# Existence and Modification of Halpern-Mann Iterations for Fixed Point and Generalized Mixed Equilibrium Problems with a Bifunction Defined on the Dual Space

Authors: Pongrus Phuangphoo; Poom Kumam
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We study and establish the existence of a solution for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space. Furthermore, we also modify Halpern-Mann iterations for finding a common solution of a generalized mixed equilibrium problem and a fixed point problem. Under suitable conditions of the purposed iterative sequences, the strong convergence theorems are established by using sunny generalized nonexpansive retraction in Banach spaces. Our results extend and improve various results existing in the current literature.


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## Body

## 1. Introduction

In the past years, variational inequalities is among the most important and interesting mathematical problems, since they have wide applications in the optimization and control, economics and equilibrium, engineering science and physical sciences.Equilibrium problem represents an important area of mathematical sciences such as optimization, operations research, game theory, financial mathematics, and mechanics. Equilibrium problems include variational inequalities, optimization problems, Nash equilibria problems, saddle point problems, and fixed point problems as special cases.Throughout this paper, we denote the strong convergence and weak convergence{xn} by xn→x, xn⇀x, respectively.LetC be a closed and convex subset of a real Banach spaceEwith the dual space E*. Let C* be a closed and convex subset of E*. We recall the following definitions.(1)
A mappingA:C→E* is said to be monotone if for each x,y∈C such that
(1)〈x-y,Ax-Ay〉≥0.(2)
A mappingA:C→E* is said to be δ-stronglymonotone, if there exists a constant δ>0 such that
(2)〈x-y,Ax-Ay〉≥δ∥x-y∥2,∀x,y∈C.(3)
A mappingA:C→E* is said to be δ-inversestronglymonotone, if there exists a constant δ>0 such that
(3)〈x-y,Ax-Ay〉≥δ∥Ax-Ay∥2,∀x,y∈C.(4)
A mappingA:C*→E is said to be skew monotone if for each x*,y*∈C* such that
(4)〈Ax*-Ay*,x*-y*〉≥0.(5)
A mappingA:D(A)⊂E*→E is said to be α-inverse strongly skew monotone if there exists a constant α>0 such that
(5)〈Ax*-Ay*,x*-y*〉≥α∥Ax*-Ay*∥2,kkkkkkkkkkkkkkkkkkkk∀x*,y*∈D(A).Definition 1.
LetEbe a Banach space. Then,(1)
E is said to be strictly convex if (∥x+y∥/2)<1 for all x,y∈UE={z∈E:∥z∥=1} with x≠y;(2)
E 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∈UE with ∥x+y∥>ϵ;(3)
E is said to be smooth if the limit
(6)limt→0∥x+ty∥-∥x∥texists, for eachx,y∈UE;(4)
E is said to be uniformly smooth if the limit (6) is attained uniformly, for all x,y∈UE;(5)
E is said to have uniformly Gâteaux differentiable norm if for all y∈U(E), the the limit (6) converges uniformly, for x∈UE.Definition 2.
LetE be a Banach space. Then, a function ρE:ℝ+→ℝ+ is said to be the modulus of smoothness of E if
(7)ρE(t)=sup{∥x+y∥+∥x-y∥2-1:∥x∥=1,∥y∥=t}.(1)
E is said to be smooth if ρE(t)>0,∀t>0.(2)
E is said to be uniformly smooth if and only if limt→0+(ρE(t)/t)=0.Definition 3.
LetE be a Banach space. Then, the modulus of convexity of E is the function δE:[0,2]→[0,1] defined by
(8)δE(ϵ)=inf{1-∥x+y2∥:∥x∥≤1,∥y∥≤1;∥x-y∥≥ϵ}.(1)
E is said to be uniformly convex if and only if δE(ϵ)>0 for all ϵ∈(0,2].(2)
Letp be a fixed real number 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].
Observe that everyp-uniformly convex is uniformly convex. One should note that no a Banach space is p-uniformly convex, for 1<p<2. It is well known that a Hilbert space is 2-uniformly convex and uniformly smooth.For anyp>1, the generalized duality mappingJp:E→2E* is defined by
(9)Jpx={f*∈E*:〈x,f*〉=∥x∥p,∥f*∥=∥x∥p-1},∀x∈E.In particular,J=J2 is called the normalized duality mapping. If E is a Hilbert space, then J=I, where I is the identity mapping. That is,
(10)J2x=Jx={f*∈E*:〈x,f*〉=∥x∥2=∥f*∥2},∀x∈E.Remark 4.
The basic properties below hold (see [1–3]).  (1)
IfE is uniformly smooth real Banach space, then J is uniformly continuous on each bounded subset of E.(2)
IfE is uniformly smooth real Banach space, then J*:E*→2E is a normalized duality mapping on E*, and then J-1=J*, (J*)J=IE, and J(J*)=IE*, where on IE and IE* are the identity mappings on E and E*, respectively.(3)
LetE be a smooth, strictly convex reflexive Banach space, and let J be the duality mapping from E into E*. Then, J-1 is also single-valued, one-to-one, and onto, and it is also the duality mapping from E* into E.(4)
IfE is a reflexive, strictly convex Banach space, then J-1 is hemicontinuous; that is, J-1 is norm-to-weak*-continuous.(5)
IfE is a reflexive, smooth, and strictly convex Banach space, then J is single-valued, one-to-one, and onto.(6)
A Banach spaceE is uniformly smooth if and only if E* is uniformly convex.(7)
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.(8)
A Banach spaceE is strictly convex if and only if J is strictly monotone; that is,
(11)〈x-y,x*-y*〉>0,wheneverx,y∈E,x≠y,x*∈Jx,y*∈Jy.(9)
Both uniformly smooth Banach space and uniformly convex Banach space are reflexive.(10)
IfE* is uniformly convex, then J is uniformly norm-to-norm continuous on each bounded subset of E.(11)
IfE* is strictly convex Banach space, then J is one-to-one; that is, x≠y implies Jx∩Jy≠∅.LetJ be the normalized duality mapping; then J is said to be weakly sequentially continuous if the strong convergence of a sequence {xn} to x∈E implies the weak* convergence of a sequence {Jxn} to Jx in E*.LetE be a smooth and strictly convex reflexive Banach space, and let C be a nonempty, closed, and convex subset of E. We assume that the Lyapunov functionalϕ:E×E→ℝ+ is defined by [3, 4]
(12)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E.LetC is nonempty, closed, and convex subset of a Banach space E. The generalized projection [3] ΠC:E→C is defined by for each x∈E,
(13)ΠC(x)=argminy∈Cϕ(x,y).Remark 5.
