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Hybrid algorithms for equilibrium and common fixed point problems with applications
Journal of Inequalities and Applications volume 2014, Article number: 221 (2014)
Abstract
In this paper, hybrid algorithms are investigated for equilibrium and common fixed point problems. Strong convergence of the algorithms is obtained in the framework of reflexive Banach spaces.
1 Introduction
Iterative algorithms have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization; see [1–11] and the references therein. The computation of solutions of nonlinear operator equations (inequalities) is important in the study of many realworld problems. Recently, the study of the convergence of various iterative algorithms for solving various nonlinear mathematical models has formed a major part of numerical mathematics. Among these iterative algorithms, the KrasnoselskiMann iterative algorithm and the Ishikawa iterative algorithm are popular and much discussed. It is well known that both the KrasnoselskiMann iterative algorithm and the Ishikawa iterative algorithm only have weak convergence even for nonexpansive mappings in infinitedimensional Hilbert spaces. In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in infinitedimensional spaces. In such problems, strong convergence is often much more desirable than weak convergence. To improve the weak convergence of the KrasnoselskiMann iterative algorithm and the Ishikawa iterative algorithm, socalled projection algorithms have been investigated in different frameworks of spaces; see [12–26] and the references therein.
The aim of this article is to investigate solutions of equilibrium and common fixed point problems in Hilbert spaces. In Section 2, we provide some necessary preliminaries. In Section 3, a projection algorithm is investigated. Strong convergence of the algorithm is obtained in a uniformly smooth and strictly convex Banach space which also has the KadecKlee property. In Section 4, applications are provided to support the main results of this paper.
2 Preliminaries
Let E be a real Banach space with the dual {E}^{\ast}. Recall that the normalized duality mapping J from E to {2}^{{E}^{\ast}} is defined by Jx=\{{f}^{\ast}\in {E}^{\ast}:\u3008x,{f}^{\ast}\u3009={\parallel x\parallel}^{2}={\parallel {f}^{\ast}\parallel}^{2}\}, where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing. Let {U}_{E}=\{x\in E:\parallel x\parallel =1\} be the unit sphere of E. E is said to be smooth iff {lim}_{t\to 0}\frac{\parallel x+ty\parallel \parallel x\parallel}{t} exists for each x,y\in {U}_{E}. It is also said to be uniformly smooth iff the above limit is attained uniformly for x,y\in {U}_{E}. E is said to be strictly convex iff \parallel \frac{x+y}{2}\parallel <1 for all x,y\in E with \parallel x\parallel =\parallel y\parallel =1 and x\ne y. It is said to be uniformly convex iff {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0 for any two sequences \{{x}_{n}\} and \{{y}_{n}\} in E such that \parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1 and {lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1. It is well known that if E is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of E. It is also well known that if E is (uniformly) smooth if and only if {E}^{\ast} is (uniformly) convex.
In what follows, we use ⇀ and → to stand for the weak and strong convergence, respectively. Recall that a space E has the KadecKlee property iff for any sequence \{{x}_{n}\}\subset E, x\in E with {x}_{n}\rightharpoonup x, and \parallel {x}_{n}\parallel \to \parallel x\parallel, we have \parallel {x}_{n}x\parallel \to 0 as n\to \mathrm{\infty}. It is known that if E is a uniformly convex Banach spaces, then E has the KadecKlee property.
