n+1},u_n)\le {\alpha }_n\varphi (x_{n+1},x_1)+(1-{\alpha }_n)\varphi (x_{n+1},x_n)+{\xi }_n.$$

(3.7)

Due to (3.5), (3.7), the assumption (ii), and Remark 2.2, we have

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

By (1.2), it follows that

$$\parallel u_n\parallel \to \parallel \overline{x}\parallel$$

(3.8)

as $n\to \mathrm{\infty }$. Since J is uniformly norm-to-norm continuous, we obtain

$$\parallel J{u}_n\parallel \to \parallel J\overline{x}\parallel$$

(3.9)

as $n\to \mathrm{\infty }$. This implies that $\{\parallel J{u}_n\parallel \}$ is bounded in $E^{\ast }$. Since $E^{\ast }$ is reflexive, we assume that $J{u}_n\rightharpoonup \overline{u}\in E^{\ast }$ as $n\to \mathrm{\infty }$. In view of $J(E)=E^{\ast }$, there exists $u\in E$ such that $Ju=\overline{u}$. This implies that $J{u}_n\rightharpoonup Ju$. We have

$$\begin{array}{rl}\varphi (x_{n+1},u_n)& ={\parallel x_{n+1}\parallel }^2-2〈x_{n+1},J{u}_n〉+{\parallel u_n\parallel }^2\\ ={\parallel x_{n+1}\parallel }^2-2〈x_{n+1},J{u}_n〉+{\parallel J{u}_n\parallel }^2.\end{array}$$

Taking ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}$ on both sides of the equality above, this yields

$$\begin{array}{rl}0& \ge {\parallel \overline{x}\parallel }^2-2〈\overline{x},\overline{u}〉+{\parallel \overline{u}\parallel }^2\\ ={\parallel \overline{x}\parallel }^2-2〈\overline{x},Ju〉+{\parallel Ju\parallel }^2\\ ={\parallel \overline{x}\parallel }^2-2〈\overline{x},Ju〉+{\parallel u\parallel }^2\\ =\varphi (\overline{x},u),\end{array}$$

which shows that $\overline{x}=u$ and so $J{u}_n\rightharpoonup J\overline{x}$. It follows from (3.9) and the Kadec-Klee property of $E^{\ast }$ that $J{u}_n\to J\overline{x}$ as $n\to \mathrm{\infty }$. Since $J^{-1}$ is norm-weak-continuous, we have

$$u_n\rightharpoonup \overline{x}.$$

(3.10)

From (3.8), (3.10), and the Kadec-Klee property of E, we have

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

(3.11)

On the other hand, we see from the weak lower semicontinuity of the norm that

$$\begin{array}{rl}\varphi (q,\overline{x})& ={\parallel q\parallel }^2-2〈q,J\overline{x}〉+{\parallel \overline{x}\parallel }^2\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}({\parallel q\parallel }^2-2〈q,J{u}_n〉+{\parallel u_n\parallel }^2)\\ =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\varphi (q,u_n)\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\varphi (q,u_n)\\ =\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\parallel q\parallel -2〈q,J{u}_n〉+{\parallel u_n\parallel }^2)\\ \le \varphi (q,\overline{x}),\end{array}$$

which implies that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (q,u_n)=\varphi (q,\overline{x}).$$

(3.12)

By (3.6) and (3.11), we obtain ${lim}_{n\to \mathrm{\infty }}\parallel x_n-u_n\parallel =0$. The uniform continuity of J on bounded sets gives

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

(3.13)

Now, using the definition of ϕ, we have, for all $q\in \mathcal{F}$,

$$\begin{array}{rl}\varphi (q,x_n)-\varphi (q,u_n)& ={\parallel x_n\parallel }^2-{\parallel u_n\parallel }^2-2〈q,J{x}_n-J{u}_n〉\\ \le \parallel x_n-u_n\parallel (\parallel x_n\parallel +\parallel u_n\parallel )+2\parallel q\parallel \parallel J{x}_n-J{u}_n\parallel .\end{array}$$

