# Strong Convergence Theorems for a Generalized Mixed Equilibrium Problem and a Family of Total Quasi-ϕ-Asymptotically Nonexpansive Multivalued Mappings in Banach Spaces

Authors: J. F. Tan; S. S. Chang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The main purpose of this paper is by using a hybrid algorithm to find a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of common fixed points for a infinite family of total quasi-ϕ-asymptotically nonexpansive multivalued mapping in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results announced by some authors.


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## Body

## 1. Introduction

Throughout this paper, we always assume thatX is a real Banach space with the dual X*, C is a nonempty closed convex subset of X, and J:X→2X is the normalized duality mappingdefined by(1.1)J(x)={f*∈X*:〈x,f*〉=‖x‖2=‖f*‖2},∀x∈E.
In the sequel, we use F(T) to denote the set of fixed points of a mapping T and use R and R+ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively. We denote by xn→x and xn⇀x the strong convergence and weak convergence of a sequence {xn}, respectively.LetΘ:C×C→R be a bifunction, ψ:C→R a real valued function, and A:C→X* a nonlinear mapping. The so-called generalized mixed equilibrium problem is to find u∈C such that(1.2)Θ(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)≥0,∀y∈C.
The set of solutions to (1.2) is denoted by Ω, that is,(1.3)Ω={u∈C:Θ(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)≥0,∀y∈C}.
Special examples are follows.(i)
IfA≡0, the problem (1.2) is equivalent to finding u∈C such that
(1.4)Θ(u,y)+ψ(y)-ψ(u)≥0,∀y∈C,
which is called the mixed equilibrium problem (MEP) [1].(ii)
IfΘ≡0, the problem (1.2) is equivalent to finding u∈C such that
(1.5)〈Au,y-u〉+ψ(y)-ψ(u)≥0,∀y∈C,
which is called the mixed variational inequality of Browder type(VI) [2].A Banach spaceX is said to be strictly convexif ∥x+y∥/2<1 for all x,y∈U={z∈X:∥z∥=1} with x≠y. X is said to be uniformly convexif, for each ϵ∈(0,2], there exists δ>0 such that ∥x+y∥/2<1-δ for all x,y∈U with ∥x-y∥≥ϵ. X is said to be smooth if the limit(1.6)lim⁡t→0‖x+ty‖-‖x‖t
exists for all x,y∈U. X is said to be uniformly smooth if the above limit is attained uniformly in x,y∈U.Remark 1.1.
The following basic properties of a Banach spaceX can be found in Cioranescu [3].(i)
IfX is uniformly smooth, then J is uniformly continuous on each bounded subset of X.(ii)
IfX is a reflexive and strictly convex Banach space, then J-1 is norm-weak-continuous.(iii)
IfX is a smooth, strictly convex, and reflexive Banach space, then J is single-valued, one-to-one and onto.(iv)
A Banach spaceX is uniformly smooth if and only if X* is uniformly convex.(v)
Each uniformly convex Banach spaceX has the Kadec-Klee property, that is, for any sequence {xn}⊂X, if xn⇀x∈X and ∥xn∥→∥x∥, then xn→x.LetX be a smooth Banach space. We always use ϕ:X×X→ℛ+ to denote the Lyapunov functional defined by(1.7)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,∀x,y∈X.
It is obvious from the definition of the function ϕ that(1.8)(‖x‖-‖y‖)2≤ϕ(x,y)≤(‖x‖+‖y‖)2,∀x,y∈X.
Following Alber [4], the generalized projectionΠC:X→C is defined by(1.9)ΠC(x)=arginf⁡y∈Cϕ(y,x),∀x∈X.Lemma 1.2 (see [4]).
LetX be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of X. Then, the following conclusions hold:(a)
ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y) for all x∈C and y∈X,(b)
ifx∈X and z∈C, then
(1.10)z=ΠCxiff〈z-y,Jx-Jz〉≥0,∀y∈C,(c)
forx,y∈X,ϕ(x,y)=0 if and only if x=y.
LetX be a smooth, strictly convex, and reflexive Banach space, C a nonempty closed convex subset of X, and T:C→C a mapping. A point p∈C is said to be an asymptotic fixed point of T  if there exists a sequence {xn}⊂C such that xn⇀p and ∥xn-Txn∥→0. We denoted the set of all asymptotic fixed points of T by F̃(T).Definition 1.3.
