Strong convergence theorems of the Halpern-Mann’s mixed iteration for a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping in Banach spaces

Abstract

In this paper, we introduce a class of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping to modify the Halpern-Mann-type iteration algorithm for a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping, which has the strong convergence under a limit condition only in the framework of Banach spaces. Our results are applied to study the approximation problem of solution to a system of equilibrium problems. Also, the results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:7864-7870, 2012) and others.

1 Introduction and preliminaries

A Banach space X is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel \le 1$ for all $x,y\in X$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. A Banach space is said to be uniformly convex if ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$ for any two sequences $\{x_n\},\{y_n\}\subset X$ with $\parallel x_n\parallel =\parallel y_n\parallel =1$ and ${lim}_{n\to \mathrm{\infty }}\parallel \frac{x_n+y_n}{2}\parallel =0$.

The norm of Banach space X is said to be Gâteaux differentiable, if, for each $x,y\in S(x)$, the limit

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

(1.1)

exists, where $S(x)=\{x:\parallel x\parallel =1,x\in X\}$. In this case, X is said to be smooth. The norm of Banach space X is said to be Fréchet differentiable, if, for each $x\in S(x)$, the limit (1.1) is attained uniformly for $y\in S(x)$ and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for $x,y\in S(x)$. In this case, X is said to be uniformly smooth.

Let D be a nonempty closed subset of a real Banach space X. A mapping $T:D\to D$ is said to be nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in D$. An element $p\in D$ is called a fixed point of a nonself multi-valued mapping $T:D\to X$ if $p\in Tp$. The set of fixed points of T is represented by $F(T)$.

A subset D of X is said to be retract of X, if there exists a continuous mapping $P:X\to D$ such that $Px=x$, for all $x\in X$. It is well known that every nonempty, closed, convex subset of a uniformly convex Banach space X is a retract of X. A mapping $P:X\to D$ is said to be a retraction, if $P^2=P$. It follows that if a mapping P is a retraction, then $Py=y$ for all y in the range of P. A mapping $P:X\to D$ is said to be a nonexpansive retraction, if it is nonexpansive and it is a retraction from X to D.

Assume that X is a real Banach space with the dual $X^{\ast }$, D is a nonempty, closed, convex subset of X. We also denote by J the normalized duality mapping from X to $2^{X^{\ast }}$ which is defined by

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

where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing.

Next we assume that X is a smooth, strictly convex and reflexive Banach space and D is a nonempty, closed, convex subset of X. In the sequel, we always use $\varphi :X\times X\to R^{+}$ to denote the Lyapunov functional defined by

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

(1.2)

It is obvious from the definition of the function ϕ that

$${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^2,$$

(1.3)

$$\varphi (y,x)=\varphi (y,z)+\varphi (z,x)+2〈z-y,Jx-Jz〉,x,y,z\in X,$$

(1.4)

and

$$\varphi (x,J^{-1}(\lambda Jy+(1-\lambda )Jz))\le \lambda \varphi (x,y)+(1-\lambda )\varphi (x,z)$$

(1.5)

for all $\lambda \in [0,1]$ and $x,y,z\in X$.

Following Alber [2], the generalized projection ${\mathrm{\Pi }}_D:X\to D$ is defined by

$${\mathrm{\Pi }}_D(x)=arg\underset{y\in D}{inf}\varphi (y,x),\mathrm{\forall }x\in X.$$

(1.6)

Lemma 1.1 (see [3])

Let X be a uniformly convex and smooth Banach space and let $\{x_n\}$ and $\{y_n\}$ be two sequences of X such that $\{x_n\}$ and $\{y_n\}$ is bounded, if $\varphi (x_n,y_n)\to 0$, then $\parallel x_n-y_n\parallel \to 0$.

Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.

In the sequel, we denote the strong convergence and weak convergence of the sequence $\{x_n\}$ by $x_n\to x$ and $x_n\rightharpoonup x$, respectively.

