# Strong Convergence Theorem for Solving Generalized Mixed Equilibrium Problems and Fixed Point Problems for Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

Authors: Zhaoli Ma; Lin Wang; Yunhe Zhao
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points for countable families of total quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results.


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## Body

## 1. Introduction

LetE be a real Banach space with the dual E* and let C be a nonempty closed convex subset of E. We denote by R+ and R the set of all nonnegative real numbers and the set of all real numbers, respectively. Also, we denote by J the normalized duality mapping from E to 2E* defined by (1.1)Jx={x*∈E*:〈x,x*〉=‖x‖2=‖x*‖2},∀x∈E,
where 〈·,·〉 denotes the generalized duality pairing. Recall that if E is smooth then J is single-valued and norm-to-weak* continuous, and that if E is uniformly smooth then J is uniformly norm-to-norm continuous on bounded subsets of E. We will denote by J the single-value duality mapping.A Banach spaceE is said to be strictly convex if ∥x+y∥/2≤1 for all x,y∈U={z∈E:∥z∥=1} with x≠y. E is said to be uniformly convex if, for each ɛ∈(0,2], there exists δ>0 such that ∥x+y∥/2≤1-δ for all x,y∈U with ∥x-y∥≥ɛ. E is said to be smooth if the limit(1.2)limt→0‖x+ty‖-‖x‖texists for all x,y∈U. E is said to be uniformly smooth if the above limit exists uniformly in x,y∈U.Remark 1.1.
The following basic properties of Banach spaceE can be founded in [1].(i)
IfE is an uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.(ii)
IfE is a reflexive and strictly convex Banach space, then J-1 is norm-weak*-continuous.(iii)
IfE is a smooth, reflexive and strictly convex Banach space, then the normalized duality mapping J:E→2E* is single-valued, one-to-one, and surjective.(iv)
A Banach spaceE is uniformly smooth if and only if E* is uniformly convex.(v)
Each uniformly convex Banach spaceE has the Kadec-Klee property, that is, for any sequence {xn}⊂E, if xn⇀x∈E and ∥xn∥→∥x∥, then xn→x. (See [1, 2]) for more details.Next, we assume thatE is a smooth, reflexive, and strictly convex Banach space. Consider the functional defined as in [3, 4] by
(1.3)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,∀x,y∈E.
It is clear that in a Hilbert space H, (1.3) reduces to ϕ(x,y)=∥x-y∥2, for all x,y∈H.It is obvious from the definition ofϕ that(1.4)(‖x‖-‖y‖)2≤ϕ(x,y)≤(‖x‖+‖y‖)2,∀x,y∈E,
and
(1.5)ϕ(x,J-1(λJy+(1-λ)Jz))≤λϕ(x,y)+(1-λ)ϕ(x,z),∀x,y∈E.Following Alber [3], the generalized projection ΠC:E→C is defined by(1.6)ΠC(x)=arginfy∈Cϕ(y,x),∀x∈E.
That is, ΠCx=x¯, where x¯ is the unique solution to the minimization problem ϕ(x¯,x)=infy∈Cϕ(y,x).The existence and uniqueness of the operatorΠC follows from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J (see, e.g., [1–5]). In Hilbert space H, ΠC=PC.LetC be a nonempty closed convex subset of E, let T be a mapping from C into itself, and let F(T) be the set of fixed points of T. A point p∈C is called an asymptotically fixed point of T [6] if there exists a sequence {xn}⊂C such that xn⇀p and ∥xn-Txn∥→0. The set of asymptotical fixed points of T will be denoted by F̂(T). A point p∈C is said to be a strong asymptotic fixed point of T, if there exists a sequence {xn}⊂C such that xn→p and ∥xn-Txn∥→0. The set of strong asymptotical fixed points of T will be denoted by F̃(T).A mappingT:C→C is said to be relatively nonexpansive [7–9], if F(T)≠∅, F(T)=F̂(T) and ϕ(p,Tx)≤ϕ(p,x), for all x∈C, p∈F(T).A mappingT:C→C is said to be quasi-ϕ-nonexpansive, if F(T)≠∅ and ϕ(p,Tx)≤ϕ(p,x), for all x∈C, p∈F(T).A mappingT:C→C is said to be quasi-ϕ-asymptotically nonexpansive, if F(T)≠∅ and there exists a real sequence {kn}⊂[1,∞) with kn→1 such that(1.7)ϕ(p,Tnx)≤knϕ(p,x),∀n≥1,x∈C,p∈F(T).A mappingT:C→C is said to be total quasi-ϕ-asymptotically nonexpansive, if F(T)≠∅ and there exists nonnegative real sequences {νn}, {μn} with νn→0, μn→0(asn→∞) and a strictly increasing continuous function ξ:R+→R+ with ξ(0)=0 such that(1.8)ϕ(p,Tnx)≤ϕ(p,x)+νnξ(ϕ(p,x))+μn,∀n≥1,x∈C,p∈F(T).A countable family of mappings{Tn}:C→C is said to be uniformly total quasi-ϕ-asymptotically nonexpansive, if ⋂i=1∞F(Ti)≠∅, and there exists nonnegative real sequences {νn}, {μn} with νn→0, μn→0(asn→∞) and a strictly increasing continuous function ξ:R+→R+ with ξ(0)=0 such that for each i≥1 and each x∈C, p∈⋂i=1∞F(Ti)(1.9)ϕ(p,Tinx)≤ϕ(p,x)+νnξ(ϕ(p,x))+μn,∀n≥1.Remark 1.2.
