Relaxed iterative algorithms for a system of generalized mixed equilibrium problems and a countable family of totally quasi-Phi-asymptotically nonexpansive multi-valued maps, with applications

Abstract

In this article, a Krasnoselskii-type and a Halpern-type algorithm for approximating a common fixed point of a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps and a solution of a system of generalized mixed equilibrium problem are constructed. Strong convergence of the sequences generated by these algorithms is proved in uniformly smooth and strictly convex real Banach spaces with the Kadec-Klee property. Several applications of our theorems are also presented. Finally, our theorems are a significant improvement of several important recent results.

1 Introduction

In what follows, we assume that X is a real Banach space with dual space $X^{*}$, K is a nonempty, closed, and convex subset of X, and → and ⇀ will, respectively, denote strong and weak convergence.

(See, e.g., Wang and Zang [1] for a similar definition for self maps.) Let $G:K \to2^{X}$ be any map. A point $u\in K$ is called a fixed point of G if and only if $u\in Gu$ and it is called an asymptotic fixed point of G if there exists a sequence $\{u_{n}\}$ in K that converges weakly to u and $\lim_{n\to\infty}d(u_{n},Gu_{n}):=\lim_{n\to\infty} \inf_{\eta_{n}\in Gu_{n}} \Vert u_{n}-\eta_{n}\Vert =0$ (see Chang et al. [2]). We denote the set of fixed points and asymptotic fixed points of G by $F(G)$ and $\hat{F}(G)$, respectively.

A subset K of X is said to be a retract of X, if there exists a continuous map $P:X\to K$ such that $Pu=u$, for all $u\in X$. It is well known that every nonempty, closed, convex subset of a uniformly convex Banach space X is a retract of X. A map $P:X\to K$ is said to be a retraction if $P^{2} =P$. A map $P:X\to K$ is said to be a nonexpansive retraction, if it is nonexpansive and it is a retraction from X to K.

Define the Lyapunov functional $\phi: X \times X\to \Bbb {R}$ by

$$\begin{aligned} \phi(u,v)=\Vert u\Vert ^{2}-2\langle u,Jv\rangle+ \Vert v\Vert ^{2} \quad \forall u,v \in X. \end{aligned}$$

(1.1)

From the definition of ϕ it is obvious that

$$\begin{aligned} \bigl(\Vert u\Vert -\Vert v\Vert \bigr)^{2}\leq \phi (u,v)\leq \bigl(\Vert u\Vert +\Vert v\Vert \bigr)^{2} \quad \forall u,v \in X. \end{aligned}$$

(1.2)

In what follows, we assume that $P:X\to K$ is a nonexpansive retraction.

Definition 1.1

A nonself multi-valued map $G:K \to2^{X}$ is said to be relatively asymptotically nonexpansive if $F(G)\ne\emptyset$, $\hat{F}(G)=F(G)$, and there exists a real sequence $\{\beta_{n}\} \subset[1,\infty)$, $\beta_{n}\downarrow1$ such that $\phi(p,\eta _{n})\leq\beta_{n}\phi(p,u)$ $\forall u\in K$, $p\in F(G)$, $\eta_{n} \in G(PG)^{n-1}u$, $n\geq1$ (see, e.g., Wang and Zang [1] for a similar definition for self maps).

The following definitions appear in Bo and Yi [3].

Definition 1.2

A nonself multi-valued map $G:K \to2^{X}$ is said to be

• quasi-ϕ-nonexpansive if $F(G)\ne\emptyset$ and $\phi(p,\eta_{n})\leq\phi(p,u)$ $\forall u\in K$, $p\in F(G)$, $\eta _{n}\in G(PG)^{n-1}u$, $n\geq1$;

• quasi-ϕ-asymptotically nonexpansive if $F(G)\ne\emptyset$ and there exists a real sequence $\{\beta_{n}\}\subset[1,\infty)$, $\beta_{n}\downarrow1$ such that $\phi(p,\eta_{n})\leq\beta_{n} \phi(p,u)$ $\forall u\in K$, $p\in F(G)$, $\eta_{n}\in G(PG)^{n-1}u$, $n \geq1$;

• totally quasi-ϕ-asymptotically nonexpansive if $F(G)\ne\emptyset$ and there exist nonnegative real sequences $\{\gamma_{n}\}$, $\{\delta_{n}\}$ with $\gamma_{n}\to0$, $\delta_{n} \to0$ ($n\to\infty$) and a strictly increasing and continuous function $\rho: \Bbb {R}^{+}\to \Bbb {R}^{+}$ with $\rho(0)=0$ such that

  $$\begin{aligned} \begin{aligned}[b] &\phi(p,\eta_{n})\leq\phi(p,u)+ \gamma_{n} \rho \bigl[\phi(p,u) \bigr]+\delta_{n} \\ &\quad \forall u\in K, p\in F(G), \eta_{n}\in G(PG)^{n-1}u, n\geq1. \end{aligned} \end{aligned}$$

  (1.3)

Remark 1

From the definitions, it is easy to see that the class of relatively asymptotically nonexpansive multi-valued nonself maps and the class of quasi-ϕ-nonexpansive multi-valued nonself maps are proper subclasses of the class of quasi-ϕ-asymptotically nonexpansive multi-valued nonself maps and that the class of quasi-ϕ-asymptotically nonexpansive multi-valued nonself maps is a proper subclass of the class of totally quasi-ϕ-asymptotically nonexpansive multi-valued nonself maps, but the converse may not be true.

Definition 1.3

A countable family of multi-valued nonself maps, $G_{i}:K\to2^{X}$, $i=1,2,3,\ldots$ , is said to be

• uniformly quasi-ϕ-asymptotically nonexpansive if $\bigcap_{i=1}^{\infty}F(G_{i})\ne\emptyset$ and there exists a sequence $\{\beta_{n}\}\subset[1,\infty)$, $\beta_{n}\downarrow1$ such that, for each $i\geq1$,

  $$\begin{aligned} \phi(p,\eta_{n})\leq\beta_{n}\phi(p,u) \quad \forall u\in K, p \in\bigcap_{i=1}^{\infty}F(G_{i}), \eta_{n}\in G _{i}(PG_{i})^{n-1}u, n\geq1 \end{aligned}$$

  (see, e.g., Chang et al. [4]);

• uniformly totally quasi-ϕ-asymptotically nonexpansive if $\bigcap_{i=1}^{\infty}F(G_{i})\ne\emptyset$ and there exist nonnegative real sequences $\{\gamma_{n}\}$, $\{\delta_{n}\}$ with $\gamma_{n} \to0$, $\delta_{n}\to0$ ($n\to\infty$) and a strictly increasing and continuous function $\rho: \Bbb {R}^{+}\to \Bbb {R}^{+}$ with $\rho(0)=0$ such that, for each $i\ge1$,

  $$\begin{aligned} \begin{gathered} \phi(p,\eta_{n})\leq\phi(p,u)+ \gamma_{n}\rho \bigl[ \phi(p,u) \bigr]+\delta_{n} \\ \quad \forall u\in K, p\in\bigcap _{i=1}^{\infty}F(G_{i}), \eta_{n}\in G _{i}(PG_{i})^{n-1}u, n \geq1 \end{gathered} \end{aligned}$$

  (see, e.g., Yi [5]).

Remark 2

From the definitions, it is easy to see that a countable family of uniformly quasi-ϕ-asymptotically nonexpansive multi-valued nonself maps is a countable family of uniformly totally quasi-ϕ-asymptotically nonexpansive multi-valued nonself maps.

Remark 3

We also remark that a collection of countable families of uniformly totally quasi-ϕ-asymptotically nonexpansive multi-valued nonself maps is a subcollection of a collection of countable families of totally quasi-ϕ-asymptotically nonexpansive multi-valued nonself maps.