From the definition ofϕ. It is easy to see that  (1)
(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2, for all x,y∈E;(2)
ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉, for all x,y,z∈E;(3)
ϕ(x,y)=〈x,Jx-Jy〉+〈y-x,Jy〉≤∥x∥∥Jx-Jy∥+∥y-x∥∥y∥, for all x,y,z∈E;(4)
ifE is a real Hilbert space H, then ϕ(x,y)=∥x-y∥2, and ΠC=PC (the metric projection of H onto C).Lemma 6 (see [3, 4]).
IfC is a nonempty, closed, and convex subset of a smooth and strictly convex reflexive real Banach space E, then(1)
forx∈E and u∈C, one has
(14)u=ΠC(x)⟺〈u-y,Jx-Ju〉≥0,∀y∈C;(2)
ϕ(x,ΠC(y))+ϕ(ΠC(y),y)≤ϕ(x,y),∀x∈C, andy∈E;(3)
ϕ(x,y)=0 if and only if x=y,  ∀x,y∈E.LetC be a nonempty, closed subset of a smooth, strictly convex, and reflexive Banach space E such that J(C) is closed and convex. For solving the equilibrium problem, let us assume that a bifunction F:J(C)×J(C)→ℝ satisfies the following conditions:(DA1)
F(x*,x*)=0,  for all x*∈J(C);(DA2)
Fis monotone, that is, F(x*,y*)+F(y*,x*)≤0,  for all x*,y*∈J(C)(DA3)
for allx*,y*,z*∈J(C),(15)limsupt↓0F(tz*+(1-t)x*,y*)≤F(x*,y*);(DA4)
for allx*∈J(C),  F(x*,·) is convex and lower semicontinuous.The following result is in Blum and Oettli [5], and see proof in [6].Letℝ be the set of real numbers, let E be a real Banach space with the norm ∥·∥, and let 〈·,·〉 that is the dual pair between E and E* by E* be the dual space of E. Let C be a nonempty, closed, and convex subset of E, let J be the duality mapping from E into E* such that J(C) is closed and convex of E*, let us assume that a bifunction F:J(C)×J(C)→ℝ satisfies suitable conditions, let A:C*→E be a skew monotone operator from J(C) into E, and let φ:J(C)→ℝ be a real-valued function.Thegeneralized mixed equilibrium problem is to find z^∈C such that
(16)F(Jz^,Jy)+〈AJz^,Jy-Jz^〉+φ(Jy)-φ(Jz^)≥0,∀y∈C.The set of solutions of (16) is denoted by GMEP(F,A,φ); that is,
(17)GMEP(F,A,φ)={z^∈C:F(Jz^,Jy)+〈AJz^,Jy-Jz^〉+φ(Jy)-φ(Jz^)≥0,∀y∈C}.IfA≡0, then the problem (16) reduces to the mixed equilibrium problem which is to find z^∈C such that
(18)F(Jz^,Jy)+φ(Jy)-φ(Jz^)≥0,∀y∈C.The set of solution of problem (18) is denoted by MEP(F,φ); that is,
(19)MEP(F,φ)={z^∈C:F(Jz^,Jy)+φ(Jy)-φ(Jz^)≥0,∀y∈C}.IfA≡0 and φ≡0, then the problem (16) reduces to the equilibrium problem which is to find z^∈C such that
(20)F(Jz^,Jy)≥0,∀y∈C.The set of solution of problem (20) is denoted by EP(F); that is,
(21)EP(F)={z^∈C:F(Jz^,Jy)≥0,∀y∈C}.The above formulation (20) was considered in Takahashi and Zembayashi [7], and they proved a strong convergence theorem for finding a solution of the equilibrium problem (20) in Banach spaces.IfF≡0 and A≡0, then the problem (16) reduces to variational inequality, which is to find z^∈C such that
(22)〈AJz^,Jy-Jz^〉≥0,∀y∈C.The set of solution of problem (22) is denoted by VI(J(C),A); that is,
(23)VI(J(C),A)={z^∈C:〈AJz^,Jy-Jz^〉≥0,∀y∈C}.In the sequel, letT:C→C be a mapping, we denote by Fix(T) the set of fixed points of T; that is,
(24)Fix(T)={x∈C:Tx=x}.In 1953, Mann [8] introduced an iterative algorithm which is defined by the initial point x0 is taken in C arbitrarily and
(25)xn+1=αnxn+(1-αn)Txn,n≥0
where the sequence αn∈[0,1]. Mann’s iteration can yield only weak convergence.In 1967, Halpern [9] introduced another iterative algorithm which is defined by the initial point x0 is taken in C arbitrarily and
(26)x0=u∈Cchosenarbitrary,xn+1=αnu+(1-αn)Txn,n≥0,
which satisfied the conditions limn→∞αn=0 and ∑n=1∞‍αn=∞. Then, the sequence {xn} is converges strongly to a fixed point of T.In 2007, Takahashi and Zembayashi [7] introduced an iterative algorithm for finding a solution of an equilibrium problem with a bifunction defined on the dual space of a Banach space by using the shrinking projection method, and they established the strong convergence of the following results.Theorem TZ.
LetE be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, and let C be a nonempty, closed and convex subset of E such that J(C) is closed, and convex of E*. Assume that a mapping F:J(C)×J(C)→ℝ satisfied the conditions (DA1)–(DA4) such that EP(F)≠∅. Let {xn} be a sequence generated by the following algorithm:
(27)x0∈C,C0=Cchosenarbitrary,un∈CsuchthatF(Jun,Jy)+1rn〈un-xn,Jy-Jun〉≥0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk∀y∈C,yn=αnxn+(1-αn)un,Cn+1={z∈Cn:ϕ(yn,z)≤ϕ(xn,z)},xn+1=RCn+1(x0),∀n∈ℕ∪{0},
where J is the duality mapping on E, the sequence {αn}⊂[0,1] such thatlimsupn→∞αn<1,rn⊂[a,∞) for some a>0, and RCn+1 is the sunny generalized nonexpansive retraction from E onto Cn+1. Then, the sequence {xn} converges strongly to some point p=REF(F)(x0), where REP(F) is the sunny generalized nonexpansive retraction from E to EP(F).In 2010, Plubtieng and Sriprad [10] proved the existence theorem of the variational inequality problem for skew monotone operator defined on the dual space of a smooth Banach space, and they established weak convergence theorem for finding a solution of the variational inequality problem using projection algorithm method with a new projection which was introduced by Ibaraki and Takahashi [11] and Iiduka and Takahashi [12] in Banach spaces.LetE be a smooth, strictly convex, and reflexive Banach space, let E* be the dual space of E, and let C be a nonempty, closed, and convex subset of E such that J(C) is closed and convex of E*, where J is the duality mapping on E. Let A be a skew monotone operator from J(C) into E. Then, the variational inequality problem is to find z∈C such that
(28)〈AJz,Jy-Jz〉≥0,∀y∈C.The set of solution of problem (28) is denoted by VI(J(C),A); that is,
(29)VI(J(C),A)={z∈C:〈AJz,Jy-Jz〉≥0,∀y∈C}.Theorem PS.