Let E be a smooth Banach space. Let us consider the functional defined by \varphi (x,y)={\parallel x\parallel}^{2}2\u3008x,Jy\u3009+{\parallel y\parallel}^{2}, \mathrm{\forall}x,y\in E. Observe that, in a Hilbert space H, the equality is reduced to \varphi (x,y)={\parallel xy\parallel}^{2}, x,y\in H. As we know, if C is a nonempty, closed, and convex subset of a Hilbert space H and {P}_{C}:H\to C is the metric projection of H onto C, then {P}_{C} is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently it is not available in more general Banach spaces. In this connection, Alber [27] recently introduced a generalized projection operator {\mathrm{\Pi}}_{C} in a Banach space E which is an analog of the metric projection {P}_{C} in Hilbert spaces. Recall that the generalized projection {\mathrm{\Pi}}_{C}:E\to C is a map that assigns to an arbitrary point x\in E the minimum point of the functional \varphi (x,y), that is, {\mathrm{\Pi}}_{C}x=\overline{x}, where \overline{x} is the solution to the minimization problem \varphi (\overline{x},x)={min}_{y\in C}\varphi (y,x). Existence and uniqueness of the operator {\mathrm{\Pi}}_{C} follow from the properties of the functional \varphi (x,y) and strict monotonicity of the mapping J. If E is a reflexive, strictly convex, and smooth Banach space, then \varphi (x,y)=0 if and only if x=y; for more details, see [27] and the references therein. In Hilbert spaces, {\mathrm{\Pi}}_{C}={P}_{C}. From the definition of the function ϕ, we also have
and
Let F be a bifunction from C\times C to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem. Find p\in C such that F(p,y)\ge 0, \mathrm{\forall}y\in C. We use EP(F) to denote the solution set of the equilibrium problem. Numerous problems in physics, optimization and economics reduce to finding a solution of the equilibrium problem.
Next, we give the following assumptions:
(A1) F(x,x)=0, \mathrm{\forall}x\in C;
(A2) F is monotone, i.e., F(x,y)+F(y,x)\le 0, \mathrm{\forall}x,y\in C;
(A3) {lim\hspace{0.17em}sup}_{t\downarrow 0}F(tz+(1t)x,y)\le F(x,y), \mathrm{\forall}x,y,z\in C;
(A4) for each x\in C, y\mapsto F(x,y) is convex and weakly lower semicontinuous.
Let C be a nonempty subset of E and let T:C\to C be a mapping. In this paper, we use F(T) to stand for the fixed point set of T. Recall that T is said to be asymptotically regular on C iff for any bounded subset K of C, {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\{\parallel {T}^{n+1}x{T}^{n}x\parallel :x\in K\}=0. Recall that T is said to be closed iff for any sequence \{{x}_{n}\}\subset C such that {lim}_{n\to \mathrm{\infty}}{x}_{n}={x}_{0} and {lim}_{n\to \mathrm{\infty}}T{x}_{n}={y}_{0}, then T{x}_{0}={y}_{0}. Recall that a point p in C is said to be an asymptotic fixed point of T iff C contains a sequence \{{x}_{n}\} which converges weakly to p such that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0. The set of asymptotic fixed points of T will be denoted by \tilde{F}(T). T is said to be relatively nonexpansive iff \tilde{F}(T)=F(T)\ne \mathrm{\varnothing} and
T is said to be relatively asymptotically nonexpansive iff \tilde{F}(T)=F(T)\ne \mathrm{\varnothing} and
where \{{\mu}_{n}\}\subset [0,\mathrm{\infty}) is a sequence such that {\mu}_{n}\to 0 as n\to \mathrm{\infty}.
Remark 2.1 The class of relatively asymptotically nonexpansive mappings, which is an extension of the class of relatively nonexpansive mappings, was first introduced in [28].
Recall that T is said to be quasiϕnonexpansive iff F(T)\ne \mathrm{\varnothing} and
Recall that T is said to be asymptotically quasiϕnonexpansive iff there exists a sequence \{{\mu}_{n}\}\subset [0,\mathrm{\infty}) with {\mu}_{n}\to 0 as n\to \mathrm{\infty} such that
Remark 2.2 The class of asymptotically quasiϕnonexpansive mappings [14, 29] which is an extension of the class of quasiϕnonexpansive mappings [8]. The class of quasiϕnonexpansive mappings and the class of asymptotically quasiϕnonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasiϕnonexpansive mappings and asymptotically quasiϕnonexpansive ones do not require the restriction F(T)=\tilde{F}(T).
In order to prove our main results, we need the following lemmas.
Lemma 2.3 [30]
Let E be a smooth and uniformly convex Banach space and let r>0. Then there exists a strictly increasing, continuous, and convex function g:[0,2r]\to R such that g(0)=0 and
for all {x}_{1},{x}_{2},\dots ,{x}_{N},\dots \in {B}_{r}:=\{x\in E:\parallel x\parallel \le r\} and {\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{N},\dots \in [0,1] such that {\sum}_{i=1}^{\mathrm{\infty}}{\alpha}_{i}=1.