A motivation for the study of the class of totally quasi-ϕ-asymptotically nonexpansive self or nonself maps is the objective to unify various definitions of classes of maps, associated with the class of relatively nonexpansive self or nonself maps, which are extensions to arbitrary real Banach spaces of nonexpansive nonself maps, with nonempty fixed point sets in Hilbert spaces. Our objective is to prove general convergence theorems applicable to all these classes.

Definition 1.4

See, e.g., Feng et al. [6] for a similar definition for self maps

A multi-valued nonself map $G:K \to2^{X}$ is said to be

• equally continuous if for $u_{n}, v_{n} \in K$ we have

  $$\begin{aligned} \begin{gathered} \lim_{n\to\infty} \Vert u_{n}-v_{n}\Vert =0 \quad \Longrightarrow\quad \lim_{n\to\infty} \Vert \eta_{n_{u}}- \eta_{n_{v}}\Vert =0 \\ \quad \forall \eta_{n_{u}}\in G(PG)^{n-1}u_{n}, \eta_{n_{v}}\in G(PG)^{n-1}v_{n}; \end{gathered} \end{aligned}$$

• uniformly continuous if for $u_{n}, v_{n} \in K$ we have

  $$\begin{aligned} \lim_{n\to\infty} \Vert u_{n}-v_{n}\Vert =0\quad \Longrightarrow\quad \lim_{n\to\infty} \Vert \eta_{n_{u}}- \eta_{n_{v}}\Vert =0 \quad \forall \eta_{n_{u}}\in Gu_{n}, \eta_{n_{v}}\in Gv_{n}; \end{aligned}$$

• uniformly L-Lipschitz continuous if there exists a constant $L>0$ such that

  $$\begin{aligned} \Vert \eta_{u}-\eta_{v}\Vert \leq L\Vert u-v\Vert \quad \forall \eta_{u}\in G(PG)^{n-1}u, \eta_{v}\in G(PG)^{n-1}v, n\ge1. \end{aligned}$$

Remark 4

It is easy to see that the class of uniformly L-Lipschitz multi-valued nonself maps is a proper subclass of the class of uniformly continuous multi-valued nonself maps and the class of uniformly continuous multi-valued nonself maps is a proper subclass of the class of equally continuous multi-valued nonself maps.

Let $\psi:K\to \Bbb {R}$ be a real-valued function, $A:K\to X^{*}$ be a nonlinear map, and $f:K\times K\to \Bbb {R}$ be a bifunction. The generalized mixed equilibrium problem is to find $u^{*} \in K$ such that

$$\begin{aligned} f \bigl(u^{*},v \bigr)+\psi(v)-\psi \bigl(u^{*} \bigr)+ \bigl\langle v-u^{*}, Au^{*} \bigr\rangle \geq0, \quad \forall v \in K. \end{aligned}$$

(1.4)

The set of solutions of the generalized mixed equilibrium problem is denoted by $\boldsymbol{\mathit {GMEP}}(f, A,\psi)$.

The class of generalized mixed equilibrium problems includes, as special cases, the class of mixed equilibrium problems ($A\equiv0$; see, e.g., Ceng and Yao [7] and the references contained therein); the class of generalized equilibrium problems ($\zeta\equiv0$; see, e.g., Takahashi and Takahashi [8]); the class of equilibrium problems ($A\equiv0$, $\zeta\equiv0$; see, e.g., Fan [9], Blum and Oettli [10], and the references contained therein); the class of variational inequality problems ($h\equiv0$, $\zeta\equiv0$; see, e.g., Stampacchia [11]); and the class of convex minimization problems ($A\equiv0$, $h\equiv0$).

The generalized mixed equilibrium problem has applications in physics, economics, finance, transportation, network and structural analysis, ecology, image reconstruction, and elasticity. It includes, as special cases, fixed point problems, variational inequality problems, complementarity problems, equilibrium problems, optimization problems, Nash equilibrium problems in noncooperative games, etc. (see, e.g., Blum and Otelli [10], Dafermos and Nagurney [12], Su [13], Barbagallo [14], Moudafi [15], and the references contained therein). In other words, the $\mathit {GMEP}(f,A,\psi)$ is a unifying model for several problems arising in physics, engineering, science, optimization, finance, economics, etc. The projection method, which was first introduced by Haugazeau [16], has been utilized to solve the mixed equilibrium problem, the generalized equilibrium problem, and equilibrium problems in Banach spaces (see, e.g., Qin et al. [17], Cholamjiak et al. [18], Cho et al. [19], Ceng and Yao [7], and the references therein). The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compactness assumptions imposed on maps or subsets of spaces.

Several strong and weak convergence theorems for asymptotically nonexpansive, relatively nonexpansive, quasi-ϕ-nonexpansive and quasi-ϕ-asymptotically nonexpansive self or nonself maps have been established by various authors in the setting of Banach spaces (see, e.g., Thianwan [20], Nilsrakoo et al. [21], Wang [22], Ma and Wang [23], Chidume et al. [24, 25], and the references contained therein).

In 2012, Chang et al. [4] considered the class of uniformly quasi-ϕ-asymptotically nonexpansive nonself maps and studied, in a uniformly convex and uniformly smooth real Banach space, the following Halpern-type algorithm:

$$\begin{aligned} \textstyle\begin{cases} u_{1} \in X, \quad \text{chosen arbitrarily},\quad\quad K_{1}=K, \\ y_{n,i}=J^{-1} (\alpha_{n} Ju_{1}+(1-\alpha_{n}) (\beta_{n}Ju _{n}+(1-\beta_{n})JG_{i}(PG_{i})^{n-1}u_{n} ) ), \quad i\ge1, \\ K_{n+1} =\{z\in K_{n}:\sup_{i\ge1}\phi(z,y_{n,i})\leq\alpha_{n} \phi(z,u_{1})+(1-\alpha_{n})\phi(z,u_{n})+\theta_{n}\}, \\ u_{n+1}=\Pi_{ K_{n+1}}u_{1},\quad n\geq1, \end{cases}\displaystyle \end{aligned}$$

(1.5)

where $\theta_{n}=(\beta_{n}-1)\sup_{u\in F}\phi(u,u_{n})$, $F:=\bigcap_{i=1}^{\infty}F(G_{i})$, and $\{G_{i}\}_{i=1}^{\infty}$ is a countable family of uniformly L-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive nonself maps. The authors prove that the sequence $\{u_{n}\}$, generated by the above iterative scheme, converges strongly to $\Pi_{F}u_{1}$, under the following conditions:

$(C1)$ :

$\lim_{n\to\infty}\alpha_{n}=0$;

$(C2)$ :

$0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to\infty} \beta_{n}<1$;

$(C3)$ :

F is a bounded and convex subset of K,

where $\alpha_{n}\in[0,1]$ and $\beta_{n}\in(0,1)$.

In the same year, Zhao and Chang [26] proved that the sequence $\{u_{n}\}$, generated by algorithm (1.5), converges strongly to $\Pi_{F}u_{1}$ under conditions $(C1)$, $(C2)$, and the following condition:

$(C3^{*})$ :

F is a nonempty bounded subset of K,

where $\{G_{i}\}_{i=1}^{\infty}$ is a countable family of uniformly L-Lipschitz continuous, closed and uniformly totally quasi-ϕ-asymptotically nonexpansive nonself maps.

Later in the same year, Yi [5] established the results in the paper of Zhao and Chang [26], when $\{G_{i}\}_{i=1}^{\infty}$ is a countable family of uniformly L-Lipschitz continuous and uniformly totally quasi-ϕ-asymptotically nonexpansive nonself maps, under conditions $(C1)$, $(C2)$, and the following condition:

$(C3^{**})$ :

F is a nonempty subset of K.

