Hybrid projection methods for a bifunction and relatively asymptotically nonexpansive mappings

Abstract

The purpose of this paper is to investigate a bifunction equilibrium problem and a fixed point problem of relatively asymptotically nonexpansive mappings based on a generalized projection method. A weak convergence theorem for common solutions is established in a uniformly smooth and uniformly convex Banach space.

1 Introduction and preliminaries

Let E be a real Banach space, $E^{\ast }$ be the dual space of E, and C be a nonempty subset of E. Let F be a bifunction from $C\times C$ to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem: Find $\overline{x}\in C$ such that

$$F(\overline{x},y)\ge 0,\mathrm{\forall }y\in C.$$

(1.1)

From now on, we use $\mathit{EP}(F)$ to denote the solution set of equilibrium problem (1.1) and assume that F satisfies the following conditions:

• (A1) $F(x,x)=0$, $\mathrm{\forall }x\in C$;

• (A2) F is monotone, i.e., $F(x,y)+F(y,x)\le 0$, $\mathrm{\forall }x,y\in C$;

• (A3)
  $$\underset{t\downarrow 0}{lim\hspace{0.17em}sup}F(tz+(1-t)x,y)\le F(x,y),\mathrm{\forall }x,y,z\in C;$$

• (A4) for each $x\in C$, $y\mapsto F(x,y)$ is convex and weakly lower semi-continuous.

Let $U_E=\{x\in E:\parallel x\parallel =1\}$ be the unit sphere of E. Then the Banach space E is said to be smooth iff

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

exists for each $x,y\in U_E$. It is also said to be uniformly smooth iff the above limit is attained uniformly for $x,y\in U_E$. It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. Recall that E is said to be uniformly convex iff ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$ for any two sequences $\{x_n\}$ and $\{y_n\}$ in E such that $\parallel x_n\parallel =\parallel y_n\parallel =1$ and ${lim}_{n\to \mathrm{\infty }}\parallel \frac{x_n+y_n}{2}\parallel =1$. It is well known that E is uniformly smooth if and only if $E^{\ast }$ is uniformly convex.

Recall that a Banach space E enjoys 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 $n\to \mathrm{\infty }$. For more details on the Kadec-Klee property, the readers can refer to [1] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.

Let $T:C\to C$ be a mapping. From now on, we use $F(T)$ to denote the fixed point set of T. Recall that T 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 ${lim}_{n\to \mathrm{\infty }}T{x}_n=y_0$, then $T{x}_0=y_0$. In this paper, we use → and ⇀ to denote the strong convergence and the weak convergence, respectively.

Recall that the normalized duality mapping J from E to $2^{E^{\ast }}$ is defined by

$$Jx=\{f^{\ast }\in E^{\ast }:〈x,f^{\ast }〉={\parallel x\parallel }^2={\parallel f^{\ast }\parallel }^2\},$$

where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing. Next, we assume that E is a smooth Banach space. Consider the functional defined by

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

Observe that in a Hilbert space H the equality is reduced to $\varphi (x,y)={\parallel x-y\parallel }^2$, $x,y\in H$. As we all know, if C is a nonempty closed convex subset of a Hilbert space H and $P_C:H\to C$ is the metric projection of H onto C, then $P_C$ is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [2] recently introduced a generalized projection operator ${\mathrm{\Pi }}_C$ in a Banach space E which is an analogue of the metric projection $P_C$ in Hilbert spaces. Recall that the generalized projection ${\mathrm{\Pi }}_C:E\to C$ is a map 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 the solution to the minimization problem

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

Existence and uniqueness of the operator ${\mathrm{\Pi }}_C$ follow from the properties of the functional $\varphi (x,y)$ and strict monotonicity of the mapping J. In Hilbert spaces, ${\mathrm{\Pi }}_C=P_C$. It is obvious from the definition of a function ϕ that

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

(1.2)

Remark 1.1 If E is a reflexive, strictly convex, and smooth Banach space, then $\varphi (x,y)=0$ if and only if $x=y$; for more details, see [2] and the references therein.

Recall that a point p in C is said to be an asymptotic fixed point of a mapping T iff C contains a sequence $\{x_n\}$ which converges weakly to p so that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T^n{x}_n\parallel =0$. The set of asymptotic fixed points of T will be denoted by $\tilde{F}(T)$.

Recall that a mapping T is said to be relatively nonexpansive iff

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

Recall that a mapping T is said to be relatively asymptotically nonexpansive iff

$$\tilde{F}(T)=F(T)\ne \mathrm{\varnothing },\varphi (p,T^{n}x)\le (1+{\mu }_n)\varphi (p,x),\mathrm{\forall }x\in C,\mathrm{\forall }p\in F(T),\mathrm{\forall }n\ge 1,$$

where $\{{\mu }_n\}\subset [0,\mathrm{\infty })$ is a sequence such that ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$.