LetE be a uniformly convex and 2-uniformly smooth Banach space whose duality mapping J is weakly sequentially continuous. Let C be a nonempty, closed and convex subset of E such that J(C) is closed, and convex, and let A be an α-inverse-strongly-skew-monotone operator of JC into E such that VI(J(C),A)≠∅, and ∥AJz∥≤∥AJy-AJz∥, for all y∈C and z∈VI(J(C),A). Let {xn} be a sequence defined by x1=x∈C and
(30)xn+1=RC(xn-λnAJxn),
for every n=1,2,3,…, where RC is the sunny generalized nonexpansive retraction of E into C,  {αn}⊂[a,b] and for some a,b with 0<a<b<(α/c), where c>0 is a constant that satisfies ∥Jx-Jy∥≤c∥x-y∥, for all x,y∈C. Then, the sequence {xn} converges weakly to some element z∈VI(J(C),A). Further z=limn→∞RVI(J(C),A)xn.In 2011, Chen et al. [13] introduced a new iterative method for finding a solution of equilibrium with a bifunction defined on the dual space of a Banach space. They established the strong convergence theorem by using the sunny generalized nonexpansive retraction in Banach spaces.Theorem CCW.
LetC be a nonempty, closed and convex subset of a uniformly convex and uniformly smooth Banach space such that J(C) is closed and convex. Assume that a bifunction F:J(C)×J(C)→ℝ satisfies the conditions (DA1)–(DA4). Define a sequence {xn} in C by the following algorithm:
(31)x0∈Cchosenarbitrary,un∈CsuchthatF(Jun,Jy)+1rn〈un-xn,Jy-Jun〉≥0,∀y∈C,xn+1=αnx0+(1-αn)(βnxn+(1-βn)un),∀n∈ℕ,
where J is the duality mapping on E, the sequences {αn}, {βn}⊂[0,1], and {rn}⊂[a,∞), for some a>0 such that
(32)∑n=0∞‍αn<∞,liminfn→∞βn(1-βn)>0,liminfn→∞rn>0.
Then, the sequence REP(F){xn} converges strongly to some point p∈EF(F), where REP(F) is the sunny generalized nonexpansive retraction from E to EP(F).In 2012, Saewan et al. [14] introduced a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problems and the set of fixed points for a -ϕ-nonexpansive mapping in Banach spaces by using sunny generalized nonexpansive retraction in Banach spaces.Theorem SCK.
LetE be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty, closed, and convex subset of E such that J(C) is closed and convex of E*. Let F:J(C)×J(C)→ℝ be a bifunction that satisfies the conditions (DA1)–(DA4), and let T:C→C be a closed and -ϕ-nonexpansive mapping. Assume that 𝔉:=Fix(T)∩MEP(F,φ)≠∅. For an initial point x0∈C and define a sequence {xn} inCby the following algorithm:
(33)x0∈Cchosenarbitrary,un∈CsuchthatF(Jun,Jy)+φ(Jy)-φ(Jun)+1rn〈un-xn,Jy-Jun〉≥0,∀y∈C,xn+1=αnx0+(1-αn)(βnxn+(1-βn)Tun),∀n∈ℕ,
where J is the duality mapping on E, the sequence {αn}, {βn}⊂[a,b], and {rn}⊂[c,∞),for somea,b∈(0,1)andc>0. If the following conditions are satisfied:
(34)∑n=0∞‍αn<∞,∑n=0∞‍βn<∞,liminfn→∞rn>0.
Then, the sequence R𝔉{xn} converges strongly to some point p∈𝔉, where R𝔉 is the sunny generalized nonexpansive retraction from E onto 𝔉.In this paper, Motivated and inspired by the previously mentioned above results, we study and investigate the existence of theorem for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space, and we construct an iterative procedure generated by the conditions for solving the common solution of a generalized mixed equilibrium problem and a fixed point problem by using the sunny generalized nonexpansive retraction. Under some suitable assumptions, the strong convergence theorem are established in Banach spaces. The results obtained in this paper extend and improve several recent results in this area.


## 2. Preliminaries

Definition 7.
LetC be a nonempty, closed subset of a smooth Banach space.  (1)
A mappingT:C→C is said to be closed if for each {xn}⊂C, xn→x and Txn→y imply Tx=y.(2)
A mappingT:C→C is said to be nonexpansive if
(35)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.(3)
A mappingT:C→C is said to be -ϕ-nonexpansiveif Fix(T)≠∅, and
(36)ϕ(Tx,Ty)≤ϕ(x,y),∀x,y∈C.(4)
A mappingT:C→C is said to be generalized nonexpansive[15] if Fix(T)≠∅ and
(37)ϕ(Tx,p)≤ϕ(x,p),∀x∈C,p∈Fix(T).Definition 8 (see [15]).
LetC be a nonempty, closed subset of a smooth Banach space E. A mapping R:E→C is called(1)
a retraction ifR2=R;(2)
a sunny ifR(Rx+t(x-Rx))=Rx, for all x∈E and t≥0.We also know that ifE is a smooth, strictly convex, and reflexive Banach space and C is nonempty, closed, and convex subset of E, then there exists a sunny generalized nonexpansive retraction RC of E onto C if and only if J(C) is closed and convex. In this case RC is given by
(38)RC=J-1ΠJ(C)J.Definition 9 (see [15]).
A nonempty, closed subsetC of a smooth Banach space E is said to be a sunny generalized nonexpansive retractionof E if there exists a sunny generalized nonexpansive R from E onto C.Lemma 10 (see [11]).
LetC be a nonempty, closed, and subset of a smooth and strictly convex Banach space E, and let R be a retraction from E onto C. Then, the following are equivalent:  (1)
R is sunny generalized nonexpansive;(2)
〈x-Rx,Jy-JRx〉≤0, for all x∈E and y∈C.Lemma 11 (see [11]).
LetC be a nonempty, closed, and sunny generalized nonexpansive retraction of a smooth and strictly convex Banach space E. Then, the sunny generalized nonexpansive retraction from E onto C is uniquely determined.Lemma 12 (see [11]).