Lemma 2.4 [27]
Let E be a reflexive, strictly convex, and smooth Banach space. Let C be a nonempty, closed, and convex subset of E and let x\in E. Then
Lemma 2.5 [27]
Let C be a nonempty, closed, and convex subset of a smooth Banach space E and let x\in E. Then {x}_{0}={\mathrm{\Pi}}_{C}x if and only if
Lemma 2.6 [31]
Let E be a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property and let C be a nonempty, closed, and convex subset of E. Let T:C\to C be an asymptotically quasiϕnonexpansive mapping. Then F(T) is closed and convex.
Let C be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space E. Let F be a bifunction from C\times C to ℝ satisfying the assumptions (A1)(A4). Let r>0 and let x\in E. Then there exists z\in C such that F(z,y)+\frac{1}{r}\u3008yz,JzJx\u3009\ge 0, \mathrm{\forall}y\in C. Define a mapping {T}_{r}:E\to C by
Then the following conclusions hold:

(1)
{S}_{r} is a singlevalued firmly nonexpansivetype mapping, i.e., for all x,y\in E, \u3008{S}_{r}x{S}_{r}y,J{S}_{r}xJ{S}_{r}y\u3009\le \u3008{S}_{r}x{S}_{r}y,JxJy\u3009;

(2)
F({S}_{r})=EP(F) is closed and convex;

(3)
{S}_{r} is quasiϕnonexpansive;

(4)
\varphi (q,{S}_{r}x)+\varphi ({S}_{r}x,x)\le \varphi (q,x), \mathrm{\forall}q\in F({S}_{r}).
3 Main results
Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also has the KadecKlee property. Let C be a nonempty, closed, and convex subset of E and let Δ be an index set. Let {F}_{j} be a bifunction from C\times C to ℝ satisfying (A1)(A4) for every j\in \mathrm{\Delta}. Let {T}_{i}:C\to C an asymptotically quasiϕnonexpansive mapping for every i\ge 1. Assume that {T}_{i} is closed asymptotically regular on C and {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where \{{\alpha}_{n,i}\} is a real number sequence in (0,1) for every i\ge 1, \{{r}_{n,j}\} is a real number sequence in [r,\mathrm{\infty}), where r is some positive real number and {M}_{n}:=sup\{\varphi (z,{x}_{n}):z\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})\}. Assume that {\sum}_{i=0}^{\mathrm{\infty}}{\alpha}_{n,i}=1 and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,i}>0 for every i\ge 1. Then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})}{x}_{0}, where {\mathrm{\Pi}}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})} is the generalized projection from E onto {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}).
Proof Using Lemma 2.6 and Lemma 2.7, we find that {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}) is closed and convex so that {\mathrm{\Pi}}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})}{x}_{0} is well defined. By induction, we easily find that the sets {C}_{n} are convex and closed.
Next, we show that {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j})\subset {C}_{n}. It suffices to claim that
for every j\in \mathrm{\Delta}. It is clear that {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j})\subset {C}_{1,j}=C. Now, we assume that {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j})\subset {C}_{m,j} for some m and for every j\in \mathrm{\Delta}. Since \mathrm{\forall}z\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j})\subset {C}_{m,j}, one finds that
which yields z\in {C}_{m+1,j}, that is, {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j})\subset {C}_{n}.