Lemma 1.2 (see [2])

Let X be a smooth, strictly convex, and reflexive Banach space and D be a nonempty, closed, convex subset of X. Then the following conclusions hold:

• (a) $\varphi (x,y)=0$ if and only if $x=y$;

• (b) $\varphi (x,{\mathrm{\Pi }}_{D}y)+\varphi ({\mathrm{\Pi }}_{D}y,y)\le \varphi (x,y)$, $\mathrm{\forall }x,y\in D$;

• (c) if $x\in X$ and $z\in D$, then $z={\mathrm{\Pi }}_{D}x$ if and only if $〈z-y,Jx-Jz〉\ge 0$, $\mathrm{\forall }y\in D$.

Remark 1.1 (see [4])

Let ${\mathrm{\Pi }}_D$ be the generalized projection from a smooth, reflexive and strictly convex Banach space X onto a nonempty, closed, convex subset D of X. Then ${\mathrm{\Pi }}_D$ is a closed and quasi-ϕ-nonexpansive from X onto D.

Remark 1.2 (see [4])

If H is a real Hilbert space, then $\varphi (x,y)={\parallel x-y\parallel }^2$, and ${\mathrm{\Pi }}_D$ is the metric projection of H onto D.

Definition 1.1 Let $P:X\to D$ be the nonexpansive retraction.

• (1) A nonself multi-valued mapping $T:D\to X$ is said to be quasi-ϕ-nonexpansive, if $F(T)\ne \mathrm{\Phi }$, and

  $$\varphi (p,z_n)\le \varphi (p,x),\mathrm{\forall }x\in D,p\in F(T),z_n\in T{(PT)}^{n-1}x,\mathrm{\forall }n\ge 1;$$

  (1.7)

• (2) A nonself multi-valued mapping $T:D\to X$ is said to be quasi-ϕ-asymptotically nonexpansive, if $F(T)\ne \mathrm{\Phi }$ and there exists a real sequence $k_n\subset [1,+\mathrm{\infty })$, $k_n\to 1$ (as $n\to \mathrm{\infty }$) such that

  $$\varphi (p,z_n)\le k_n\varphi (p,x),\mathrm{\forall }x\in D,p\in F(T),z_n\in T{(PT)}^{n-1}x,\mathrm{\forall }n\ge 1;$$

  (1.8)

• (3) A nonself multi-valued mapping $T:D\to X$ is said to be totally quasi-ϕ-asymptotically nonexpansive, if $F(T)\ne \mathrm{\Phi }$ and there exist nonnegative real sequences $\{v_n\}$, $\{{\mu }_n\}$, with $v_n,{\mu }_n\to 0$ (as $n\to \mathrm{\infty }$) and a strictly increasing continuous function $\zeta :R^{+}\to R^{+}$ with $\zeta (0)=0$ such that

  $$\begin{array}{r}\varphi (p,z_n)\le \varphi (p,x)+v_n\zeta [\varphi (p,x)]+{\mu }_n,\\ \mathrm{\forall }x\in D,p\in F(T),z_n\in T{(PT)}^{n-1}x,\mathrm{\forall }n\ge 1.\end{array}$$

  (1.9)

Remark 1.3 From the definitions, it is obvious that a quasi-ϕ-nonexpansive nonself multi-valued mapping is a quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping, and a quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping is a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping, but the converse is not true.

Now, we give an example of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping.

Example 1.1 (see [4])

Let D be a unit ball in a real Hilbert space $l^2$ and let $T:D\to l^2$ be a nonself multi-valued mapping defined by

$$T:(x_1,x_2,\dots )\to (0,x_1^2,a_2{x}_2,a_3{x}_3,\dots )\in l^2,\mathrm{\forall }(x_1,x_2,\dots )\in D,$$

where $\{a_i\}$ is a sequence in $(0,1)$ such that ${\prod }_{i=2}^{\mathrm{\infty }}{a}_i=\frac{1}{2}$.

It is proved in [5] that

• (i) $\parallel Tx-Ty\parallel \le 2\parallel x-y\parallel$, $\mathrm{\forall }x,y\in D$;

• (ii) $\parallel T^{n}x-T^{n}y\parallel \le 2{\prod }_{j=2}^n{a}_j$, $\mathrm{\forall }x,y\in D$, $n\ge 2$.