In 2014, Bo and Yi [3] proved that the sequence $\{u_{n}\}$, generated by the iterative algorithm

$$\begin{aligned} \textstyle\begin{cases} u_{1} \in X,\quad \text{chosen arbitrarily},\quad\quad K_{1}=K, \\ y_{n}=J^{-1} (\alpha_{n} Ju_{1}+(1-\alpha_{n}) (\beta_{n}Ju _{n}+(1-\beta_{n})J\eta_{n} ) ),\quad\eta_{n}\in G(PG)^{n-1}u _{n}, \\ K_{n+1} =\{z\in K_{n}:\phi(z,y_{n})\leq\alpha_{n} \phi(z,u_{1})+(1- \alpha_{n})\phi(z,u_{n})+\theta_{n}\}, \\ u_{n+1}=\Pi_{ K_{n+1}}u_{1},\quad n\geq1, \end{cases}\displaystyle \end{aligned}$$

converges strongly to $\Pi_{F}u_{1}$ under conditions $(C1)$, $(C2)$, and $(C3^{**})$, where $F:=F(G)$ and G is a uniformly L-Lipschitz continuous and totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued map, while $\theta_{n}=\gamma_{n}\sup_{u\in F}\phi(u,u_{n})+\delta_{n}$.

The results of Bo and Yi [3], Yi [5], Zhao and Chang [26], and Chang et al. [4] are important generalizations and improvements of important known results.

Motivated by these authors, it is our purpose in this paper to study the following Krasnoselskii-type and Halpern-type algorithms:

$$\begin{aligned}& \textstyle\begin{cases} u_{0} \in X,\quad \text{chosen arbitrarily},\quad\quad K_{1}=K, \quad \quad u_{1}=\Pi_{K_{1}}u_{0}, \\ y_{n}=J^{-1} (\sigma Ju_{n}+(1-\sigma)J\eta_{m_{n}}^{(i_{n})} ), \quad (\eta_{m_{n}}^{(i_{n})}\in G_{i_{n}}(PG_{i_{n}})^{m_{n}-1}u_{n} ), \\ z_{n}=\Lambda_{r_{i_{n}}}y_{n}, \\ K_{n+1} =\{v\in K_{n}:\phi(v,z_{n})\leq\phi(v,u_{n})+\omega_{n}\}, \\ u_{n+1}=\Pi_{K_{n+1}}u_{0},\quad n\geq1, \end{cases}\displaystyle \end{aligned}$$

(1.6)

$$\begin{aligned}& \textstyle\begin{cases} u_{0} \in X, \quad \text{chosen arbitrarily}, \quad\quad K_{1}=K, \quad \quad u_{1}=\Pi_{K_{1}}u_{0}, \\ y_{n}=J^{-1} (\sigma_{n} Ju_{0}+(1-\sigma_{n})J\eta_{m_{n}}^{(i _{n})} ), \quad (\eta_{m_{n}}^{(i_{n})}\in G_{i_{n}}(PG_{i_{n}})^{m_{n}-1}u_{n} ), \\ z_{n}=\Lambda_{r_{i_{n}}}y_{n}, \\ K_{n+1} =\{v\in K_{n}:\phi(v,z_{n})\leq\sigma_{n}\phi(v,u_{0})+(1- \sigma_{n})\phi(v,u_{n})+ \omega_{n}\}, \\ u_{n+1}=\Pi_{K_{n+1}}u_{0},\quad n\geq1. \end{cases}\displaystyle \end{aligned}$$

(1.7)

We also aim to prove, in a uniformly smooth and strictly convex real Banach space X with Kadec-Klee property, that the sequences generated by these algorithms converge strongly to an element in $W:= (\bigcap_{i=1}^{\infty}F(G_{i}) )\cap (\bigcap_{i=1}^{\infty} \mathit {GMEP}(h_{i},A _{i},\zeta_{i}) )$, where $\{G_{i}\}_{i=1}^{\infty}$ is a countable family of equally continuous and totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps; $\{A_{i}\}_{i=1}^{\infty}$, $A_{i}:K \to X^{*}$ is a sequence of continuous and monotone maps; $\{h_{i}\} _{i=1}^{\infty}$, $h_{i}:K\times K\to \Bbb {R}$ is a sequence of bifunctions satisfying appropriate conditions and $\{\zeta_{i}\}_{i=1}^{\infty}$, $\zeta_{i}:K\to \Bbb {R}$ is a sequence of lower-semicontinuous and convex functions. Our theorems are significant improvements and generalizations of numerous results for this class of nonlinear problems (in particular, the results of Bo and Yi [3], Yi [5], Zhao and Chang [26], Chang et al. [4], Lv [27], Wang and Zhang [28], Dadashi and Postolache [29], Yao and Postolache [30], and the results of a host of other authors [see Remark 6 below]).

2 Preliminaries

A map $J:X\rightarrow2^{ X^{*}}$ defined by $Ju:= \{u^{*}\in X^{*}: \langle u,u^{*}\rangle=\Vert u\Vert \Vert u^{*}\Vert , \Vert u\Vert =\Vert u^{*}\Vert \}$ is called a normalized duality map on X, where $\langle\cdot,\cdot \rangle$ denotes the duality pairing between elements of X and $X^{*}$.

We now present some lemmas that will be used in the sequel.

Lemma 2.1

See Bo and Yi [3]

Let X be a smooth, strictly convex and reflexive Banach space and K be a nonempty, closed, convex subset of X. Let $G:K \to X$ be a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with $\delta_{1} = 0$. Then $F(G)$ is a closed and convex subset of K.

Lemma 2.2

See Chang et al. [2]

Let X be a uniformly smooth and strictly convex real Banach space with Kadec-Klee property and let K be a nonempty closed convex subset of X. Let $\{u_{n}\}$ and $\{y_{n}\}$ be two sequences in K such that $u_{n}\to u^{*}$ and $\phi(u_{n},y_{n})\to0$, where ϕ is the function defined by (1.1). Then $y_{n}\to u^{*}$.

Lemma 2.3

See Alber [31]

Let K be a nonempty closed and convex subset of a reflexive strictly convex and smooth Banach space X. Then

$$\begin{aligned} \phi (u,\Pi_{K}y )+\phi (\Pi_{K}y,y )\leq \phi(u,v), \quad \forall u\in K, v\in X. \end{aligned}$$

(2.1)

Let K be a nonempty closed and convex subset of a Banach space X. For solving the generalized mixed equilibrium problem (1.4), we assume that a bifunction $h:K\times K\to \Bbb {R}$ satisfies the following conditions:

$(B1)$ :

$h(u,u)=0$, $\forall u\in X$,

$(B2)$ :

h is monotone, that is, $h(u,v)+h(v,u)\leq0$, $\forall u,v \in X$,

$(B3)$ :

for all $u,y,z\in X$, $\limsup_{t\downarrow0}h(tz+(1-t)u,v) \leq h(u,v)$,

$(B4)$ :

for all $u\in K$, $h(u,\cdot):K\to \Bbb {R}$ is convex and lower-semicontinuous.

Lemma 2.4

See Zhang [32]

Let X be a smooth, strictly convex and reflexive Banach space and let K be a nonempty closed convex subset of X. Let $A:K\to X^{*}$ be a continuous and monotone mapping, $\zeta:K \to \Bbb {R}$ be a lower-semicontinuous and convex function, and $h:K\times K\to \Bbb {R}$ be a bifunction satisfying the conditions $(B1)$-$(B4)$. Let $r>0$ be any given number and $u \in X$ be any given point. Then the following observations hold:

• (1) There exists $z\in K$ such that

  $$\begin{aligned} h(z,v) +\zeta(v)-\zeta(z)+\langle v-z, Az\rangle+\frac{1}{r} \langle v-z,Jz-Ju \rangle\geq0,\quad\forall v\in K. \end{aligned}$$

• (2) If we define a mapping $\Lambda_{r}:K\to K$ by

  $$\begin{aligned} \begin{aligned} \Lambda_{r}(u)&= \biggl\{ z\in K: h(z,v) +\zeta(v)-\zeta(z)+\langle v-z, Az\rangle+\frac{1}{r} \langle v-z,Jz-Ju \rangle\geq0, \\ &\quad \forall v \in K \biggr\} , \quad u\in K, \end{aligned} \end{aligned}$$

  then the mapping $G_{r}$ has the following properties:

  • (a) $G_{r}$ is single-valued;

  • (b) $F(\Lambda_{r})=\mathit {GMEP}(h,A,\zeta)=\hat{F}(\Lambda_{r})$;

  • (c) $\mathit {GMEP}(h,A,\zeta)$ is a closed convex set of K;

  • (d) $\phi(q,\Lambda_{r}u)+\phi(\Lambda_{r}u,u)\leq\phi(q,u)$ $\forall q\in F(\Lambda_{r})$, $u\in X$.