Next, we prove that {x}_{n}\to p, where p\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j}). It follows from Lemma 2.4 that \varphi ({x}_{n},{x}_{0})\le \varphi (w,{x}_{0})\varphi (w,{x}_{n})\le \varphi (w,{x}_{0}), for \mathrm{\forall}w\in {\bigcap}_{i=1}^{N}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j}). This shows that the sequence \varphi ({x}_{n},{x}_{0}) is bounded. Hence \{{x}_{n}\} is also bounded. Since the space is reflexive, we may, without loss of generality, assume that {x}_{n}\rightharpoonup p, where p\in {C}_{n}. Using \varphi ({x}_{n},{x}_{0})\le \varphi (p,{x}_{0}), we find that
It follows that {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{0})=\varphi (p,{x}_{0}). Hence, we have {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}\parallel =\parallel p\parallel. Since E has the KadecKlee property, one sees that {x}_{n}\to p as n\to \mathrm{\infty}. Next, we prove p\in {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j}). In view of {C}_{n+1}\subset {C}_{n} and {x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{0}\in {C}_{n}, we have \varphi ({x}_{n+1},{x}_{n})\le \varphi ({x}_{n+1},{x}_{0})\varphi ({x}_{n},{x}_{0}). Letting n\to \mathrm{\infty}, we obtain \varphi ({x}_{n+1},{x}_{n})\to 0. Since {x}_{n+1}\in {C}_{n+1}, we see that \varphi ({x}_{n+1},{u}_{n,j})\le \varphi ({x}_{n+1},{x}_{n})+{\sum}_{i=1}^{\mathrm{\infty}}{\mu}_{n,i}{M}_{n}. We, therefore, obtain {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n+1},{u}_{n,j})=0. Hence {lim}_{n\to \mathrm{\infty}}\parallel {u}_{n,j}\parallel =\parallel p\parallel. It follows that {lim}_{n\to \mathrm{\infty}}\parallel J{u}_{n,j}\parallel =\parallel Jp\parallel. This shows that \{J{u}_{n,j}\} is a bounded sequence. Since E is reflexive ({E}^{\ast} is also reflexive), we may assume that J{u}_{n,j}\rightharpoonup {u}^{\ast ,j}\in {E}^{\ast}. Using the reflexivity of E, we also see that J(E)={E}^{\ast}. This shows that there exists an {u}^{j}\in E such that J{u}^{j}={u}^{\ast ,j}. It follows that \varphi ({x}_{n+1},{u}_{n})={\parallel {x}_{n+1}\parallel}^{2}2\u3008{x}_{n+1},J{u}_{n}\u3009+{\parallel J{u}_{n}\parallel}^{2}. Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} in both sides of the equality above yields
that is, p={u}^{j}. This yields Jp={u}^{\ast ,j}. It follows that J{u}_{n,j}\rightharpoonup Jp\in {E}^{\ast}. Since {E}^{\ast} has the KadecKlee property, we obtain J{u}_{n,j}Jp\to 0 as n\to \mathrm{\infty}. Since {J}^{1}:{E}^{\ast}\to E is demicontinuous. It follows that {u}_{n,j}\rightharpoonup p. Since the space E has the KadecKlee property, one finds that {u}_{n,j}\to p as n\to \mathrm{\infty}. In view of \parallel {x}_{n}{u}_{n,j}\parallel \le \parallel {x}_{n}p\parallel +\parallel p{u}_{n,j}\parallel, we have {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{u}_{n,j}\parallel =0. It follows that {lim}_{n\to \mathrm{\infty}}\parallel J{x}_{n}J{u}_{n,j}\parallel =0. Notice that
It follows that {lim}_{n\to \mathrm{\infty}}\varphi (z,{x}_{n})\varphi (z,{u}_{n,j})=0. By virtue of (3.1), we find that \varphi (z,{y}_{n})\le \varphi (z,{x}_{n})+{\sum}_{i=1}^{N}{\mu}_{n,j}{M}_{n}, where z\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j}). In view of {u}_{n,j}={S}_{{r}_{n,j}}{y}_{n}, we find from Lemma 2.7 that
It follows that {lim}_{n\to \mathrm{\infty}}\varphi ({u}_{n,j},{y}_{n})=0. This in turn yields that \parallel {u}_{n,j}\parallel \parallel {y}_{n}\parallel \to 0 as n\to \mathrm{\infty}. Since {u}_{n,j}\to p as n\to \mathrm{\infty}, one finds that {lim}_{n\to \mathrm{\infty}}\parallel {y}_{n}\parallel =\parallel p\parallel. It follows that {lim}_{n\to \mathrm{\infty}}\parallel J{y}_{n}\parallel =\parallel Jp\parallel. Since {E}^{\ast} is also reflexive, we may assume that J{y}_{n}\rightharpoonup {y}^{\ast}\in {E}^{\ast}. In view of J(E)={E}^{\ast}, we see that there exists y\in E such that Jy={y}^{\ast}. It follows that
