# A New Hybrid Algorithm for Solving a System of Generalized Mixed Equilibrium Problems, Solving a Family of Quasi-ϕ-Asymptotically Nonexpansive Mappings, and Obtaining Common Fixed Points in Banach Space

Authors: J. F. Tan; S. S. Chang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

The main purpose of this paper is to introduce a new hybrid iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi-ϕ-asymptotically nonexpansive mappings, and null spaces of finite family of γ-inverse strongly monotone mappings in a
2-uniformly convex and uniformly smooth real Banach space. The results presented in the paper improve and extend the corresponding results announced by some authors.


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## Body

## 1. Introduction

Throughout this paper, we assume thatE is a real Banach space with a dual E*, C is a nonempty closed convex subset of E, and 〈·,·〉 is the duality pairing between members of E and E*. The mapping J:E→2E* defined by (1.1)J(x)={f*∈E*:〈x,f*〉=‖x‖2;‖f*‖=‖x‖},x∈E
is called the normalized duality mapping.LetF:C×C→ℝ be a bifunction, let B:C→E* be a nonlinear mapping, and let Φ:C→ℝ¯ be a proper extended real-valued function. The “so-called” generalized mixed equilibrium problem forF,B,Φ is to find x*∈C such that (1.2)F(x*,y)+〈y-x*,Bx*〉+Φ(y)-Φ(x*)≥0,∀y∈C.
The set of solutions of (1.2) is denoted by GMEP(F,B,Φ), that is, (1.3)GMEP⁡(F,B,Φ)={x∈C:F(x*,y)+〈y-x*,Bx*〉+Φ(y)-Φ(x*)≥0,∀y∈C}.Special Examples
(1)
IfΦ≡0, then the problem (1.2) is reduced to the generalized equilibrium problem (GEP), and the set of its solutions is denoted by
(1.4)GEP⁡(F,B)={x∈C:F(x*,y)+〈y-x*,Bx*〉≥0,∀y∈C}.(2)
IfB≡0, then the problem (1.2) is reduced to the mixed equilibrium problem (MEP), and the set of its solutions is denoted by
(1.5)MEP⁡(F,B)={x∈C:F(x*,y)+Φ(y)-Φ(x*)≥0,∀y∈C}.These show that the problem (1.2) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.2). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach space (see, e.g., [1–3]).LetE be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Throughout this paper, the Lyapunov function ϕ:E×E→ℝ+ is defined by (1.6)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,∀x,y∈E.
Following Alber [4], the generalized projection ΠC:E→C is defined by (1.7)ΠC(x)=arg⁡min⁡y∈C⁡ϕ(y,x),∀x∈E.LetC be a nonempty closed convex subset of E, let S:C→C be a mapping, and let F(S) be the set of fixed points of S. 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 ||xn-Sxn||→0. We denoted the set of all asymptotic fixed points of S by F̃(S). A point p∈C is said to be a strong asymptotic fixed point ofS if there exists a sequence {xn}⊂C such that xn→p and ||xn-Sxn||→0. We denoted the set of all strongly asymptotic fixed points of S by F̂(S).A mappingS:C→C is said to be nonexpansive if (1.8)‖Sx-Sy‖≤‖x-y‖,∀x,y∈C.A mappingS:C→C is said to be relatively nonexpansive [5] if F(S)≠∅, F(S)=F̃(S) and (1.9)ϕ(p,Sx)≤ϕ(p,x),∀x∈C,p∈F(S).A mappingS:C→C is said to be weak relatively nonexpansive [6] if F(S)≠∅, F(S)=F̂(S) and (1.10)ϕ(p,Sx)≤ϕ(p,x),∀x∈C,p∈F(S).A mappingS:C→C is said to be closed if for any sequence {xn}⊂C with xn→x and Sxn→y, then Sx=y.A mappingS:C→C is said to be quasi-ϕ-nonexpansive if F(S)≠∅ and (1.11)ϕ(p,Sx)≤ϕ(p,x),∀x∈C,p∈F(S).A mappingS:C→C is said to be quasi-ϕ-asymptotically nonexpansive, if F(S)≠∅ and there exists a real sequence {kn}⊂[1,∞) with kn→1 such that (1.12)ϕ(p,Snx)≤knϕ(p,x),∀n≥1,x∈C,p∈F(S).From the definition, 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. The class of quasi-ϕ-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, and the class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.A mappingA:C→E* is said to be α-inverse strongly monotone if there exists α>0 such that (1.13)〈x-y,Ax-Ay〉≥α‖Ax-Ay‖2.
