# Strong Convergence Theorems for Countable Families of Uniformly Quasi-ϕ-Asymptotically Nonexpansive Mappings and a System of Generalized Mixed Equilibrium Problems

Authors: Siwaporn Saewan; Poom Kumam
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The purpose of this paper is to present a new hybrid block iterative scheme by the generalizedf-
projection method for finding a common element of the fixed point set for a countable family of uniformly
quasi-ϕ-asymptotically nonexpansive mappings and the set of solutions of the system of generalized mixed
equilibrium problems in a strictly convex and uniformly smooth Banach space with the Kadec-Klee
property. Furthermore, we prove that our new iterative scheme converges strongly to a common element
of the aforementioned sets. The results presented in this paper improve and extend important recent
results in the literature.


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## Body

## 1. Introduction

LetE be a Banach space with it’s dual space E*, and let C be a nonempty closed convex subset of E. It is well known that the metric projection operator PC:E→C plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming problems, game theory, variational inequality, complementarity problems, and so forth (see, e.g., [1, 2] and the references therein). In 1994, Alber [3] introduced and studied the generalized projections ΠE:E→C and πC:E*→C from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections for approximately solving the variational inequalities and von Neumann’s intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solving the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized f-projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by Abler [3] and proved some properties of the generalized f-projection operator. In 2009, Fan et al. [5] presented some basic results for the generalized f-projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces.Block iterative method is a method which is often used by many authors to solve theconvex feasibility problem(CFP) (see, [6, 7], etc.). In 2008, Plubtieng and Ungchittrakool [8] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Later, Saewan and Kumam [9, 10] introduced a new modified block hybrid projection algorithm for finding a common element of the set of solutions of the generalized equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property.On the other hand, let{θi}i∈Γ:C×C→ℝ be a bifunction, let {φi}i∈Γ:C→ℝ be a real-valued function, and let {Ai}i∈Γ:C→E* be a monotone mapping, where Γ is an arbitrary index set. The system of generalized mixed equilibrium problems is to find x∈C such that (1.1)θi(x,y)+〈Aix,y-x〉+φi(y)-φi(x)≥0,i∈Γ,∀y∈C.
If Γ is a singleton, then problem (1.1) reduces into the generalized mixed equilibrium problem, which is to find x∈C such that(1.2)θ(x,y)+〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
The set of solutions (1.2) is denoted by GMEP(θ,A,φ), that is,(1.3)GMEP⁡(θ,A,φ)={x∈C:θ(x,y)+〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C}.
If A≡0, the problem (1.2) reduces into the mixed equilibrium problem forθ, denoted by MEP(θ,φ), which is to find x∈C such that(1.4)θ(x,y)+φ(y)-φ(x)≥0,∀y∈C.
If θ≡0, the problem (1.2) reduces into the mixed variational inequality of Browder type, denoted by VI(C,A,φ), which is to find x∈C such that(1.5)〈Ax,y-x〉+φ(y)-φ(x)≥0,∀y∈C.
If A≡0 and φ≡0, the problem (1.2) reduces into the equilibrium problem forθ, denoted by EP(θ), which is to find x∈C such that(1.6)θ(x,y)≥0,∀y∈C.
If θ≡0, the problem (1.4) reduces into the minimize problem, denoted by argmin(φ), which is to find x∈C such that(1.7)φ(y)-φ(x)≥0,∀y∈C.
The above formulation (1.5) was shown in [11] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In other words, the EP(θ) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. Some solution methods have been proposed to solve the EP(θ); see, for example, [11–24] and references therein.A pointx∈C is a fixed pointof a mapping S:C→C if Sx=x, by F(S) denote the set of fixed points of S; that is, F(S)={x∈C:Sx=x}. Recall that S is said to be nonexpansive if(1.8)‖Sx-Sy‖≤‖x-y‖,∀x,y∈C.S is said to be quasi-nonexpansive if F(S)≠∅ and(1.9)‖x-Sy‖≤‖x-y‖,∀x∈F(S),y∈C.S is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that(1.10)‖Snx-Sny‖≤kn‖x-y‖,∀x,y∈C,∀n≥1.S is said to be asymptotically quasi-nonexpansive if F(S)≠∅ and there exists a sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that(1.11)‖x-Sny‖≤kn‖x-y‖,∀x∈F(S),y∈C,∀n≥1.Recall that a pointp in C is said to be an asymptotic fixed point of S [25] if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Sxn∥=0. The set of asymptotic fixed points of S will be denoted by F̃(S).LetE be a real Banach space with norm ∥·∥, let C be a nonempty closed convex subset of E, and let E* denote the dual of E. Let 〈·,·〉 denote the duality pairing of E* and E. If E is a Hilbert space, 〈·,·〉 denotes an inner product on E. Consider the functional defined by(1.12)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,forx,y∈E,
where J:E→2E* is the normalized duality mapping.A mappingS from C into itself is said to be relatively nonexpansive [26–28] if F̃(S)=F(S)≠∅ and(1.13)ϕ(p,Sx)≤ϕ(p,x),∀x∈C,p∈F(S).S is said to be relatively asymptotic nonexpansive [29] if F̃(S)=F(S)≠∅ and there exists a sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that(1.14)ϕ(p,Snx)≤knϕ(p,x),∀x∈C,p∈F(S),n≥1.
The asymptotic behavior of a relatively nonexpansive mapping was studied in [30–32].S is said to be ϕ-nonexpansive if(1.15)ϕ(Sx,Sy)≤ϕ(x,y),∀x,y∈C.S is said to be quasiϕ-nonexpansive [17, 33, 34] if F(S)≠∅ and(1.16)ϕ(p,Sx)≤ϕ(p,x),∀x∈C,p∈F(S).S is said to be ϕ-asymptotically nonexpansive if there exists a real sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that(1.17)ϕ(Snx,Sny)≤knϕ(x,y),∀x,y∈C.S is said to be quasiϕ-asymptotically nonexpansive [34, 35] if F(S)≠∅ and there exists a real sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that(1.18)ϕ(p,Snx)≤knϕ(p,x),∀x∈C,p∈F(S),n≥1.