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} in both sides of the equality above yields
That is, p=y, which in turn implies that {y}^{\ast}=Jp. It follows that J{y}_{n}\rightharpoonup Jp\in {E}^{\ast}. Since the space {E}^{\ast} has the KadecKlee property, we arrive at J{y}_{n}Jp\to 0 as n\to \mathrm{\infty}. Note that {J}^{1}:{E}^{\ast}\to E is demicontinuous. It follows that {y}_{n}\rightharpoonup p. Since E has the KadecKlee property, we obtain {y}_{n}\to p as n\to \mathrm{\infty}. In view of \parallel {u}_{n,j}{y}_{n}\parallel \le \parallel {u}_{n,j}p\parallel +\parallel p{y}_{n}\parallel, we find that {lim}_{n\to \mathrm{\infty}}\parallel {u}_{n,i}{y}_{n}\parallel =0. Since J is uniformly normtonorm continuous on any bounded sets, we have {lim}_{n\to \mathrm{\infty}}\parallel J{u}_{n,j}J{y}_{n}\parallel =0. From the assumption {r}_{n,j}\ge {r}_{j}, we see that {lim}_{n\to \mathrm{\infty}}\frac{\parallel J{u}_{n,j}J{y}_{n}\parallel}{{r}_{n,j}}=0. Notice that
From the condition (A2), we find that
Taking the limit as n\to \mathrm{\infty}, we find that {F}_{j}(y,p)\le 0, \mathrm{\forall}y\in C. For 0<{t}_{j}<1 and y\in C, define {y}_{{t}_{j}}={t}_{j}y+(1{t}_{j})p. It follows that {y}_{t,j}\in C, which yields {F}_{j}({y}_{t,j},p)\le 0. It follows from the conditions (A1) and (A4) that
This yields {F}_{j}({y}_{t,j},y)\ge 0. Letting {t}_{j}\to 0, we find from the condition (A3) that {F}_{j}(p,y)\ge 0, \mathrm{\forall}y\in C. This implies that p\in EP({F}_{j}) for every j\in \mathrm{\Delta}.
Now, we are in a position to show that p\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}). It follows from Lemma 2.3 that
From {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}(1{\alpha}_{n,i})>0, we find that {lim}_{n\to \mathrm{\infty}}g(\parallel J{x}_{n}J{T}_{i}^{n}{x}_{n}\parallel )=0. Hence, we have
Since {x}_{n}\to p as n\to \mathrm{\infty} and J:E\to {E}^{\ast} is demicontinuous, we obtain J{x}_{n}\rightharpoonup Jp\in {E}^{\ast}. Note that \parallel J{x}_{n}\parallel \parallel Jp\parallel =\parallel {x}_{n}\parallel \parallel p\parallel \le \parallel {x}_{n}p\parallel. This implies that \parallel J{x}_{n}\parallel \to \parallel Jp\parallel as n\to \mathrm{\infty}. Since {E}^{\ast} has the KadecKlee property, we see that
On the other hand, we have \parallel J{T}_{i}^{n}{x}_{n}Jp\parallel \le \parallel J{T}_{i}^{n}{x}_{n}J{x}_{n}\parallel +\parallel J{x}_{n}Jp\parallel. Combining (3.2) with (3.3), one obtains that {lim}_{n\to \mathrm{\infty}}\parallel J{T}_{i}^{n}{x}_{n}Jp\parallel =0. Since {J}^{1}:{E}^{\ast}\to E is demicontinuous, one sees that {T}_{i}^{n}{x}_{n}\rightharpoonup p. Notice that
This yields {lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}^{n}{x}_{n}\parallel =\parallel p\parallel. Since the space E has the KadecKlee property, we obtain {lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}^{n}{x}_{n}p\parallel =0. Since \parallel {T}^{n+1}{x}_{n}p\parallel \le \parallel {T}^{n+1}{x}_{n}{T}^{n}{x}_{n}\parallel +\parallel {T}^{n}{x}_{n}p\parallel and T is asymptotically regular, we find that {lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}^{n+1}{x}_{n}p\parallel =0. That is, {T}_{i}{T}_{i}^{n}{x}_{n}p\to 0 as n\to \mathrm{\infty}. It follows from the closedness of {T}_{i} that {T}_{i}p=p for every i\ge 1.