If A is an α-inverse strongly monotone mapping, then it is 1/α-Lipschitzian.Iterative approximation of fixed points for relatively nonexpansive mappings in the setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushita and Takahashi [5] obtained weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Su et al. [6, 7],  Zegeye and Shahzad [8], Wattanawitoon and Kumam [9], and Zhang [10] extend the notion from relatively nonexpansive mappings or quasi-ϕ-nonexpansive mappings to quasi-ϕ-asymptotically nonexpansive mappings and also prove some convergence theorems to approximate a common fixed point of quasi-ϕ-nonexpansive mappings or quasi-ϕ-asymptotically nonexpansive mappings.Motivated and inspired by these facts, the purpose of this paper is to introduce a hybrid iterative scheme for finding a common element of null spaces of finite family of inverse strongly monotone mappings, set of common fixed points of an infinite family of quasi-ϕ-asymptotically nonexpansive mappings, and the set of solutions of generalized mixed equilibrium problem.


## 2. Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will be needed in proving our main results.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. It is said to be uniformly convex if 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. The convexity modulus ofE is the function δE:(0,2]→[0,1] defined by (2.1)δE(ϵ)=inf⁡{1-‖12(x+y)‖:x,y∈U,‖x-y‖≥ϵ},
for all ϵ∈(0,2]. It is well known that δE(ϵ) is a strictly increasing and continuous function with δE(0)=0, and δE(ϵ)/ϵ is nondecreasing for all ϵ∈(0,2]. Let p>1, then E is said to be p-uniformly convex if there exists a constant c>0 such that δE(ϵ)≥cϵp, for all ϵ∈(0,2]. The space E is said to be smooth if the limit (2.2)lim⁡t→0‖x+ty‖-‖x‖t
exists for all x,y∈U. And E is said to be uniformly smooth if the limit exists uniformly in x,y∈U.In the sequel, we will make use of the following lemmas.Lemma 2.1 (see [11]).
LetE be a 2-uniformly convex real Banach space, then for all x,y∈E, the inequality ||x-y||≤(2/c2)||Jx-Jy|| holds, where 0<c≤1, and c is called the 2-uniformly convex constant of  E.Lemma 2.2 (see [12]).
LetE be a smooth, strict convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E, then the following conclusions hold:(i)
ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y), for all x∈C, y∈E,(ii)
letx∈E and z∈C, then
(2.3)z=ΠCx⟺〈z-y,Jx-Jz〉≥0,∀y∈C.Lemma 2.3 (see [12]).
LetE be a uniformly convex and smooth Banach space, and let {xn}, {yn} be sequences of E. If ϕ(xn,yn)→0(asn→∞) and either {xn} or {yn} is bounded, then xn-yn→0(as n→∞).Lemma 2.4 (see [10]).
LetE be a uniformly convex Banach space, let r be a positive number, and let Br(0) be a closed ball of E. For any given points {x1,x2,…,xn,…}⊂Br(0) and for any given positive numbers {λ1,λ2,…} 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 i,j∈{1,2,…}, i<j,
(2.4)‖∑n=1∞λnxn‖2≤∑n=1∞λn‖xn‖2-λiλjg(‖xi-xj‖).For solving the generalized mixed equilibrium problem, let us assume that the bifunctionF:C×C→ℝ satisfies the following conditions:(A1)
F(x,x)=0 for all x∈C,(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0, for all x,y∈C,(A3)
limsupt↓0F(x+t(z-x),y)≤F(x,y), for all x,y,z∈C,(A4)
the functiony↦F(x,y) is convex and lower semicontinuous.Lemma 2.5 (see [13]).