Remark 1.2 The class of relatively nonexpansive mappings was first considered in Butnariu et al. [3]. The class of relatively asymptotically nonexpansive mappings was first considered in Agarwal et al. [4] and the references therein.

Recently, many authors investigated fixed point problems of a (relatively) nonexpansive mapping based on hybrid projection methods; for more details, see [5–37] and the references therein. However, most of the results are on strong convergence. In this article, we investigate a bifunction equilibrium problem and a fixed point problem of relatively asymptotically nonexpansive mappings based on a generalized projection method. A weak convergence theorem for common solutions is established in a uniformly smooth and uniformly convex Banach space.

The following lemmas play an important role in this paper.

Lemma 1.3 [37, 38]

Let C be a closed convex subset of a uniformly smooth and uniformly convex Banach space E. Let F be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4). Let $r>0$ and $x\in E$. Then there exists $z\in C$ such that $F(z,y)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0$, $\mathrm{\forall }y\in C$. Define a mapping $S_r:E\to C$ by $S_{r}x=\{z\in C:F(z,y)+\frac{1}{r}〈y-z,Jz-Jx〉,\mathrm{\forall }y\in C\}$. Then the following conclusions hold:

• (a) $S_r$ is single-valued;

• (b) $S_r$ is a firmly nonexpansive-type mapping, i.e., for all $x,y\in E$,

  $$〈S_{r}x-S_{r}y,J{S}_{r}x-J{S}_{r}y〉\le 〈S_{r}x-S_{r}y,Jx-Jy〉;$$

• (c) $F(S_r)=\mathit{EP}(F)$ is closed and convex;

• (d) $S_r$ is relatively nonexpansive;

• (e) $\varphi (q,S_{r}x)+\varphi (S_{r}x,x)\le \varphi (q,x)$, $\mathrm{\forall }q\in F(S_r)$.

Lemma 1.4 [4]

Let E be a uniformly smooth and uniformly convex Banach space. Let C be a nonempty closed and convex subset of E. Let $T:C\to C$ be a relatively asymptotically nonexpansive mapping. Then $F(T)$ is a closed convex subset of C.

Lemma 1.5 [2]

Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty, closed, and convex subset of E, and let $x\in E$. Then

$$\varphi (y,{\mathrm{\Pi }}_{C}x)+\varphi ({\mathrm{\Pi }}_{C}x,x)\le \varphi (y,x),\mathrm{\forall }y\in C.$$

Lemma 1.6 [2]

Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and let $x\in E$. Then $x_0={\mathrm{\Pi }}_{C}x$ if and only if

$$〈x_0-y,Jx-J{x}_0〉\ge 0,\mathrm{\forall }y\in C.$$

Lemma 1.7 [39]

Let E be a smooth and uniformly convex Banach space, and let $r>0$. Then there exists a strictly increasing, continuous, and convex function $g:[0,2r]\to R$ such that $g(0)=0$ and

$${\parallel tx+(1-t)y\parallel }^2\le t{\parallel x\parallel }^2+(1-t){\parallel y\parallel }^2-t(1-t)g(\parallel x-y\parallel )$$

for all $x,y\in B_r=\{x\in E:\parallel x\parallel \le r\}$ and $t\in [0,1]$.

Lemma 1.8 [40]

Let $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be three nonnegative sequences satisfying the following condition:

$$a_{n+1}\le (1+b_n)a_n+c_n,\mathrm{\forall }n\ge n_0,$$

where $n_0$ is some nonnegative integer. If ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{c}_n<\mathrm{\infty }$, then the limit of the sequence $\{a_n\}$ exists. If, in addition, there exists a subsequence $\{{\alpha }_{n_i}\}\subset \{{\alpha }_n\}$ such that ${\alpha }_{n_i}\to 0$, then ${\alpha }_n\to 0$ as $n\to \mathrm{\infty }$.

Lemma 1.9 [41]

Let E be a smooth and uniformly convex Banach space, and let $r>0$. Then there exists a strictly increasing, continuous, and convex function $g:[0,2r]\to R$ such that $g(0)=0$ and $g(\parallel x-y\parallel )\le \varphi (x,y)$ for all $x,y\in B_r$.