(1) A mapping T:C→C is said to be relatively nonexpansive[5] if F(T)≠∅,F(T)=F(T̃) and
(1.11)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).
(2) A mapping T:C→C is said to be closed if, for any sequence {xn}⊂C with xn→x and Txn→y, Tx=y.
LetC be a nonempty closed convex subset of a Banach space X. Let N(C) be the family of nonempty subsets of C.Definition 1.4.
(1) Let T:C→N(C) be a multivalued mapping and q a point in C. The definitions of Tq,T2q,T3q,…,Tnq,n≥1 are as follows:
(1.12)Tq:={q1:q1∈T(q)},T2q=T(T(q)):=⋃q1∈T(q)T(q1),T3q=T(T2(q)):=⋃q2∈T2(q)T(q2),⋮Tnq=T(Tn-1(q)):=⋃qn-1∈Tn-1(q)T(qn-1),n≥1.
(2) Let T:C→N(C) be a multivalued mapping. A point p∈C is said to be an asymptotic fixed point ofT if there exists a sequence {xn}⊂C such that xn⇀p and limn→∞d(xn,T(xn))=0. We denoted the set of all asymptotic fixed points of T by F̃(T).
(3) A multivalued mapping T:C→N(C) is said to be relatively nonexpansive[5] if F(T)≠∅,F(T)=F̃(T) and
(1.13)ϕ(p,w)≤ϕ(p,x),∀x∈C,w∈Tx,p∈F(T).
(4) A multivalued mapping T:C→N(C) is said to be closed if, for any sequence {xn}⊂C with xn→x and wn∈T(xn) with wn→y, then y∈Tx.Definition 1.5.
(1) A multivalued mapping T:C→N(C) is said to be quasi-ϕ-nonexpansive if F(T)≠∅ and
(1.14)ϕ(p,w)≤ϕ(p,x),∀x∈C,w∈Tx,p∈F(T).
(2) A multivalued mapping T:C→N(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
(1.15)ϕ(p,wn)≤knϕ(p,x),∀n≥1,x∈C,wn∈Tnx,p∈F(T).
(3) A multivalued mapping T:C→N(C) is said to be total quasi-ϕ-asymptotically nonexpansive if F(T)≠∅ and there exist nonnegative real sequences {νn},{μn} with νn→0,μn→0 (as n→∞) and a strictly increasing continuous function ζ:ℛ+→ℛ+ with ζ(0)=0 such that for all x∈C,p∈F(T)(1.16)ϕ(p,wn)≤ϕ(p,x)+νnζ(ϕ(p,x))+μn,∀n≥1,wn∈Tnx.Definition 1.6.
(1) Let {Ti}i=1∞:C→N(C) be a sequence of mappings. {Ti}i=1∞ is said to be a family of uniformly total quasi-ϕ-asymptotically nonexpansive multivalued mappings if ∩i=1∞F(Ti)≠∅ and there exist nonnegative real sequences {νn},{μn} with νn→0,μn→0 (as n→∞) and a strictly increasing continuous function ζ:ℛ+→ℛ+ with ζ(0)=0 such that for all i≥1,x∈C,p∈∩i=1∞F(Ti)(1.17)ϕ(p,wn,i)≤ϕ(p,x)+νnζ(ϕ(p,x))+μn,∀wn,i∈Tinx,∀n≥1.
(2) A total quasi-ϕ-asymptotically nonexpansive multivalued mapping T:C→N(C) is said to be uniformly L-Lipschitz continuous if there exists a constant L>0 such that
(1.18)‖wn-sn‖≤L‖x-y‖,∀x,y∈C,wn∈Tnx,sn∈Tny,n≥1.In 2005, Matsushita and Takahashi [5] proved weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping in a uniformly convex and uniformly smooth Banach space X. In 2008, Plubtieng and Ungchittrakool [6] proved the strong convergence theorems to approximate a fixed point of two relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space X. In 2010, Chang et al. [7] obtained the strong convergence theorem for an infinite family of quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space X with Kadec-Klee property. In 2011, Chang et al. [8] proved some approximation theorems of common fixed points for countable families of total quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space X with Kadec-Klee property. In 2011, Homaeipour and Razani [9] proved weak and strong convergence theorems for a single relatively nonexpansive multivalued mapping in a uniformly convex and uniformly smooth Banach space X. On the other hand, In 2009, Zhang [10] proved the strong convergence theorem for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a finite family of quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and uniformly convex Banach space. Recently, Tang [11], Cho et al. [12–21], and Noor et al. [22–26] extended the finite family of quasi-ϕ-asymptotically nonexpansive mappings to infinite family of quasi-ϕ-asymptotically nonexpansive mappings.Motivated and inspired by the researches going on in this direction, the purpose of this paper is by using the hybrid iterative algorithm to find a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasi-ϕ-asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. In order to get the strong convergence theorems, the hybrid algorithms are presented and used to approximate the fixed point. The results presented in the paper improve and extend some recent results announced by some authors.