From the definition, it is easy to know that:(i)
each relatively nonexpansive mapping is closed;(ii)
takingξ(t)=t, t≥0, νn=(kn-1) and μn=0 then νn→0  (asn→∞) and (1.7) can be rewritten as
(1.10)ϕ(p,Tnx)≤ϕ(p,x)+νnξ(ϕ(p,x))+μn,∀n≥1,x∈C,p∈F(T),
this implies that each quasi-ϕ-asymptotically nonexpansive mapping must be a total quasi-ϕ-asymptotically nonexpansive mapping, but the converse is not true;(iii)
the class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true;(iv)
the class of quasi-ϕ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true. (See more details [10–14]).Letf:C×C→R be a bifunction, where R is the set of real numbers. The equilibrium problem (For short, EP) is to find x*∈C such that(1.11)f(x*,y)≥0,∀y∈C.
The set of solutions of EP (1.11) is denoted by EP(f).LetB:C→H be a nonlinear mapping. The generalized equilibrium problem (for short, GEP) is to find x*∈C such that
(1.12)f(x*,y)+〈Bx*,y-x*〉≥0,∀y∈C.
The set of solutions of GEP (1.12) is denoted by GEP(f,B), that is,
(1.13)GEP(f,B)={x*∈C:f(x*,y)+〈Bx*,y-x*〉≥0,∀y∈C}.Letφ:C→R∪{+∞} be a function. The mixed equilibrium problem (for short, MEP) is to find x*∈C such that(1.14)f(x*,y)+φ(y)-φ(x*)≥0,∀y∈C.
The set of solutions of MEP (1.14) is denoted by MEP(f).The concept generalized mixed equilibrium problem (for short,GMEP) was introduced by Peng and Yao [15] in 2008. GMEP is to find x*∈C such that(1.15)f(x*,y)+φ(y)-φ(x*)+〈Bx*,y-x*〉≥0,∀y∈C.
The set of solutions of GMEP (1.15) is denoted by GMEP(f,B,φ), that is,
(1.16)GMEP(f,B,φ)={x*∈C:f(x*,y)+φ(y)-φ(x*)+〈Bx*,y-x*〉≥0,∀y∈C}.The equilibrium problem is an unifying model for several problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems (e. g., [16, 17]). Many authors have proposed some useful methods to solve the EP, GEP, MEP, GMEP; see, for instance, [15–23] and the references therein.In 2005, Matsushita and Takahashi [13] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping T in a Banach space E:
(1.17)x0∈Cchosen arbitrary,yn=J-1(αnJxn+(1-αn)Jxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,n≥0.