LetE be a smooth, strict convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Let F:C×C→ℝ be a bifunction satisfying conditions (A1)–(A4). Let r>0 and x∈E, then there exists z∈C such that
(2.5)F(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.By the same way as given in the proofs of [14, Lemmas  2.8 and 2.9], we can prove that the bifunction (2.6)Γ(x,y)=F(x,y)+Φ(y)-Φ(x)+〈y-x,Bx〉,∀x,y∈C
satisfies conditions (A1)–(A4) and the following conclusion holds.Lemma 2.6.
LetE be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Let F:C×C→ℝ be a bifunction satisfying conditions (A1)–(A4), let B:C→E* be a β-inverse strongly monotone mapping, and let Φ:C→ℝ be a lower semicontinuous and convex function. For given r>0 and x∈E, define a mapping KrΓ:E→C by
(2.7)KrΓ(x)={z∈C:F(z,y)+Φ(y)-Φ(z)+〈y-z,Bz〉+1r〈y-z,Jz-Jx〉≥0,∀y∈C},
then the following hold:(i)
KrΓ is single valued,(ii)
KrΓ is a firmly nonexpansive-type mapping, that is, for all x,y,∈E,(2.8)〈KrΓ(x)-KrΓ(y),JKrΓ(x)-JKrΓ(y)〉≤〈KrΓ(x)-KrΓ(y),Jx-Jy〉,(iii)
F(KrΓ)=GMEP(F,Φ,B),(iv)
GMEP(F,Φ,B) is closed and convex,(v)
ϕ(p,KrΓ(x))+ϕ(KrΓ(x),x)≤ϕ(p,x), for all p∈F(KrΓ).In the sequel, we make use of the functionV:E×E*→ℝ defined by (2.9)V(x,x*)=‖x‖2-2〈x,x*〉+‖x*‖2,
for all x∈E and x*∈E*. Observe that V(x,x*)=ϕ(x,J-1x*) for all x∈E and x*∈E*. The following lemma is well known.Lemma 2.7 (see [4]).
LetE be a smooth, strict convex, and reflexive Banach space with E* as its dual, then
(2.10)V(x,x*)+2〈J-1x*-x,y*〉≤V(x,x*+y*),
for all x∈E and x*,y*∈E*.


## 3. Main Results

In this section, we will propose the following new iterative scheme{xn} for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, the set of common fixed points of a family of quasi-ϕ-asymptotically nonexpansive mappings, and null spaces of finite family of γ-inverse strongly monotone mappings in the setting of 2-uniformly convex and uniformly smooth real Banach spaces:(3.1)x0∈C0=C,yn=ΠCJ-1(Jxn-λAn+1xn),zn=J-1(αn,0Jxn+∑i=1∞αn,iJTinyn),un=KrM,nΓMKrM-1,nΓM-1⋯Kr2,nΓ2Kr1,nΓ1zn,Cn+1={v∈Cn:ϕ(v,un)≤ϕ(v,xn)+ξn},xn+1=ΠCn+1x0,n≥0,
where Krk,nΓk:C→C, k=1,2,…,M is the mapping defined by (2.7), An=An(modN), rk,n∈[d,∞) for some d>0 and 0<λ<c2γ/2, where c is the 2-uniformly convex constant of E, for each n≥1, αn,0+∑i=1∞αn,i=1 and for each j≥1,liminfn→∞αn,0αn,j>0.Definition 3.1.
A countable family of mappings{Ti:C→C} is said to be uniformly quasi-φ-asymptotically nonexpansive mappings if there exists a sequence {kn}⊂[1,∞) with kn→1 such that for each i≥1(3.2)‖Tinx-Tiny‖≤kn‖x-y‖,∀x,y∈C,andforeachn≥1.Theorem 3.2.