## 2. Preliminaries

Lemma 2.1 (see [8]).
LetX be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and C a nonempty closed convex set of X. 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 (1.7), and then yn→p.Lemma 2.2.
LetX and C be as in Lemma 2.1. Let T:C→N(C) be a closed and total quasi-ϕ-asymptotically nonexpansive multivalued mapping with nonnegative real sequences {νn},{μn} and a strictly increasing continuous function ζ:ℛ+→ℛ+ such that νn→0,μn→0 (as n→∞), and ζ(0)=0. If μ1=0, then the fixed point set F(T) is a closed and convex subset of C.Proof.
Letting{xn} be a sequence in F(T) with xn→p (as n→∞), we prove that p∈F(T). In fact, by the assumption that T is a total quasi-ϕ-asymptotically nonexpansive multivalued mapping and μ1=0, we have
(2.1)ϕ(xn,u)≤ϕ(xn,p)+ν1ζ(ϕ(xn,p)),∀u∈Tp.
Furthermore, we have
(2.2)ϕ(p,u)=lim⁡n→∞ϕ(xn,u)≤lim⁡n→∞(ϕ(xn,p)+ν1ζ(ϕ(xn,p)))=0,∀u∈Tp.
By Lemma 1.2(c), p=u. Hence, p∈Tp. This implies that p∈F(T), that is, F(T) is closed.
Next, we prove thatF(T) is convex. For any x,y∈F(T),t∈(0,1), putting q=tx+(1-t)y, we prove that q∈F(T). Indeed, let {un} be a sequence generated by
(2.3)u1∈Tq,u2∈Tu1⊂T2q,u3∈Tu2⊂T3q,⋮un∈Tun-1⊂Tnq,⋮
In view of the definition of ϕ(x,y), for all un∈Tun-1⊂Tnq, we have
(2.4)ϕ(q,un)=‖q‖2-2〈q,Jun〉+‖un‖2=‖q‖2-2t〈x,Jun〉-2(1-t)〈y,Jun〉+‖un‖2=‖q‖2+tϕ(x,un)+(1-t)ϕ(y,un)-t‖x‖2-(1-t)‖y‖2
since
(2.5)tϕ(x,un)+(1-t)ϕ(y,un)≤t(ϕ(x,q)+νnζ(ϕ(x,q))+μn)+(1-t)(ϕ(y,q)+νnζ(ϕ(y,q))+μn)=t(‖x‖2-2〈x,Jq〉+‖q‖2+νnζ(ϕ(x,q))+μn)+(1-t)(‖y‖2-2〈y,Jq〉+‖q‖2+νnζ(ϕ(y,q))+μn)=t‖x‖2+(1-t)‖y‖2-‖q‖2+tνnζ(ϕ(x,q))+(1-t)νnζ(ϕ(y,q))+μn.
Substituting (2.5) into (2.4) and simplifying it, we have
(2.6)ϕ(q,un)≤tνnζ(ϕ(x,q))+(1-t)νnζ(ϕ(y,q))+μn⟶0(n⟶∞).
By Lemma 2.1, we have un→q (as n→∞). This implies that un+1→q (as n→∞). Since T is closed, we have q∈Tq, that is, q∈F(T).
This completes the proof of Lemma2.2.Lemma 2.3 (see [7]).
LetX be a uniformly convex Banach space, r>0, a positive number, and Br(0) a closed ball of X. Then, for any given sequence {xn}n=1∞⊂Br(0) and for any given sequence {λn}n=1∞ of positive numbers with Σn=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 integers i,j with i<j,
(2.7)‖∑n=1∞λnxn‖2≤∑n=1∞λn‖xn‖2-λiλjg(‖xi-xj‖).