Let $\sqrt{k_1}=2$, $\sqrt{k_n}=2{\prod }_{j=2}^n{a}_j$, $n\ge 2$. Then ${lim}_{n\to \mathrm{\infty }}{k}_n=1$. Letting ${\nu }_n=k_n-1$ ($n\ge 2$), $\zeta (t)=t$ ($t\ge 0$) and $\{{\mu }_n\}$ be a nonnegative real sequence with ${\mu }_n\to 0$, then from (i) and (ii) we have

$${\parallel T^{n}x-T^{n}y\parallel }^2\le {\parallel x-y\parallel }^2+{\nu }_n\zeta \left({\parallel x-y\parallel }^2\right)+{\mu }_n,\mathrm{\forall }x,y\in D.$$

Since D is a unit ball in a real Hilbert space $l^2$, it follows from Remark 1.2 that $\varphi (x,y)={\parallel x-y\parallel }^2$, $\mathrm{\forall }x,y\in D$. The inequality above can be written as

$$\varphi (T^{n}x,T^{n}y)\le \varphi (x,y)+{\nu }_n\zeta (\varphi (x,y))+{\mu }_n,\mathrm{\forall }x,y\in D.$$

Again since $0\in D$ and $0\in F(T)$, this implies that $F(T)\ne \mathrm{\Phi }$. From the inequality above, we get

$$\varphi (p,z_n)\le \varphi (p,x)+{\nu }_n\zeta (\varphi (p,x))+{\mu }_n,\mathrm{\forall }p\in F(T),x\in D,z_n\in T{(PT)}^{n-1}x,$$

where P is the nonexpansive retraction. This shows that the mapping T defined above is a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping.

Lemma 1.3 Let X be a smooth, strictly convex and reflexive Banach space and D be a nonempty, closed, convex subset of X. Let $T:D\to X$ be a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping with ${\mu }_1=0$. Then $F(T)$ is a closed and convex subset of D.

Proof Let $\{x_n\}$ be a sequence in $F(T)$ such that $x_n\to p$. Since T is a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping, we have

$$\varphi (x_n,z)\le \varphi (x_n,p)+v_1\zeta (\varphi (x_n,p)),z\in Tp,\mathrm{\forall }n\in N.$$

Therefore,

$$\varphi (p,z)=\underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,z)\le \underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,p)+v_1\zeta (\varphi (x_n,p))=\varphi (p,p)=0.$$

By Lemma 1.2, we obtain $p=z\in Tp$. So we have $p\in F(T)$. This implies that $F(T)$ is closed.

Let $p,q\in F(T)$ and $t\in (0,1)$, and put $w=tp+(1-t)q$. We prove that $w\in F(T)$. Indeed, in view of the definition of ϕ, let $\{u_n\}$ be a sequence generated by $u_1\in Tw,u_2\in T(PT)w,u_3\in T{(PT)}^{2}w,\dots ,u_n\in T{(PT)}^{n-1}w\subset TP{u}_{n-1}$, we have

$$\begin{array}{rcl}\varphi (w,u_n)& =& {\parallel w\parallel }^2-2〈w,J{u}_n〉+{\parallel u_n\parallel }^2\\ =& {\parallel w\parallel }^2-2〈tp+(1-t)q,J{u}_n〉+{\parallel u_n\parallel }^2\\ =& {\parallel w\parallel }^2+t\varphi (p,u_n)+(1-t)\varphi (q,u_n)-t{\parallel p\parallel }^2-(1-t){\parallel q\parallel }^2.\end{array}$$

(1.10)

Since

$$\begin{array}{r}t\varphi (p,u_n)+(1-t)\varphi (q,u_n)\\ \le t[\varphi (p,w)+v_n\zeta [\varphi (p,w)]+{\mu }_n]+(1-t)[\varphi (q,w)+v_n\zeta [\varphi (q,w)]+{\mu }_n]\\ =t\{{\parallel p\parallel }^2-2〈p,Jw〉+{\parallel w\parallel }^2+v_n\zeta [\varphi (p,w)]+{\mu }_n\}\\ +(1-t)\{{\parallel q\parallel }^2-2〈q,Jw〉+{\parallel w\parallel }^2+v_n\zeta [\varphi (q,w)]+{\mu }_n\}\\ =t{\parallel p\parallel }^2+(1-t){\parallel q\parallel }^2-{\parallel w\parallel }^2+t{v}_n\zeta [\varphi (p,w)]+(1-t)v_n\zeta [\varphi (q,w)]+{\mu }_n.\end{array}$$