Finally, we prove that p={\mathrm{\Pi}}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j})}{x}_{0}. Since {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{0}, we see that
Since {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})\subset {C}_{n}, we find that
Letting n\to \mathrm{\infty}, we arrive at \u3008pw,J{x}_{0}Jp\u3009\ge 0, \mathrm{\forall}w\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}). In view of Lemma 2.5, we find that p={\mathrm{\Pi}}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EF({F}_{j})}{x}_{0}. This completes the proof. □
Remark 3.2 Theorem 3.1 mainly improves the corresponding results in Kim [11] and Qin et al. [33].
If T is quasiϕnonexpansive, we have the following result.
Corollary 3.3 Let E be a uniformly smooth and strictly convex Banach space which also has the KadecKlee property. Let C be a nonempty, closed, and convex subset of E and let Δ be an index set. Let {F}_{j} be a bifunction from C\times C to ℝ satisfying (A1)(A4) for every j\in \mathrm{\Delta}. Let {T}_{i}:C\to C a quasiϕnonexpansive mapping for every i\ge 1. Assume that {T}_{i} is closed on C and {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}) is nonempty. Let \{{x}_{n}\} be a sequence generated in the following manner:
where \{{\alpha}_{n,i}\} is a real number sequence in (0,1) for every i\ge 1, \{{r}_{n,j}\} is a real number sequence in [r,\mathrm{\infty}), where r is some positive real number. Assume that {\sum}_{i=0}^{N}{\alpha}_{n,i}=1 and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,i}>0 for every i\ge 1. Then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})}{x}_{0}, where {\mathrm{\Pi}}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})} is the generalized projection from E onto {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}).
Remark 3.4 Corollary 3.3 mainly improves the corresponding results in Qin et al. [8]. Indeed, the space is extended from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space; the mapping is extended from a pari of mappings to infinite many mappings. We also remark here that the framework of the space in this paper can be applicable to {L}^{p}, p\ge 1.
Finally for a single mapping and bifunction, we have the following result.
Corollary 3.5 Let E be a uniformly smooth and strictly convex Banach space which also has the KadecKlee property. Let C be a nonempty, closed, and convex subset of E and let F be a bifunction from C\times C to ℝ satisfying (A1)(A4). Let T:C\to C an asymptotically quasiϕnonexpansive mapping. Assume that T is closed asymptotically regular on C and F(T)\cap EP(F) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where \{{\alpha}_{n}\} is a real number sequence in (0,1), \{{r}_{n}\} is a real number sequence in [r,\mathrm{\infty}), where r is some positive real number and {M}_{n}:=sup\{\varphi (z,{x}_{n}):z\in F(T)\cap EP(F)\}. Assume {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0. Then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)\cap EP(F)}{x}_{0}, where {\mathrm{\Pi}}_{F(T)\cap EP(F)} is the generalized projection from E onto F(T)\cap EP(F).
4 Applications
First, we give the corresponding results in the framework of Hilbert spaces.
Theorem 4.1 Let E be a Hilbert space. Let C be a nonempty, closed, and convex subset of E and let Δ be an index set. Let {F}_{j} be a bifunction from C\times C to ℝ satisfying (A1)(A4) for every j\in \mathrm{\Delta}. Let {T}_{i}:C\to C an asymptotically quasinonexpansive mapping for every i\ge 1. Assume that {T}_{i} is closed asymptotically regular on C and {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where \{{\alpha}_{n,i}\} is a real number sequence in (0,1) for every i\ge 1, \{{r}_{n,j}\} is a real number sequence in [r,\mathrm{\infty}), where r is some positive real number and {M}_{n}:=sup\{{\parallel z{x}_{n}\parallel}^{2}:z\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})\}. Assume that {\sum}_{i=0}^{\mathrm{\infty}}{\alpha}_{n,i}=1 and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,i}>0 for every i\ge 1. Then the sequence \{{x}_{n}\} converges strongly to {Proj}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})}{x}_{0}, where {Proj}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})} is the metric projection from E onto {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\cap {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}).