2 Main results

Theorem 2.1 Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed and convex subset of E. Let F be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4). Let $T:C\to C$ be a relatively asymptotically nonexpansive mapping with the sequence $\{{\mu }_{n,1}\}$, and let $S:C\to C$ be a relatively asymptotically nonexpansive mapping with the sequence $\{{\mu }_{n,2}\}$. Assume that $\mathrm{\Phi }:=F(T)\cap F(S)\cap \mathit{EP}(F)$ is nonempty. Let $\{x_n\}$ be a sequence generated in the following manner:

$$\{\begin{array}{c}{y}_0\in E\mathit{\text{chosen arbitrarily}},\hfill \\ x_n\in C\mathit{\text{ such that }}F(x_n,x)+\frac{1}{r_n}〈x-x_n,J{x}_n-J{y}_n〉\ge 0,\mathrm{\forall }x\in C,\hfill \\ y_{n+1}=J^{-1}({\alpha }_{n}J{x}_n+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{S}^n{x}_n),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

where $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$ are real sequences in $[0,1]$ and $\{r_n\}$ is a real number sequence in $[r,\mathrm{\infty })$, where $r>0$ is some real number. Assume that J is weakly sequentially continuous and the following restrictions hold:

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

• (ii) ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$;

• (iii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_n>0$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_n>0$.

Then the sequence $\{x_n\}$ converges weakly to $\overline{x}\in \mathrm{\Phi }$, where $\overline{x}={lim}_{n\to \mathrm{\infty }}{\mathrm{\Pi }}_{\mathrm{\Phi }}{x}_n$.

Proof Set ${\mu }_n=max\{{\mu }_{n,1},{\mu }_{n,2}\}$. Fixing $p\in \mathrm{\Phi }$, we find that

$$\begin{array}{rcl}\varphi (p,x_{n+1})& =& \varphi (p,S_{r_{n+1}}{y}_{n+1})\\ \le & \varphi (p,y_{n+1})\\ =& {\parallel p\parallel }^2-2〈p,{\alpha }_{n}J{x}_n+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{S}^n{x}_n〉\\ +{\parallel {\alpha }_{n}J{x}_n+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{S}^n{x}_n\parallel }^2\end{array}$$

(2.1)

$$\begin{array}{rc}\le & {\parallel p\parallel }^2-2{\alpha }_n〈p,J{x}_n〉-2{\beta }_n〈p,J{T}^n{x}_n〉-2{\gamma }_n〈p,J{S}^n{x}_n〉\\ +{\alpha }_n{\parallel x_n\parallel }^2+{\beta }_n{\parallel T^n{x}_n\parallel }^2+{\gamma }_n{\parallel S^n{x}_n\parallel }^2\\ =& {\alpha }_n\varphi (p,x_n)+{\beta }_n\varphi (p,T^n{x}_n)+{\gamma }_n\varphi (p,S^n{x}_n)\\ \le & \varphi (p,x_n)+{\beta }_n{\mu }_n\varphi (p,x_n)+{\gamma }_n{\mu }_n\varphi (p,x_n)\\ \le & (1+{\mu }_n)\varphi (p,x_n).\end{array}$$

(2.2)

In view of Lemma 1.8, we obtain that ${lim}_{n\to \mathrm{\infty }}\varphi (p,x_n)$ exits. This implies that the sequence $\{x_n\}$ is bounded. In the light of Lemma 1.7, we find that

$$\begin{array}{rcl}\varphi (p,x_{n+1})& =& \varphi (p,S_{r_{n+1}}{y}_{n+1})\\ \le & {\parallel p\parallel }^2-2〈p,{\alpha }_{n}J{x}_n+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{S}^n{x}_n〉\\ +{\parallel {\alpha }_{n}J{x}_n+{\beta }_{n}J{T}^n{x}_n+{\gamma }_{n}J{S}^n{x}_n\parallel }^2\\ \le & {\parallel p\parallel }^2-2{\alpha }_n〈p,J{x}_n〉-2{\beta }_n〈p,J{T}^n{x}_n〉-2{\gamma }_n〈p,J{S}^n{x}_n〉\\ +{\alpha }_n{\parallel x_n\parallel }^2+{\beta }_n{\parallel T^n{x}_n\parallel }^2+{\gamma }_n{\parallel S^n{x}_n\parallel }^2-{\alpha }_n{\beta }_{n}g\left(\parallel J{T}^n{x}_n-J{x}_n\parallel \right)\\ \le & \varphi (p,x_n)+{\beta }_n{\mu }_n\varphi (p,x_n)+{\gamma }_n{\mu }_n\varphi (p,x_n)-{\alpha }_n{\beta }_{n}g\left(\parallel J{T}^n{x}_n-J{x}_n\parallel \right)\\ \le & (1+{\mu }_n)\varphi (p,x_n)-{\alpha }_n{\beta }_{n}g\left(\parallel J{T}^n{x}_n-J{x}_n\parallel \right).\end{array}$$