LetC be a nonempty, closed, and subset of a smooth and strictly convex Banach space E such that there exists a sunny generalized nonexpansive retraction R from E onto C. Let x∈E and z∈C. Then, the following hold:  (1)
z=Rx if and only if 〈x-z,Jy-Jz〉≤0, for all y∈C;(2)
ϕ(x,Rx)+ϕ(Rx,z)≤ϕ(x,z).Lemma 13 (see [16]).
LetC be a nonempty, closed, and subset of a smooth, strictly convex, and reflexive Banach space E. Then, the following are equivalent:  (1)
C is sunny generalized nonexpansive retraction of E;(2)
J(C) is closed and convex.Remark 14.
LetE be a Hilbert space. By the Lemmas 11 and 13, a sunny generalized nonexpansive retraction from E onto C reduces to a metric projection operator P from E onto C.Lemma 15 (see [16]).
LetE be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty, closed, and sunny generalized nonexpansive retraction of E, and let R be the sunny generalized nonexpansive retraction from E onto C. Let x∈E and z∈C. Then, the following are equivalent:  (1)
z=Rx;(2)
ϕ(x,z)=miny∈Cϕ(x,y).Lemma 16 (see [4]).
Let E be a uniformly smooth and strictly convex real Banach space, and let{xn} and {yn} be two sequences of E. If ϕ(xn,yn)→0 and either {xn} or {yn} is bounded, then ∥xn-yn∥→0.Lemma 17 (see [17]).
Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and letC be a nonempty, closed, and convex subset of E. Let {xn} and {yn} be two sequences in C and p∈E. If xn→p and ϕ(xn,yn)→0, then yn→p.Lemma 18 (see [18]).
Let{an} and {bn} be two sequences of nonnegative real numbers satisfying the inequality
(39)an+1≤an+bn,∀n≥1.
If ∑n=0∞‍bn<∞, then limn→∞an exists.Lemma 19 (see [19]).
LetE be a uniformly convex Banach space. Then, for any r>0, there exists a strictly increasing, continuous, and convex function h:[0,2r]→ℝ such that h(0)=0 and
(40)∥tx+(1-t)y∥2≤t∥x∥2+(1-t)∥y∥2-t(1-t)h(∥x-y∥),∀x,y∈Br(z),t∈[0,1],
where Br(z)={z∈E:∥z∥≤r}.Lemma 20 (see [4]).
LetE be a smooth and uniformly convex Banach space. Then, for any r>0, there exists a strictly increasing, continuous, and convex function h:[0,2r]→ℝ such that h(0)=0 and
(41)h(∥x-y∥)≤ϕ(x,y),∀x,y∈Br(z),t∈[0,1],
where Br(z)={z∈E:∥z∥≤r}.Now, let us recall the following well-known concept and result.Definition 21 (see [20]).
LetB be a subset of a topological vector space X. A mapping G:B→2X is called a KKM mapping if conv{x1,x2,x3,…,xm}⊂⋃i=1m‍G(xi) for xi∈B and i=1,2,3,…,m, where convA denotes the convex hull of the set A.In [21], Fan gave the following famous infinite-dimensional generalization of Knaster, Kuratowski, and Mazurkiewicz’s classical finite-dimensional result.Lemma 22 (see [21]).
LetB be a subset of a Hausdorff topological vector space X, and let G:B→2X be a KKM mapping. If G(x) is closed, for all x∈B and is compact for at least one x∈B, then ⋂x∈B‍G(x)≠∅.Lemma 23 (see [14]).
LetC be a nonempty, closed, and convex subset of a smooth and strictly convex Banach space, and let T:C→C be a closed and -ϕ-nonexpansive mapping. Then, Fix(T) is a closed and convex subset of C.


## 3. Existence Theorem

In this section, we prove the existence theorem of a solution for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space.Lemma 24.
LetC be a nonempty, compact, and convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E, let J be the duality mapping from E into E* such that J(C) is closed and convex, let us assume that a bifunction F:J(C)×J(C)→ℝ satisfies the following conditions (DA1)–(DA4), let C* be a nonempty, closed, and convex subset of E*, let A:C*→E be an α-inverse strongly skew monotone and let φ:J(C)→ℝ be a convex and lower semicontinuous. Letr>0 be given real number and x∈E be any point. Then, there exists z∈C such that
(42)F(Jz,Jy)+〈AJz,Jy-Jz〉+1r〈z-x,J(y-z)〉+φ(Jy)-φ(Jz)≥0,∀y∈C.Proof.
Letx0 be any point in E. For each y∈C, we define the mapping G:C→2E as follows:
(43)G(y)={z∈C:F(Jz,Jy)+〈AJz,Jy-Jz〉+1r〈z-x0,J(y-z)〉+φ(Jy)-φ(Jz)≥0,∀y∈C}.
It is easy to see that y∈G(y), and hence G(y)≠∅.(a)
First, we will show thatG is a KKM mapping.
Suppose thatG is not a KKM mapping. Then, there exists a finite subset {y1,y2,y3,…,ym} of C and αi>0 with ∑i=1m‍αi=1 such that x^=∑i=1m‍αiyi∉⋃i=1m‍G(yi) for all i=1,2,3,…,m.
It follows from the definition of a mappingG that
(44)F(Jx^,Jyi)+〈AJx^,Jyi-Jx^〉+1r〈x^-x0,J(yi-x^)〉+φ(Jyi)-φ(Jx^)<0,∀i=1,2,3,…,m.
By the assumptions of (DA1) and (DA4), we get
(45)0=F(Jx^,Jx^)+〈AJx^,Jx^-Jx^〉+1r〈x^-x0,J(x^-x^)〉+φ(Jx^)-φ(Jx^)≤∑i=1m‍αi(1rF(Jx^,Jyi)+〈AJx^,Jyi-Jx^〉+1r〈x^-x0,J(yi-x^)〉+φ(Jyi)-φ(Jx^))<0,
which is a contradiction. Thus, G is a KKM mapping on C.(b)
Next, we will show thatG(y) is closed, for all y∈C.Let {zn} be a sequence in G(y) such that zn→z0 as n→∞.
It then follows fromzn∈G(y) that
(46)F(Jzn,Jy)+〈AJzn,Jy-Jzn〉+1r〈zn-x0,J(y-zn)〉+φ(Jy)-φ(Jzn)≥0.