LetC be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth real Banach space E with a dual E*. Let {Ti:C→C}i=1∞ be a countable family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞) such that kn→1. Suppose further that for each i≥1, Ti is uniformly Li-Lipschitzian. Let An:C→E*, n=1,2,…,N be a finite family of γn-inverse strongly monotone mappings, and let γ=min{γn,n=1,2,…,N}. Let {Fm:C×C→ℝ,m=1,2,…,M} be a finite family of equilibrium functions satisfying conditions (A1)–(A4), and {Φm:C→ℝ,m=1,2,…,M} be a finite family of lower semicontinuous convex function, and let {Bm:C→E*,m=1,2,…,M} be a finite family of βm-inverse strongly monotone mappings. If
(3.3)Ω=⋂i=1∞F(Ti)∩⋂n=1NAn-1(0)∩⋂m=1MGMEP⁡(Fm,Bm,Φm)
is a nonempty and bounded subset in C and ξn=supp∈Ω(kn-1)ϕ(p,xn), then the sequence {xn} defined by (3.1) converges strongly to some point x*∈Ω.Proof.
We divide the proof of Theorem3.2 into five steps. (I) Sequences{xn}, {yn}, and {Tinyn} all are bounded.
In fact, sincexn=ΠCnx0, for any p∈Ω, from Lemma 2.2, we have
(3.4)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0).
This implies that the sequence {ϕ(xn,x0)} is bounded, and so {xn} is bounded.
On the other hand, by Lemmas2.1 and 2.7, we have that
(3.5)ϕ(p,yn)=ϕ(p,ΠCJ-1(Jxn-λAn+1xn))≤ϕ(p,J-1(Jxn-λAn+1xn))=V(p,Jxn-λAn+1xn)≤V(p,(Jxn-λAn+1xn)+λAn+1xn)-2〈J-1(Jxn-λAn+1xn)-p,λAn+1xn〉=V(p,Jxn)-2λ〈J-1(Jxn-λAn+1xn)-p,An+1xn〉=ϕ(p,xn)-2λ〈xn-p,An+1xn〉-2λ〈J-1(Jxn-λAn+1xn)-xn,An+1xn〉=ϕ(p,xn)-2λ〈xn-p,An+1xn-An+1p〉(sinceAnp=0,∀n≥1)-2λ〈J-1(Jxn-λAn+1xn)-xn,An+1xn〉≤ϕ(p,xn)-2λγ‖An+1xn‖2+2λ‖J-1(Jxn-λAn+1xn)-J-1Jxn‖×‖An+1xn‖≤ϕ(p,xn)-2λγ‖An+1xn‖2+4λ2c2‖An+1xn‖2(by Lemma2.1)≤ϕ(p,xn)+2λ(2λc2-γ)‖An+1xn‖2.
Thus, using the fact that λ≤(c2/2)γ, we have that
(3.6)ϕ(p,yn)≤ϕ(p,xn).
Moreover, by the assumption that {Ti:C→C}i=1∞ is a countable family of uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {kn,i}⊂[1,∞) such that kn=supi≥1kn,i→1 (n→∞), hence for any given p∈Ω, from (3.6) we have that
(3.7)φ(p,Tinyn)≤knφ(p,yn)≤knφ(p,xn),∀n≥1,i≥1.
Hence, for each i≥1, {Tinyn} is also bounded, denoted by
(3.8)M=sup⁡n≥0,i≥1{‖xn‖,‖yn‖,‖Tinyn‖}<∞.
By the way, from the definition of {ξn}, it is easy to see that
(3.9)ξn=sup⁡p∈Ω(kn-1)ϕ(p,xn)≤sup⁡p∈Ω(kn-1)(‖p‖+M)2⟶0(n⟶∞).(II) For eachn≥0, Cn is a closed and convex subset of C and Ω⊂Cn.
It is obvious thatC0=C is closed and convex. Suppose that Cn is closed and convex for some n≥1. Since the inequality ϕ(v,un)≤ϕ(v,xn)+ξn is equivalent to
(3.10)2〈v,Jxn-Jun〉≤‖xn‖2-‖un‖2+ξn,
therefore, we have
(3.11)Cn+1={v∈Cn:2〈v,Jxn-Jun〉≤‖xn‖2-‖un‖2+ξn}.