It follows that

$${\alpha }_n{\beta }_{n}g\left(\parallel J{T}^n{x}_n-J{x}_n\parallel \right)\le (1+{\mu }_n)\varphi (p,x_n)-\varphi (p,x_{n+1}).$$

This finds from the restrictions (ii) and (iii) that

$$\underset{n\to \mathrm{\infty }}{lim}g\left(\parallel J{T}^n{x}_n-J{x}_n\parallel \right)=0.$$

This implies that

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

Since $J^{-1}$ is uniformly norm-to-norm continuous on bounded sets, we find that

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

In the same way, we find that

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

Since $\{x_n\}$ is bounded, we see that there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $\{x_{n_i}\}$ converges weakly to $p\in C$. It follows that $p\in F(T)\cap F(S)$. Next, we prove that $p\in \mathit{EP}(F)$. Let $r={sup}_{n\ge 1}\{\parallel x_n\parallel ,\parallel y_n\parallel \}$. In view of Lemma 1.9, we find that there exists a continuous, strictly increasing and convex function h with $h(0)=0$ such that

$$h(x,y)\le \varphi (x,y),\mathrm{\forall }x,y\in B_r.$$

It follows from (2.1) that

$$\begin{array}{rl}h(\parallel x_n-y_n\parallel )& \le \varphi (x_n,y_n)\\ \le \varphi (p,y_n)-\varphi (p,x_n)\\ \le \varphi (p,x_{n-1})-\varphi (p,x_n)+{\mu }_{n-1}\varphi (p,x_{n-1}).\end{array}$$

This implies that

$$\underset{n\to \mathrm{\infty }}{lim}h(\parallel x_n-y_n\parallel )=0.$$

It follows from the property of h that

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

Since J is uniformly norm-to-norm continuous on bounded sets, one has

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

Since $\{r_n\}$ is a real number sequence in $[r,\mathrm{\infty })$, where $r>0$ is some real number, one finds that

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

Notice that $x_n=S_{r_n}{y}_n$, one sees that

$$F(x_n,x)+\frac{1}{r_n}〈x-x_n,J{x}_n-J{y}_n〉\ge 0,\mathrm{\forall }x\in C.$$

By replacing n by $n_i$, one finds from (A2) that

$$\begin{array}{rl}\parallel x-x_{n_i}\parallel \frac{\parallel J{x}_{n_i}-J{y}_{n_i}\parallel }{r_{n_i}}& \ge \frac{1}{r_{n_i}}〈x-x_{n_i},J{x}_{n_i}-J{y}_{n_i}〉\\ \ge F(x,x_{n_i}).\end{array}$$

Letting $i\to \mathrm{\infty }$ in the above inequality, one obtains from (A4) that

$$F(x,p)\le 0,\mathrm{\forall }x\in C.$$

For $0<t<1$ and $y\in C$, define $x_t=tx+(1-t)p$. It follows that $x_t\in C$, which yields that $F(x_t,p)\le 0$. It follows from (A1) and (A4) that

$$0=F(x_t,x_t)\le tF(x_t,x)+(1-t)F(x_x,p)\le tF(x_t,x).$$

That is,

$$F(x_t,x)\ge 0.$$

Letting $t\downarrow 0$, we obtain from (A3) that $F(p,x)\ge 0$, $\mathrm{\forall }x\in C$. This implies that $p\in \mathit{EP}(F)$. This completes the proof that $p\in F(T)\cap F(S)\cap \mathit{EP}(F)$. Define $z_n={\mathrm{\Pi }}_{F(T)\cap F(S)\cap \mathit{EP}(F)}{x}_n$. It follows from (2.1) that

$$\varphi (z_n,x_{n+1})\le (1+{\mu }_n)\varphi (z_n,x_n).$$

(2.3)

This in turn implies from Lemma 1.5 that

$$\begin{array}{rl}\varphi (z_{n+1},x_{n+1})& =\varphi ({\mathrm{\Pi }}_{F(T)\cap F(S)\cap \mathit{EP}(F)}{x}_{n+1},x_{n+1})\\ \le \varphi (z_n,x_{n+1})-\varphi (z_n,{\mathrm{\Pi }}_{F(T)\cap F(S)\cap \mathit{EP}(F)}{x}_{n+1})\\ \le \varphi (z_n,x_{n+1})-\varphi (z_n,z_{n+1})\\ \le \varphi (z_n,x_{n+1}).\end{array}$$

