Relaxed iterative algorithms for a system of generalized mixed equilibrium problems and a countable family of totally quasiPhiasymptotically nonexpansive multivalued maps, with applications
 CE Chidume^{1}Email author,
 OM Romanus^{1} and
 UV Nnyaba^{1}
https://doi.org/10.1186/s136630170616x
© The Author(s) 2017
Received: 24 June 2017
Accepted: 19 September 2017
Published: 25 November 2017
Abstract
In this article, a Krasnoselskiitype and a Halperntype algorithm for approximating a common fixed point of a countable family of totally quasiϕasymptotically nonexpansive nonself multivalued 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 KadecKlee property. Several applications of our theorems are also presented. Finally, our theorems are a significant improvement of several important recent results.
Keywords
MSC
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.
Definition 1.1
A nonself multivalued 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 \(\phi(p,\eta _{n})\leq\beta_{n}\phi(p,u)\) \(\forall u\in K\), \(p\in F(G)\), \(\eta_{n} \in G(PG)^{n1}u\), \(n\geq1\) (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

quasiϕnonexpansive if \(F(G)\ne\emptyset\) and \(\phi(p,\eta_{n})\leq\phi(p,u)\) \(\forall u\in K\), \(p\in F(G)\), \(\eta _{n}\in G(PG)^{n1}u\), \(n\geq1\);

quasiϕasymptotically nonexpansive if \(F(G)\ne\emptyset \) and there exists a real sequence \(\{\beta_{n}\}\subset[1,\infty)\), \(\beta_{n}\downarrow1\) such that \(\phi(p,\eta_{n})\leq\beta_{n} \phi(p,u)\) \(\forall u\in K\), \(p\in F(G)\), \(\eta_{n}\in G(PG)^{n1}u\), \(n \geq1\);

totally quasiϕasymptotically 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$$\begin{aligned} \begin{aligned}[b] &\phi(p,\eta_{n})\leq\phi(p,u)+ \gamma_{n} \rho \bigl[\phi(p,u) \bigr]+\delta_{n} \\ &\quad \forall u\in K, p\in F(G), \eta_{n}\in G(PG)^{n1}u, n\geq1. \end{aligned} \end{aligned}$$(1.3)
Remark 1
From the definitions, it is easy to see that the class of relatively asymptotically nonexpansive multivalued nonself maps and the class of quasiϕnonexpansive multivalued nonself maps are proper subclasses of the class of quasiϕasymptotically nonexpansive multivalued nonself maps and that the class of quasiϕasymptotically nonexpansive multivalued nonself maps is a proper subclass of the class of totally quasiϕasymptotically nonexpansive multivalued nonself maps, but the converse may not be true.
Definition 1.3

uniformly quasiϕasymptotically 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 \(i\geq1\),(see, e.g., Chang et al. [4]);$$\begin{aligned} \phi(p,\eta_{n})\leq\beta_{n}\phi(p,u) \quad \forall u\in K, p \in\bigcap_{i=1}^{\infty}F(G_{i}), \eta_{n}\in G _{i}(PG_{i})^{n1}u, n\geq1 \end{aligned}$$

uniformly totally quasiϕasymptotically 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 \(i\ge1\),(see, e.g., Yi [5]).$$\begin{aligned} \begin{gathered} \phi(p,\eta_{n})\leq\phi(p,u)+ \gamma_{n}\rho \bigl[ \phi(p,u) \bigr]+\delta_{n} \\ \quad \forall u\in K, p\in\bigcap _{i=1}^{\infty}F(G_{i}), \eta_{n}\in G _{i}(PG_{i})^{n1}u, n \geq1 \end{gathered} \end{aligned}$$
Remark 2
From the definitions, it is easy to see that a countable family of uniformly quasiϕasymptotically nonexpansive multivalued nonself maps is a countable family of uniformly totally quasiϕasymptotically nonexpansive multivalued nonself maps.
Remark 3
We also remark that a collection of countable families of uniformly totally quasiϕasymptotically nonexpansive multivalued nonself maps is a subcollection of a collection of countable families of totally quasiϕasymptotically nonexpansive multivalued nonself maps.
A motivation for the study of the class of totally quasiϕasymptotically 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

equally continuous if for \(u_{n}, v_{n} \in K\) we have$$\begin{aligned} \begin{gathered} \lim_{n\to\infty} \Vert u_{n}v_{n}\Vert =0 \quad \Longrightarrow\quad \lim_{n\to\infty} \Vert \eta_{n_{u}} \eta_{n_{v}}\Vert =0 \\ \quad \forall \eta_{n_{u}}\in G(PG)^{n1}u_{n}, \eta_{n_{v}}\in G(PG)^{n1}v_{n}; \end{gathered} \end{aligned}$$