3 Main results

In what follows, ${i_{n}}$ and ${m_{n}}$ are the unique solutions to the positive integer equation $n=i+\frac{(m-1)m}{2}$ ($m\geq i$, $n=1,2,\ldots$). That is, for each $n\geq1$, there exist unique $i_{n}$ and $m_{n}$ such that

$$\begin{aligned}& i_{1}=1,\qquad i_{2}=1, \qquad i_{3}=2,\qquad i_{4}=1, \\& i_{5}=2, \qquad i_{6}=3,\qquad i_{7}=1, \qquad i_{8}=2,\ldots; \\& m_{1}=1,\qquad m_{2}=2, \qquad m_{3}=2,\qquad m_{4}=3, \\& m_{5}=3, \qquad m_{6}=3,\qquad m_{7}=4, \qquad m_{8}=4,\ldots. \end{aligned}$$

See Deng [33]. We now prove the following strong convergence theorem using a Krasnoselskii-type algorithm.

Theorem 3.1

Let X be a uniformly smooth and strictly convex real Banach space with Kadec-Klee property and let $X^{*}$ be its dual space. Let K be a nonempty closed and convex subset of X and $h_{i}:K \times K\to \Bbb {R}$, $i=1,2,3,\ldots$ , be a sequence of bifunctions satisfying $(B1)$-$(B4)$. Let $A_{i}:K\to X^{*}$, $i=1,2,3,\ldots$ , be a sequence of continuous monotone maps and $G_{i}:K \to{2^{X}}$, $i=1,2,\ldots$ , be an infinite family of equally continuous and totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps with nonnegative real sequences $\{\gamma_{n}^{(i)}\}$, $\{\delta_{n}^{(i)} \}$ and a sequence of strictly increasing and continuous functions $\{\rho_{i}\}$, $\rho_{i}:\Bbb {R}^{+}\to \Bbb {R}^{+}$, such that $\gamma_{n} ^{(i)}\to0$, $\delta_{n}^{(i)}\to0$ and $\rho_{i}(0)=0$. Let $\zeta_{i}:K\to \Bbb {R}$, $i=1,2,3,\ldots$ , be a sequence of convex and lower-semicontinuous functions. Suppose for each i, $\delta_{1}^{(i)}=0$ and $\Omega:= (\bigcap_{i=1}^{\infty}F(G_{i}) )\cap (\bigcap_{i=1}^{\infty} \mathit {GMEP}(h_{i}, A_{i},\zeta_{i}) )$ is a nonempty subset of K. Then the sequence $\{u_{n}\}$ generated by algorithm (1.6) converges strongly to $\Pi_{\Omega }u_{0}$, where $\sigma\in(0,1)$, $r_{i_{n}}\in[a,\infty)$ for some $a>0$, and $\omega_{n}=\gamma_{m_{n}}^{(i_{n})}\rho_{i_{n}} [ \phi(p,u_{n}) ]+ \delta_{m_{n}}^{(i_{n})}$, $p\in\Omega$.

Proof

The proof is presented in a number of steps.

Step 1: $K_{n}$ is closed and convex for all $n\geq1$.

Clearly, $K_{1}=K$ is closed and convex. Assume $K_{n}$ is closed and convex for some $n\geq1$. It is easy to see that $K_{n+1}=\{v\in K _{n}:2\langle v, Ju_{n}-Jz_{n}\rangle\leq \Vert u_{n}\Vert -\Vert z_{n}\Vert +\omega _{n}\}$. Consequently, it is closed and convex. Hence, Step 1 is completed.

Step 2: $\Omega\subset K_{n}$ for all $n\geq1$.

Again, we proceed by induction. Clearly, $\Omega\subset K_{1}$. Suppose $\Omega\subset K_{n}$ for some $n\geq1$. Let $q\in\Omega$. Then, by applying Lemma 2.4(d), the definition of ϕ, the convexity of $\Vert \cdot \Vert ^{2}$, and the totally quasi-ϕ-asymptotically nonexpansiveness of $G_{i}$, we have

$$\begin{aligned} \phi(q,z_{n}) =&\phi (q,\Lambda_{r_{i_{n}}}y_{n} )\leq\phi (q,y_{n} ) = \phi \bigl(q,J^{-1} \bigl(\sigma Ju_{n}+(1-\sigma)J \eta_{m_{n}}^{(i_{n})} \bigr) \bigr) \\ =& \Vert q\Vert ^{2}-2 \bigl\langle q, \sigma Ju_{n}+(1-\sigma)J\eta_{m_{n}} ^{(i_{n})} \bigr\rangle + \bigl\Vert \sigma Ju_{n}+(1-\sigma)J\eta_{m _{n}}^{(i_{n})} \bigr\Vert ^{2} \\ \leq& \Vert q\Vert ^{2}-2\sigma\langle q,Ju_{n}\rangle -2(1-\sigma) \bigl\langle q,J\eta_{m_{n}}^{(i_{n})} \bigr\rangle + \sigma \Vert u_{n}\Vert ^{2}+(1-\sigma) \bigl\Vert \eta_{m_{n}}^{(i_{n})} \bigr\Vert ^{2} \\ =& \sigma\phi(q,u_{n})+(1-\sigma)\phi \bigl(q,\eta_{m_{n}}^{(i_{n})} \bigr) \\ \leq& \sigma\phi(q,u_{n})+(1-\sigma) \bigl[\phi(q,u_{n})+ \gamma_{m_{n}}^{(i_{n})}\rho_{i_{n}} \bigl( \phi(q,u_{n}) \bigr)+ \delta_{m_{n}}^{(i_{n})} \bigr] \\ \leq&\phi(q,u_{n})+\gamma_{m_{n}}^{(i_{n})} \rho_{i_{n}} \bigl(\phi(q,u _{n}) \bigr)+\delta_{m_{n}}^{(i_{n})} =\phi(q,u_{n})+\omega_{n}, \end{aligned}$$

which implies that $q\in K_{n+1}$. Therefore, $\Omega\subset K_{n}$ for all $n\geq1$.

Step 3: $\{\phi(u_{n},u_{0})\}$ is convergent and $\lim_{n\to\infty}\omega_{n}=0$.

Since $u_{n}=\Pi_{K_{n}}u_{0}$ and $K_{n+1}\subset K_{n}$ for all $n\geq1$, we have $\phi(u_{n},u_{0})\leq\phi(u_{n+1},u_{0})$, which implies that $\{\phi(u_{n},u_{0})\}$ is nondecreasing. Furthermore, by Lemma 2.3, we have

$$ \phi(u_{n},u_{0})=\phi (\Pi_{K_{n}}u_{0},u_{0} ) \leq\phi(q,u _{0})-\phi(q,u_{n}) \leq \phi(q,u_{0}), $$

for all $n\geq1$ and $q\in\Omega$. Thus, $\{\phi(u_{n},u_{0})\}$ is bounded. Hence, $\{\phi(u_{n},u_{0})\}$ is convergent. Furthermore, by inequality (1.2), $\{u_{n}\}$ is bounded. Now, for each $i\geq1$, define $K_{i}:=\{k\ge1:k=i+\frac{(m-1)m}{2}, m\ge i, m \in \Bbb {N}\}$. Observe that if, for each $i\ge1$, $k\in K_{i}$, then $\gamma_{m_{k}}^{(i_{k})}=\gamma_{m_{k}}^{(i)}$, $\delta_{m_{k}}^{(i _{k})}=\delta_{m_{k}}^{(i)}$, and $\rho_{i_{k}}=\rho_{i}$. Also, $m_{k}\to\infty$ as $k\to\infty$, $k\in K_{i}$. Therefore, $\lim_{n\to\infty} \omega_{n}=0$.