Proof Note that \varphi (x,y)={\parallel xy\parallel}^{2}, J=I, the identity mapping, and the generalized projection is reduced to the metric projection. In the framework of Hilbert spaces, the class of asymptotically quasiϕnonexpansive mappings is reduced to the class of asymptotically quasinonexpansive mappings. Using Theorem 3.1, we easily find the desired conclusion. □
Next, we give a results on common solutions of a family of equilibrium problems.
Corollary 4.2 Let E be a Hilbert space. Let C be a nonempty, closed, and convex subset of E and let Δ be an index set. Let {F}_{j} be a bifunction from C\times C to ℝ satisfying (A1)(A4) for every j\in \mathrm{\Delta}. Assume that {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}) is nonempty. Let \{{x}_{n}\} be a sequence generated in the following manner:
where \{{r}_{n,j}\} is a real number sequence in [r,\mathrm{\infty}), where r is some positive real number. Then the sequence \{{x}_{n}\} converges strongly to {Proj}_{{\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})}{x}_{0}, where {Proj}_{{\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j})} is the metric projection from E onto {\bigcap}_{j\in \mathrm{\Delta}}EP({F}_{j}).
If {F}_{j}(x,y)=0, then we find from Theorem 4.1 the following result.
Corollary 4.3 Let E be a Hilbert space and let C be a nonempty, closed, and convex subset of E. Let {T}_{i}:C\to C an asymptotically quasinonexpansive mapping for every i\ge 1. Assume that {T}_{i} is closed asymptotically regular on C and {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where \{{\alpha}_{n,i}\} is a real number sequence in (0,1) for every i\ge 1, and {M}_{n}:=sup\{{\parallel z{x}_{n}\parallel}^{2}:z\in {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\}. Assume that {\sum}_{i=0}^{\mathrm{\infty}}{\alpha}_{n,i}=1 and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n,0}{\alpha}_{n,i}>0 for every i\ge 1. Then the sequence \{{x}_{n}\} converges strongly to {Proj}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})}{x}_{0}, where {Proj}_{{\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})} is the metric projection from E onto {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}).
Remark 4.4 Comparing with Theorem 2.2 in Kim and Xu [34], we have the following: (1) one mapping is extended to an infinite family of mappings; (2) the set {Q}_{n} is relaxed; (3) the mapping is extended from asymptotically nonexpansive mappings to asymptotically quasinonexpansive mappings, which may not be continuous.
From Corollary 4.3, the following result is not hard to derive.
Corollary 4.5 Let E be a Hilbert space. Let C be a nonempty, closed, and convex subset of E. Let T:C\to C be a closed quasinonexpansive mapping. Let \{{x}_{n}\} be a sequence generated in the following manner:
where \{{\alpha}_{n}\} is a real number sequence in (0,1) such that {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0. Then the sequence \{{x}_{n}\} converges strongly to {Proj}_{F(T)}{x}_{0}, where {Proj}_{F(T)} is the metric projection from E onto F(T).
Remark 4.6 Corollary 4.5 is a shrinking version of the corresponding results in Nakajo and Takahashi [35]. It deserves mentioning that the mapping in our result is quasinonexpansive. The restriction of the demiclosed principal is relaxed.
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Acknowledgements
This paper is dedicated to Professor Chang with respect and admiration. The authors are grateful to the editor and the three anonymous reviewers for useful suggestions which improved the contents of the article.
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Zhang, Q., Wu, H. Hybrid algorithms for equilibrium and common fixed point problems with applications. J Inequal Appl 2014, 221 (2014). https://doi.org/10.1186/1029242X2014221
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DOI: https://doi.org/10.1186/1029242X2014221
Keywords
 asymptotically quasiϕnonexpansive mapping
 quasiϕnonexpansive mapping
 algorithm
 equilibrium problem
 fixed point