For solving the generalized mixed equilibrium problem, let us assume that the functionψ:C→ℛ is convex and lower semicontinuous, the nonlinear mapping A:C→X* is continuous and monotone, and the bifunction Θ:C×C→ℛ satisfies the following conditions:(A1)
Θ(x,x)=0,forallx∈C,(A2)
Θ is monotone, that is, Θ(x,y)+Θ(y,x)≤0,forallx,y∈C,(A3)
limsupt↓0Θ(x+t(z-x),y)≤Θ(x,y),forallx,y,z∈C,(A4)
the functiony↦Θ(x,y) is convex and lower semicontinuous.Lemma 2.4.
LetX be a smooth, strictly convex and reflexive Banach space and C a nonempty closed convex subset of X. Let Θ:C×C→ℛ be a bifunction satisfying conditions (A1)–(A4). Let r>0 and x∈X. Then, the following hold.(i)
[27] There exists z∈C such that
(2.8)Θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.(ii)
[28] Define a mapping Tr:X→C by
(2.9)Trx={z∈C:Θ(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},x∈X. Then, the following conclusions hold:(a)
Tr is single-valued,(b)
Tr is a firmly nonexpansive-type mapping, that is, forallz,y∈X,
(2.10)〈Tr(z)-Tr(y),JTr(z)-JTr(y)〉≤〈Tr(z)-Tr(y),Jz-Jy〉,(c)
F(Tr)=EP(Θ)=F(Tr)̂,(d)
EP(Θ) is closed and convex,(e)
ϕ(q,Tr(x))+ϕ(Tr(x),x)≤ϕ(q,x),forallq∈F(Tr).Lemma 2.5 (see [10]).
LetX be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of X. Let A:C→X* be a continuous and monotone mapping, ψ:C→ℛ a lower semicontinuous and convex function, and Θ:C×C→ℛ a bifunction satisfying conditions (A1)–(A4). Let r>0 be any given number and x∈X any given point. Then, the following hold.
(i) There existsu∈C such that forally∈C(2.11)Θ(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)+1r〈y-u,Ju-Jx〉≥0.
(ii) If one defines a mappingKr:C→C by
(2.12)Kr(x)={1ru∈C:Θ(u,y)+〈Au,y-u〉+ψ(y)-ψ(u)+1r〈y-u,Ju-Jx〉≥0,∀y∈C},x∈C,
then, the mapping Kr has the following properties:(a)
Kr is single-valued,(b)
Kr is a firmly nonexpansive-type mapping, that is, forallz,y∈X(2.13)〈Kr(z)-Kr(y),JKr(z)-JKr(y)〉≤〈Kr(z)-Kr(y),Jz-Jy〉,(c)
F(Kr)=Ω=F(Kr)̂,(d)
Ω is a closed convex set of C,(e)
ϕ(p,Kr(z))+ϕ(Kr(z),z)≤ϕ(p,z),forallp∈F(Kr),z∈X.Remark 2.6.
It follows from Lemma2.4 that the mapping Kr:C→C defined by (2.12) is a relatively nonexpansive mapping. Thus, it is quasi-ϕ-nonexpansive.


## 3. Main Results

In this section, we will use the hybrid iterative algorithm to find a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasi-ϕ-asymptotically nonexpansive multivalued mappings in a uniformly smooth and strictly convex Banach space with Kadec-Klee property.Theorem 3.1.
LetX be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and C a nonempty closed and convex subset of X. Let Θ:C×C→ℛ be a bifunction satisfying conditions (A1)–(A4), A:C→X* a continuous and monotone mapping, and ψ:C→ℛ a lower semicontinuous and convex function. Let {Ti}i=1∞:C→N(C) be an infinite family of closed and uniformly total quasi-ϕ-asymptotically nonexpansive multivalued 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. Let x0∈C, C0=C, and let {xn} be a sequence generated by
(3.1)xn+1=∏Cn+1x0,Cn+1={ν∈Cn:ϕ(ν,un)≤ϕ(ν,xn)+ξn},∀n≥0,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn,0Jxn+∑i=1∞βn,iJwn,i),un∈Csuchthat,∀y∈C,Θ(un,y)+〈Aun,y-un〉+ψ(y)-ψ(un)+1rn〈y-un,Jun-Jyn〉≥0,
where wn,i∈Tinxn,foralln≥1,i≥1,ξn=νnsupp∈ℱζ(ϕ(p,xn))+μn, ΠCn+1 is the generalized projection of X onto Cn+1, and {αn} and {βn,0,βn,i} are sequences in [0,1] satisfying the following conditions:(a)
for eachn≥0, βn,0+Σi=1∞βn,i=1,(b)
liminfn→∞βn,0βni>0 for any i≥1,(c)
0≤αn≤α<1 for some α∈(0,1).