They prove that {xn} converges strongly to ΠF(T)x0, where ΠF(T) is the generalized projection from C onto F(T).Recently, Qin et al. [24] proposed a shrinking projection method to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of quasi-ϕ-nonexpansive mappings in the framework of Banach spaces:(1.18)x0=xchosen arbitrary,C1=C,x1=ΠC1x0,yn=J-1(αn,0Jxn+∑i=1Nαn,iJTixn),un∈Csuch thatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x0,n≥0,
where ΠCn+1 is the generalized projection from E onto Cn+1. They prove that the sequence {xn} converges strongly to Π∩i=1NF(Ti)∩EP(f)x0.In [25], Saewan and Kumam introduced a modified new hybrid projection method to find a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings in an uniformly smooth and strictly convex Banach spaces E with Kadec-Klee property:
(1.19)x0∈Cchosen arbitrary,x1=ΠC1x0,C1=C,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn,0Jxn+∑i=1∞βn,iJTinxn),un∈Csuch thatun=Krnyn,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+ξn},xn+1=ΠCn+1x0,n≥0,
where ξn=supp∈F(kn-1)ϕ(p,xn), ΠCn+1 is the generalized projection of E onto Cn+1. They prove that the sequence {xn} converges strongly to Π∩i=1∞F(Ti)∩GMEP(f)x0.Very recently, Chang et al. [26] proposed the following iterative algorithm for solving fixed point problems for total quasi-ϕ-asymptotically nonexpansive mappings:
(1.20)x0∈Cchosen arbitrary,C0=C,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn,0Jxn+∑i=1∞βn,iJTinxn),Cn+1={z∈Cn:ϕ(ν,yn)≤ϕ(ν,xn)+ξn},xn+1=ΠCn+1x0,n≥0,
where ξn=νnsupp∈Fξ(ϕ(p,xn))+μn, ΠCn+1 is the generalized projection of E onto Cn+1. They prove that the sequence {xn} converges strongly to Π∩i=1∞F(Ti)x0.Inspired and motivated by the recent work of Matsushita and Takahashi [13], Qin et al. [24], Saewan and Kumam [25], Chang et al. [26], and so forth, we introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points of a countable families of total quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results in [13, 24–29].


## 2. Preliminaries

Throughout this paper, letE be a real Banach space with the dual E* and let C be a nonempty closed convex subset of E. We denote the strong convergence, weak convergence of a sequence {xn} to a point x∈E by xn→x, xn⇀x, respectively, and F(T) is the fixed point set of a mapping T.In this paper, for solving generalized mixed equilibrium problems, we assume that bifunctionf:C×C→R satisfies the following conditions:(A1)
f(x,x)=0, for all x∈C;(A2)
f(x,y)+f(y,x)≤0, for all x,y∈C;(A3)
for allx,y,z∈C, limt↓0f(tz+(1-t)x,y)≤f(x,y);(A4)
for eachx∈C, the function y↦f(x,y) is convex and lower semicontinuous.Lemma 2.1 (see [16]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space  E, let  f be a bifunction from  C×C to  R satisfying  (A1)–(A4), and let  r>0 and  x∈E, then there exists  z∈C such that
(2.1)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.2 (see [30]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space  E. Let  B:C→E* be a continuous and monotone mapping, let  φ:C→R be convex and lower semicontinuous and let  f be a bifunction from  C×C to  R satisfying  (A1)–(A4). For  r>0 and  x∈E, then there exists  u∈C such that
(2.2)f(u,y)+φ(y)-φ(u)+〈Bu,y-u〉+1r〈y-u,Ju-Jx〉≥0,∀y∈C.
Define a mapping Kr:C→C as follows:
(2.3)Kr(x)={u∈C:f(u,y)+φ(y)-φ(u)+〈Bu,y-u〉+1r〈y-u,Ju-Jx〉≥0,∀y∈C}
for all x∈C. Then, the following hold:(1)
Kr is single-valued;(2)
Kr is firmly nonexpansive, that is, for allx,y∈E,
(2.4)〈Krx-Kry,JKrx-JKry〉≤〈Krx-Kry,Jx-Jy〉;(3)
F(Kr)=GMEP(f,B,φ);(4)
GMEP(f,B,φ) is closed and convex;(5)
ϕ(p,Krz)+ϕ(Krz,z)≤ϕ(p,z), for all p∈F(Kr) and z∈E.Lemma 2.3 (see [28]).
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  {xn} and  {yn} be two sequences in  C such that  xn→p and  ϕ(xn,yn)→0, where  ϕ is the function defined by (1.3), then  yn→p.Lemma 2.4 (see [3]).
LetE be a smooth, strictly convex and reflexive Banach space and let  C be a nonempty closed convex subset of  E. Then, the following conclusions hold: (a)
ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y), for all x∈C,y∈E;(b)
ifx∈E  and z∈C, then z=ΠCx if and only if 〈z-y,Jx-Jz〉≥0, for all y∈C;(c)
forx,y∈E, ϕ(x,y)=0  if and only if x=y.Lemma 2.5 (see [28]).
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  T:C→C be a closed and total  quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences  {νn},  {μn}, and a strictly increasing continuous functions  ξ:R+→R+ such that  νn→0,  μn→0 (as  n→∞) and  ξ(0)=0. If  μ1=0, then the fixed point set  F(T) of  T is a closed and convex subset of  C.Lemma 2.6 (see [31]).