(1.11)

Substituting (1.11) into (1.10) and simplifying it, we have

$$\varphi (w,u_n)\le t{v}_n\zeta [\varphi (p,w)]+(1-t)v_n\zeta [\varphi (q,w)]+{\mu }_n\to 0(\text{as }n\to \mathrm{\infty }).$$

Hence, $u_n\to w$ holds, which yields $u_{n+1}\to w$. Since TP is closed and $u_{n+1}\in T{(PT)}^{n}w\subset TP{u}_n$, we have $w\in TPw$. It follows from $w\in D$ that $w\in Tw$, i.e., $w\in F(T)$. This implies that $F(T)$ is convex. This completes the proof of Lemma 1.3. □

Definition 1.2 (see [1])

A nonself mapping $T:D\to X$ is said to be uniformly L-Lipschitz continuous, if there exists a constant $L>0$, such that

$$\parallel T{(PT)}^{n-1}x-T{(PT)}^{n-1}y\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in D,\mathrm{\forall }n\ge 1.$$

(1.12)

Definition 1.3 A nonself multi-valued mapping $T:D\to X$ is said to be uniformly L-Lipschitz continuous, if there exists a constant $L>0$, such that

$$d(T{(PT)}^{n-1}x,T{(PT)}^{n-1}y)\le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in D,\mathrm{\forall }n\ge 1,$$

(1.13)

where $d(\cdot ,\cdot )$ is Hausdorff metric.

Strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasi-ϕ-nonexpansive and quasi-ϕ-asymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see [1–4, 6–24]). In recent years, by hybrid projection methods, strong and weak convergence problems for totally quasi-ϕ quasi-ϕ-asymptotically nonexpansive nonself and multi-valued mapping, respectively, was also studied by Kim et al. (see [6, 7]), Li et al. (see [8]), Chang et al. (see [9]) and Yang et al. (see [10]).

Inspired by specialists above, the purpose of this paper is to modify the Halpern-Mann’s mixed type iteration algorithm for a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping, which has the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. [1, 11–13], Hao et al. [14], Guo et al. [15], Yildirim et al. [16], Thianwan [17], Nilsrakoo et al. [18], Pathak et al. [19], Qin et al. [20], Su et al. [21], Wang [22, 23], Yang et al. [24] and others.

2 Main results

Theorem 2.1 Let X be a real uniformly smooth and uniformly convex Banach space, D be a nonempty, closed, convex subset of X. Let $P:X\to D$ be the nonexpansive retraction. Let $T:D\to X$ be a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping with sequence $\{v_n\}$, $\{{\mu }_n\}$ (${\mu }_1=0$), with $v_n,{\mu }_n\to 0$ (as $n\to \mathrm{\infty }$) and a strictly increasing continuous function $\zeta :R^{+}\to R^{+}$ with $\zeta (0)=0$ such that T is uniformly L-Lipschitz continuous. Let $\{{\alpha }_n\}$ be a sequence in $[0,1]$ and $\{{\beta }_n\}$ be a sequence in $(0,1)$ satisfying the following conditions:

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

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

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

$$\{\begin{array}{c}{x}_1\in X\mathit{\text{is arbitrary}};D_1=D,\hfill \\ y_n=J^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)],z_n\in T{(PT)}^{n-1}{x}_n,\hfill \\ D_{n+1}=\{z\in D_n:\varphi (z,y_n)\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{D_{n+1}}{x}_1(n=1,2,\dots ),\hfill \end{array}$$

(2.1)

where ${\xi }_n=v_n{sup}_{p\in F(T)}\zeta (\varphi (p,x_n))+{\mu }_n$, ${\mathrm{\Pi }}_{D_{n+1}}$ is the generalized projection of X onto $D_{n+1}$. If $F(T)\ne \mathrm{\varnothing }$, then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{F(T)}{x}_1$.

Proof (I) First, we prove that $D_n$ are closed and convex subsets in D.