IfG:=ℱ∩Ω=∩i=1∞F(Ti)∩Ω is a nonempty and bounded subset of C, then the sequence {xn} converges strongly to ΠGx0.Proof.
First, we define two functionsH:C×C→ℛ and Kr:C→C by
(3.2)H(x,y)=Θ(x,y)+〈Ax,y-x〉+ψ(y)-ψ(x),∀x,y∈C,Kr(x)={u∈C:H(u,y)+1r〈y-u,Ju-Jx〉≥0,∀y∈C},x∈C.
By Lemma2.5, we know that the function H satisfies conditions (A1)–(A4) and Kr has properties (a)–(e). Therefore, (3.1) is equivalent to
(3.3)xn+1=∏Cn+1x0,Cn+1={ν∈Cn:ϕ(ν,un)≤ϕ(ν,xn)+ξn},∀n≥0,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn,0Jxn+∑i=1∞βn,iJwn,i),un∈Csuchthat,∀y∈C,H(un,y)+1rn〈y-un,Jun-Jyn〉≥0.
Now we divide the proof of Theorem3.1 into six steps.
(i)ℱ and Cn are closed and convex for each n≥0.
In fact, it follows from Lemma2.2 that F(Ti),i≥1, is a closed and convex subset of C. Therefore, ℱ is a closed and convex subset C.
Again by the assumption,C0=C is closed and convex. Suppose that Cn is closed and convex for some n≥1. Since the condition ϕ(ν,yn)≤ϕ(ν,xn)+ξn is equivalent to
(3.4)2〈ν,Jxn-Jyn〉≤‖xn‖2-‖yn‖2+ξn,n=1,2,…,
the set
(3.5)Cn+1={ν∈Cn:2〈ν,Jxn-Jyn〉≤‖xn‖2-‖yn‖2+ξn}
is closed and convex. Therefore, Cn is closed and convex for each n≥0.
(ii){xn} is bounded and {ϕ(xn,x0)} is a convergent sequence.
Indeed, it follows from (3.1) and Lemma 1.2(a) that for all n≥0,u∈F(T)(3.6)ϕ(xn,x0)=ϕ(∏Cnx0,x0)≤ϕ(u,x0)-ϕ(u,∏Cnx0)≤ϕ(u,x0).
This implies that {ϕ(xn,x0)} is bounded. By virtue of (1.3), we know that {xn} is bounded.
In view of the structure of{Cn}, we have Cn+1⊂Cn,xn=ΠCnx0 and xn+1=ΠCn+1x0. This implies that xn+1∈Cn and
(3.7)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥0.
Therefore, {ϕ(xn,x0)} is a convergent sequence.
(iii)G:=ℱ∩Ω⊂Cn for all n≥0.
Indeed, it is obvious thatG⊂C0=C. Suppose that G⊂Cn for some n∈𝒩. Since un=Krnyn, by Lemma 2.5 and Remark 2.6, Krn is quasi-ϕ-nonexpansive. Hence, for any given u∈G⊂Cn and n≥1 we have
(3.8)ϕ(u,un)=ϕ(u,Krnyn)≤ϕ(u,yn)=ϕ(u,J-1(αnJxn+(1-αn)Jzn))=‖u‖2-2〈u,αnJxn+(1-αn)Jzn〉+‖αnJxn+(1-αn)Jzn‖2≤‖u‖2-2αn〈u,Jxn〉-2(1-αn)〈u,Jzn〉+αn‖xn‖2+(1-αn)‖zn‖2=αnϕ(u,xn)+(1-αn)ϕ(u,zn).