A mapping S is said to be closed if for any sequence {xn}⊂C with xn→x and Sxn→y, then Sx=y.Remark 1.1.
It is easy to know that each relatively nonexpansive mapping is closed. The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, and the class of quasi-ϕ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true (see for more detail [30–32, 36]).As well known ifC is a nonempty closed convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. In this connection, Alber [1] recently introduced the generalized projectionΠC:E→C is a map that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(x,y); that is, ΠCx=x¯, where x¯ is the solution to the minimization problem(1.19)ϕ(x¯,x)=inf⁡y∈Cϕ(y,x).
The existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ(y,x) and the strict monotonicity of the mapping J (see, e.g., [1, 37–40]). It is obvious from the definition of function ϕ that(1.20)(‖y‖-‖x‖)2≤ϕ(y,x)≤(‖y‖-‖x‖)2,∀x,y∈E.
If E is a Hilbert space, then ϕ(y,x)=∥y-x∥2 and ΠC becomes the metric projection of E onto C.Next we recall the concept of the generalizedf-projection operator. Let G:C×E*→ℝ∪{+∞} be a functional defined as follows:(1.21)G(ξ,ϖ)=‖ξ‖2-2〈ξ,ϖ〉+‖ϖ‖2+2ρf(ξ),
where ξ∈C,ϖ∈E*,ρ is positive number, and f:C→ℝ∪{+∞} is proper, convex, and lower semicontinuous. From definitions of G and f, it is easy to see the following properties:  (1)
G(ξ,ϖ) is convex and continuous with respect to ϖ when ξ is fixed;(2)
G(ξ,ϖ) is convex and lower semicontinuous with respect to ξ when ϖ is fixed.LetE be a real Banach space with its dual E*. Let C be a nonempty closed convex subset of E. We say that πCf:E*→2C is generalizedf-projection operator if(1.22)πCfϖ={u∈C:G(u,ϖ)=inf⁡ξ∈CG(ξ,ϖ),∀ϖ∈E*}.In 2005, Matsushita and Takahashi [36] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping S in a Banach space E:(1.23)x0∈C,chosenarbitrarily,yn=J-1(αnJxn+(1-αn)JSxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0.They proved that{xn} converges strongly to ΠF(S)x0, where ΠF(S) is the generalized projection from C onto F(S).Motivated by the results of Takahashi and Zembayashi [41], Cholamjiak and Suantai [12] proved the following strong convergence theorem by the hybrid iterative scheme for approximation of common fixed point of countable families of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space: x0∈E, x1=ΠC1x0,C1=C,(1.24)yn,i=J-1(αnJxn+(1-αn)JSxn),un,i=Trm,nθmTrm-1,nθm-1⋯Tr1,nθ1yn,i,Cn+1={z∈Cn:sup⁡i>1ϕ(z,Jun,i)≤ϕ(z,Jxn)},xn+1=ΠCn+1x0,n≥1.
Then, they proved that under certain appropriate conditions imposed on {αn} and {rn,i}, the sequence {xn} converges strongly to ΠCn+1x0.In 2010, Li et al. [42] introduced the following hybrid iterative scheme for the approximation of fixed point of relatively nonexpansive mapping using the properties of generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex: x0∈C,(1.25)yn=J-1(αnJxn+(1-αn)JSxn),Cn+1={w∈Cn:G(w,Jyn)≤G(w,Jxn)},xn+1=ΠCn+1fx0,n≥0,
where ΠCf:E→2C is generalized f-projection operator. They proved the strong convergence theorem for finding an element in the fixed point set of S. We remark here that the results of Li et al. [42] extended and improved on the results of Matsushita and Takahashi [36].Recently, Shehu [43] introduced a new iterative scheme by hybrid methods and proved strong convergence theorem for the approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a solution to a system of generalized mixed equilibrium problems in a uniformly convex and uniformly smooth Banach space by using the properties of the generalized f-projection operator. Chang et al. [44] used the modified block iterative method to propose an iterative algorithm for solving the convex feasibility problems for an infinite family of quasi-ϕ-asymptotically nonexpansive mappings. Very recently, Kim [45] and Saewan and Kumam [46] considered the shrinking projection methods for asymptotically quasi-ϕ-nonexpansive mappings in a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property.In this paper, we introduce a new hybrid block iterative scheme of the generalizedf-projection operator for finding a common element of the fixed point set of uniformly quasi-ϕ-asymptotically nonexpansive mappings and the set of solutions of the system of generalized mixed equilibrium problems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Then, we prove that our new iterative scheme converges strongly to a common element of the aforementioned sets. The results presented in this paper improve and extend the results of Shehu [43], Chang et al. [44], Li et al. [42], Takahashi and Zembayashi [41], Cholamjiak and Suantai [12], and many authors.


## 2. Preliminaries

A Banach spaceE is said to be strictly convex if ∥(x+y)/2∥<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. Let U={x∈E:∥x∥=1} be the unit sphere of E. Then a Banach space E is said to be smooth if the limit limt→0(∥x+ty∥-∥x∥)/t exists for each x,y∈U. It is also said to be uniformly smooth if the limit exists uniformly in x,y∈U. Let E be a Banach space. The modulus of smoothness of E is the function ρE:[0,∞)→[0,∞) defined by ρE(t)=sup{((∥x+y∥+∥x-y∥)/2)-1:∥x∥=1,∥y∥≤t}. The modulus of convexity of E is the function δE:[0,2]→[0,1] defined by δE(ɛ)=inf{1-∥(x+y)/2∥:x,y∈E,∥x∥=∥y∥=1,∥x-y∥≥ɛ}. The normalized duality mappingJ:E→2E* is defined by J(x)={x*∈E*:〈x,x*〉=∥x∥2,∥x*∥=∥x∥}. If E is a Hilbert space, then J=I, where I is the identity mapping.Remark 2.1.
IfE is a reflexive, strictly convex and smooth Banach space, then for x,y∈E, ϕ(x,y)=0 if and only if x=y. It is sufficient to show that if ϕ(x,y)=0, then x=y. From (1.12), we have ∥x∥=∥y∥. This implies that 〈x,Jy〉=∥x∥2=∥Jy∥2. From the definition of J, one has Jx=Jy. Therefore, we have x=y; see [38, 40] for more details.Recall that a Banach spaceE has the Kadec-Klee property [38, 40, 47], if for any sequence {xn}⊂E and x∈E with xn⇀x and ∥xn∥→∥x∥, then ∥xn-x∥→0 as n→∞. It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.Remark 2.2.