In fact, by Lemma 1.3, $F(T)$ is closed and convex in D. By the assumption, $D_1=D$ is closed and convex. Suppose that $D_n$ is closed and convex for some $n\ge 1$. In view of the definition of ϕ, we have

$$\begin{array}{rcl}{D}_{n+1}& =& \{z\in D_n:\varphi (z,y_n)\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\}\\ =& \{z\in D:\varphi (z,y_n)\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\}\cap D_n\\ =& \{z\in D:2{\alpha }_n〈z,J{x}_1〉+2(1-{\alpha }_n)〈z,J{x}_n〉-2〈z,J{y}_n〉\\ \phantom{\{}\le {\alpha }_n{\parallel x_1\parallel }^2+(1-{\alpha }_n){\parallel x_n\parallel }^2-{\parallel y_n\parallel }^2\}\cap D_n.\end{array}$$

This shows that $D_{n+1}$ is closed and convex. The conclusions are proved.

• (II) Next, we prove that $F(T)\subset D_n$, for all $n\ge 1$.

It is obvious that $F(T)\subset D_1$. Suppose that $F(T)\subset D_n$, $w_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)$ and $z_n\in T{(PT)}^{n-1}{x}_n$. Hence, for any $u\in F(T)\subset D_n$, by (1.5), we have

$$\begin{array}{rcl}\varphi (u,y_n)& =& \varphi (u,J^{-1}({\alpha }_{n}J{x}_1+(1-{\alpha }_n)J{w}_n))\\ \le & {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)\varphi (u,w_n),\end{array}$$

(2.2)

and

$$\begin{array}{rcl}\varphi (u,w_n)& =& \varphi (u,J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n))\\ \le & {\beta }_n\varphi (u,x_n)+(1-{\beta }_n)\varphi (u,z_n)\\ \le & {\beta }_n\varphi (u,x_n)+(1-{\beta }_n)\{\varphi (u,x_n)+v_n\zeta [\varphi (u,x_n)]+{\mu }_n\}\\ =& \varphi (u,x_n)+(1-{\beta }_n)v_n\zeta [\varphi (u,x_n)]+(1-{\beta }_n){\mu }_n.\end{array}$$

(2.3)

Therefore, we have

$$\begin{array}{rcl}\varphi (u,y_n)& \le & {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)[\varphi (u,x_n)+(1-{\beta }_n)v_n\zeta [\varphi (u,x_n)]+(1-{\beta }_n){\mu }_n]\\ \le & {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)\varphi (u,x_n)+v_n\underset{p\in F(T)}{sup}\zeta [\varphi (p,x_n)]\\ =& {\alpha }_n\varphi (u,x_1)+(1-{\alpha }_n)\varphi (u,x_n)+{\xi }_n,\end{array}$$

(2.4)

where ${\xi }_n=v_n{sup}_{p\in F(T)}\zeta (\varphi (p,x_n))+{\mu }_n$. This shows that $u\in D_{n+1}$ and so $F(T)\subset D_n$. The conclusion is proved.

• (III) Now we prove that $\{x_n\}$ converges strongly to some point $p^{\ast }$.

Since $x_n={\mathrm{\Pi }}_{D_n}{x}_1$, from Lemma 1.2(c), we have

$$〈x_n-y,J{x}_1-J{x}_n〉\ge 0,\mathrm{\forall }y\in D_n.$$

Again since $F(T)\subset D_n$, we have

$$〈x_n-u,J{x}_1-J{x}_n〉\ge 0,\mathrm{\forall }u\in F(T).$$

It follows from Lemma 1.2(b) that, for each $u\in F(T)$ and for each $n\ge 1$,

$$\varphi (x_n,x_1)=\varphi ({\mathrm{\Pi }}_{D_n}{x}_1,x_1)\le \varphi (u,x_1)-\varphi (u,x_n)\le \varphi (u,x_1).$$

(2.5)

Therefore, $\{\varphi (x_n,x_1)\}$ is bounded, and so is $\{x_n\}$. Since $x_n={\mathrm{\Pi }}_{D_n}{x}_1$ and $x_{n+1}={\mathrm{\Pi }}_{D_{n+1}}{x}_1\in D_{n+1}\subset D_n$, we have $\varphi (x_n,x_1)\le \varphi (x_{n+1},x_1)$. This implies that $\{\varphi (x_n,x_1)\}$ is nondecreasing. Hence ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_1)$ exists.