By the assumption (DA3), the continuity of J, and the lower semicontinuity of φ and ∥·∥2, it follows from (46) that
(47)φ(z0)≤liminfn→∞φ(zn)≤liminfn→∞(1rF(Jzn,Jy)+〈AJzn,Jy-Jzn〉+1r〈zn-x0,J(y-zn)〉+φ(Jy))≤limsupn→∞(1rF(Jzn,Jy)+〈AJzn,Jy-Jzn〉+1r〈zn-x0,J(y-zn)〉+φ(Jy))=limsupn→∞(F(Jzn,Jy)+〈AJzn,Jy-Jzn〉+1r〈(zn-y)+(y-x0),J(y-zn)〉+φ(Jy))=limsupn→∞(1rF(Jzn,Jy)+〈AJzn,Jy-Jzn〉+1r〈zn-y,J(y-zn)〉+1r〈y-x0,J(y-zn)〉+φ(Jy))=limsupn→∞(1rF(Jzn,Jy)+〈AJzn,Jy-Jzn〉+1r〈y-x0,J(y-zn)〉-1r〈y-zn,J(y-zn)〉+φ(Jy))=limsupn→∞(1rF(Jzn,Jy)+〈AJzn,Jy-Jzn〉+1r〈y-x0,J(y-zn)〉-1r∥y-zn∥2+φ(Jy))≤limsupn→∞F(Jzn,Jy)+limsupn→∞〈AJzn,Jy-Jzn〉+1rlimsupn→∞〈y-x0,J(y-zn)〉-liminfn→∞1r∥y-zn∥2+φ(Jy)≤F(Jz0,Jy)+〈AJz0,Jy-Jz0〉+1r〈y-x0,J(y-z0)〉-1r∥y-z0∥2+φ(Jy)=F(Jz0,Jy)+〈AJz0,Jy-Jz0〉+1r〈y-x0,J(y-z0)〉-1r〈y-z0,J(y-z0)〉+φ(Jy)=F(Jz0,Jy)+〈AJz0,Jy-Jz0〉+1r〈y-x0,J(y-z0)〉+1r〈z0-y,J(y-z0)〉+φ(Jy)=F(Jz0,Jy)+〈AJz0,Jy-Jz0〉+1r〈(y-x0)+(z0-y),J(y-z0)〉+φ(Jy)=F(Jz0,Jy)+〈AJz0,Jy-Jz0〉+1r〈z0-x0,J(y-z0)〉+φ(Jy).
Now, we get
(48)F(Jz0,Jy)+〈AJz0,Jy-Jz0〉+1r〈z0-x0,J(y-z0)〉+φ(Jy)-φ(Jz0)≥0.
Therefore, z0∈G(y), and so G(y) is closed, for all y∈C.(c)
We will show thatG(y) is weakly compact.
Now, we know thatG(y) is closed and subset of C.
SinceC is compact. Therefore, G(y) is compact, and then G(y) is weakly compact.
By using (a), (b), and (c) and Lemma22, we can conclude that ⋂y∈C‍G(y)≠∅.
Therefore, there existsz∈C such that
(49)F(Jz,Jy)+〈AJz,Jy-Jz〉+1r〈z-x,J(y-z)〉+φ(Jy)-φ(Jz)≥0,∀y∈C.Theorem 25.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex real Banach space E such that J(C) is closed and convex, let us assume that a bifunction F:J(C)×J(C)→ℝ satisfies the following conditions (DA1)–(DA4), let C* be a nonempty, closed, and convex subset of E*, let A:C*→E be an α-inverse strongly skew monotone and let φ:J(C)→ℝ be a convex and lower semicontinuous. Letr>0 be given real number and x∈E be any point. We define a mapping Sr:E→C as follows:
(50)Sr(x)={z∈C:F(Jz,Jy)+〈AJz,Jy-Jz〉+1r〈z-x,J(y-z)〉+φ(Jy)-φ(Jz)≥0,∀y∈C}.
Then, the following conclusion hold:(1)
Sr is single-valued;(2)
〈Srx-Sry,J(Srx-Sry)〉≤〈x-y,J(Srx-Sry)〉,
∀x,y∈E;(3)
Fix(Sr)=GMEP(F,A,φ);(4)
J(GMEP(F,A,φ)) is closed and convex;Proof.
We will complete this proof by the following four items.
(1)We will show thatSris single-valued.
Indeed, for anyx∈E and r>0, let z1,z2∈Sr(x). Then,
(51)F(Jz1,Jz2)+〈AJz1,Jz2-Jz1〉+1r〈z1-x,J(z2-z1)〉+φ(Jz2)-φ(Jz1)≥0,F(Jz2,Jz1)+〈AJz2,Jz1-Jz2〉+1r〈z2-x,J(z1-z2)〉+φ(Jz1)-φ(Jz2)≥0.
Adding the two inequalities, we have
(52)0≤F(Jz1,Jz2)+F(Jz2,Jz1)+〈AJz1,Jz2-Jz1〉+〈AJz2,Jz1-Jz2〉+1r〈z1-x,J(z2-z1)〉+1r〈z2-x,J(z1-z2)〉+φ(Jz2)-φ(Jz1)+φ(Jz1)-φ(Jz2)=F(Jz1,Jz2)+F(Jz2,Jz1)+〈AJz1,Jz2-Jz1〉-〈AJz2,Jz2-Jz1〉+1r〈z1-x,J(z2-z1)〉-1r〈z2-x,J(z2-z1)〉=F(Jz1,Jz2)+F(Jz2,Jz1)+〈AJz1-AJz2,Jz2-Jz1〉+1r〈(z1-x)-(z2-x),J(z2-z1)〉=F(Jz1,Jz2)+F(Jz2,Jz1)+〈AJz1-AJz2,Jz2-Jz1〉+1r〈z1-z2,J(z2-z1)〉=F(Jz1,Jz2)+F(Jz2,Jz1)-〈AJz1-AJz2,Jz1-Jz2〉+1r〈z1-z2,J(z2-z1)〉.
Therefore,
(53)F(Jz1,Jz2)+F(Jz2,Jz1)-〈AJz1-AJz2,Jz1-Jz2〉+1r〈z1-z2,J(z2-z1)〉≥0.
From the condition (DA2), A is an α-inverse strongly skew monotone, and we have
(54)0≤F(Jz1,Jz2)+F(Jz2,Jz1)-〈AJz1-AJz2,Jz1-Jz2〉+1r〈z1-z2,J(z2-z1)〉≤-α∥AJz1-AJz2∥2+1r〈z1-z2,J(z2-z1)〉≤1r〈z1-z2,J(z2-z1)〉.
Since r>0, J is monotone, E is strictly convex, and we obtain
(55)z1=z2.
This implies that Sr is single-valued.