Furthermore, it follows from Lemma 2.3 that for any u∈G⊂Cn,wn,i∈Tinxn, and i≥1 we have
(3.9)ϕ(u,zn)=ϕ(u,J-1(βn,0Jxn+∑i=1∞βn,iJwn,i))=‖u‖2-2〈u,βn,0Jxn+∑i=1∞βn,iJwn,i〉+‖βn,0Jxn+∑i=1∞βn,iJwn,i‖2≤‖u‖2-2βn,0〈u,Jxn〉-2∑i=1∞βn,i〈u,Jwn,i〉+βn,0‖xn‖2+∑i=1∞βn,i‖wn,i‖2-βn,0βn,lg(‖Jxn-Jwn,l‖)=βn,0ϕ(u,xn)+∑i=1∞βn,iϕ(u,wn,i)-βn,0βn,lg(‖Jxn-Jwn,l‖)≤βn,0ϕ(u,xn)+∑i=1∞βn,i(ϕ(u,xn)+νnζ(ϕ(u,xn))+μn)-βn,0βn,lg(‖Jxn-Jwn,l‖)≤ϕ(u,xn)+νnsup⁡p∈Fζ(ϕ(p,xn))+μn-βn,0βn,lg(‖Jxn-Jwn,l‖)=ϕ(u,xn)+ξn-βn,0βn,lg(‖Jxn-Jwn,l‖).
Substituting (3.9) into (3.8) and simplifying it, we have for all u∈G(3.10)ϕ(u,un)≤ϕ(u,yn)≤ϕ(u,xn)+(1-αn)ξn-(1-αn)βn,0βn,lg(‖Jxn-Jwn,l‖)≤ϕ(u,xn)+ξn-(1-αn)βn,0βn,lg(‖Jxn-Jwn,l‖)≤ϕ(u,xn)+ξn,
that is, u∈Cn+1 and so G⊂Cn+1 for all n≥0.
By the way, in view of the assumption on{νn},{μn} we have
(3.11)ξn=νnsup⁡p∈Fζ(ϕ(p,xn))+μn⟶0(n⟶∞).
(iv){xn} converges strongly to some point p*∈C.
In fact, since{xn} is bounded and X 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. In view of xni=ΠCnix0, we have
(3.12)ϕ(xni,x0)≤ϕ(p*,x0),∀ni≥0.
Since the norm ∥·∥ is weakly lower semicontinuous, we have
(3.13)liminf⁡ni→∞ϕ(xni,x0)=liminf⁡ni→∞(‖xni‖2-2〈xni,Jx0〉+‖x0‖2)≥‖p*‖2-2〈p*,Jx0〉+‖x0‖2=ϕ(p*,x0),
and so
(3.14)ϕ(p*,x0)≤liminf⁡ni→∞ϕ(xni,x0)≤limsup⁡ni→∞ϕ(xni,x0)≤ϕ(p*,x0).
This implies that limni→∞ϕ(xni,x0)=ϕ(p*,x0), and so ∥xni∥→∥p*∥. Since xni⇀p*, by virtue of Kadec-Klee property of X, we obtain that
(3.15)lim⁡ni→∞xni=p*.
Since {ϕ(xn,x0)} is convergent, this together with limni→∞ϕ(xni,x0)=ϕ(p*,x0), shows that limn→∞ϕ(xn,x0)=ϕ(p*,x0). If there exists some sequence {xnj}⊂{xn} such that xnj→q, then from Lemma 1.2(a) we have that
(3.16)ϕ(p*,q)=lim⁡ni,nj→∞ϕ(xni,xnj)=lim⁡ni,nj→∞ϕ(xni,∏Cnjx0)≤lim⁡ni,nj→∞(ϕ(xni,x0)-ϕ(∏Cnjx0,x0))=lim⁡ni,nj→∞(ϕ(xni,x0)-ϕ(xnj,x0))=ϕ(p*,x0)-ϕ(p*,x0)=0.
This implies that p*=q and
(3.17)lim⁡n→∞xn=p*.
(v) Now we prove thatp*∈G=ℱ∩Ω.
First, we prove thatp*∈ℱ. In fact, since xn+1∈Cn+1⊂Cn, it follows from (3.1) and (3.17) that
(3.18)ϕ(xn+1,yn)≤ϕ(xn+1,xn)+ξn⟶0(n⟶∞).
By the virtue of Lemma 2.1, we have
(3.19)lim⁡n→∞yn=p*.
From (3.10), for any u∈ℱ and wn,i∈Tinxn, we have
(3.20)ϕ(u,yn)≤ϕ(u,xn)+ξn-(1-αn)βn,0βn,lg(‖Jxn-Jwn,l‖),
that is,
(3.21)(1-αn)βn,0βn,lg(‖Jxn-Jwn,l‖)≤ϕ(u,xn)+ξn-ϕ(u,yn)⟶0(n⟶∞).