LetE be an uniformly convex Banach space, let  r be a positive number, and let  Br(0) be a closed ball of  E. Then, for any sequence  {xi}i=1∞⊂Br(0) and for any sequence  {λi}i=1∞ of positive numbers with  ∑n=1∞λn=1, there exists a continuous, strictly increasing, and convex function  g:[0,2r]→[0,∞),  g(0)=0 such that, for any positive integer  i≠1, the following holds:
(2.5)‖∑n=1∞λnxn‖2≤∑n=1∞λn‖xn‖2-λ1λig(‖x1-xi‖).


## 3. Main Results

Theorem 3.1.
LetC be a nonempty, closed, and convex subset of an uniformly smooth and strictly convex Banach Banach space  E with Kadec-Klee property. Let  B:C→E* be a continuous and monotone mapping and let  φ:C→R be a lower semicontinuous and convex function. Let  f be a bifunction from  C×C to  R satisfying  (A1)–(A4). Let  {Ti}i=1∞:C→C be a countable family of closed and uniformly total  quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences  {νn},  {μn} and a strictly increasing continuous function  ζ:R+→R+ such that  μ1=0,  νn→0,  μn→0 (as  n→∞), and  ζ(0)=0, and for each  i≥1,  Ti is uniformly  Li-Lipschitz continuous.  {xn} is defined by
(3.1)x0∈Cchosen arbitrary,C0=C,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn,0Jxn+∑i=1∞βn,iJTinxn),un∈Csuch thatun=Krnyn,Cn+1={ν∈Cn:ϕ(ν,un)≤ϕ(ν,xn)+ξn},xn+1=ΠCn+1x0,n≥0,
where ξn=νnsupq∈Θζ(ϕ(q,xn))+μn, ΠCn+1is the generalized projection of E  onto Cn+1, {rn}⊂[a,∞)  for some a>0, {βn,0,βn,i}  and {αn}  are sequences in  [0,1] satisfying the following conditions: (1)
for eachn≥0,βn,0+∑i=1∞βn,i=1;(2)
liminfn→∞βn,0βn,i>0 for any i≥1;(3)
0≤αn≤α<1 for some α∈(0,1). If Θ:=⋂i=1∞F(Ti)∩GMEP(f,B,φ) is a nonempty and bounded subset in C, then the sequence {xn}  converges strongly to p∈F, where p=ΠΘx0.Proof.
We will divide the proof into seven steps.Step 1.
We first show thatΘ and Cn are closed and convex for each n≥0.
It follows from Lemma2.5 that F(Ti) is closed and convex subset of C for each i≥1. Therefore, Θ is closed and convex in C.
Again by the assumption,C0=C is closed and convex. Suppose that Cn is closed and convex for some n≥1. Since for any z∈Cn, we know that
(3.2)ϕ(z,un)≤ϕ(z,xn)+ξn⟺2〈z,Jxn-Jun〉≤‖xn‖2-‖un‖2+ξn.
Hence, the set Cn+1={z∈Cn:2〈z,Jxn-Jun〉≤∥xn∥2-∥un∥2+ξn} is closed and convex. Therefore, ΠCnx0 and ΠΘx0 are well defined.Step 2.
We show thatΘ⊂Cn for all n≥0.
It is obvious thatΘ⊂C0=C. Suppose that Θ⊂Cn for some n≥1. Since E is uniformly smooth, E* is uniformly convex. By the convexity of ∥·∥2, property of ϕ, for any given q∈Θ⊂Cn, we observe that
(3.3)ϕ(q,un)=ϕ(q,Krnyn)≤ϕ(q,yn)=ϕ(q,J-1(αnJxn+(1-αn)Jzn))=‖q‖2-2〈q,αnJxn+(1-αn)Jzn〉+‖αnJxn+(1-αn)Jzn‖2≤‖q‖2-2αn〈q,Jxn〉-2(1-αn)〈q,Jzn〉+αn‖xn‖2+(1-αn)‖zn‖2=αnϕ(q,xn)+(1-αn)ϕ(q,zn).