LetE be a Banach space. Then we know that  (1)
ifE is an arbitrary Banach space, then J is monotone and bounded;(2)
ifE is strictly convex, then J is strictly monotone;(3)
ifE is smooth, then J is single valued and semicontinuous;(4)
ifE is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E;(5)
E is reflexive, smooth, and strictly convex, then the normalized duality mapping J=J2 is single valued, one-to-one, and onto;(6)
ifE is uniformly smooth, then E is smooth and reflexive;(7)
E is uniformly smooth if and only if E* is uniformly convex; see [38] for more details.We also need the following lemmas for the proof of our main results.For solving the equilibrium problem for a bifunctionθ:C×C→ℝ, let us assume that θ satisfies the following conditions.  (A1)
θ(x,x)=0,for allx∈C.(A2)
θ is monotone; that is, θ(x,y)+θ(y,x)≤0,for allx,y∈C.(A3)
for eachx,y,z∈C,(2.1)lim⁡t↓0θ(tz+(1-t)x,y)≤θ(x,y).(A4)
for eachx∈C, y↦θ(x,y) is convex and lower semicontinuous.For example, letA be a continuous and monotone operator of C into E* and define(2.2)θ(x,y)=〈Ax,y-x〉,∀x,y∈C.Then,θ satisfies (A1)–(A4). The following result is in Blum and Oettli [11].Motivated by Combettes and Hirstoaga [13] in a Hilbert space and Takahashi and Zembayashi [48] in a Banach space, Zhang [49] obtained the following lemma.Lemma 2.3 (Liu et al. [50], Zhang [49, Lemma 1.5]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space  E. Let  θ be a bifunction from  C×C to  ℝ satisfying  (A1)–(A4), let  A:C→E* be a continuous and monotone mapping, let  φ:C→ℝ be a semicontinuous and convex functional, for  r>0, and let  x∈E. Then, there exists  z∈C such that
(2.3)Q(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C,
where Q(z,y)=θ(z,y)+〈Az,y-z〉+φ(y)-φ(z). Furthermore, define a mapping Tr:E→C as follows:
(2.4)Trx={z∈C:Q(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then the following holds.  (1)
Tr is single-valued.(2)
Tr is firmly nonexpansive; that is, for all x,y∈E,〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉.(3)
F(Tr)=F̃(Tr)=GMEP(θ,A,φ).(4)
GMEP(θ,A,φ) is closed and convex.(5)
ϕ(p,Trz)+ϕ(Trz,z)≤ϕ(p,z),for allp∈F(Tr) and z∈E.For the generalizedf-projection operator, Wu and Huang [4] proved the following basic properties.Lemma 2.4 (Wu and Huang [4]).
LetE be a reflexive Banach space with its dual E* and let C be a nonempty closed convex subset of E. The following statements hold.  (1)
πCfϖ is nonempty closed convex subset of C for all ϖ∈E*.(2)
IfE is smooth, then for all ϖ∈E*, x∈πCfϖ if and only if(2.5)〈x-y,ϖ-Jx〉+ρf(y)-ρf(x)≥0,∀y∈C.(3)
IfE is strictly convex and f:C→ℝ∪{+∞} is positive homogeneous (i.e., f(tx)=tf(x) for all t>0 such that tx∈C where x∈C), then πCf is single-valued mapping.Recently, Fan et al. [5] have shown that the condition f which is positive homogeneous and appeared in [5, Lemma 2.1(iii)] can be removed.Lemma 2.5 (Fan et al. [5]).
LetE be a reflexive Banach space with its dual E*, and let C be a nonempty closed convex subset of E. If E is strictly convex, then πCfϖ is single valued.Recall thatJ is single value mapping when E is a smooth Banach space. There exists a unique element ϖ∈E* such that ϖ=Jx where x∈E. This substitution for (1.21) gives(2.6)G(ξ,Jx)=‖ξ‖2-2〈ξ,Jx〉+‖x‖2+2ρf(ξ).Now we consider the second generalizedf-projection operator in Banach spaces (see [42]).Definition 2.6.
LetE be a real smooth Banach space, and let C be a nonempty closed convex subset of E. We say that ΠCf:E→2C is generalized f-projection operator if
(2.7)ΠCfx={u∈C:G(u,Jx)=inf⁡ξ∈CG(ξ,Jx),∀x∈E}.Lemma 2.7 (Deimling [51]).
LetE be a Banach space, and let f:E→ℝ∪{+∞} be a lower semicontinuous convex functional. Then there exist x*∈E* and α∈ℝ such that
(2.8)f(x)≥〈x,x*〉+α,∀x∈E.Lemma 2.8 (Li et al. [42]).
LetE be a reflexive smooth Banach space, and C let be a nonempty closed convex subset of E. The following statements hold.(1)
ΠCfx is nonempty closed convex subset of C for all x∈E.(2)
For allx∈E, x̂∈ΠCfx if and only if
(2.9)〈x̂-y,Jx-Jx̂〉+ρf(y)-ρf(x̂)≥0,∀y∈C.(3)
IfE is strictly convex, then ΠCf is single-valued mapping.Lemma 2.9 (Li et al. [42]).
LetE be a reflexive smooth Banach space and let C be a nonempty closed convex subset of E, and let x∈E, x̂∈ΠCfx. Then
(2.10)ϕ(y,x̂)+G(x̂,Jx)≤G(y,Jx),∀y∈C.Remark 2.10.
LetE be a uniformly convex and uniformly smooth Banach space, and let f(x)=0 for all x∈E. Then Lemma 2.9 reduces to the property of the generalized projection operator considered by Alber [1].Lemma 2.11 (Li et al. [42]).
LetE be a Banach space, and let f:E→ℝ∪{+∞} be a proper, convex, and lower semicontinuous mapping with convex domain D(f). If {xn} is a sequence in D(f) such that xn⇀x̂∈D(f) and limn→∞G(xn,Jy)=G(x̂,Jy), then limn→∞∥xn∥=∥x̂∥.Lemma 2.12 (Chang et al. [44]).