By the construction of $\{D_n\}$, for any $m\ge n$, we have $D_m\subset D_n$ and $x_m={\mathrm{\Pi }}_{D_m}{x}_1\in D_n$. This shows that

$$\varphi (x_m,x_n)=\varphi (x_m,{\mathrm{\Pi }}_{D_n}{x}_1)\le \varphi (x_m,x_1)-\varphi (x_n,x_1)\to 0(\text{as }n\to \mathrm{\infty }).$$

It follows from Lemma 1.1 that ${lim}_{n\to \mathrm{\infty }}\parallel x_m-x_n\parallel =0$. Hence $\{x_n\}$ is a Cauchy sequence in D. Since D is complete, without loss of generality, we can assume that ${lim}_{n\to \mathrm{\infty }}{x}_n=p^{\ast }$ (some point in D).

By the assumption, it is easy to see that

$$\underset{n\to \mathrm{\infty }}{lim}{\xi }_n=\underset{n\to \mathrm{\infty }}{lim}[v_n\underset{p\in F(T)}{sup}\zeta (\varphi (p,x_n))+{\mu }_n]=0.$$

(2.6)

• (IV) Now we prove that $p^{\ast }\in F(T)$.

Since $x_{n+1}\in D_{n+1}$, from (2.1) and (2.6), we have

$$\varphi (x_{n+1},y_n)\le {\alpha }_n\varphi (x_{n+1},x_1)+(1-{\alpha }_n)\varphi (x_{n+1},x_n)+{\xi }_n\to 0.$$

(2.7)

Since $x_n\to p^{\ast }$, it follows from (2.7) and Lemma 1.1 that

$$y_n\to p^{\ast }.$$

(2.8)

Since $\{x_n\}$ is bounded and T is a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping, we have

$$\varphi (p,z_n)\le \varphi (p,x_n)+v_n\zeta [\varphi (p,x_n)]+{\mu }_n,\mathrm{\forall }x\in D,\mathrm{\forall }n,i\ge 1,p\in F(T).$$

This implies that $\{z_n\}$ is also bounded.

By condition (ii), we have

$$\begin{array}{rcl}\parallel w_n\parallel & =& \parallel J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)\parallel \\ \le & {\beta }_n\parallel x_n\parallel +(1-{\beta }_n)\parallel z_n\parallel \\ \le & \parallel x_n\parallel +\parallel z_n\parallel ,\end{array}$$

this implies that $\{w_n\}$ is also bounded.

In view of ${\alpha }_n\to 0$, from (2.1), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{y}_n-J{w}_n\parallel =\underset{n\to \mathrm{\infty }}{lim}{\alpha }_n\parallel J{x}_1-J{w}_n\parallel =0.$$

(2.9)

Since $J^{-1}$ is uniformly continuous on each bounded subset of $X^{\ast }$, it follows from (2.8) and (2.9) that

$$w_n\to p^{\ast }.$$

(2.10)

Since J is uniformly continuous on each bounded subset of X, we have

$$\begin{array}{rcl}0& =& \underset{n\to \mathrm{\infty }}{lim}\parallel J{w}_n-J{p}^{\ast }\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}\parallel ({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)-J{p}^{\ast }\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}\parallel {\beta }_n(J{x}_n-J{p}^{\ast })+(1-{\beta }_n)(J{z}_n-J{p}^{\ast })\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}(1-{\beta }_n)\parallel J{z}_n-J{p}^{\ast }\parallel .\end{array}$$

(2.11)

By condition (ii), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{z}_n-J{p}^{\ast }\parallel =0.$$

Since J is uniformly continuous, this shows that

$$\underset{n\to \mathrm{\infty }}{lim}{z}_n=p^{\ast }.$$

(2.12)

Again by the assumptions that $T:D\to X$ be uniformly L-Lipschitz continuous, thus we have

$$\begin{array}{r}d(T{(PT)}^n{x}_n,T{(PT)}^{n-1}{x}_n)\\ \le d(T{(PT)}^n{x}_n,T{(PT)}^n{x}_{n+1})+d(T{(PT)}^n{x}_{n+1},x_{n+1})\\ +\parallel x_{n+1}-x_n\parallel +d(x_n,T{(PT)}^{n-1}{x}_n)\\ \le (L+1)\parallel x_{n+1}-x_n\parallel +d(T{(PT)}^n{x}_{n+1},x_{n+1})+d(x_n,T{(PT)}^{n-1}{x}_n).\end{array}$$

(2.13)

We get ${lim}_{n\to \mathrm{\infty }}d(T{(PT)}^n{x}_n,T{(PT)}^{n-1}{x}_n)=0$, since ${lim}_{n\to \mathrm{\infty }}{z}_n=p^{\ast }$ and ${lim}_{n\to \mathrm{\infty }}{x}_n=p^{\ast }$.