Furthermore, it follows from Lemma 2.6 that, for any positive integers l>1 and for any q∈Θ, we have
(3.4)ϕ(q,zn)=ϕ(q,J-1(βn,0Jxn+∑i=1∞βn,iJTinxn))=‖q‖2-2〈q,βn,0Jxn+∑i=1∞βn,iJTinxn〉+‖βn,0Jxn+∑i=1∞βn,iJTinxn‖2≤‖q‖2-2βn,0〈q,Jxn〉-2∑i=1∞βn,i〈q,JTinxn〉+βn,0‖xn‖2+∑i=1∞βn,i‖Tinxn‖2-βn,0βn,lg(‖Jxn-JTlnxn‖)=βn,0ϕ(q,xn)+∑i=1∞βn,iϕ(q,Tinxn)-βn,0βn,lg(‖Jxn-JTlnxn‖)≤βn,0ϕ(q,xn)+∑i=1∞βn,i{ϕ(q,xn)+νnζ(ϕ(q,xn))+μn}-βn,0βn,lg(‖Jxn-JTlnxn‖)≤ϕ(q,xn)+νnsupp∈Θϕ(p,xn)+μn-βn,0βn,lg(‖Jxn-JTlnxn‖)=ϕ(q,xn)+ξn-βn,0βn,lg(‖Jxn-JTlnxn‖).
Substituting (3.4) into (3.3), we get
(3.5)ϕ(q,un)≤αnϕ(q,xn)+(1-αn)ϕ(q,zn)≤αnϕ(q,xn)+(1-αn)[ϕ(q,xn)+ξn-βn,0βn,lg(‖Jxn-JTlnxn‖)]≤ϕ(q,xn)+(1-αn)ξn.
This shows that q∈Cn+1. Further, this implies that Θ⊂Cn+1 and hence Θ⊂Cn for all n≥0. Since Θ is nonempty, Cn is a nonempty closed convex subset of E, and hence ΠCn exists for all n≥0. This implies that the sequence {xn} is well defined.
Moreover, by the assumption of{νn}, {μn}, and Θ, from (1.4), we have
(3.6)ξn=νnsupp∈Θζ(ϕ(p,xn))+μn⟶0,n⟶∞.Step 3.
{xn} is bounded and {ϕ(xn,x0)} is a convergent sequence.
It follows from (3.1) and Lemma 2.4 that
(3.7)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0),∀p∈Cn+1,∀n≥0.
From definition of Cn+1 that xn=ΠCnx0 and xn+1=ΠCn+1x0, we have
(3.8)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥0.
Therefore, {ϕ(xn,x0)} is nondecreasing and bounded. So, {ϕ(xn,x0)} is a convergent sequence, without loss of generality, we can assume that limn→∞ϕ(xn,x0)=d≥0. In particular, by (1.4), the sequence {(∥xn∥-∥x0∥)2} is bounded. This implies {xn} is also bounded.Step 4.
We prove that{xn} converges strongly to some point p∈C.
Since{xn} is bounded and E is reflexive, there exists a subsequence {xni}⊂{xn} such that xni⇀p (some point in C). Since Cn is closed and convex and Cn+1⊂Cn, this implies that Cn is weakly closed and p∈Cn for each n≥0. From xni=ΠCnix0, we have
(3.9)ϕ(xni,x0)≤ϕ(p,x0),∀ni≥0.
Since the norm ∥·∥ is weakly lower semicontinuous, we have
(3.10)liminfni→∞ϕ(xni,x0)=liminfni→∞{‖xni‖2-2〈xni,Jx0〉+‖x0‖2}≥‖p‖2-2〈p,Jx0〉+‖x0‖2=ϕ(p,x0),
and so
(3.11)ϕ(p,x0)≤liminfni→∞ϕ(xni,x0)≤limsupni→∞ϕ(xni,x0)≤ϕ(p,x0).
This implies that limni→∞ϕ(xni,x0)→ϕ(p,x0), and so ∥xn∥→∥p∥. Since xni⇀p, by virtue of the Kadec-Klee property of E, we obtain that
(3.12)limni→∞xni=p.
Since {ϕ(xn,x0)} is convergent, this together with limni→∞ϕ(xni,x0)=ϕ(p,x0), we have limn→∞ϕ(xn,x0)=ϕ(p,x0). If there exists some subsequence {xnj}⊂{xn} such that xnj→q, then from Lemma 2.4, we have that
(3.13)ϕ(p,q)=limni,nj→∞ϕ(xni,xnj)=limni,nj→∞ϕ(xni,ΠCnjx0)≤limni,nj→∞(ϕ(xni,x0)-ϕ(ΠCnjx0,x0))=limni,nj→∞(ϕ(xni,x0)-ϕ(xnj,x0))=ϕ(p,x0)-ϕ(p,x0)=0.
This implies that p=q and
(3.14)limn→∞xn=p.Step 5.
We prove thatlimn→∞∥Jxn-Jun∥=0.
By definition ofΠCnx0, we have
(3.15)ϕ(xn+1,xn)=ϕ(xn+1,ΠCnx0)≤ϕ(xn+1,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0).