From (3.13), we obtain

$$\varphi (q,x_n)-\varphi (q,u_n)\to 0$$

as $n\to \mathrm{\infty }$. By (3.12), it follows that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (q,x_n)=\varphi (q,\overline{x}).$$

(3.14)

Hence, for any $q\in \mathcal{F}\subset C_n$, it follows from the convexity of ${\parallel \cdot \parallel }^2$ and Lemma 2.3 that

$$\begin{array}{rl}\varphi (q,u_n)=& \varphi (q,T_{r_n}^{\theta }{y}_n)\\ \le & \varphi (q,y_n)\\ =& \varphi (q,J^{-1}({\alpha }_{n}J{x}_1+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{x}_n))\\ =& {\parallel q\parallel }^2-2〈q,{\alpha }_{n}J{x}_1+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{x}_n〉\\ +{\parallel {\alpha }_{n}J{x}_1+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{x}_n\parallel }^2\\ \le & \parallel q\parallel -2{\alpha }_n〈q,J{x}_1〉-2{\beta }_n〈q,J{T}^n{x}_n〉-2{\gamma }_n〈q,J{x}_n〉\\ +{\alpha }_n{\parallel J{x}_1\parallel }^2+{\beta }_n{\parallel J{T}^n{x}_n\parallel }^2+{\gamma }_n{\parallel J{x}_n\parallel }^2\\ =& {\alpha }_n\varphi (q,x_1)+{\beta }_n\varphi (q,T^n{x}_n)+{\gamma }_n\varphi (q,x_n)\\ \le & {\alpha }_n\varphi (q,x_1)+{\beta }_n(\varphi (q,x_n)+{\xi }_n)+{\gamma }_n\varphi (q,x_n)\\ \le & {\alpha }_n\varphi (q,x_1)+(1-{\alpha }_n)\varphi (q,x_n)+{\xi }_n.\end{array}$$

(3.15)

From (3.12), (3.14), (3.15), Remark 2.2, and the assumption (ii), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (q,y_n)=\varphi (q,\overline{x}).$$

From Lemma 2.3, we see that for any $q\in \mathcal{F}$ and $u_n=T_{r_n}^{\theta }{y}_n$,

$$\begin{array}{rl}\varphi (u_n,y_n)& =\varphi (T_{r_n}^{\theta }{y}_n,y_n)\\ \le \varphi (q,y_n)-\varphi (q,T_{r_n}^{\theta }{y}_n)\\ =\varphi (q,y_n)-\varphi (q,u_n).\end{array}$$

Taking $n\to \mathrm{\infty }$ on both sides of the inequality above, we have

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (u_n,y_n)=0.$$

From (1.2), we have ${(\parallel u_n\parallel -\parallel y_n\parallel )}^2\to 0$ as $n\to \mathrm{\infty }$. By (3.8), we have

$$\parallel y_n\parallel \to \parallel \overline{x}\parallel$$

(3.16)

as $n\to \mathrm{\infty }$, and so

$$\parallel J{y}_n\parallel \to \parallel J\overline{x}\parallel$$

(3.17)

as $n\to \mathrm{\infty }$. That is, $\{\parallel J{y}_n\parallel \}$ is bounded in $E^{\ast }$. Since $E^{\ast }$ is reflexive, we can assume that $J{y}_n\rightharpoonup y^{\ast }\in E^{\ast }$ as $n\to \mathrm{\infty }$. In view of $J(E)=E^{\ast }$, there exists $y\in E$ such that $Jy=y^{\ast }$. It follows that

$$\begin{array}{rl}\varphi (u_n,y_n)& ={\parallel u_n\parallel }^2-2〈u_n,J{y}_n〉+{\parallel y_n\parallel }^2\\ ={\parallel u_n\parallel }^2-2〈u_n,J{y}_n〉+{\parallel J{y}_n\parallel }^2.\end{array}$$