This implies that Cn+1 is closed and convex. Thus, for each n≥0, Cn is a closed and convex subset of C.
Next, we prove thatΩ⊂Cn for all n≥0. Indeed, it is obvious that Ω⊂C0=C. Suppose that Ω⊂Cn for some n≥1. Since E is uniformly smooth, E* is uniformly convex. For any given p∈Ω⊂Cn and for any positive integer j>0, from Lemma 2.4, we have
(3.12)ϕ(p,un)=ϕ(p,KrM,nΓMKrM-1,nΓM-1⋯Kr2,nΓ2Kr1,nΓ1zn)≤ϕ(p,zn)=ϕ(p,J-1(αn,0Jxn+∑i=1∞αn,iJTinyn))=‖p‖2-2〈p,αn,0Jxn+∑i=1∞αn,iJTinyn〉+‖αn,0Jxn+∑i=1∞αn,iJTinyn‖2≤‖p‖2-2αn,0〈p,Jxn〉-2∑i=1∞αn,i〈p,JTinyn〉+αn,0‖xn‖2+∑i=1∞αn,i‖Tinyn‖2-αn,0αn,jg(‖Jxn-JTjnyn‖)=αn,0ϕ(p,xn)+∑i=1∞αn,iϕ(p,Tinyn)-αn,0αn,jg(‖Jxn-JTjnyn‖)≤αn,0ϕ(p,xn)+∑i=1∞αn,iknϕ(p,yn)-αn,0αn,jg(‖Jxn-JTjnyn‖).
Having this together with (3.6), we have
(3.13)ϕ(p,un)≤ϕ(p,zn)≤αn,0ϕ(p,xn)+∑i=1∞αn,iknϕ(p,xn)-αn,0αn,jg(‖Jxn-JTjnyn‖)≤knϕ(p,xn)-αn,0αn,jg(‖Jxn-JTjnyn‖)≤ϕ(p,xn)+sup⁡z∈Ω(kn-1)ϕ(z,xn)-αn,0αn,jg(‖Jxn-JTjnyn‖)=ϕ(p,xn)+ξn-αn,0αn,jg(‖Jxn-JTjnyn‖)≤ϕ(p,xn)+ξn.
Hence, p∈Cn+1 and Ω⊂Cn for all n≥0.(III){xn} is a Cauchy sequence.
Sincexn=ΠCnx0 and xn+1=ΠCn+1x0∈Cn+1⊂Cn, we have that
(3.14)ϕ(xn,x0)≤ϕ(xn+1,x0),
which implies that the sequence {ϕ(xn,x0)} is nondecreasing and bounded, and so limn→∞ϕ(xn,x0) exists. Hence, for any positive integer m, using Lemma 2.2 we have
(3.15)ϕ(xn+m,xn)=ϕ(xn+m,ΠCnx0)≤ϕ(xn+m,x0)-ϕ(xn,x0),
for all n≥0. Since limn→∞ϕ(xn,x0) exists, we obtain that
(3.16)ϕ(xn+m,xn)⟶0(n⟶∞),∀m∈Z+.
Thus, by Lemma 2.3, we have that ∥xn+m-xn∥→0 as n→∞. This implies that the sequence {xn} is a Cauchy sequence in C. Since C is a nonempty closed subset of Banach space E, it is complete. Hence, there exists an x* in C such that
(3.17)xn⟶x*(n⟶∞).(IV) We show thatx*∈⋂i=1∞F(Ti).
Sincexn+1∈Cn+1 by the structure of Cn+1, we have that
(3.18)ϕ(xn+1,un)≤ϕ(xn+1,xn)+ξn.
Again by (3.16) and Lemma 2.3, we get that limn→∞||xn+1-un||=0. But
(3.19)‖xn-un‖≤‖xn-xn+1‖+‖xn+1-un‖.
Thus,
(3.20)lim⁡n→∞‖xn-un‖=0.
This implies that un→x* as n→∞. Since J is norm-to-norm uniformly continuous on bounded subsets of E, we have that
(3.21)lim⁡n→∞‖Jxn-Jun‖=0.