Since limn→∞ϕ(xn,x0) exists, we have
(3.16)limn→∞ϕ(xn+1,xn)=0.
Since xn+1=ΠCn+1x0∈Cn+1⊂Cn and the definition of Cn+1, we get
(3.17)ϕ(xn+1,un)≤ϕ(xn+1,xn)+ξn.
It follows from (3.6) and (3.16) that
(3.18)limn→∞ϕ(xn+1,un)=0.
From (1.4), we have
(3.19)limn→∞‖un‖=p.
So,
(3.20)limn→∞‖Jun‖=Jp.
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 know that J(E)=E*. Hence, there exist x∈C such that Jx=x*. It follows that
(3.21)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 the both sides of equality above and by the weak lower semicontinuity of norm ∥·∥, we have
(3.22)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 (1.4) and the Kadec-Klee property of E, we have
(3.23)limn→∞un=p.
Since ∥xn-un∥≤∥xn-p∥+∥p-un∥, so,
(3.24)liminfn→∞‖xn-un‖=0.
Since J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain
(3.25)liminfn→∞‖Jxn-Jun‖=0,Step 6.
We show thatp∈Θ:=⋂i=1∞F(Ti)⋂GMEP(f,B,φ).
First, we show thatp∈⋂i=1∞F(Ti).
Sincexn+1∈Cn+1, it follows from (3.1) and (3.14) that
(3.26)ϕ(xn+1,un)≤ϕ(xn+1,xn)+ξn⟶0(asn⟶∞).
Since xn→p, by Lemma 2.3,
(3.27)limn→∞un=p.
By (3.3) and (3.4), for any q∈Θ, we have
(3.28)ϕ(q,un)≤ϕ(q,xn)+ξn-(1-αn)βn,0βn,lg(‖Jxn-JTlnxn‖).
So,
(3.29)(1-αn)βn,0βn,lg‖Jxn-JTlnxn‖≤ϕ(q,xn)+ξn-ϕ(q,un)⟶0(asn⟶∞).
Therefore,
(3.30)limn→∞(1-αn)βn,0βn,lg‖Jxn-JTlnxn‖=0.
In view of the property of g, we have
(3.31)‖Jxn-JTlnxn‖⟶0(asn⟶∞).
Since Jxn→Jp, this implies that limn→∞JTlnxn=Jp. From Remark 1.1(ii), it yields
(3.32)Tlnxn⇀p(asn⟶∞).
Again since
(3.33)‖Tlnxn‖-‖p‖=‖J(Tlnxn)‖-‖Jp‖≤‖J(Tlnxn)-Jp‖⟶0(asn⟶∞),
this together with (3.32) and the Kadec-Klee-property of E shows that
(3.34)limn→∞Tlnxn=p.
By the assumption that Tl is uniformly Ll-Lipschitz continuous, we have
(3.35)‖Tln+1xn-Tlnxn‖≤‖Tln+1xn-Tln+1xn+1‖+‖Tln+1xn+1-xn+1‖+‖xn+1-xn‖+‖xn-Tlnxn‖≤(Ll+1)‖xn+1-xn‖+‖Tln+1xn+1-xn+1‖+‖xn-Tlnxn‖.
This together with (3.34) and xn→p shows that limn→∞∥Tln+1xn-Tlnxn∥=0 and limn→∞Tln+1xn=p, that is, limn→∞TlTlnxn=p. In view of the closeness of Tl, it follows that Tlp=p, that is, p∈F(Tl). By the arbitrariness of l≥1, we have p∈∩i=1∞F(Ti).
Now, we show thatp∈GMEP(f,B,φ).
It follows from (3.2), (3.3), (3.6), Lemma 2.4, and un=Krnyn that
(3.36)ϕ(un,yn)=ϕ(Krnyn,yn)≤ϕ(p,yn)-ϕ(p,Krnyn)≤ϕ(p,xn)-ϕ(p,Krnyn)+ξn=ϕ(p,xn)-ϕ(p,un)+ξn⟶0,(asn⟶∞).
By (1.4), we have
(3.37)‖un‖⟶‖yn‖,(asn⟶∞).
Since un→p as n→∞, so
(3.38)‖yn‖⟶‖p‖,(asn⟶∞).
Therefore,
(3.39)‖Jun‖⟶‖Jp‖,(asn⟶∞).
Since E* is reflexive, we may assume that Jyn⇀z*∈E*. In view of the reflexive of E, we have J(E)=E*. Hence, there exist z∈E such that Jz=z*. It follows that
(3.40)ϕ(un,yn)=‖un‖2-2〈un,Jyn〉+‖yn‖2=‖un‖2-2〈un,Jyn〉+‖Jyn‖2.