Taking ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}$ on both sides of the equality above, it follows that

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

From Remark 1.1, $\overline{x}=y$, i.e., $y^{\ast }=J\overline{x}$. It follows that $J{y}_n\rightharpoonup J\overline{x}\in E^{\ast }$ as $n\to \mathrm{\infty }$. From (3.17) and the Kadec-Klee property of $E^{\ast }$, we have

$$J{y}_n\to J\overline{x}$$

as $n\to \mathrm{\infty }$. Since $J^{-1}$ is norm-weak^∗-continuous, $y_n\rightharpoonup \overline{x}$ as $n\to \mathrm{\infty }$. From (3.16) and the Kadec-Klee property of E, we have

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

Step 5. We show that $\overline{x}\in \mathcal{F}$.

By Step 4, we get

$$\underset{n\to \mathrm{\infty }}{lim}\parallel u_n-y_n\parallel =0.$$

The uniform continuity of J on bounded sets gives

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{u}_n-J{y}_n\parallel =0.$$

(3.18)

From the assumption $r_n\ge a$ and (3.18), we see that $\frac{\parallel J{u}_n-J{y}_n\parallel }{r_n}\to 0$ as $n\to \mathrm{\infty }$. But from (A2) and (3.1), we note that

$$\frac{1}{r_n}〈y-u_n,J{u}_n-J{y}_n〉\ge -\theta (u_n,y)\ge \theta (y,u_n),\mathrm{\forall }y\in C$$

and hence

$$\parallel y-u_n\parallel \frac{\parallel J{u}_n-J{y}_n\parallel }{r_n}\ge \theta (y,u_n),\mathrm{\forall }y\in C,$$

which implied that $\theta (y,\overline{x})\le 0$ for all $y\in C$. Put $y_t=ty+(1-t)\overline{x}$ for all $t\in (0,1]$ and $y\in C$. Then we get $y_t\in C$ and $\theta (y_t,\overline{x})\le 0$. Therefore, from (A1) and (A4), we obtain

$$\begin{array}{rl}0& =\theta (y_t,y_t)\le t\theta (y_t,y)+(1-t)\theta (y_t,\overline{x})\\ \le t\theta (y_t,y).\end{array}$$

Thus, $\theta (y_t,y)\ge 0$ for all $y\in C$. Furthermore, as $t\to \mathrm{\infty }$, we have from (A3) that $\theta (\overline{x},y)\ge 0$ for all $y\in C$. This implies that $\overline{x}\in EP(\theta )$.

Finally, we show that $\overline{x}\in F(T)$. In view of $y_n=J^{-1}({\alpha }_{n}J{x}_1+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{x}_n)$, we find that

$$J{u}_n-J{y}_n={\alpha }_n(J{u}_n-J{x}_1)+{\beta }_n(J{u}_n-J{T}^n{x}_n)+{\gamma }_n(J{u}_n-J{x}_n).$$

Hence we have

$$\begin{array}{rl}{\beta }_n\parallel J{u}_n-J{T}^n{x}_n\parallel \le & \parallel J{u}_n-J\overline{x}\parallel +\parallel J\overline{x}-J{y}_n\parallel +{\alpha }_n\parallel J{u}_n-J{x}_1\parallel \\ +{\gamma }_n\parallel J{u}_n-J{x}_n\parallel .\end{array}$$

From the assumptions (ii), (iii), and (3.13), we have

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

(3.19)

Notice that

$$\parallel J{T}^n{x}_n-J\overline{x}\parallel \le \parallel J{T}^n{x}_n-J{u}_n\parallel +\parallel J{u}_n-J\overline{x}\parallel .$$

This implies from (3.19) that

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

(3.20)

The demicontinuity of $J^{-1}:E^{\ast }\to E$ implies that $T^n{x}_n\rightharpoonup \overline{x}$ as $n\to \mathrm{\infty }$. We have

$$|\parallel T^n{x}_n\parallel -\parallel \overline{x}\parallel |=|\parallel J{T}^n{x}_n\parallel -\parallel J\overline{x}\parallel |\le \parallel J{T}^n{x}_n-J\overline{x}\parallel .$$