(2)We will show that〈Srx-Sry,J(Srx-Sry)〉≤〈x-y,J(Srx-Sry)〉, for all x,y∈E.
Indeed, for anyx,y∈C and r>0, we have
(56)F(JSrx,JSry)+〈AJSrx,JSry-JSrx〉+1r〈Srx-x,J(Sry-Srx)〉+φ(JSry)-φ(JSrx)≥0,F(JSry,JSrx)+〈AJSry,JSrx-JSry〉+1r〈Sry-y,J(Srx-Sry)〉+φ(JSrx)-φ(JSry)≥0.
Adding the two inequalities, we have
(57)0≤F(JSrx,JSry)+F(JSry,JSrx)+〈AJSrx,JSry-JSrx〉+〈AJSry,JSrx-JSry〉+1r〈Srx-x,J(Sry-Srx)〉+1r〈Sry-y,J(Srx-Sry)〉+φ(JSry)-φ(JSrx)+φ(JSrx)-φ(JSry)=F(JSrx,JSry)+F(JSry,JSrx)+〈AJSrx,JSry-JSrx〉-〈AJSry,JSry-JSrx〉+1r〈Srx-x,J(Sry-Srx)〉-1r〈Sry-y,J(Sry-Srx)〉=F(JSrx,JSry)+F(JSry,JSrx)+〈AJSrx-AJSry,JSry-JSrx〉+1r〈(Srx-x)-(Sry-y),J(Sry-Srx)〉=F(JSrx,JSry)+F(JSry,JSrx)+〈AJSrx-AJSry,JSry-JSrx〉+1r〈(Srx-Sry)-(x-y),J(Sry-Srx)〉=F(JSrx,JSry)+F(JSry,JSrx)-〈AJSrx-AJSry,JSrx-JSry〉+1r〈(Srx-Sry)-(x-y),J(Sry-Srx)〉.
Therefore,
(58)F(JSrx,JSry)+F(JSry,JSrx)-〈AJSrx-AJSry,JSrx-JSry〉+1r〈(Srx-Sry)-(x-y),J(Sry-Srx)〉≥0.
From the condition (DA2), A is an α-inverse strongly skew monotone, and we have
(59)0≤F(JSrx,JSry)+F(JSry,JSrx)-〈AJSrx-AJSry,JSrx-JSry〉+1r〈(Srx-Sry)-(x-y),J(Sry-Srx)〉≤-α∥AJSrx-AJSry∥2+1r〈(Srx-Sry)-(x-y),J(Sry-Srx)〉≤1r〈(Srx-Sry)-(x-y),J(Sry-Srx)〉≤-1r〈(Srx-Sry)-(x-y),J(Srx-Sry)〉.
Since r>0, we have
(60)〈(Srx-Sry)-(x-y),J(Srx-Sry)〉≤0.
Therefore,
(61)〈Srx-Sry,J(Srx-Sry)〉-〈x-y,J(Srx-Sry)〉≤0.
This implies that
(62)〈Srx-Sry,J(Srx-Sry)〉≤〈x-y,J(Srx-Sry)〉.
(3)We will show thatFix(Sr)=GMEP(F,A,φ).
It is easy to see that(63)z∈Fix(Sr)⟺z=Srz⟺F(Jz,Jy)+〈AJz,Jy-Jz〉+1r〈z-z,J(y-z)〉+φ(Jy)-φ(Jz)≥0⟺F(Jz,Jy)+〈AJz,Jy-Jz〉+φ(Jy)-φ(Jz)≥0⟺z∈GMEP(F,A,φ).
This implies that Fix(Sr)=GMEP(F,A,φ).
(4)We will show thatJ(GMEP(F,A,φ))is closed and convex.
For eachy∈C, we define the mappingH:C→2Eas follows:
(64)H(y)={z∈C:F(Jz,Jy)+〈AJz,Jy-Jz〉+φ(Jy)-φ(Jz)≥0}.
It is easy to see that y∈H(y), so that H(y)≠∅.
Next, we will show thatH is a KKM mapping.
Suppose that there exists a finite subset{z1,z2,…,zm} of C and βi>0 with ∑i=1m‍βi=1 such that z^=∑i=1m‍βizi∉H(zi), for all i=1,2,3,…,m. Then,
(65)F(Jz^,Jzi)+〈AJz^,Jzi-Jz^〉+φ(Jzi)-φ(Jz^)<0,i=1,2,3,…,m.
It follows from (DA1) and (DA4) that
(66)0=F(Jz^,Jz^)+〈AJz^,Jz^-Jz^〉+φ(Jz^)-φ(Jz^)≤∑i=1m‍βi(F(Jz^,Jzi)+〈AJz^,Jzi-Jz^〉+φ(Jzi)-φ(Jz^))<0.
which is the contradiction. Hence, H is a KKM mapping on C.
(4.1) Next, we will show thatH(y) is closed, for each y∈C.
For anyy∈C, let {zn} be any sequence in H(y) such that zn→z0 as n→∞.
Hence,zn-x0→z-x0 as n→∞. Next, we will show that z0∈H(y). Then, for each y∈C, we have
(67)F(Jzn,Jy)+〈AJzn,Jy-Jzn〉+φ(Jy)-φ(Jzn)≥0.
It follows from the assumption (DA3) that(68)F(Jz0,Jy)+〈AJz0,Jy-Jz0〉+φ(Jy)-φ(Jz0)≥limsupn→∞F(Jzn,Jy)+limn→∞(〈AJzn,Jy-Jzn〉+φ(Jy)-φ(Jzn))≥0.
This implies thatz0∈H(y), and hence H(y) is closed, for each y∈C.
Therefore,⋂y∈C‍H(y)=J(GMEP(F,A,φ)) is closed.
(4.2) Next, we will show thatJ(GMEP(F,A,φ)) is convex.
Letz1*,z2*∈J(GMEP(F,A,φ)); then, we have z1*=Jz1∈J(C) and z2*=Jz2∈J(C), where z1,z2∈C.
Fork,t∈(0,1), let z*=kz1*+(1-k)z2*, and for any y∈C, we set xt*=tJy+(1-t)z*.