It follows from (2.3) that

$$\varphi (z_{n+1},x_{n+1})\le (1+{\mu }_n)\varphi (z_n,x_n).$$

This finds from Lemma 1.8 that the sequence $\{\varphi (z_n,x_n)\}$ is a convergence sequence. It follows from (2.1) that

$$\varphi (p,x_{n+m})\le \varphi (p,x_n)+L\left(\sum _{i=1}^m{\mu }_{n+m-i}\right),$$

(2.4)

where $L={sup}_{n\ge 1}\varphi (p,x_n)$. Since $z_n\in F(T)\cap F(S)\cap \mathit{EP}(F)$, we find that

$$\varphi (z_n,x_{n+m})\le \varphi (z_n,x_n)+M\left(\sum _{i=1}^m{\mu }_{n+m-i}\right),$$

where $M={sup}_{n\ge 1}\varphi (z_n,x_n)$. Since $z_{n+m}={\mathrm{\Pi }}_{F(T)\cap F(S)\cap \mathit{EP}(F)}{x}_{n+m}$, we find from Lemma 1.5 that

$$\varphi (z_n,z_{n+m})+\varphi (z_{n+m},x_{n+m})\le \varphi (z_n,x_{n+m})\le \varphi (z_n,x_n)+M\left(\sum _{i=1}^m{\mu }_{n+m-i}\right).$$

It follows that

$$\varphi (z_n,z_{n+m})\le \varphi (z_n,x_n)-\varphi (z_{n+m},x_{n+m})+M\left(\sum _{i=1}^m{\mu }_{n+m-i}\right).$$

In view of Lemma 1.9, we find that there exists a continuous, strictly increasing, and convex function g with

$$g(\parallel z_n-z_m\parallel )\le \varphi (z_n,z_m)\le \varphi (z_n,x_n)-\varphi (z_{n+m},x_{n+m})+M\left(\sum _{i=1}^m{\mu }_{n+m-i}\right).$$

This shows that $\{z_n\}$ is a Cauchy sequence. Since $F(T)\cap F(S)\cap \mathit{EP}(F)$ is closed, one sees that $\{z_n\}$ converges strongly to $z\in F(T)\cap F(S)\cap \mathit{EP}(F)$. Since $p\in F(T)\cap F(S)\cap \mathit{EP}(F)$, we find from Lemma 1.6 that

$$〈z_{n_k}-p,J{x}_{n_k}-J{z}_{n_k}〉\ge 0.$$

Notice that J is weakly sequentially continuous. Letting $k\to \mathrm{\infty }$, we find that $〈z-p,Jp-Jz〉\ge 0$. It follows from the monotonicity of J that $〈z-p,Jp-Jz〉\le 0$. Since the space is uniformly convex, we find that $z=p$. This completes the proof. □

Remark 2.2 Theorem 2.1 improves Theorem 2.5 in Qin et al. [36] on the mappings from the class of relatively nonexpansive mappings to the class of relatively asymptotically nonexpansive mappings.

If $T=S$, then Theorem 2.1 is reduced to the following.

Corollary 2.3 Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed and convex subset of E. Let F be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4). Let $T:C\to C$ be a relatively asymptotically nonexpansive mapping with the sequence $\{{\mu }_n\}$. Assume that $\mathrm{\Phi }:=F(T)\cap \mathit{EP}(F)$ is nonempty. Let $\{x_n\}$ be a sequence generated in the following manner:

$$\{\begin{array}{c}{y}_0\in E\mathit{\text{chosen arbitrarily}},\hfill \\ x_n\in C\mathit{\text{ such that }}F(x_n,x)+\frac{1}{r_n}〈x-x_n,J{x}_n-J{y}_n〉\ge 0,\mathrm{\forall }x\in C,\hfill \\ y_{n+1}=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n)J{T}^n{x}_n),\mathrm{\forall }n\ge 0,\hfill \end{array}$$

where $\{{\alpha }_n\}$ is a real sequence in $[0,1]$ and $\{r_n\}$ is a real number sequence in $[r,\mathrm{\infty })$, where $r>0$ is some real number. Assume that J is weakly sequentially continuous and the following restrictions hold:

• (i) ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$;

• (ii) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n(1-{\alpha }_n)>0$.

Then the sequence $\{x_n\}$ converges weakly to $\overline{x}\in \mathrm{\Phi }$, where $\overline{x}={lim}_{n\to \mathrm{\infty }}{\mathrm{\Pi }}_{\mathrm{\Phi }}{x}_n$.

Remark 2.4 Corollary 2.3 is an improvement of Theorem 4.1 in Zembayashi and Takahashi [37] on the nonlinear mapping.