In view of the continuity of TP, it yields $p^{\ast }\in TP{p}^{\ast }$. We have $p^{\ast }\in C$, which implies that $p^{\ast }\in T{p}^{\ast }$. We have $p^{\ast }\in F(T)$.

• (V) Finally, we prove that $p^{\ast }={\mathrm{\Pi }}_{F(T)}{x}_1$ and so $x_n\to {\mathrm{\Pi }}_{F(T)}{x}_1=p^{\ast }$.

Let $w={\mathrm{\Pi }}_{F(T)}{x}_1$. Since $w\in F(T)\subset D_n$ and $x_n={\mathrm{\Pi }}_{D_n}{x}_1$, we have $\varphi (x_n,x_1)\le \varphi (w,x_1)$. This implies that

$$\varphi (p^{\ast },x_1)=\underset{n\to \mathrm{\infty }}{lim}\varphi (x_n,x_1)\le \varphi (w,x_1),$$

(2.14)

which yields $p^{\ast }=w={\mathrm{\Pi }}_{F(T)}{x}_1$. Therefore, $x_n\to {\mathrm{\Pi }}_{F(T)}{x}_1$. The proof of Theorem 3.1 is completed. □

By Remark 1.3, the following corollary is obtained.

Corollary 2.1 Let X, D, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ be the same as in Theorem  2.1. Let $T:D\to X$ be a quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping with sequence $k_n\subset [1,+\mathrm{\infty })$, $k_n\to 1$, $T:D\to X$ be uniformly L-Lipschitz continuous.

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

$$\{\begin{array}{c}{x}_1\in X\mathit{\text{is arbitrary}};D_1=D,\hfill \\ y_n=J^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)](i\ge 1),z_n\in T{(PT)}^{n-1}{x}_n,\hfill \\ D_{n+1}=\{z\in D_n:\varphi (z,y_n)\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{D_{n+1}}{x}_1(n=1,2,\dots ),\hfill \end{array}$$

(2.15)

where ${\xi }_n=(k_n-1){sup}_{p\in F(T)}\varphi (p,x_n)$, ${\mathrm{\Pi }}_{D_{n+1}}$ is the generalized projection of X onto $D_{n+1}$. If $F(T)\ne \mathrm{\varnothing }$, then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{F(T)}{x}_1$.

Corollary 2.2 Let X, D, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ be the same as in Theorem  2.1. Let $T:D\to X$ be a quasi-ϕ-nonexpansive nonself multi-valued mapping, $T:D\to X$ be uniformly L-Lipschitz continuous.

Suppose $\{x_n\}$ is a sequence generated by

$$\{\begin{array}{c}{x}_1\in X\mathit{\text{is arbitrary}};D_1=D,\hfill \\ y_n=J^{-1}[{\alpha }_{n}J{x}_1+(1-{\alpha }_n)({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)](i\ge 1),z_n\in T{(PT)}^{n-1}{x}_n,\hfill \\ D_{n+1}=\{z\in D_n:\varphi (z,y_n)\le {\alpha }_n\varphi (z,x_1)+(1-{\alpha }_n)\varphi (z,x_n)+{\xi }_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{D_{n+1}}{x}_1(n=1,2,\dots ),\hfill \end{array}$$

(2.16)

where ${\xi }_n=(k_n-1){sup}_{p\in F(T)}\varphi (p,x_n)$, ${\mathrm{\Pi }}_{D_{n+1}}$ is the generalized projection of X onto $D_{n+1}$. If $F(T)\ne \mathrm{\varnothing }$, then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{F(T)}{x}_1$.

3 Application

First, we present an example of a quasi-ϕ-nonexpansive nonself multi-valued mapping.