It follows from (DA1) and (DA4) that(69)0=F(xt*,xt*)+φ(xt*)-φ(xt*)≤F(xt*,tJy+(1-t)z*)+φ(tJy+(1-t)z*)-φ(xt*)≤tF(xt*,Jy)+(1-t)F(xt*,z*)+tφ(xt*)+(1-t)φ(Jy)-φ(xt*)≤tF(xt*,Jy)+(1-t)φ(Jy)-(1-t)φ(xt*)≤F(xt*,Jy)+φ(Jy)-φ(xt*),(70)0=〈Axt*,xt*-xt*〉=〈Axt*,(xt*-Jy)+(Jy-xt*)〉=〈Axt*,xt*-Jy〉+〈Axt*,Jy-xt*〉=〈Axt*,xt*-Jy〉-〈Axt*,xt*-Jy〉≤〈Axt*,xt*-Jy〉=〈Axt*,tJy+(1-t)z*-Jy〉=〈Axt*,tJy+(1-t)z*-(t+(1-t))Jy〉=〈Axt*,(1-t)(z*-Jy)〉=(1-t)〈Axt*,z*-Jy〉=(t-1)〈Axt*,Jy-z*〉≤〈Axt*,Jy-z*〉.
Adding two inequalities (69) and (70), we get
(71)0≤F(xt*,Jy)+〈Axt*,Jy-z*〉+φ(Jy)-φ(xt*).
Letting t↓0. It follows from (DA3) and the hemicontinuous of A that
(72)F(z*,Jy)+〈Az*,Jy-z*〉+φ(Jy)-φ(z*)≥0,∀y∈C.
Hence, z*∈J(GMEP(F,A,φ)), and thus J(GMEP(F,A,φ)) is convex.
This completes the proof.


## 4. Iterative Algorithm for Strong Convergence Theorem

In this section, we modify Halpern-Mann iteration to find the common solution of a generalized mixed equilibrium problem and a fixed point problem in Banach spaces.Theorem 26.
LetE be a uniformly smooth and strictly convex real Banach space, let C be a nonempty, closed and convex subset of E such that J(C) is closed, and convex of E*, let us assume that a bifunction F:J(C)×J(C)→ℝ satisfies the following conditions (DA1)–(DA4), let C* be a nonempty, closed and convex subset of E*, let A:C*→E be an α-inverse strongly skew monotone, let φ:J(C)→ℝ be a convex, and lower semicontinuous, and let T:C→C be a closed and -ϕ-nonexpansive mapping. Assume that Ω:=GMEP(F,A,φ)∩Fix(T) is nonempty. Let {xn} be a sequence generated by
(73)x0∈Cchosenarbitrary,un∈CsuchthatF(Jun,Jy)+〈AJun,Jy-Jun〉+1rn〈un-xn,J(y-un)〉+φ(Jy)-φ(Jun)≥0,∀y∈C,xn+1=αnx0+(1-αn)(βnxn+(1-βn)Tun),∀n∈ℕ,
where J is the duality mapping on E and {αn} is a sequence in [0,1], {βn}⊂[a,b], for some 0<a<b<1 and {rn}⊂[c,∞), for some c>0 such that
(74)∑n=0∞‍αn<∞,∑n=0∞‍βn<∞,liminfn→∞rn>0.
Then, the sequence {RΩ(xn)} converges strongly to some point in Ω, where RΩ is the sunny generalized nonexpansive retraction from E onto Ω.Proof.
We will complete this proof by the following three steps.
Step
1.We will show that the sequences{xn}and{un}are bounded.
Letun=Srnxn and yn=βnxn+(1-βn)Tun, for any n≥1. Then,
(75)xn+1=αnx0+(1-αn)yn.
From Theorem 25 and Lemma 23, we know that GMEP(F,A,φ) and Fix(T) are closed, and convex subset of E. Therefore, Ω is nonempty, closed and convex subset of E.
For anyp∈Ω, T is a closed and -ϕ-nonexpansive mapping, we compute
(76)ϕ(yn,p)≤ϕ(βnxn+(1-βn)Tun,p)=∥βnxn+(1-βn)Tun∥2-2〈βnxn+(1-βn)Tun,Jp〉+∥p∥2=∥βnxn+(1-βn)Tun∥2-2βn〈xn,Jp〉-2(1-βn)〈Tun,Jp〉+(βn+(1-βn))∥p∥2≤βn∥xn∥2+(1-βn)∥Tun∥2-2βn〈xn,Jp〉-2(1-βn)〈Tun,Jp〉+(βn+(1-βn))∥p∥2=βn(∥xn∥2-2〈xn,Jp〉+∥p∥2)+(1-βn)(∥Tun∥2-2〈Tun,Jp〉+∥p∥2)=βnϕ(xn,p)+(1-βn)ϕ(Tun,p)≤βnϕ(xn,p)+(1-βn)ϕ(un,p)=βnϕ(xn,p)+(1-βn)ϕ(Srnxn,p)≤βnϕ(xn,p)+(1-βn)ϕ(xn,p)=ϕ(xn,p),
and we have
(77)ϕ(xn+1,p)≤ϕ(αnx0+(1-αn)yn,p)=∥αnx0+(1-αn)yn∥2-2〈αnx0+(1-αn)yn,Jp〉+∥p∥2=∥αnx0+(1-αn)yn∥2-2αn〈x0,Jp〉-2(1-αn)〈yn,Jp〉+(αn+(1-αn))∥p∥2≤αn∥x0∥2+(1-α)∥yn∥2-2αn〈x0,Jp〉-2(1-αn)〈yn,Jp〉+(αn+(1-αn))∥p∥2=αn(∥x0∥2-2〈x0,Jp〉+∥p∥2)+(1-αn)(∥yn∥2-2〈yn,Jp〉+∥p∥2)=αnϕ(x0,p)+(1-αn)ϕ(yn,p)≤αnϕ(x0,p)+(1-αn)ϕ(xn,p)≤αnϕ(x0,p)+ϕ(xn,p).
By virtue of ∑n=0∞‍αn<∞, it follows from Lemma 16 that limn→∞ϕ(xn,p) exists.
Therefore,{ϕ(xn,p)} is bounded, and so {xn} is bounded. Hence, {un} and {yn} are also bounded.
Step
2.We will show thatRΩ(xn)is bounded.
Letzn=RΩ(xn)andp∈Ω. Then, zn∈Ω.
It follows from Lemma12(2) that
(78)ϕ(xn,zn)=ϕ(xn,RΩ(xn))≤ϕ(xn,p)-ϕ(RΩ(xn),p)≤ϕ(xn,p).
Since {xn} is bounded. Therefore, {zn} is bounded.
Hence,RΩ(xn) is bounded.
Step
3.We will show that{RΩ(xn)}converges strongly to some point inΩ.
From (78), we haveϕ(xn,zn)≤ϕ(xn,p). Replacing xn by x0, we get ϕ(x0,zn)≤ϕ(x0,p).
Therefore,{ϕ(x0,zn)} is bounded.