LetE be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let S:C→C be a closed and quasi-ϕ-asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞), kn→1. Then F(S) is a closed convex subset of C.Lemma 2.13 (Chang et al. [44]).
LetE be a uniformly convex Banach space, let r>0 be a positive number, and let Br(0) be a closed ball of E. Then, for any given sequence {xi}i=1∞⊂Br(0) and for any given sequence {λi}i=1∞ of positive number with ∑n=1∞λn=1, there exists a continuous, strictly increasing, and convex function g:[0,2r)→[0,∞) with g(0)=0 such that, for any positive integer i,j with i<j,
(2.11)‖∑n=1∞λnxn‖2≤∑n=1∞λn‖xn‖2-λiλjg(‖xi-xj‖).Definition 2.14.
(Chang et al. [44]).(1)
Let{Si}i=1∞:C→C be a sequence of mappings. {Si}i=1∞ is said to be a family of uniformly quasi-ϕ-asymptotically nonexpansive mappings, if ℱ:=∩i=1∞F(Si)≠∅, and there exists a sequence {kn}⊂[1,∞) with kn→1 such that for each i≥1(2.12)ϕ(p,Sinx)≤knϕ(p,x),∀p∈F,x∈C,∀n≥1.(2)
A mappingS:C→C is said to be uniformlyL-Lipschitz continuous, if there exists a constant L>0 such that(2.13)‖Snx-Sny‖≤L‖x-y‖,∀x,y∈C.Iff(x)≥0, it is clearly by the definition of mappings {Si}i=1∞ is a family of uniformly quasi-ϕ-asymptotically nonexpansive is equivalent to if ∩i=1∞F(Si)≠∅ and there exists a sequence {kn}⊂[1,∞) with kn→1 such that for each i≥1,(2.14)G(p,JSinx)≤knG(p,Jx),∀p∈F,x∈C,∀n≥1.


## 3. Strong Convergence Theorem

Now we state and prove our main result.Theorem 3.1.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let {Si}i=1∞:C→C be an infinite family of closed uniformly Li-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞), kn→1, and let f:E→ℝ be a convex lower semicontinuous mapping with C⊂int(D(f)). For each j=1,2,…,m, let θj be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4), let Aj:C→E* be a continuous and monotone mapping, and let φj:C→ℝ be a lower semicontinuous and convex function. Assume that F:=(∩i=1∞F(Si))∩(∩j=1mGMEP(θj,Aj,φj))≠∅. For an initial point x0∈E with x1=ΠC1fx0 and C1=C, we define the sequence {xn} as follows:
(3.1)zn=J-1(αn,0Jxn+∑i=1∞αn,iJSinxn),yn=J-1(βnJxn+(1-βn)Jzn),un=Trm,nQmTrm-1,nQm-1⋯Tr2,nQ2Tr1,nQ1yn,Cn+1={z∈Cn:G(z,Jun)≤G(z,Jxn)+(kn-1)Mn},xn+1=ΠCn+1fx0,∀n≥1,
where Mn=supq∈F{G(q,Jxn)}, {αn,i},{βn} are sequences in [0,1], and ∑i=0∞αn,i=1, for all n≥0, satisfy the following conditions.(i)
{rj,n}⊂[d,∞) for some d>0.(ii)
liminfn→∞αn,0αn,i>0 for alli≥1, and liminfn→∞(1-βn)>0.(iii)
f(x)≥0for allx∈C. and f(0)=0. Then {xn} converges strongly to p∈F, where p=ΠFfx0.Proof.
We split the proof into six steps.
Step 1.
We first show thatCn+1 is closed and convex for each n≥1.
ClearlyC1=C is closed and convex. Suppose that Cn is closed and convex for each n∈ℕ. Since for any z∈Cn, we know that G(z,Jun)≤G(z,Jxn)+(kn-1)Mn is equivalent to
(3.2)2〈z,Jxn-Jun〉≤‖xn‖2-‖un‖2+(kn-1)Mn,
it follow that
(3.3)Cn+1={z∈Cn:2〈z,Jxn-Jun〉≤‖xn‖2-‖un‖2+(kn-1)Mn}.
So, Cn+1 is closed and convex. This implies that ΠCn+1fx0 is well defined for all n≥1.
Step 2.
We show thatF⊂Cn for all n≥1.
We show by induction thatF⊂Cn for all n∈ℕ. It is obvious that F⊂C1=C. Suppose that F⊂Cn for some n≥1. Let q∈F⊂Cn, by the convexity of ∥·∥2, Lemma 2.13, and the uniformly quasi-ϕ-asymptotically nonexpansive of Si, we compute
(3.4)G(q,Jzn)=G(q,(αn,0Jxn+∑i=1∞αn,iJSinxn))=‖q‖2-2〈q,αn,0Jxn+∑i=1∞αn,iJSinxn〉+‖αn,0Jxn+∑i=1∞αn,iJSinxn‖2+2ρf(q)=‖q‖2-2αn,0〈q,Jxn〉-2∑i=1∞αn,i〈q,JSinxn〉+‖αn,0Jxn+∑i=1∞αn,iJSinxn‖2+2ρf(q)≤‖q‖2-2αn,0〈q,Jxn〉-2∑i=1∞αn,i〈q,JSinxn〉+αn,0‖Jxn‖2+∑i=1∞αn,i‖JSinxn‖2-αn,0αn,jg‖Jxn-JSjnxn‖+2ρf(q)=∥q∥2-2αn,0〈q,Jxn〉+αn,0‖Jxn‖2-2∑i=1∞αn,i〈q,JSinxn〉+∑i=1∞αn,i‖JSinxn‖2-αn,0αn,jg‖Jxn-JSjnxn‖+2ρf(q)=αn,0G(q,Jxn)+∑i=1∞αn,iG(q,JSinxn)-αn,0αn,jg‖Jxn-JSjnxn‖≤αn,0knG(q,Jxn)+∑i=1∞αn,iknG(q,Jxn)-αn,0αn,jg‖Jxn-JSjnxn‖≤knG(q,Jxn).