Step 4: $u_{n}\to u^{*}$, $z_{n}\to u^{*}$, and $y_{n}\to u ^{*}$ as $n\to\infty$, for some $u^{*}\in K$.

By the boundedness of $\{u_{n}\}$ and reflexivity of X, there exists a subsequence $\{u_{n_{k}}\}$ of $\{u_{n}\}$ such that $u_{n_{k}} \rightharpoonup u^{*}$ as $k\to\infty$. Since $K_{n_{k}}$ is weakly closed, $u^{*}\in K_{n_{k}}\subset K$. Thus, $u_{n_{k}}=\Pi_{K_{n_{k}}}u _{0}$ implies that $\phi(u_{n_{k}},u_{0})\leq\phi(u^{*},u_{0})$ $\forall k\geq1$. By the weak lower-semicontinuity of $\Vert \cdot \Vert$, we have

$$\begin{aligned} \liminf_{k\to\infty}\phi(u_{n_{k}},u_{0}) =& \liminf_{k\to\infty} \bigl\{ \Vert u_{n_{k}}\Vert ^{2}-2\langle u_{n_{k}},Ju_{0}\rangle+\Vert u_{0}\Vert ^{2} \bigr\} \\ \geq& \bigl\Vert u^{*} \bigr\Vert ^{2}-2 \bigl\langle u^{*},Ju_{0} \bigr\rangle +\Vert u_{0}\Vert ^{2}= \phi \bigl(u^{*},u_{0} \bigr), \end{aligned}$$

which implies that $\phi(u^{*},u_{0})\leq\liminf_{k\to\infty} \phi(u_{n_{k}},u_{0})\leq\limsup_{k\to\infty}\phi(u_{n_{k}},u _{0})\leq\phi(u^{*},u_{0})$. Hence, $\lim_{k\to\infty} \phi(u_{n_{k}},u_{0})=\phi(u^{*},u_{0})$. Therefore, $\lim_{k\to\infty} \Vert u_{n_{k}}\Vert =\Vert u^{*}\Vert$. By the Kadec-Klee property of X, we have $\lim_{k\to\infty} u_{n_{k}}=u^{*}$. Since $\{\phi(u_{n},u_{0})\}$ is convergent and $\lim_{k\to\infty}\phi(u_{n_{k}},u_{0})=\phi(u^{*},u_{0})$, we have $\lim_{n\to\infty}\phi(u_{n},u_{0})=\phi(u^{*},u_{0})$.

Claim: $u_{n}\to u^{*}$.

Suppose there exists a subsequence $\{u_{n_{j}}\}$ of $\{u_{n}\}$ such that $u_{n_{j}}\to p$ as $j\to\infty$. By applying Lemma 2.3, we have

$$\begin{aligned} \phi \bigl(u^{*},p \bigr) =&\lim_{j,k\to\infty} \phi(u_{n_{k}},u_{n_{j}})= \lim_{j,k\to\infty}\phi (u_{n_{k}},\Pi_{K_{n_{j}}}u_{0} ) \\ \leq& \lim_{j,k\to\infty} \bigl(\phi(u_{n_{k}},u_{0})- \phi ( \Pi_{K_{n_{j}}}u_{0},u_{0} ) \bigr) \\ =&\lim_{j,k\to\infty} \bigl(\phi(u_{n_{k}},u_{0})- \phi(u _{n_{i}},u_{0}) \bigr) =\phi \bigl(u^{*},u_{0} \bigr)-\phi \bigl(u^{*},u_{0} \bigr)=0, \end{aligned}$$

which implies $u^{*}=p$. Hence, the claim holds. Again, by Lemma 2.3 we have

$$\begin{aligned} \phi(u_{n+1},u_{n}) =&\phi (u_{n+1}, \Pi_{K_{n}}u_{0} ) \leq \phi(u_{n+1},u_{0})- \phi (\Pi_{K_{n}}u_{0},u_{0} ) =\phi(u _{n+1},u_{0})-\phi(u_{n},u_{0}), \end{aligned}$$

which implies that $\phi(u_{n+1},u_{n})\to0$ as $n\to\infty$. Since $u_{n+1}\in K_{n+1}$, we have $\phi(u_{n+1},z_{n})\leq\phi(u_{n+1},u _{n})+\omega_{n}$. Consequently, $\phi(u_{n+1},z_{n})\to0$ as $n\to\infty$. Therefore, by Lemma 2.2, $z_{n}\to u^{*}$. From $\phi(q,z_{n})\leq\phi(q,y_{n})\leq\phi(q,u_{n})+\omega_{n}$, we have $\phi(q,y_{n})\leq\phi(q,u_{n})+\omega_{n}$. Combining this with the fact that $z_{n}=\Lambda_{r_{i_{n}}}y_{n}$ and Lemma 2.4(d), we have

$$\begin{aligned} \phi(z_{n},y_{n}) =&\phi(\Lambda_{r_{i_{n}}}y_{n},y_{n}) \leq \phi(q,y_{n})-\phi(q,\Lambda_{r_{i_{n}}}y_{n}) \\ \leq& \phi(q,u_{n})-\phi(q,\Lambda_{r_{i_{n}}}y_{n}) + \omega_{n} = \phi(q,u_{n})-\phi(q,z_{n})+ \omega_{n}, \end{aligned}$$

for any $q\in\Omega$. This implies that $\lim_{n\to\infty} \phi(z_{n},y_{n})=0$. Again, by Lemma 2.2 we have $y_{n}\to u ^{*}$.

Step 5: $u^{*}\in\Omega$.

We first show that $u^{*}\in\bigcap_{i=1}^{\infty}F(G_{i})$. Note that $G_{i_{k}}(PG_{i_{k}})^{m_{k}-1}=G_{i}(PG_{i})^{m_{k}-1}$ whenever $k\in K_{i}$ for each $i\geq1$. From Step 4 and by the uniform continuity of J on bounded subsets of X, we have $\Vert Jy_{k}-Ju^{*} \Vert \to0$, $\Vert Ju_{k}-Ju^{*}\Vert \to0$ as $k\to \infty$. Thus, for each $i\geq1$, $\eta_{m_{k}}^{(i)}\in G_{i_{k}}(PG_{i_{k}})^{m_{k}-1}u _{k}$, we have

$$\begin{aligned} \bigl\Vert Jy_{k}-Ju^{*} \bigr\Vert =& \bigl\Vert \sigma Ju_{k}+(1-\sigma)J\eta_{m_{k}}^{(i)}-Ju ^{*} \bigr\Vert \\ =& \bigl\Vert (1-\sigma) \bigl(J\eta_{m_{k}}^{(i)}-Ju^{*} \bigr)- \sigma \bigl(Ju^{*}-Ju _{k} \bigr) \bigr\Vert \\ \geq&(1-\sigma) \bigl\Vert J\eta_{m_{k}}^{(i)}-Ju^{*} \bigr\Vert - \sigma \bigl\Vert Ju ^{*}-Ju_{k} \bigr\Vert , \end{aligned}$$

which implies that $\lim_{k\to\infty} \Vert J\eta_{m_{k}}^{(i)}-Ju ^{*}\Vert =0$ for each $i\geq1$. By the norm-to-weak continuity of $J^{-1}$, we have, for each $i\geq1$, $\eta_{m_{k}}^{(i)} \rightharpoonup u^{*}$ as $k\to\infty$. Furthermore,

$$\begin{aligned} \bigl\vert \bigl\Vert \eta_{m_{k}}^{(i)} \bigr\Vert - \bigl\Vert u^{*} \bigr\Vert \bigr\vert = \bigl\vert \bigl\Vert J \eta_{m_{k}} ^{(i)} \bigr\Vert - \bigl\Vert Ju^{*} \bigr\Vert \bigr\vert \leq \bigl\Vert J \eta_{m_{k}}^{(i)}-Ju^{*} \bigr\Vert \to0. \end{aligned}$$