With the aid of (3.20), we see that ${lim}_{n\to \mathrm{\infty }}\parallel T^n{x}_n\parallel =\parallel \overline{x}\parallel$. Since E has the Kadec-Klee property, we find that

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

(3.21)

Since

$$\parallel T^{n+1}{x}_n-\overline{x}\parallel \le \parallel T^{n+1}{x}_n-T^n{x}_n\parallel +\parallel T^n{x}_n-\overline{x}\parallel ,$$

we find from (3.21) and the asymptotic regularity of T that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel T^{n+1}{x}_n-\overline{x}\parallel =0,$$

i.e., $T{T}^n{x}_n-\overline{x}\to 0$ as $n\to \mathrm{\infty }$. It follows from the closedness of T that $T\overline{x}=\overline{x}$. So, $\overline{x}\in F(T)$ and hence $\overline{x}\in \mathcal{F}=F(T)\cap EP(\theta )$.

Step 6. We show that $\overline{x}={\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$ and so $x_n\to {\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$ as $n\to \mathrm{\infty }$.

Since ℱ is a closed convex set, it follows from Lemma 2.2 that ${\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$ is single-valued, which is denoted by $\tilde{x}$. By the definition of $x_n={\mathrm{\Pi }}_{C_n}^f{x}_1$ and $\tilde{x}\in \mathcal{F}\subset C_n$, we also have

$$G(x_n,J{x}_1)\le G(\tilde{x},J{x}_1)$$

for all $n\ge 1$. By the definition of G, we know that for any $x\in E$, $G(u,Jx)$ is convex and lower semicontinuous with respect to u and so

$$\begin{array}{rl}G(\overline{x},J{x}_1)& \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}G(x_n,J{x}_1)\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}G(x_n,J{x}_1)\\ \le G(\tilde{x},J{x}_1).\end{array}$$

From the definition of ${\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$ and $\overline{x}\in \mathcal{F}$, we conclude that

$$\overline{x}=\tilde{x}={\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$$

and $x_n\to \overline{x}={\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$ as $n\to \mathrm{\infty }$. This completes the proof. □

Remark 3.1

• (i) If $f=0$, then $G(x,Jy)=\varphi (x,y)$ and ${\mathrm{\Pi }}_{C_n}^f={\mathrm{\Pi }}_{C_n}$.

• (ii) If we take $f=0$, $\theta =0$, $u_n=y_n$, and ${\alpha }_n=0$ for all $n\in \mathbb{N}$, then the iterative scheme (3.1) reduces to the following scheme:

  $$\{\begin{array}{c}{x}_0\in E\text{ chosen arbitrarily},\hfill \\ C_1=C,\hfill \\ y_n=J^{-1}({\beta }_{n}J{T}^n{x}_n+(1-{\beta }_n)J{x}_n),\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,y_n)\le \varphi (z,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

where ${\xi }_n=max\{0,{sup}_{p\in F(T),x\in C}(\varphi (p,T^{n}x)-\varphi (p,x))\}$, which is the algorithm introduced by Hao [10] and an improvement to (1.3).

If T is quasi-ϕ-nonexpansive, then Theorem 3.1 is reduced to following without the boundedness of $F(T)$ and the asymptotically regularity of T.

Corollary 3.1 Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let θ be a bifunction from $C\times C$ to ℝ satisfying the conditions (A1)-(A4). Let $T:C\to C$ be a closed and quasi-ϕ-nonexpansive mapping. Assume that $\mathcal{F}=F(T)\cap EP(\theta )$ is nonempty. Let $f:E\to {\mathbb{R}}^{+}$ be a convex and lower semicontinuous function with $C\subset int(D(f))$ and $f(0)=0$. Let $\{{\alpha }_n\}$ be a sequence in $[0,1]$ and $\{{\beta }_n\}$, $\{{\gamma }_n\}$ be sequences in $(0,1)$ satisfying the following conditions:

• (i) ${\alpha }_n+{\beta }_n+{\gamma }_n=1$;

• (ii) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$;

• (iii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$.