Since un=Ωnmyn, when Ωnj=Trj,nQjTrj-1,nQj-1⋯Tr2,nQ2Tr1,nQ1, j=1,2,3,…,m, Ωn0=I, it follows from (3.4) that
(3.5)G(q,Jun)=G(q,JΩnmyn)≤G(q,Jyn)=G(q,βnJxn+(1-βn)Jzn)=‖q‖2-2〈q,βnJxn+(1-βn)Jzn〉+‖βnJxn+(1-βn)Jzn‖2+2ρf(q)≤‖q‖2-2βn〈q,Jxn〉-2(1-βn)〈q,Jzn〉+βn‖xn‖2+(1-βn)‖zn‖2+2ρf(q)=βnG(q,Jxn)+(1-βn)G(q,Jzn)≤βnG(q,Jxn)+(1-βn)knG(q,Jxn)≤G(q,Jxn)+(1-βn)(kn-1)G(q,Jxn)=G(q,Jxn)+(kn-1)Mn.
This shows that q∈Cn+1 which implies that F⊂Cn+1, and hence, F⊂Cn for all n≥1. Since F is nonempty, Cn is a nonempty closed convex subset of E, and hence, ΠCnf exist for all n≥0. This implies that the sequence {xn} is well defined.
Step 3.
We show that{xn} is bounded.
Sincef:E→ℝ is convex and lower semicontinuous mapping, from Lemma 2.7, we have known that there exist x*∈E* and α∈ℝ such that
(3.6)f(x)≥〈x,x*〉+α,∀x∈E.
Since xn∈E, it follows that
(3.7)G(xn,Jx0)=‖xn‖2-2〈xn,Jx0〉+‖x0‖2+2ρf(xn)≥‖xn‖2-2〈xn,Jx0〉+‖x0‖2+2ρ〈xn,x*〉+2ρα=‖xn‖2-2〈xn,Jx0-ρx*〉+‖x0‖2+2ρα≥‖xn‖2-2∥xn∥∥Jx0-ρx*∥+‖x0‖2+2ρα=(‖xn‖-‖Jx0-ρx*‖)2+‖x0‖2-‖Jx0-ρx*‖2+2ρα.
For each q∈F⊂Cn and by the definition of Cn that xn=ΠCnfx0, it follows from (3.7) that
(3.8)G(q,Jx0)≥G(xn,Jx0)≥(‖xn‖-‖Jx0-ρx*‖)2+‖x0‖2-‖Jx0-ρx*‖2+2ρα.
This implies that {xn} is bounded and so are {G(xn,Jx0)}.
Step 4.
We show thatlimn→∞∥un-xn∥=0 and limn→∞∥Jun-Jxn∥=0.
By the fact thatxn+1=ΠCn+1fx0∈Cn+1⊂Cn and xn=ΠCnfx0, followed by Lemma 2.9, we get
(3.9)0≤(‖xn+1‖-‖xn‖)2≤ϕ(xn+1,xn)≤G(xn+1,Jx0)-G(xn,Jx0).
This implies that {G(xn,Jx0)} is nondecreasing. So, we obtain that limn→∞G(xn,Jx0) exist, and taking n→∞, we obtain that
(3.10)lim⁡n→∞ϕ(xn+1,xn)=0.
Since {xn} is bounded in C and E is reflexive, we can assume that xn⇀p. From the fact that xn=ΠCnfx0 when Cn is closed and convex for each n≥1, it is easy to see that p∈Cn and we get
(3.11)G(xn,Jx0)≤G(p,Jx0),∀n≥1.
Since f is convex and lower semicontinuous, we have
(3.12)liminf⁡n→∞G(xn,Jx0)=liminf⁡n→∞{‖xn‖2-2〈xn,Jx0〉+‖x0‖2+2ρf(xn)}≥∥p∥2-2〈p,Jx0〉+∥x0∥2+2ρf(p)=G(p,Jx0).
By (3.11) and (3.12), we get
(3.13)G(p,Jx0)≤liminf⁡n→∞G(xn,Jx0)≤limsup⁡n→∞G(xn,Jx0)≤G(p,Jx0).
That is limn→∞G(xn,Jx0)=G(p,Jx0), by Lemma 2.11, we have ∥xn∥→∥p∥, from the Kadec-Klee property of E, we obtain that
(3.14)lim⁡n→∞xn=p,
and we also have
(3.15)lim⁡n→∞xn+1=p.
Since xn+1=ΠCn+1fx0∈Cn+1⊂Cn and from the definition of Cn+1, we have
(3.16)G(xn+1,Jun)≤G(xn+1,Jxn)+(kn-1)Mn,∀n∈N,
is equivalent to
(3.17)ϕ(xn+1,un)≤ϕ(xn+1,xn)+(kn-1)Mn,∀n∈N.
By (3.10) and in view of limn→∞(kn-1)Mn=0, we also have
(3.18)lim⁡n→∞ϕ(xn+1,un)=0.
From (1.20), it follow that
(3.19)(‖xn+1‖-‖un‖)2⟶0.
Since ∥xn+1∥→∥p∥, we also have
(3.20)‖un‖⟶‖p‖asn⟶∞.
It follows that
(3.21)‖Jun‖⟶‖Jp‖asn⟶∞.
This implies that {∥Jun∥} is bounded in E*. Note that E is reflexive and E* is also reflexive; we can assume that Jun⇀x*∈E*. In view of the reflexive of E, we see that J(E)=E*. Hence, there exists x∈E such that Jx=x*. It follows that
(3.22)ϕ(xn+1,un)=‖xn+1‖2-2〈xn+1,Jun〉+‖un‖2=‖xn+1‖2-2〈xn+1,Jun〉+‖Jun‖2.
Taking liminfn→∞ on both sides of the equality above and in view of the weak lower semicontinuity of norm ∥·∥, it yields that
(3.23)0≥‖p‖2-2〈p,x*〉+‖x*‖2=‖p‖2-2〈p,Jx〉+‖Jx‖2=‖p‖2-2〈p,Jx〉+‖x‖2=ϕ(p,x).
That is p=x, which implies that x*=Jp. It follows that Jun⇀Jp∈E*. From (3.21) and the Kadec-Klee property of E*; that, is Jun→Jp as n→∞, we known that J-1:E*→E is norm-weak*-continuous, that is, un⇀p. From (3.20) and the Kadec-Klee property of E, we have
(3.24)lim⁡n→∞un=p.