From (3.13), (3.20), and (3.21), we have that
(3.22)αn,0αn,jg(‖Jxn-JTjnyn‖)≤ϕ(p,xn)-ϕ(p,un)+ξn=‖xn‖2-‖un‖2+2〈p,Jun-Jxn〉+ξn≤‖xn-un‖(‖un‖+‖xn‖)+2〈p,Jun-Jxn〉+ξn⟶0(n⟶∞).
In view of condition liminfn→∞αn,0αn,j>0, we see that
(3.23)g(‖Jxn-JTjnyn‖)⟶0(n⟶∞).
It follows from the property of g that
(3.24)‖Jxn-JTjnyn‖⟶0(n⟶∞).
Since xn→x* and J is uniformly continuous, it yields Jxn→Jx*. Hence, from (3.24), we have
(3.25)JTjnyn⟶Jx*(n⟶∞).
Since E* is uniformly smooth and J-1 is uniformly continuous, it follows that
(3.26)Tjnyn⟶x*(n⟶∞),∀j≥1.
Moreover, using inequalities (3.12) and (3.5), we obtain that
(3.27)ϕ(p,un)≤αn,0ϕ(p,xn)+∑i=1∞αn,iknϕ(p,xn)+∑i=1∞αn,ikn2λ(2c2λ-γ)‖An+1xn‖2≤ϕ(p,xn)+ξn+kn2λ(2c2λ-γ)‖An+1xn‖2,∀p∈Ω.
This implies that
(3.28)kn2λ(γ-2c2λ)‖An+1xn‖2≤ϕ(p,xn)-ϕ(p,un)+ξn,
that is,
(3.29)lim⁡n→∞‖An+1xn‖2=0.
It follows from (3.1) and (3.29) that we have
(3.30)lim⁡n→∞‖yn-x*‖=lim⁡n→∞‖ΠCJ-1(Jxn-λAn+1xn)-x*‖≤lim⁡n→∞‖J-1(Jxn-λAn+1xn)-x*‖=0.
Furthermore, by the assumption that for each j≥1, Tj is uniformly Li-Lipschitz continuous, hence, we have
(3.31)‖Tjn+1yn-Tjnyn‖≤‖Tjn+1yn-Tjn+1yn+1‖+‖Tjn+1yn+1-yn+1‖+‖yn+1-yn‖+‖yn-Tjnyn‖≤(Lj+1)‖yn+1-yn‖+‖Tjn+1yn+1-yn+1‖+‖yn-Tjnyn‖.
This together with (3.26) and (3.30) yields
(3.32)lim⁡n→∞‖Tjn+1yn-Tjnyn‖=0.
Hence, from (3.26), we have
(3.33)lim⁡n→∞Tjn+1yn=x*,
that is,
(3.34)lim⁡n→∞TjTjnyn=x*.
In view of (3.26) and the closeness of Tj, it yields that Tjx*=x* for all j≥1. This implies that x*∈⋂j=1∞F(Tj).(IV) Now, we prove thatx*∈⋂n=1NAn-1(0).
It follows from (3.29) that
(3.35)lim⁡n→∞‖An+1xn‖=0.
Since limn→∞xn=x*, we have that for every subsequence {xnj}j≥1 of {xn}n≥0,limj→∞xnj=x* and
(3.36)lim⁡j→∞Anj+1xnj=0.
Let {nq}q≥1⊂ℕ be an increasing sequence of natural numbers such that Anq+1=A1, for all q∈ℕ, then limp→∞||xnq-x*||=0 and
(3.37)0=lim⁡q→∞Anq+1xnq=lim⁡q→∞A1xnq.
Since A1 is γ-inverse strongly monotone, it is Lipschitz continuous, and thus
(3.38)A1x*=A1(lim⁡q→∞xnq)=lim⁡q→∞A1xnq=0.
Hence,
(3.39)x*∈A1-1(0).