By conditions (b) and (c) it is shown that limn→∞g(∥Jxn-Jwn,l∥)=0. In view of property of g, we have
(3.22)‖Jxn-Jwn,l‖⟶0(n⟶∞).
Since Jxn→Jp*, this implies that Jwn,l→Jp*. From Remark 1.1(ii) it yields
(3.23)wn,l⇀p*(n⟶∞),∀l≥1.
Again since
(3.24)|‖wn,l‖-‖p*‖|=|‖Jwn,l‖-‖Jp*‖|≤‖Jwn,l-Jp*‖⟶0(n⟶∞),
this together with (3.23) and the Kadec-Klee property of X shows that
(3.25)lim⁡n→∞wn,l=p*,∀l≥1.
Let {sn,l} be a sequence generated by
(3.26)s2,l∈Tlw1,l⊂Tl2x1,s3,l∈Tlw2,l⊂Tl3x2,⋮sn+1,l∈Tlwn,l⊂Tln+1xn,⋮
By the assumption that each Ti is uniformly Li-Lipschitz continuous, for any wn,l∈Tlnxn and sn+1,l∈Tlwn⊂Tln+1xn we have
(3.27)‖sn+1,l-wn,l‖≤‖sn+1,l-wn+1,l‖+‖wn+1,l-xn+1‖+‖xn+1-xn‖+‖xn-wn,l‖≤(Ll+1)‖xn+1-xn‖+‖wn+1,l-xn+1‖+‖xn-wn,l‖.
This together with (3.17) and (3.27) shows that limn→∞∥sn+1,l-wn,l∥=0 and limn→∞sn+1,l=p*. In view of the closeness of Tl, it yields that p*∈Tp*, that is, p*∈F(Tl). By the arbitrariness of l≥1, we have
(3.28)p*∈F=⋂i=1∞F(Ti).
Next, we prove thatp*∈Ω. Since xn+1=ΠCn+1x0∈Cn, it follows from (3.1) and (3.17) that
(3.29)ϕ(xn+1,un)≤ϕ(xn+1,xn)+ξn⟶0(n⟶∞).
Since xn→p*, by virtue of Lemma 2.1 we have
(3.30)lim⁡n→∞un=p*.
This together with (3.19) shows that ∥un-yn∥→0 and limn→∞∥Jun-Jyn∥→0. By the assumption that rn≥a,foralln≥0, we have
(3.31)lim⁡n→∞‖Jun-Jyn‖rn=0.
Since H(un,y)+(1/rn)〈y-un,Jun-Jyn〉≥0,forally∈C, by condition (A1), we have
(3.32)1rn〈y-un,Jun-Jyn〉≥-H(un,y)≥H(y,un),∀y∈C.
By the assumption that y↦H(x,y) is convex and lower semicontinuous, letting n→∞ in (3.32), from (3.30) and (3.31), we have H(y,p*)≤0,forally∈C.
Fort∈(0,1] and y∈C, letting yt=ty+(1-t)p*, there are yt∈C and H(yt,p*)≤0. By conditions (A1) and (A4), we have
(3.33)0=H(yt,yt)≤tH(yt,y)+(1-t)H(yt,p*)≤tH(yt,y).
Dividing both sides of the above equation by t, we have H(yt,y)≤0,forally∈C. Letting t↓0, from condition (A3), we have H(p*,y)≤0,forally∈C, that is, p*∈Ω, and p*∈G=ℱ∩Ω.
(vi) We prove thatxn→p*=ΠGx0.
Letq=ΠGx0. Since q∈G⊂Cn and xn=ΠCnx0, we have
(3.34)ϕ(xn,x0)≤ϕ(q,x0),∀n≥0.
This implies that
(3.35)ϕ(p*,x0)=lim⁡n→∞ϕ(xn,x0)≤ϕ(q,x0).
In view of the definition of ΠGx0, from (3.35) we have p*=q. Therefore, xn→p*=ΠGx0.
This completes the proof of Theorem3.1.


## 4. Conclusions

Recently theextended general variational inequalities have been introduced and studied in Noor [24, 25]. We would like to point out that the results and the methods presented in this paper will be used to study this kind of extended general variational inequalities and its multivalued version.



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*Source: 147915-2012-05-16.xml*