Since ∥xn-un∥≤∥xn-p∥+∥p-un∥, it follows that
(3.25)lim⁡n→∞‖xn-un‖=0.
From J that is uniformly norm-to-norm continuous on bounded subsets of E, we obtain
(3.26)lim⁡n→∞‖Jun-Jxn‖=0.
Step 5.
We will show thatp∈F:=(∩i=1∞F(Si))∩(∩j=1mGMEP(θj,Aj,φj)).
(a) We show that p∈∩i=1∞F(Si).
Forq∈F, we note that
(3.27)ϕ(q,xn)-ϕ(q,un)+(kn-1)Mn=‖xn‖2-‖un‖2-2〈q,Jxn-Jun〉+(kn-1)Mn≤‖xn-un‖(‖xn‖+‖un‖)+2‖q‖‖Jxn-Jun‖+(kn-1)Mn.
It follows from ∥xn-un∥→0, ∥Jxn-Jun∥→0 and (kn-1)Mn→0 as n→∞ that
(3.28)ϕ(q,xn)-ϕ(q,un)+(kn-1)Mn⟶0asn⟶∞.
For any i≥1 and any q∈F, it follows from (3.4) and (3.5) that
(3.29)G(q,Jun)≤βnG(q,Jxn)+(1-βn)G(q,Jzn)≤βnG(q,Jxn)+(1-βn)(∑i=1∞αn,iknG(q,Jxn)αn,0knG(q,Jxn)+∑i=1∞αn,iknG(q,Jxn)-αn,0αn,jg‖Jxn-JSjnxn‖∑)=βnG(q,Jxn)+(1-βn)(knG(q,Jxn)-αn,0αn,jg‖Jxn-JSjnxn‖)=βnG(q,Jxn)+(1-βn)knG(q,Jxn)-(1-βn)αn,0αn,jg‖Jxn-JSjnxn‖≤G(q,Jxn)+(kn-1)Mn-(1-βn)αn,0αn,jg‖Jxn-JSjnxn‖.
It follows that
(3.30)αn,0αn,jg‖Jxn-JSjnxn‖≤1(1-βn)(G(q,Jxn)-G(q,Jun)+(kn-1)Mn),
is in equivalence to
(3.31)αn,0αn,jg‖Jxn-JSjnxn‖≤1(1-βn)(ϕ(q,xn)-ϕ(q,un)+(kn-1)Mn).
From (3.28), liminfn→∞αn,0αn,i>0 and liminfn→∞(1-βn)>0, we see that
(3.32)g(‖Jxn-JSjnxn‖)⟶0,n⟶∞.
It follows from the property of g that
(3.33)lim⁡n→∞‖Jxn-JSjnxn‖=0,∀j≥1.
Since xn→p and J is uniformly continuous, it yields that Jxn→Jp. Thus from (3.33), we have
(3.34)JSinxn⟶Jp,∀i≥1.
Since J-1:E*→E is norm-weak*-continuous, we also have
(3.35)Sinxn⇀p,∀i≥1.
On the other hand, for each i≥1, we observe that
(3.36)|‖Sinxn‖-‖p‖|=|‖J(Sinxn)‖-‖Jp‖|≤‖J(Sinxn)-Jp‖.
In view of (3.34), we obtain ∥Sinxn∥→∥p∥  for each i≥1. Since E has the Kadec-Klee property, we get
(3.37)Sinxn⟶pforeachi≥1,n∈N.
By the assumption that for each i≥1, Si is uniformly Li-Lipschitz continuous, so we have
(3.38)‖Sin+1xn-Sinxn‖≤‖Sin+1xn-Sin+1xn+1‖+‖Sin+1xn+1-xn+1‖+‖xn+1-xn‖+‖xn-Sinxn‖≤(Li+1)‖xn+1-xn‖+‖Sin+1xn+1-xn+1‖+‖xn-Sinxn‖.
By (3.14), (3.15), and (3.37), it yields that ∥Sin+1xn-Sinxn∥→0,n→∞,for alli≥1. From Sinxn→p, we get Sin+1xn→p, that is, SiSinxn→p. In view of the closeness of Si, we have Sip=p, for all i≥1. This imply that p∈∩i=1∞F(Si).
(b) We show that p∈∩j=1mGMEP(θj,Aj,φj).
Sincexn+1=ΠCn+1fx0∈Cn+1⊂Cn and from (3.5), we have
(3.39)G(xn+1,Jyn)≤G(xn+1,Jxn)+(kn-1)Mn
is in equivalence to
(3.40)ϕ(xn+1,yn)≤ϕ(xn+1,xn)+(kn-1)Mn.
From (3.10) and (kn-1)Mn→0, as n→∞, we see that
(3.41)lim⁡n→∞ϕ(xn+1,yn)=0.
From (1.20), it follows that
(3.42)(‖xn+1‖-‖yn‖)2⟶0.
Since ∥xn+1∥→∥p∥, we have
(3.43)‖yn‖⟶‖p‖,asn⟶∞.
It follow that
(3.44)‖Jyn‖⟶‖Jp‖asn⟶∞.
This implies that {∥Jyn∥} is bounded in E* and E* is reflexive; we can assume that Jyn⇀y*∈E*. In view of J(E)=E*. Hence, there exists y∈E such that Jy=y*. It follows that
(3.45)ϕ(xn+1,yn)=‖xn+1‖2-2〈xn+1,Jyn〉+‖yn‖2=‖xn+1‖2-2〈xn+1,Jyn〉+‖Jyn‖2.
Taking liminfn→∞ on both sides of the equality above and in view of the weak lower semicontinuous of norm ∥·∥, it yields that
(3.46)0≥‖p‖2-2〈p,y*〉+‖y*‖2=‖p‖2-2〈p,Jy〉+‖Jy‖2=‖p‖2-2〈p,Jy〉+‖y‖2=ϕ(p,y).
That is p=y, which implies that y*=Jp. It follows that Jyn⇀Jp∈E*. From (3.44) and the Kadec-Klee property of E*, that is, Jun→Jp as n→∞, note that J-1:E*→E is norm-weak*-continuous, that is, yn⇀p. From (3.43) and the Kadec-Klee property of E, we have
(3.47)lim⁡n→∞yn=p.