Thus, for each $i\geq1$, $\lim_{k\to\infty} \Vert \eta_{m_{k}} ^{(i)}\Vert =\Vert u^{*}\Vert$. Hence, by the Kadec-Klee property of X, we have

$$\begin{aligned} \lim_{k\to\infty}\eta_{m_{k}}^{(i)}=u^{*} \quad \forall i\geq1. \end{aligned}$$

(3.1)

We now consider the sequence $\{w_{m_{k}}^{(i)}\}_{k\in K_{i}}$, generated by

$$\begin{aligned} w_{m_{k+1}}^{(i)}\in G_{i}P\eta_{m_{k}}^{(i)} \subset G_{i_{k}}(PG_{i _{k}})^{m_{k}}u_{k}, \quad k\in K_{i} \ \forall i\ge1. \end{aligned}$$

By the continuity of $G_{i}P$, we have, for each $i\geq1$, $\lim_{k\to\infty}w_{m_{k+1}}^{(i)}=w^{*}$, $w^{*}\in G_{i}Pu ^{*}=G_{i}u^{*}$, since $u^{*}\in K$. Using the continuity of both $G_{i}$ and (3.1), we obtain for each $i\geq1$

$$\begin{aligned} \begin{aligned} \bigl\Vert w_{m_{k+1}}^{(i)}-\eta_{m_{k}}^{(i)} \bigr\Vert &\leq \bigl\Vert w_{m _{k+1}}^{(i)}- \eta_{m_{k+1}}^{(i)} \bigr\Vert + \bigl\Vert \eta_{m_{k+1}}^{(i)}-u _{k+1} \bigr\Vert +\Vert u_{k+1}-u_{k}\Vert \\ &\quad {}+ \bigl\Vert u_{k}-\eta_{m_{k}}^{(i)} \bigr\Vert \to0,\quad k\to\infty. \end{aligned} \end{aligned}$$

Therefore, for each $i\geq1$, $w_{m_{k+1}}^{(i)}\to u^{*}$ as $k\to\infty$. Hence, by the uniqueness of the limit, we have $w^{*}=u^{*}$. Thus, $u^{*}\in\bigcap_{i=1}^{\infty}F(G_{i})$.

We now show that $u^{*}\in \bigcap_{i=1}^{\infty} \mathit {GMEP}(h_{i},A_{i}, \zeta_{i})$. Let $i\ge1$. Define a function $\Psi_{i}:K\times K \to \Bbb {R}$ by

$$\begin{aligned} \Psi_{i}(u,v)=h_{i}(u,v)+\zeta_{i}(v)- \zeta_{i}(u)+ \langle v-u, A _{i}u\rangle\quad\forall u, v \in K. \end{aligned}$$

Then, as shown by Zhang [32], for each i, $\Psi_{i}$ satisfies $(B1)$-$(B4)$. Note that if $k\in K_{i}$, for each $i\geq1$, then $\Psi_{i_{k}}=\Psi_{i}$ and $\Lambda_{r_{i_{k}}}=\Lambda_{r_{i}}$. Now, the equation $z_{k} =\Lambda_{r_{i_{k}}}y_{k}$ implies that, for each $i\geq1$,

$$\begin{aligned} \Psi_{i}(z_{k},v)+\frac{1}{r_{i}} \langle v-z_{k},Jz_{k}-Jy_{k} \rangle\geq0 \quad\forall v\in K. \end{aligned}$$

(3.2)

By applying $(B2)$, we have, for each $i\ge1$,

$$\begin{aligned} \frac{1}{r_{i}} \langle v-z_{k},Jz_{k}-Jy_{k} \rangle \geq-\Psi_{i}(z_{k},v)\geq\Psi_{i}(v,z_{k}) \quad\forall v\in K, \end{aligned}$$

(3.3)

which implies that

$$\begin{aligned} \Psi_{i}(v,z_{k})\leq\frac{1}{r_{i}} \langle v-z_{k},Jz_{k}-Jy _{k} \rangle\leq \frac{1}{r_{i}}\Vert v-z_{k}\Vert \Vert Jz _{k}-Jy_{k} \Vert \leq\frac{1}{r_{i}} \bigl(\Vert v\Vert +M \bigr)\Vert Jz _{k}-Jy_{k}\Vert , \end{aligned}$$

for all $v\in K$, $i\ge1$, and some $M>0$. This implies that $\liminf_{k\to\infty}\Psi_{i}(v,z_{k})\leq0$ for all $v\in K$, $i \geq1$. From $(B4)$, we obtain, for $i\ge1$, $\Psi_{i}(v,u^{*}) \leq\liminf_{k\to\infty} \Psi_{i}(v,z_{k})\leq0$ $\forall v \in K$. Let $t\in(0,1)$ and $v\in K$. Then $v_{t}=tv+(1-t)u^{*} \in K$. Therefore, for each $i\geq1$, $\Psi_{i}(v_{t},u^{*})\leq0$. From conditions $(B1)$ and $(B4)$ we have, for each $i\ge1$,

$$\begin{aligned} 0=\Psi_{i}(v_{t},v_{t})\leq t \Psi_{i}(v_{t},v)+(1-t)\Psi_{i} \bigl(v_{t},u ^{*} \bigr)\leq t\Psi_{i}(v_{t},v) \quad \Longrightarrow\quad \Psi_{i}(v_{t},v)\geq0 \quad \forall v\in K. \end{aligned}$$

By $(B3)$, we have, for each $i\ge1$, $\Psi_{i}(u^{*},v)\geq \limsup_{t\downarrow0}\Psi_{i}(v_{t},v)\geq0$ $\forall v\in K$. Therefore, $u^{*}\in \mathit {GMEP}(h_{i},\zeta_{i},A_{i})$ for each $i\ge1$. Hence, $u^{*}\in\bigcap_{i=1}^{\infty} \mathit {GMEP}(h_{i},\zeta_{i},A_{i})$.

Step 6: $u^{*}=\Pi_{\Omega}u_{0}$.

Let $v=\Pi_{\Omega}u_{0}$. Since $u^{*}\in\Omega$, we have

$$\begin{aligned} \phi(v,u_{0})\leq\phi \bigl(u^{*},u_{0} \bigr). \end{aligned}$$

(3.4)

Also, since $u_{n}=\Pi_{K_{n}}u_{0}$ and $v\in\Omega\subset K_{n}$, we have

$$\begin{aligned} \phi(u_{n},u_{0})\leq\phi(v,u_{0}). \end{aligned}$$

Since $u_{n}\to u^{*}$, we have

$$\begin{aligned} \phi \bigl(u^{*},u_{0} \bigr)\leq \phi(v,u_{0}). \end{aligned}$$

(3.5)

From inequalities (3.4) and (3.5) we obtain $\phi(u^{*},u _{0})= \phi(v,u_{0})$. Thus, $u^{*}=v=\Pi_{\Omega}u_{0}$. This completes the proof. □

We now prove the following strong convergence theorem using a Halpern-type algorithm.