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 LLipschitz continuous if there exists a constant \(L>0\) such that$$\begin{aligned} \Vert \eta_{u}\eta_{v}\Vert \leq L\Vert uv\Vert \quad \forall \eta_{u}\in G(PG)^{n1}u, \eta_{v}\in G(PG)^{n1}v, n\ge1. \end{aligned}$$
Remark 4
It is easy to see that the class of uniformly LLipschitz multivalued nonself maps is a proper subclass of the class of uniformly continuous multivalued nonself maps and the class of uniformly continuous multivalued nonself maps is a proper subclass of the class of equally continuous multivalued nonself maps.
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, quasiϕnonexpansive and quasiϕasymptotically 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).
 \((C1)\) :

\(\lim_{n\to\infty}\alpha_{n}=0\);
 \((C2)\) :

\(0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to\infty} \beta_{n}<1\);
 \((C3)\) :

F is a bounded and convex subset of K,
 \((C3^{*})\) :

F is a nonempty bounded subset of K,
 \((C3^{**})\) :

F is a nonempty subset of K.
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.
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 quasiϕasymptotically nonexpansive multivalued 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 KadecKlee 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]
 \((B1)\) :

\(h(u,u)=0\), \(\forall u\in X\),
 \((B2)\) :

h is monotone, that is, \(h(u,v)+h(v,u)\leq0\), \(\forall u,v \in X\),
 \((B3)\) :

for all \(u,y,z\in X\), \(\limsup_{t\downarrow0}h(tz+(1t)u,v) \leq h(u,v)\),
 \((B4)\) :

for all \(u\in K\), \(h(u,\cdot):K\to \Bbb {R}\) is convex and lowersemicontinuous.
Lemma 2.4
See Zhang [32]
 (1)There exists \(z\in K\) such that$$\begin{aligned} h(z,v) +\zeta(v)\zeta(z)+\langle vz, Az\rangle+\frac{1}{r} \langle vz,JzJu \rangle\geq0,\quad\forall v\in K. \end{aligned}$$
 (2)If we define a mapping \(\Lambda_{r}:K\to K\) bythen the mapping \(G_{r}\) has the following properties:$$\begin{aligned} \begin{aligned} \Lambda_{r}(u)&= \biggl\{ z\in K: h(z,v) +\zeta(v)\zeta(z)+\langle vz, Az\rangle+\frac{1}{r} \langle vz,JzJu \rangle\geq0, \\ &\quad \forall v \in K \biggr\} , \quad u\in K, \end{aligned} \end{aligned}$$
 (a)
\(G_{r}\) is singlevalued;
 (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)
\(\phi(q,\Lambda_{r}u)+\phi(\Lambda_{r}u,u)\leq\phi(q,u)\) \(\forall q\in F(\Lambda_{r})\), \(u\in X\).
 (a)
3 Main results
Theorem 3.1
Let X be a uniformly smooth and strictly convex real Banach space with KadecKlee 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 quasiϕasymptotically nonexpansive nonself multivalued 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 lowersemicontinuous 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 \(n\geq1\).
Step 3: \(\{\phi(u_{n},u_{0})\}\) is convergent and \(\lim_{n\to\infty}\omega_{n}=0\).
Step 4: \(u_{n}\to u^{*}\), \(z_{n}\to u^{*}\), and \(y_{n}\to u ^{*}\) as \(n\to\infty\), for some \(u^{*}\in K\).
Claim: \(u_{n}\to u^{*}\).
Step 5: \(u^{*}\in\Omega\).
Step 6: \(u^{*}=\Pi_{\Omega}u_{0}\).
We now prove the following strong convergence theorem using a Halperntype algorithm.
Theorem 3.2
Let X be a uniformly smooth and strictly convex real Banach space with KadecKlee 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 quasiϕasymptotically nonexpansive nonself multivalued 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 lowersemicontinuous 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 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: \(u^{*}\in\Omega\).
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 quasiϕasymptotically nonexpansive nonself multivalued 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 quasiϕasymptotically nonexpansive nonself multivalued 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 quasiϕasymptotically nonexpansive nonself multivalued 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
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 multivalued maps is considered, whereas in Bo and Yi [3] a single nonself multivalued map is considered.

The requirement that G is uniformly LLipschitz 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 KadecKlee 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 quasiϕasymptotically nonexpansive singlevalued nonself maps to the slightly more general class of countable family of totally quasiϕasymptotically nonexpansive nonself multivalued 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 KadecKlee 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 Krasnoselskiitype and the Halperntype for approximating a common point in the set of common fixed points of a countable family of totally quasiϕasymptotically nonexpansive nonself multivalued 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 KadecKlee property. Finally, our theorems are important improvements of several important recent results on these classes of nonlinear problems.
Declarations
Acknowledgements
The authors thank the referees for their time and comments. We also express our deepest gratitude to The African Capacity Building Foundation for funding this research work.
Funding
Research supported from ACBF Research Grant Funds to AUST.
Authors’ contributions
All the authors carried out the work in this paper with the consultation of each other. All authors read and approved the final manuscript.
Competing interests
The authors have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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