For q∈F⊂Cn, by nonexpansiveness, we observe that
(3.48)ϕ(q,un)=ϕ(q,Ωnmyn)≤ϕ(q,Ωnm-1yn)≤ϕ(q,Ωnm-2yn)⋮≤ϕ(q,Ωnjyn).
By Lemma (2.12)(5), we have for j=1,2,3,…,m(3.49)ϕ(Ωnjyn,yn)≤ϕ(q,yn)-ϕ(q,Ωnjyn)≤ϕ(q,xn)-ϕ(q,Ωnjyn)+(kn-1)Mn≤ϕ(q,xn)-ϕ(q,un)+(kn-1)Mn.
From (3.28), we get ϕ(Ωnjyn,yn)→0asn→∞, for j=1,2,3,…,m. From (1.20), it follow that
(3.50)(‖Ωnjyn‖-‖yn‖)2⟶0.
Since ∥yn∥→∥p∥, we also have
(3.51)‖Ωnjyn‖⟶‖p‖asn⟶∞.
Since {Ωnjyn} is bounded and E is reflexive, without loss of generality, we may assume that Ωnjyn⇀h. From the first step, we have known that Cn is closed and convex for each n≥1, it is obvious that h∈Cn. Again since
(3.52)ϕ(Ωnjyn,yn)=‖Ωnjyn‖2-2〈Ωnjyn,Jyn〉+‖yn‖2,
taking liminfn→∞ on both sides of the equality above, we have
(3.53)0≥∥h∥2-2〈h,Jp〉+‖p‖2=ϕ(h,p).
This implies that h=p,for allj=1,2,3,…m, then it follow that
(3.54)Ωnjyn⇀p,
from (3.51), (3.54) and the Kadec-Klee property, we have
(3.55)lim⁡n→∞Ωnjyn=p∀j=1,2,3,…,m.
By using the triangle inequality, we obtain
(3.56)‖Ωnjyn-Ωnj-1yn‖≤‖Ωnjyn-p‖+‖p-Ωnj-1yn‖.
Hence, we obtain that
(3.57)lim⁡n→∞‖Ωnjyn-Ωnj-1yn‖=0,∀j=1,2,3,…,m.
Since {rj,n}⊂[d,∞) and J is uniformly norm-to-norm continuous on bounded subsets, so
(3.58)lim⁡n→∞‖JΩnjyn-JΩnj-1yn‖rj,n=0,∀j=1,2,3,…,m.
From Lemma 2.3, we get for j=1,2,3,…,m(3.59)Qj(Ωnjyn,y)+1rj,n〈y-Ωnjyn,JΩnjyn-JΩnj-1yn〉≥0,∀y∈C.
From (A2), we get
(3.60)1rj,n〈y-Ωnjyn,JΩnjyn-JΩnj-1yn〉≥Qj(y,Ωnjyn),∀y∈C,∀j=1,2,3,…,m.
From (3.55) and (3.58), we have
(3.61)0≥Qj(y,p),∀y∈C,∀j=1,2,3,…,m.
For t with 0<t≤1 and y∈C, let yt=ty+(1-t)p. Then, we get that yt∈C. From (3.61), it follows that
(3.62)Qj(yt,p)≤0,∀y∈C,∀j=1,2,3,…,m.
By the conditions (A1) and (A4), we have for j=1,2,3,…,m(3.63)0=Qj(yt,yt)≤tQj(yt,y)+(1-t)Qj(yt,p)≤tQj(yt,y)≤Qj(yt,y).
From (A3), we get
(3.64)0≤Qj(yt,y)=Qj(ty+(1-t)p,y),
and letting t→0, we have
(3.65)0=lim⁡n→∞0≤lim⁡n→∞Qj(ty+(1-t)p,y)≤Qj(p,y),∀y∈C,∀j=1,2,3,…m.
This implies that p∈GMEP(θj,Aj,φj),  for allj=1,2,3,…m. Therefore, p∈∩j=1mGMEP(θj,Aj,φj). Hence, from (a) and (b), we obtain p∈F.
Step 6.
We show thatp=ΠFfx0.
SinceF is closed and convex set from Lemma 2.8, we have ΠFfx0 is single value, denote by v. From xn=ΠCnfx0 and v∈F⊂Cn, we also have
(3.66)G(xn,Jx0)≤G(v,Jx0),∀n≥1.
By definition of G and f, we know that, for each given x,G(ξ,Jx) is convex and lower semicontinuous with respect to ξ. So
(3.67)G(p,Jx0)≤liminf⁡n→∞G(xn,Jx0)≤limsup⁡n→∞G(xn,Jx0)≤G(v,Jx0).
From definition of ΠFfx0 and p∈F, we can conclude that v=p=ΠFfx0 and xn→p as n→∞. This completes the proof.For a special case thati=1,2, we can obtain the following results on a pair of quasi-ϕ-asymptotically nonexpansive mappings immediately from Theorem 3.1.Corollary 3.2.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property, and let f:E→ℝ be a convex lower semicontinuous mapping with C⊂int(D(f)). Let S1,S2:C→C be closed uniformly L1,L2-Lipschitz continuous and quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn1}⊂[1,∞), kn1→1 and {kn2}⊂[1,∞), kn2→1, respectively, and let {kn}=sup{{kn1},{kn2}}. For each j=1,2,…,m, let θj be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4), let Aj:C→E* be a continuous and monotone mapping and let φj:C→ℝ be a lower semicontinuous and convex function. Assume that F:=(F(S1)∩F(S2))∩(∩j=1mGMEP(θj,Aj,φj))≠∅. For an initial point x0∈E with x1=ΠC1fx0 and C1=C, we define the sequence {xn} as follows:
(3.68)zn=J-1(αn,0Jxn+αn,1JS1nxn+αn,2JS2nxn),yn=J-1(βnJxn+(1-βn)Jzn),un=Trm,nQmTrm-1,nQm-1⋯Tr2,nQ2Tr1,nQ1yn,Cn+1={z∈Cn:G(z,Jun)≤G(z,Jxn)+(kn-1)Mn},xn+1=ΠCn+1fx0,∀n≥1,
where Mn=supq∈F{G(q,Jxn)}, {αn,i}i=02 and {βn} are sequences in [0,1] such that ∑i=02αn,i=1 for all n≥0 and satisfy the following conditions.  (i)
{rj,n}⊂[d,∞) for some d>0.(ii)
liminfn→∞αn,0αn,i>0 for all i=1,2, and liminfn→∞(1-βn)>0.(iii)
f(x)≥0 for all x∈C, and f(0)=0.Then {xn} converges strongly to p∈F, where p=ΠFfx0.Remark 3.3.