Theorem 3.2

Let X be a uniformly smooth and strictly convex real Banach space with Kadec-Klee property and let $X^{*}$ be its dual space. Let K be a nonempty closed and convex subset of X and $h_{i}:K \times K\to \Bbb {R}$, $i=1,2,3,\ldots$ , be a sequence of bifunctions satisfying $(B1)$-$(B4)$. Let $A_{i}:K\to X^{*}$, $i=1,2,3,\ldots$ , be a sequence of continuous monotone maps and $G_{i}:K \to{2^{X}}$, $i=1,2,\ldots$ , be an infinite family of equally continuous and totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps with nonnegative real sequences $\{\gamma_{n}^{(i)}\}$, $\{\delta_{n}^{(i)} \}$ and a sequence of strictly increasing and continuous functions $\{\rho_{i}\}$, $\rho_{i}:\Bbb {R}^{+}\to \Bbb {R}^{+}$ such that $\gamma_{n}^{(i)} \to0$, $\delta_{n}^{(i)}\to0$, and $\rho_{i}(0)=0$. Let $\zeta_{i}:K \to \Bbb {R}$, $i=1,2,3,\ldots$ , be a sequence of convex and lower-semicontinuous functions. Suppose, for each i, $\delta_{1}^{(i)}=0$ and $\Omega:= (\bigcap_{i=1}^{\infty}F(G_{i}) )\cap (\bigcap_{i=1}^{\infty} \mathit {GMEP}(h _{i}, A_{i},\zeta_{i}) )$ is a nonempty subset of K. Then the sequence $\{u_{n}\}$, generated by algorithm (1.7), converges strongly to $\Pi_{\Omega}u_{0}$, where $\{\sigma_{n}\}\subset(0,1)$ with $\lim_{n\to\infty}\sigma_{n}=0$, $r_{i_{n}}\in[a, \infty)$ for some $a>0$, and $\omega_{n}=\gamma_{m_{n}}^{(i_{n})} \rho_{i_{n}} [\phi(p,u_{n}) ]+ \delta_{m_{n}}^{(i_{n})}$, $p\in\Omega$.

Proof

As in the proof of Theorem 3.1, the proof of this theorem is presented in six steps.

Step 1: $K_{n}$ is closed and convex for all $n\geq1$.

This follows easily by induction, just as in the proof of Theorem 3.1.

Step 2: $\Omega\subset K_{n}$ for all $n\ge1$. Clearly, $\Omega\subset K_{1}$. Suppose $\Omega\subset K_{n}$ for some $n\geq1$. Let $q\in\Omega$. Then by applying Lemma 2.4(d), the definition of ϕ, the convexity of $\Vert \cdot \Vert ^{2}$, and the totally quasi-ϕ-asymptotically nonexpansiveness of $G_{i}$, as in Step 2 of the proof of Theorem 3.1, we have

$$\begin{aligned} \phi(q,z_{n}) =&\phi (q, \Lambda_{r_{i_{n}}}y_{n}) \\ \leq& \phi (q,y_{n} ) \\ =&\phi \bigl(q,J^{-1} \bigl(\sigma_{n} Ju_{0}+(1-\sigma_{n})J\eta_{m _{n}}^{(i_{n})} \bigr) \bigr) \\ =& \Vert q\Vert ^{2}-2 \bigl\langle q, \sigma_{n} Ju_{0}+(1-\sigma_{n})J \eta_{m_{n}}^{(i_{n})} \bigr\rangle + \bigl\Vert \sigma_{n} Ju_{0}+(1- \sigma_{n})J\eta_{m_{n}}^{(i_{n})} \bigr\Vert ^{2} \\ \leq& \Vert q\Vert ^{2}-2\sigma_{n}\langle q,Ju_{0}\rangle -2(1-\sigma_{n}) \bigl\langle q,J \eta_{m_{n}}^{(i_{n})} \bigr\rangle +\sigma_{n}\Vert u_{0}\Vert ^{2}+(1- \sigma_{n}) \bigl\Vert \eta_{m_{n}}^{(i_{n})} \bigr\Vert ^{2} \\ =& \sigma_{n}\phi(q,u_{0})+(1- \sigma_{n})\phi \bigl(q,\eta_{m_{n}}^{(i _{n})} \bigr) \\ \leq& \sigma_{n}\phi(q,u_{0})+(1- \sigma_{n}) \bigl[\phi(q,u_{n})+ \gamma_{m_{n}}^{(i_{n})} \rho_{i_{n}} \bigl(\phi(q,u_{n}) \bigr)+ \delta_{m_{n}}^{(i_{n})} \bigr] \\ \leq&\sigma_{n}\phi(q,u_{0})+(1- \sigma_{n}) \phi(q,u_{n})+ \gamma_{m_{n}}^{(i_{n})} \rho_{i_{n}} \bigl(\phi(q,u_{n}) \bigr)+ \delta_{m_{n}}^{(i_{n})} \\ =&\sigma_{n}\phi(q,u_{0})+(1-\sigma_{n}) \phi(q,u_{n})+ \omega_{n}, \end{aligned}$$

which implies that $q\in K_{n+1}$. Therefore, by induction, $\Omega\subset K_{n}$ for all $n\geq1$.

Step 3: $\{\phi(u_{n},u_{0})\}$ is convergent and $\lim_{n\to\infty}\omega_{n}=0$.

This follows just as in Step 3 of the proof of Theorem 3.1.

Step 4: $u_{n}\to u^{*}$, $z_{n}\to u^{*}$, and $y_{n}\to u ^{*}$ as $n\to\infty$, for some $u^{*}\in K$.

The verification of this step follows the same pattern as in the verification of Step 4 in the proof of Theorem 3.1.

Step 5: $u^{*}\in\Omega$.

We first show that $u^{*}\in\bigcap_{i=1}^{\infty}F(G_{i})$. Note that $G_{i_{k}}(PG_{i_{k}})^{m_{k}-1}=G_{i}(PG_{i})^{m_{k}-1}$ whenever $k\in K_{i}$ for each $i\geq1$. From Step 4 and the uniform continuity of J on bounded subsets of X, we have $\Vert Jy_{k}-Ju^{*}\Vert \to0$ as $k\to\infty$. Thus, for each $i\geq1$, $\eta_{m_{k}}^{(i)}\in G _{i_{k}}(PG_{i_{k}})^{m_{k}-1}u_{k}$, we have

$$\begin{aligned} \bigl\Vert Jy_{k}-Ju^{*} \bigr\Vert =& \bigl\Vert \sigma_{n} Ju_{0}+(1-\sigma_{n})J \eta_{m _{k}}^{(i)}-Ju^{*} \bigr\Vert \\ =& \bigl\Vert (1-\sigma_{n}) \bigl(J\eta_{m_{k}}^{(i)}-Ju^{*} \bigr)- \sigma_{n} \bigl(Ju ^{*}-Ju_{0} \bigr) \bigr\Vert \\ \geq&(1-\sigma_{n}) \bigl\Vert J\eta_{m_{k}}^{(i)}-Ju^{*} \bigr\Vert - \sigma_{n} \bigl\Vert Ju^{*}-Ju_{0} \bigr\Vert , \end{aligned}$$

which implies that $\lim_{k\to\infty} \Vert J\eta_{m_{k}}^{(i)}-Ju ^{*}\Vert =0$ for each $i\geq1$. The rest of the justification of this step follows the same pattern as in the justification of Step 5 in the proof of Theorem 3.1.

Step 6: $u^{*}=\Pi_{\Omega}u_{0}$.

This is the same as Step 6 of the proof of Theorem 3.1. Hence, the proof is completed. □

A prototype for the control parameter in Theorem 3.2 is the canonical choice, $\sigma_{n}=\frac{1}{n}$.

4 Applications

In this section, we present some applications of Theorem 3.1. Similar applications of Theorem 3.2 also follow.