Taking liminfn→∞ on the both sides of equality above yields that
(3.41)0≥‖p‖2-2〈p,z*〉+‖z*‖2=‖p‖2-2〈p,Jz〉+‖Jz‖2=‖p‖2-2〈p,Jz〉+‖z‖2=ϕ(p,x).
That is, p=z, which implies that z*=Jp. It follows that Jyn⇀Jp∈E*. Since J-1 is norm-weak*-continuous, it follows that yn⇀p. From (3.38) and E with the Kadec-Klee property, we obtain
(3.42)yn⟶p(asn⟶∞).
It follows from (3.23) and (3.42) that
(3.43)limn→∞‖un-yn‖=0.
Since J is uniformly norm-to-norm continuous, we have
(3.44)limn→∞‖Jun-Jyn‖=0.
By Lemma 2.2, we have
(3.45)f(un,y)+φ(y)-φ(un)+〈Byn,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C.
From (A2), we have
(3.46)φ(y)-φ(un)+〈Byn,y-un〉+1rn〈y-un,Jun-Jyn〉≥-f(un,y)≥f(y,un),∀y∈C.
Putzt=ty+(1-t)p for all t∈(0,1] and y∈C. Consequently, we get zt∈C. It follows from (3.46) that
(3.47)〈Bzt,zt-un〉≥〈Bzt,zt-un〉-φ(zt)+φ(un)-〈Byn,zt-un〉+f(zt,un)-〈zt-un,Jun-Jynrn〉=〈Bzt-Bun,zt-un〉-φ(zt)+φ(un)+〈Bun-Byn,zt-un〉+f(zt,un)-〈zt-un,Jun-Jynrn〉.
Since B is continuous, and from (3.43), and un→p, yn→p, as n→∞, therefore ∥Bun-Byn∥→0. Since B is monotone, we know that 〈Bzt-Bun,zt-un〉≥0. Further, limn→∞∥Jun-Jyn∥/rn=0. So, it follows from (A4), and the weak lower semicontinuity of φ and (3.43) that
(3.48)f(zt,p)-φ(zt)+φ(p)≤limn→∞〈Bzt,zt-un〉=〈Bzt,zt-p〉.
From (A1) and (3.48), we have
(3.49)0=f(zt,zt)-φ(zt)+φ(zt)≤tf(zt,y)+(1-t)f(zt,p)+tφ(y)+(1-t)φ(p)-φ(zt)=t[f(zt,y)+φ(y)-φ(zt)]+(1-t)[f(zt,p)+φ(p)-φ(zt)]≤t[f(zt,y)+φ(y)-φ(zt)]+(1-t)〈Bzt,zt-p〉≤t[f(zt,y)+φ(y)-φ(zt)]+(1-t)t〈Bzt,y-p〉,
and hence
(3.50)f(zt,y)+φ(y)-φ(zt)+(1-t)〈Bzt,y-p〉≥0.
Letting t→0, we have
(3.51)f(p,y)+φ(y)-φ(p)+〈Bp,y-p〉≥0.
This implies that p∈GMEP(f,B,φ). Hence, p∈∩i=1∞F(Ti)∩GMEP(f,B,φ).Step 7.
We prove thatxn→p=ΠΘx0.
Letq=ΠΘx0. From xn=ΠCnx0 and q∈Θ⊂Cn, we have
(3.52)ϕ(xn,x0)≤ϕ(q,x0),∀n≥0.
This implies that
(3.53)ϕ(p,x0)=limn→∞ϕ(xn,x0)≤ϕ(q,x0).
By definition of p=ΠΘx0, we have p=q. Therefore, xn→p=ΠΘx0. This completes the proof.Takingφ=0, Ti=T for each i∈N in Theorem 3.1, we have the following result.Corollary 3.2.