Corollary3.2 improves and extends the result of Shehu [43, Theorem  3.1] following senses:  (i)
for the mappings, we extend the mappings from two closed weak relatively nonexpansive mappings (or relatively quasi-nonexpansive) mappings to a countable infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings;(ii)
for the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.
IfSi=S for each i∈ℕ, then Theorem 3.1 is reduced to the following corollary.Corollary 3.4.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let S:C→C be an infinite family of closed uniformly L-Lipschitz continuous and quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞), kn→1, and let f:E→ℝ be a convex lower semicontinuous mapping with C⊂int(D(f)). For each j=1,2,…,m, let θj be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4), let Aj:C→E* be a continuous and monotone mapping, and let φj:C→ℝ be a lower semicontinuous and convex function. Assume that F:=F(S)∩(∩j=1mGMEP(θj,Aj,φj))≠∅. For an initial point x0∈E with x1=ΠC1fx0 and C1=C, we define the sequence {xn} as follows:
(3.69)zn=J-1(αnJxn+(1-αn)JSnxn),yn=J-1(βnJxn+(1-βn)Jzn),un=Trm,nQmTrm-1,nQm-1⋯Tr2,nQ2Tr1,nQ1yn,Cn+1={z∈Cn:G(z,Jun)≤G(z,Jxn)+(kn-1)Mn},xn+1=ΠCn+1fx0,∀n≥1,
where Mn=supq∈F{G(q,Jxn)}, {αn} and {βn} are sequences in [0,1] and satisfy the following conditions:  (i)
{rj,n}⊂[d,∞) for some d>0.(ii)
liminfn→∞αn(1-αn)>0, and liminfn→∞(1-βn)>0.(iii)
f(x)≥0 for all x∈C, and f(0)=0.Then {xn} converges strongly to p∈F, where p=ΠFfx0.If takingθj≡0,Aj≡0,φj≡0,rj,n=1 and βn≡0,for alln∈ℕ in algorithm (3.69) in Corollary 3.4, we obtain the following corollary.Corollary 3.5.
(Li et al. [42]) Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed and convex subset of E,let S:C→C be a weak relative nonexpansive mapping, and let f:E→ℝ be a convex lower semicontinuous mapping with C⊂int(D(f)). Assume that {αn}n=0∞ is sequence in [0,1) such that limsupn→∞(αn)<1. Define a sequence {xn} in C by the following algorithm:
(3.70)xn=x0∈C,C0=C,yn=J-1(αnJxn+(1-αn)JSxn),Cn+1={w∈Cn:G(w,Jyn)≤G(w,Jxn)},xn+1=ΠCn+1fx0,n≥1.
If F(S) is nonempty, then {xn} converges to ΠF(S)fx0.Takingf(x)=0 for all x∈E, we have G(ξ,Jx)=ϕ(ξ,x) and ΠCfx=ΠCx. By Theorem 3.1, then we obtain the following Corollaries.Corollary 3.6.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let {Si}i=1∞:C→C be an infinite family of closed uniformly Li-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞), kn→1. For each j=1,2,…,m, let θj be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4), let Aj:C→E* be a continuous and monotone mapping, and let φj:C→ℝ be a lower semicontinuous and convex function. Assume that F:=(∩i=1∞F(Si))∩(∩j=1mGMEP(θj,Aj,φj))≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.71)zn=J-1(αn,0Jxn+∑i=1∞αn,iJSinxn),yn=J-1(βnJxn+(1-βn)Jzn),un=Trm,nQmTrm-1,nQm-1⋯Tr2,nQ2Tr1,nQ1yn,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+(kn-1)Mn},xn+1=ΠCn+1x0,∀n≥1,
where Mn=supq∈Fϕ(q,xn), {αn,i} and {βn} are sequences in [0,1] such that ∑i=0∞αn,i=1 for all n≥0. If {xn} is satisfying the following conditions.  (i)
{rj,n}⊂[d,∞) for some d>0.(ii)
liminfn→∞αn,0αn,i>0 for all i≥1 and liminfn→∞(1-βn)>0.Then {xn} converges strongly to p∈F, where p=ΠFx0.Corollary 3.7.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let S1,S2:C→C be closed uniformly L1,L2-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn1}⊂[1,∞), kn1→1 and {kn2}⊂[1,∞), kn2→1, respectively, such that {kn}=sup{{kn1},{kn2}}. For each j=1,2,…,m let θj be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4), let Aj:C→E* be a continuous and monotone mapping, and let φj:C→ℝ be a lower semicontinuous and convex function. Assume that F:=(F(S1)∩F(S2))∩(∩j=1mGMEP(θj,Aj,φj))≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.72)zn=J-1(αn,0Jxn+αn,1JS1nxn+αn,2JS2nxn),yn=J-1(βnJxn+(1-βn)Jzn),un=Trm,nQmTrm-1,nQm-1⋯Tr2,nQ2Tr1,nQ1yn,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+(kn-1)Mn},xn+1=ΠCn+1x0,∀n≥1,
where Mn=supq∈Fϕ(q,xn), {αn,i}i=02 and {βn} are sequences in [0,1] such that ∑i=02αn,i=1 for all n≥0 with the following conditions.  (i)
{rj,n}⊂[d,∞) for some d>0.(ii)
liminfn→∞αn,0αn,i>0 for all i=1,2 and liminfn→∞(1-βn)>0.Then {xn} converges strongly to p∈F, where p=ΠFx0.Remark 3.8.
Corollary3.6 and Corollary 3.7 extend and improve the results of many authors in the literature works.



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*Source: 701675-2011-08-02.xml*