4.1 Countable family of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps and system of equilibrium problems

By setting $A\equiv0$, $\zeta\equiv0$ in Theorem 3.1, the sequence $\{u_{n}\}$, defined in Theorem 3.1, converges strongly to $\Pi_{\Omega}u_{0}$, where $\Omega:= (\bigcap_{i=1}^{\infty}F(G _{i}) )\cap (\bigcap_{i=1}^{\infty} \mathit{EP}(h_{i}) )$ and $\mathit{EP}(h)$ is the set of solutions of the equilibrium problem for h.

4.2 Countable family of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps and system of convex optimization problems

By setting $A\equiv0$, $h\equiv0$ in Theorem 3.1, the sequence $\{u_{n}\}$, defined in Theorem 3.1, converges strongly to $\Pi_{\Omega}u_{0}$, where $\Omega:= (\bigcap_{i=1}^{\infty}F(G _{i}) )\cap (\bigcap_{i=1}^{\infty} \mathit {CMP}(\zeta_{i}) )$ and $\mathit {CMP}(\zeta)$ is the set of solutions of the convex minimization problem for ζ.

4.3 Countable family of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps and system of variational inequality problems

By setting $h\equiv0$, $\zeta\equiv0$ in Theorem 3.1, the sequence $\{u_{n}\}$, defined in Theorem 3.1, converges strongly to $\Pi_{\Omega}u_{0}$, where $\Omega:= (\bigcap_{i=1}^{\infty}F(G _{i}) )\cap (\bigcap_{i=1}^{\infty} \mathit {VIP}(K, A_{i}) )$ and $\mathit {VIP}(K,A)$ is the set of solutions of the variational inequality problem for A over K.

4.4 Application in classical Banach spaces

Let $X=L_{p}, l_{p}$, or $W_{p}^{m}(\Omega)$, $1< p<\infty$, where $W_{p}^{m}(\Omega)$ denotes the usual Sobolev space, and let $X^{*}$ be the dual space of X. Clearly, X is uniformly convex and uniformly smooth. Consequently, Theorem 3.1 is applicable in these spaces.

Remark 5

See, e.g., Alber and Ryazantseva [34], p.36

The analytical representations of duality maps are known in $l^{p}$, $L^{p}(G)$, and Sobolev spaces $W^{p}_{m}(G)$, $p \in(1,\infty)$, and $p^{-1}+ q^{-1}=1$.

4.5 Application in Hilbert spaces

The following theorem follows immediately from Theorem 3.1.

Theorem 4.1

Let H be a real Hilbert space and K be a nonempty closed and convex subset of H. Let $h_{i}:K \times K\to \Bbb {R}$, $i=1,2,3,\ldots$ , be a sequence of bifunctions satisfying $(B1)$-$(B4)$. Let $A_{i}:K\to H$ be a sequence of continuous monotone maps and $G_{i}:K \to{2^{H}}$, $i=1,2,\ldots$ , be an infinite family of equally continuous and totally quasi-asymptotically nonexpansive nonself multi-valued maps with nonnegative real sequences $\{\gamma_{n}^{(i)}\}$, $\{\delta_{n}^{(i)} \}$ and a sequence of strictly increasing and continuous functions $\{\rho_{i}\}$, $\rho_{i}:\Bbb {R}^{+}\to \Bbb {R}^{+}$, such that $\gamma_{n} ^{(i)}\to0$, $\delta_{n}^{(i)}\to0$, and $\rho_{i}(0)=0$. Let $\zeta_{i}:K\to \Bbb {R}$, $i=1,2,3,\ldots$ , be a sequence of convex and lower-semicontinuous functions. Suppose, for each i, $\delta_{1} ^{(i)}=0$ and $\Omega:= (\bigcap_{i=1}^{\infty}F(G_{i}) )\cap (\bigcap_{i=1}^{\infty} \mathit {GMEP}(h_{i}, A_{i},\zeta_{i}) )$ is a nonempty subset of K. Then the sequence $\{u_{n}\}$, generated by

$$\begin{aligned} \textstyle\begin{cases} u_{0} \in H,\quad \textit{chosen arbitrarily},\quad\quad K_{1}=K, \quad \quad u_{1}=\Pi_{K_{1}}u_{0}, \\ y_{n}=\sigma u_{n}+(1-\sigma)\eta_{m_{n}}^{(i_{n})}, \quad (\eta_{m_{n}}^{(i_{n})}\in G_{i_{n}}(PG_{i_{n}})^{m_{n}-1}u_{n}), \\ z_{n}=G_{r_{i_{n}}}y_{n}, \\ K_{n+1} =\{v\in K_{n}:\Vert v-z_{n}\Vert ^{2} \leq \Vert v-u_{n}\Vert ^{2} +\omega _{n}\}, \\ u_{n+1}=P_{K_{n+1}}u_{0},\quad n\geq1, \end{cases}\displaystyle \end{aligned}$$

converges strongly to $P_{\Omega}u_{0}$, where $P_{K}$ is the metric projection of H onto K, $\sigma\in(0,1)$, $r_{i_{n}}\in[a, \infty)$ for some $a>0$, and $\omega_{n}=\gamma_{m_{n}}^{(i_{n})} \rho_{i_{n}} (\Vert u_{n}-p\Vert ^{2} )+ \delta _{m_{n}}^{(i_{n})}$.

Remark 6

• (1) Theorem 3.2 improves the results of Bo and Yi [3] in the following ways:

  • In Theorem 3.2, a countable family of nonself multi-valued maps is considered, whereas in Bo and Yi [3] a single nonself multi-valued map is considered.

  • The requirement that G is uniformly L-Lipschitz continuous in Bo and Yi [3] is weakened to: for each i, $G_{i}$ is equally continuous in Theorem 3.2.

  • The algorithm in Theorem 3.2 involves only one control parameter $\{\sigma_{n}\}\subset(0,1)$ satisfying condition $(C1)$, whereas the algorithm of Bo and Yi [3] contains two control parameters $\{\beta_{n}\}\subset(0,1)$ and $\{\sigma_{n}\}\subset[0,1]$, satisfying conditions $(C1)$ and $(C2)$.

  • The Banach spaces considered in Theorem 3.2 are uniformly smooth and strictly convex real Banach spaces with Kadec-Klee property, which include uniformly smooth and uniformly convex real Banach spaces studied in Bo and Yi [3].

• (2) Theorem 3.2 improves and generalizes the results in Zhao and Chang [26] in a number of ways:

  • The class of maps considered in Zhao and Chang [26] is extended from the class of uniformly quasi-ϕ-asymptotically nonexpansive single-valued nonself maps to the slightly more general class of countable family of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps.

  • The requirement that, for each i, $G_{i}$ is uniformly $L_{i}$-Lipschitz continuous in Zhao and Chang [26] is weakened to the following statement: for each i, $G_{i}$ is equally continuous in Theorem 3.2.

  • The results of Zhao and Chang [26] are proved in uniformly smooth and uniformly convex real Banach spaces, while Theorem 3.2 is proved in the more general uniformly smooth and strictly convex real Banach spaces with the Kadec-Klee property.

  • The control parameter in the algorithm considered in Theorem 3.2 is $\{\sigma_{n}\}\subset(0,1)$ satisfying condition $(C1)$, whereas the algorithm of Zhao and Chang [26] contains two control parameters $\{\beta_{n}\}\subset(0,1)$ and $\{\sigma_{n}\}\subset[0,1]$, satisfying conditions $(C1)$ and $(C2)$.

5 Conclusion

In this article, iterative schemes of the Krasnoselskii-type and the Halpern-type for approximating a common point in the set of common fixed points of a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued maps and the set of solutions of a system of generalized mixed equilibrium problems are constructed. Strong convergence of the sequences generated by these algorithms is established in certain Banach spaces. Among other applications, our theorems are applied to solve convex feasibility problems, a system of convex minimization problems, a system of variational inequality problems, and a system of generalized equilibrium problems in uniformly smooth and strictly convex real Banach spaces with the Kadec-Klee property. Finally, our theorems are important improvements of several important recent results on these classes of nonlinear problems.