Let $\{x_n\}$ be a sequence generated by

$$\{\begin{array}{c}{x}_1\in E\mathit{\text{ chosen arbitrarily}},\hfill \\ C_1=C,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_1+{\beta }_{n}JT{x}_n+{\gamma }_{n}J{x}_n)\hfill \\ u_n\in C\mathit{\text{ such that }}\theta (u_n,y)+\frac{1}{r_n}〈y-u_n,J{u}_n-J{y}_n〉\ge 0,\mathrm{\forall }y\in C,\hfill \\ C_{n+1}=\{z\in C_n:G(z,J{u}_n)\le {\alpha }_{n}G(z,J{x}_1)+(1-{\alpha }_n)G(z,J{x}_n)\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}^f{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

where $\{r_n\}$ is a real sequence in $[a,\mathrm{\infty })$ for some $a>0$ and ${\mathrm{\Pi }}_{C_{n+1}}^f$ is the generalized f-projection operator. Then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$.

Remark 3.2

• (i) By Remark 3.1, Theorem 3.1 extends Theorem 2.1 of Hao [10].

• (ii) Theorem 3.1 generalizes Theorem 3.1 of Matsushita and Takahashi [9] in the following respects:

• from the relatively nonexpansive mapping to the asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense;

• from a uniformly convex and uniformly smooth Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property;

• (iii) in view of the mappings and the frame work of the spaces, Theorem 3.1 generalizes and improves Theorem 3.1 of Ma et al. [16], Theorem 3.1 of Qin et al. [17], Theorem 3.1 of Qing and Lv [18] and Theorem 3.1 of Saewan [19].

We now provide a nontrivial family of mappings satisfying the conditions of Theorem 3.1.

Example 3.1 Let $E=\mathbb{R}$ with the standard norm $\parallel \cdot \parallel =|\cdot |$ and $C=[0,1]$. Let $T:C\to C$ be a mapping defined by

$$Tx=\{\begin{array}{cc}\frac{1}{2}x,\hfill & x\in [0,\frac{1}{2}],\hfill \\ 0,\hfill & x\in (\frac{1}{2},1].\hfill \end{array}$$

We first show that T is an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense with $F(T)=\{0\}\ne \varphi$. In fact, for $p=0\in F(T)$, we have

$$\begin{array}{rl}\varphi (p,T^{n}x)& ={|0-T^{n}x|}^2\\ =\frac{1}{2^{2n}}{|x|}^2\\ \le {|0-x|}^2=\varphi (p,x),\mathrm{\forall }x\in [0,\frac{1}{2}]\end{array}$$

and

$$\begin{array}{rl}\varphi (p,T^{n}x)& ={|0-T^{n}x|}^2\\ =0\\ \le {|0-x|}^2=\varphi (p,x),\mathrm{\forall }x\in (\frac{1}{2},1].\end{array}$$

Therefore, we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{p\in F(T),x\in C}{sup}(\varphi (p,T^{n}x)-\varphi (p,x))\le 0.$$

Next, we define a bifunction $\theta :C\times C\to \mathbb{R}$ satisfying the conditions (A1)-(A4) by

$$\theta (x,y)=y^2-x^2.$$

Then the set of solutions $EP(\theta )$ to the equilibrium problem for θ is obviously $\{0\}$. Since $\mathcal{F}=F(T)\cap EP(\theta )\ne \varphi$ and $F(T)$ is bounded, it follows from Theorem 3.1 that the sequence defined by (3.1) converges strongly to ${\mathrm{\Pi }}_{\mathcal{F}}^f{x}_1$.

Author’s contributions

JUJ conceived of the study, its design, and its coordination. The author read and approved the final manuscript.