Example 3.1 (see [4])

Let H be a real Hilbert space, D be a nonempty closed and convex subset of H and $f:D\times D\to R$ be a bifunction satisfying the conditions: (A1) $f(x,x)=0$, $\mathrm{\forall }x\in D$; (A2) $f(x,y)+f(y,x)\le 0$, $\mathrm{\forall }x,y\in D$; (A3) for each $x,y,z\in D$, ${lim}_{t\to 0}f(tz+(1-t)x,y)\le f(x,y)$; (A4) for each given $x\in D$, the function $y\mapsto f(x,y)$ is convex and lower semicontinuous. The ‘so-called’ equilibrium problem for f is to find a $x^{\ast }\in D$ such that $f(x^{\ast },y)\ge 0$, $\mathrm{\forall }y\in D$. The set of its solutions is denoted by $EP(f)$.

Let $r>0$, $x\in H$ and define a mapping $T_r:D\to D\subset H$ as follows:

$$T_r(x)=\{z\in D,f(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in D\},\mathrm{\forall }x\in D\subset H.$$

(3.1)

Then (1) $T_r$ is single-valued, so $z=T_r(x)$; (2) $T_r$ is a relatively nonexpansive nonself mapping, therefore it is a closed quasi-ϕ-nonexpansive nonself mapping; (3) $F(T_r)=EP(f)$ and $F(T_r)$ is a nonempty and closed convex subset of D; (4) $T_r:D\to D$ is a nonexpansive. Since $F(T_r)$ nonempty, it is a quasi-ϕ-nonexpansive nonself mapping from D to H, where $\varphi (x,y)={\parallel x-y\parallel }^2$, $x,y\in H$.

In this section we utilize Corollary 2.1 to study a modified Halpern iterative algorithm for a system of equilibrium problems. We have the following result.

Theorem 3.1 Let H be a real Hilbert space, D be a nonempty closed and convex subset of H, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ be the same as in Theorem  2.1. Let $f:D\times D\to R$ be a bifunction satisfying conditions (A1)-(A4) as given in Example  3.1. Let $T_r:D\to D\subset H$ be mapping defined by (3.1), i.e.,

$$T_r(x)=\{z\in D,f(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in D\},\mathrm{\forall }x\in D\subset H.$$

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

$$\{\begin{array}{c}{x}_1\in D\mathit{\text{is arbitrary}};D_1=D,\hfill \\ f(u_n,y)+\frac{1}{r}〈y-u_n,u_n-x_n〉\ge 0,\mathrm{\forall }y\in D,r>0,\hfill \\ y_n={\alpha }_n{x}_1+(1-{\alpha }_n)[{\beta }_n{x}_n+(1-{\beta }_n)u_n],\hfill \\ D_{n+1}=\{z\in D_n:{\parallel z-y_{n,i}\parallel }^2\le {\alpha }_n{\parallel z-x_1\parallel }^2+(1-{\alpha }_n){\parallel z-x_n\parallel }^2\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{D_{n+1}}{x}_1(n=1,2,\dots ).\hfill \end{array}$$

(3.2)

If $F(T_r)\ne \mathrm{\varnothing }$, then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_{F(T_r)}{x}_1$, which is a common solution of the system of equilibrium problems for f.

Proof In Example 3.1, we have pointed out that $u_n=T_r(x_n)$, $F(T_r)=EP(f)$ is nonempty and convex, $T_r$ is a quasi-ϕ-nonexpansive nonself mapping. Since $F(T_r)$ is nonempty, and so $T_r$ is a quasi-ϕ-nonexpansive mapping and $T_r$ is uniformly 1-Lipschitzian mapping. Hence (3.1) can be rewritten as follows:

$$\{\begin{array}{c}{x}_1\in H\text{is arbitrary};D_1=D,\hfill \\ y_n={\alpha }_n{x}_1+(1-{\alpha }_n)[{\beta }_n{x}_n+(1-{\beta }_n)z_n],z_n\in T_r{x}_n,\hfill \\ D_{n+1}=\{z\in D_n:{\parallel z-y_n\parallel }^2\le {\alpha }_n{\parallel z-x_1\parallel }^2+(1-{\alpha }_n){\parallel z-x_n\parallel }^2\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{D_{n+1}}{x}_1(n=1,2,\dots ).\hfill \end{array}$$

(3.3)

Therefore, the conclusion of Theorem 3.1 can be obtained from Corollary 2.1. □