Continuing this process, we obtain that x*∈Ai-1(0), for all i=1,2,…,N. Hence,
(3.40)x*∈⋂n=1NAn-1(0).(V) Next, we prove thatx*∈⋂m=1MGMEP(Fm,Bm,Φm).
Putting𝒮nm=Krm,nΓmKrm-1,nΓm-1⋯Kr2,nΓ2Kr1,nΓ1 for m∈{1,2,…,M} and 𝒮n0=I for all n∈ℕ. For any p∈Ω, we have
(3.41)ϕ(Snmzn,Snm-1zn)≤ϕ(p,Snm-1zn)-ϕ(p,Snmzn)≤ϕ(p,zn)-ϕ(p,Snmzn)≤ϕ(p,xn)+ξn-ϕ(p,Snmzn)(by(3.13))=ϕ(p,xn)+ξn-ϕ(p,un).
It follows from (3.22) that limn→∞ϕ(𝒮nmzn,𝒮nm-1zn)=0. Since E is 2-uniformly convex and uniformly smooth Banach space and {zn} is bounded, we have that
(3.42)lim⁡n→∞‖Snmzn-Snm-1zn‖=0,m=1,2,…,M.
Since xn→x* and un→x*, now we prove that for each m=1,2,…,M, 𝒮nmzn→x* as n→∞. In fact, if m=M, then we have
(3.43)lim⁡n→∞‖SnMzn-SnM-1zn‖=lim⁡n→∞‖un-SnM-1zn‖=0,
that is, 𝒮nM-1zn→x*. By induction, the conclusion can be obtained. Since J is norm-to-norm uniformly continuous on bounded subsets of E, we get
(3.44)lim⁡n→∞‖JSnmzn-JSnm-1zn‖=0,
and since rk,n∈[d,∞) for some d>0, we have that
(3.45)lim⁡n→∞‖JSnmzn-JSnm-1zn‖rm,n=0.
Next, since Γm(𝒮nmzn,y)+(1/rm,n)〈y-𝒮nmzn,J𝒮nmzn-J𝒮nm-1zn〉≥0, for all y∈C, this implies that
(3.46)1rm,n〈y-Snmzn,JSnmzn-JSnm-1zn〉≥-Γm(Snmzn,y)≥Γm(y,Snmzn),∀y∈C.
This implies that
(3.47)Γm(y,Snmzn)≤1rm,n〈y-Snmzn,JSnmzn-JSnm-1zn〉≤(M1+‖y‖)‖JSnmzn-JSnm-1zn‖rm,n,
for some M1≥0. Since y↦Γm(x,y) is a convex and lower semicontinuous, we obtain from (3.45) and (3.47) that
(3.48)Γm(y,x*)≤liminf⁡n→∞Γm(y,Snmzn)≤0,∀y∈C.
For any t∈(0,1] and y∈C, then yt=ty+(1-t)x*∈C. Since Γm satisfies conditions (A1) and (A4), from (3.48), we have
(3.49)0=Γm(yt,yt)≤tΓm(yt,y)+(1-t)Γm(yt,x*)≤tΓm(yt,y),∀m=1,2,…M.
Delete t, and then let t→0, by condition (A3), we have
(3.50)0≤Γm(x*,y),∀y∈C,∀m=1,2,…M,
that is, for each m=1,2,…,M, we have
(3.51)Fm(x*,y)+〈y-x*,Bmx*〉+Ψm(y)-Ψm(x*)≥0,∀y∈C.
Therefore, we have that
(3.52)x*∈⋂m=1MGMEP⁡(Fm,Bm,Φm).
This completes the proof.Remark 3.3.
(1) Theorem3.2 not only improves and extends the main results in [3, 6–10] but also improves and extends the corresponding results of Chang et al. [1, 15], Wang et al. [16], Su et al. [17], and Kang et al. [18].
(2) It should be pointed out that the results presented in the paper can be used directly to study the existence problems and approximal problems of solutions to optimization problems, monotone variational inequality problems, variational inclusion problems, and equilibrium problems in some Banach spaces. For saving space, we will give them in another paper.



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*Source: 106323-2011-06-02.xml*