LetC be a nonempty, closed, and convex subset of an uniformly smooth and strictly convex Banach Banach space  E with Kadec-Klee property. Let  B:C→E* be a continuous and monotone mapping. Let  f be a bifunction from  C×C to  R satisfying  (A1)–(A4). Let  T:C→C be a closed uniformly  L-Lipschitz continuous and uniformly total  quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences  {νn},  {μn} and a strictly increasing continuous function  ζ:R+→R+ such that  μ1=0,  νn→0,  μn→0 (as  n→∞), and  ζ(0)=0. Let  {xn} be the sequence generated by
(3.54)x0∈Cchosen arbitrary,C0=C,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βnJxn+(1-βn)JTnxn),un∈Csuch thatf(un,y)+〈Byn,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={ν∈Cn:ϕ(ν,un)≤ϕ(ν,xn)+ξn},xn+1=ΠCn+1x0,n≥0,
where ξn=νnsupq∈Θζ(ϕ(q,xn))+μn, ΠCn+1is the generalized projection of E  onto Cn+1, {βn}  and {αn}are sequences in [0,1], liminfn→∞βn(1-βn)>0, {rn}⊂[a,∞) for some a>0. If Θ:=∩i=1∞F(Ti)∩GEP(f,B)  is a nonempty and bounded subset in C, then the sequence {xn} converges strongly to p∈Θ, where p=ΠΘx0.In Theorem3.1, as φ=0, B=0, Ti=T for each i∈N, we can obtain the following corollary.Corollary 3.3.
LetC be a nonempty, closed and convex subset of an uniformly smooth and strictly convex Banach Banach space  E with Kadec-Klee property. Let  f be a bifunction from  C×C to  R satisfying  (A1)–(A4), and  T:C→C be a closed uniformly  L-Lipschitz continuous and uniformly total  quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences  {νn},  {μn} and a strictly increasing continuous function  ζ:R+→R+ such that  μ1=0,  νn→0,  μn→0 (as  n→∞) and  ζ(0)=0. Let  {xn} be the sequence generated by
(3.55)x0∈Cchosen arbitrary,C0=C,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βnJxn+(1-βn)JTnxn),un∈Csuch thatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={ν∈Cn:ϕ(ν,un)≤ϕ(ν,xn)+ξn},xn+1=ΠCn+1x0,n≥0,
where ξn=νnsupq∈Θξ(ϕ(q,xn))+μn, ΠCn+1is the generalized projection of E  onto Cn+1, {βn} and {αn} are sequences in [0,1], liminfn→∞βn(1-βn)>0, {rn}⊂[a,∞)  for some a>0. If Θ:=∩i=1∞F(Ti)∩EP(f) is a nonempty and bounded subset in C, then the sequence {xn} converges strongly to  p∈Θ, where p=ΠΘx0.Definition 3.4.
A countable family of mapping{Tn}:C→C is said to be uniformly quasi-ϕ-asymptotically nonexpansive, if ∩i=1∞F(Ti)≠∅ and there exist real sequences {kn}⊂[1,∞),kn→1 such that for each i≥1,
(3.56)ϕ(p,Tinx)≤knϕ(p,x),∀x∈C,p∈⋂i=1∞F(Ti).
The following Corollary can be directly obtained from Theorem 3.1.Corollary 3.5.
LetC be a nonempty, closed and convex subset of an uniformly smooth and strictly convex Banach Banach space E with Kadec-Klee property. Let B:C→E* be a continuous and monotone mapping and let φ:C→R be a lower semicontinuous and convex function. Let f be a bifunction from C×C to R satisfying (A1)–(A4). Let {Ti}i=1∞:C→C be an infinite family of closed and uniformly Li-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞), kn→1 such that Θ:=∩i=1∞F(Ti)∩GMEP(f,B,φ) is a nonempty and bounded subset in C. Let {xn} be the sequence generated by
(3.57)x0∈Cchosen arbitrary,C0=C,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn,0Jxn+∑i=1∞βn,iJTinxn),un∈Csuch thatun=Krnyn,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+ξn},xn+1=ΠCn+1x0,n≥0,
where ξn=supp∈F(kn-1)ϕ(p,xn), ΠCn+1is the generalized projection of E  onto Cn+1, {rn}⊂[a,∞) for some a>0, {βn,0,βn,i} and {αn} are sequences in [0,1]. If ∑i=1∞βn,i=1 for all n≥0, and liminfn→∞βn,0βn,i>0 for all i≥1, then the sequence {xn} converges strongly to p∈Θ, where p=ΠΘx0.Remark 3.6.
Theorem3.1 improves and extend the corresponding results in [13, 24–29] in the following aspects:(1)
for the mappings, extend the mappings from relatively nonexpansive mappings, quasi-ϕ-nonexpansive mappings, and quasi-ϕ-asymptotically nonexpansive mappings to a countable family of total quasi-ϕ-asymptotically nonexpansive mappings;(2)
for the framework of spaces, extend the space from an uniformly smooth and uniformly convex Banach space to an uniformly smooth and strictly convex Banach space with the Kadec-Klee property.



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*Source: 506210-2012-06-12.xml*