Now, we know thatϕ(xn+1,zn)≤αnϕ(x0,zn)+ϕ(xn,zn). By Lemma 12(2), we get
(79)ϕ(xn+1,zn+1)=ϕ(xn+1,RΩ(xn+1))≤ϕ(xn+1,zn)-ϕ(RΩ(xn+1),zn)≤ϕ(xn+1,zn)≤αnϕ(x0,zn)+ϕ(xn,zn).
Since{ϕ(x0,zn)} is bounded. There exists M>0 such that |ϕ(x0,zn)|≤M.
By the assumption∑n=0∞‍αn<∞, we have
(80)∑n=0∞‍ϕ(x0,zn)≤M∑n=0∞‍αn<∞.
It follows from Lemma 16 that limn→∞ϕ(xn,zn) exists.
For anym∈ℕ, we get
(81)ϕ(xn+m,p)≤ϕ(xn,p)+∑j=0m-1‍αn+jϕ(x0,p),ϕ(xn+m,zn)≤ϕ(xn,zn)+∑j=0m-1‍αn+jϕ(x0,zn).
Since zn+m=RΩ(xn+m) and from Lemma 12(2), we have
(82)ϕ(xn+m,zn+m)+ϕ(zn+m,zn)≤ϕ(xn+m,zn)≤ϕ(xn,zn)+∑j=0m-1‍αn+jϕ(x0,p).
Hence,
(83)ϕ(zn+m,zn)≤ϕ(xn,zn)-ϕ(xn+m,zn+m)+∑j=0m-1‍αn+jϕ(x0,p).
We set r=sup{∥zn∥:n∈ℕ}. From Lemma 20, it follows that there exists a strictly increasing, continuous, and convex function h:[0,2r]→ℝ such that h(0)=0 and
(84)h(∥zn-zn+m∥)≤ϕ(zn+m,zn)≤ϕ(xn,zn)-ϕ(xn+m,zn+m)+∑j=0m-1‍αn+jϕ(x0,zn).
Since {ϕ(xn,zn)} is convergent sequence, {ϕ(x0,zn)} is bounded, and ∑n=0∞‍αn<∞, then, we obtain
(85)limn→∞∥zn-zn+m∥=0,∀m∈ℕ.
This implies that {zn} is a Cauchy sequence.
Note thatΩ is closed.
Thus, there existsp∈Ω such that zn→p.
Therefore, the sequenceRΩ(xn) converges strongly to some point p∈Ω.
This completes the proof.If we setA≡0 in Theorem 26, then Theorem 26 reduces to the following corollary which extends and improves the following result of Saewan et al. [14].Corollary 27.
LetE be a uniformly smooth and strictly convex real Banach space, let C be a nonempty, closed, and convex subset of E such that J(C) is closed and convex of E*, let us assume that a bifunction F:J(C)×J(C)→ℝ satisfies the following conditions (DA1)–(DA4), let C* be a nonempty, closed, and convex subset of E*, let φ:J(C)→ℝ be a convex and lower semicontinuous, and let T:C→C be a closed and -ϕ-nonexpansive mapping. Assume that Ω:=MEP(F,φ)∩Fix(T) is nonempty. Let {xn} be a sequence generated by
(86)x0∈Cchosenarbitrary,un∈CsuchthatF(Jun,Jy)+1rn〈un-xn,J(y-un)〉+φ(Jy)-φ(Jun)≥0,∀y∈C,xn+1=αnx0+(1-αn)(βnxn+(1-βn)Tun),∀n∈ℕ,
where J is the duality mapping on E and {αn} is a sequence in [0,1], {βn}⊂[a,b], for some 0<a<b<1 and {rn}⊂[c,∞), for some c>0 such that
(87)∑n=0∞‍αn<∞,∑n=0∞‍βn<∞,liminfn→∞rn>0.
Then, the sequence {RΩ(xn)} converges strongly to some point in Ω, where RΩ is the sunny generalized nonexpansive retraction from E onto Ω.If we setA≡0 and φ≡0 in Theorem 26, then Theorem 26 reduces to the following corollary which extends and improves the following result of Chen et al. [13].Corollary 28.
LetE be a uniformly smooth and strictly convex real Banach space, let C be a nonempty, closed, and convex subset of E such that J(C) is closed and convex of E*, let us assume that a bifunction F:J(C)×J(C)→ℝ satisfies the following conditions (DA1)–(DA4), let C* be a nonempty, closed, and convex subset of E*, and let T:C→C be a closed and -ϕ-nonexpansive mapping. Assume that Ω:=EP(F)∩Fix(T) is nonempty. Let {xn} be a sequence generated by
(88)x0∈Cchosenarbitrary,un∈CsuchthatF(Jun,Jy)+1rn〈un-xn,J(y-un)〉≥0,∀y∈C,xn+1=αnx0+(1-αn)(βnxn+(1-βn)Tun),∀n∈ℕ,
where J is the duality mapping on E and {αn} is a sequence in [0,1], {βn}⊂[a,b], for some 0<a<b<1 and {rn}⊂[c,∞), for some c>0 such that
(89)∑n=0∞‍αn<∞,∑n=0∞‍βn<∞,liminfn→∞rn>0.
Then, the sequence {RΩ(xn)} converges strongly to some point in Ω, where RΩ is the sunny generalized nonexpansive retraction from E onto Ω.If we setA≡0, φ≡0, and T≡I (identity mapping) in Theorem 26, then Theorem 26 reduces to the following corollary which extends and improves the following result of Chen et al. [13].Corollary 29.
LetE be a uniformly smooth and strictly convex real Banach space, let C be a nonempty, closed, and convex subset of E such that J(C) is closed and convex of E*, let us assume that a bifunction F:J(C)×J(C)→ℝ satisfies the following conditions (DA1)–(DA4), and let C* be a nonempty, closed, and convex subset of E*. Assume that EP(F) is nonempty. Let {xn} be a sequence generated by
(90)x0∈Cchosenarbitrary,un∈CsuchthatF(Jun,Jy)+1rn〈un-xn,J(y-un)〉≥0,∀y∈C,xn+1=αnx0+(1-αn)(βnxn+(1-βn)un),∀n∈ℕ,
where J is the duality mapping on E and {αn} is a sequence in [0,1], {βn}⊂[a,b], for some 0<a<b<1 and {rn}⊂[c,∞), for some c>0 such that
(91)∑n=0∞‍αn<∞,∑n=0∞‍βn<∞,liminfn→∞rn>0.
Then, the sequence {REP(F)(xn)} converges strongly to some point in Ω, where REP(F) is the sunny generalized nonexpansive retraction from E onto EP(F).



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*Source: 753096-2013-10-30.xml*