# New Generalized Mixed Equilibrium Problem with Respect to Relaxed Semi-Monotone Mappings in Banach Spaces

Authors: Rabian Wangkeeree; Pakkapon Preechasilp
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We introduce the new generalized mixed equilibrium problem with respect to relaxed semimonotone
mappings. Using the KKM technique, we obtain the existence of solutions for the generalized mixed equilibrium
problem in Banach spaces. Furthermore, we also introduce a hybrid projection algorithm for finding a common element
in the solution set of a generalized mixed equilibrium problem and the fixed point set of an asymptotically nonexpansive
mapping. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main
results extend various results existing in the current literature.


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## Body

## 1. Introduction

LetE be a Banach space with the dual E* and let E** denote the dual space of E*. If E=E**, then E is called reflexive. We denote by 𝒩 and ℛ the sets of positive integers and 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 if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets of E. We shall still denote by J the single-valued duality mapping.LetC be a nonempty subset of E**,η:C×C→E** be a mapping and let ξ:E**→ℛ a function with ξ(tz)=tpξ(z) for all t>0 and z∈E**, where p>1 is a constant. A mapping A:C×C→E* is said to be relaxedη-ξsemimonotone[1] if the following two conditions hold:(i)
for each fixedu∈C,A(u,·) is relaxed η-ξ monotone; that is,(1.2)〈A(u,v)-A(u,w),η(v,w)〉≥ξ(v-w),∀v,w∈C;(ii)
for each fixedv∈C, A(·,v) is completely continuous; that is, for any net {uj} in C, uj→u0 in weak* topology of E**, then {A(uj,v)} has a subsequence {A(ujk,v)}→A(u0,v) in norm topology of E*.In caseη(x,y)=x-y for all x,y∈C and ξ≡0, A is called semi-monotone[2]. The following is an example of η-ξ semi-monotone mapping.Example 1.1.
LetC=(-∞,∞),A(x,y)=x+y, and
(1.3)η(x,y)={-c(x-y),x≥y,c(x-y),x<y,
where c>0 is a constant. Then, A is relaxed η-ξ semi-monotone with
(1.4)ξ(z)={-cz2,z≥0,cz2,z<0.
Letf:C×C→ℛ be a bifunction, η:C×C→E** a mapping, and ξ:E**→ℛ, φ:C→ℛ two real-valued functions, and let A:C×C→E* be a η-ξ semi-monotone mapping. We consider the problem of finding u∈C such that(1.5)f(u,v)+〈A(u,u),η(v,u)〉+φ(v)≥φ(u),∀v∈C,
which is called the generalized mixed equilibrium problem with respect to relaxedη-ξsemi-monotone mapping(GMEP(f,A,η,φ)). The set of such u∈C is denoted by GMEP(f,A,η,φ), that is,
(1.6)GMEP⁡(f,A,η,φ)={u∈C:f(u,v)+〈A(u,u),η(v,u)〉+φ(v)≥φ(u),∀v∈C}.Now, let us consider some special cases of the problem (1.5).(a)
In the case off≡0, (1.5) is deduced to the following variational-like inequality problem:(1.7)findu∈Csuchthat〈A(u,u),η(v,u)〉+φ(v)-φ(u)≥0,∀v∈C.
The problem (1.7) was studied by Fang and Huang [1]. Using the KKM technique and η-ξ monotonicity of the mapping φ, they [1] obtained the existence of solutions of the variational-like inequality problem (1.7) in a real Banach space.(b)
In the case off≡0,φ≡0 and η(v,u)=v-u for all v,u∈C, the problem (1.5) is deduced to the following variational inequality problem:(1.8)Findu∈Csuchthat〈A(u,u),v-u〉≥0,∀v∈C.
The problem (1.8) was studied by Chen [2]. They obtained the existence results of solutions in a real Banach space.WhenE is a reflexive Banach space, we know E**=j(E), where j:E→E** is the duality mapping defined by 〈jx,f〉=〈f,x〉, for all x∈E,f∈E*, which is an isometric mapping, so we may regard E=E** under an isometry. The following problems can be derived as special cases of the problem (1.5).(c)
In caseE is reflexive (i.e., E=E**), f≡0 and η(v,u)=v-u for all v,u∈C, the problem (1.5) is deduced to the following variational inequality problem:(1.9)findu∈Csuchthat〈A(u,u),v-u〉+φ(v)-φ(u)≥0,∀v∈C.
The problem (1.9) was studied by Chen [2].(d)
IfE is reflexive (i.e., E=E**) and A≡0, (1.5) is deduced to the mixed equilibrium problem:(1.10)findu∈Csuchthatf(u,v)+φ(v)≥φ(u),∀v∈C.
The problem (1.10) was considered and studied by Ceng and Yao [3]; Cholamjiak and Suantai [4].(e)
In the case ofA≡0 and φ≡0, (1.5) is deduced to the following classical equilibrium problem:(1.11)findu∈Csuchthatf(u,v)≥0,∀v∈C.
The set of all solution of (1.11) is denoted by EP(f), that is,(1.12)EP⁡(f)={u∈C:f(u,v)≥0,∀v∈C}.
Numerous problems in physics, optimization, and economics can be reduced to find a solution of the equilibrium problem, variational inequality problem, and related optimization problems; see, for instance, [5–11]. Some methods have been proposed to solve the equilibrium problem in a Hilbert space; see, for instance, Blum and Oettli [12]; Combettes and Hirstoaga [13]; Moudafi [14].LetC be a nonempty, closed convex subset of E. A mapping S:C→E is called nonexpansive if ∥Sx-Sy∥≤∥x-y∥ for all x,y∈C. Also a mapping S:C→C is called asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 as n→∞ such that ∥Snx-Sny∥≤kn∥x-y∥ for all x,y∈C and for each n≥1. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [15] as an important generalization of nonexpansive mappings. Denote by F(S) the set of fixed points of S, that is, F(S)={x∈C:Sx=x}. There are several methods for approximating fixed points of a nonexpansive mapping; see, for instance, [16–21]. Furthermore, since 1972, a host of authors have studied weak and strong convergence problems of the iterative processes for the class of asymptotically nonexpansive mappings; see, for instance, [22–25]. In 1953, Mann [16] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping S in a Hilbert space H:(1.13)xn+1=αnxn+(1-αn)Sxn,∀n∈N,
where the initial point x0 is taken in C arbitrarily and {αn} is a sequence in [0,1]. However, we note that Mann's iteration process (1.13) has only weak convergence, in general; for instance, see [26–28]. In 2003, Nakajo and Takahashi [29] introduced the following iterative algorithm for the nonexpansive mapping S in the framework of Hilbert spaces:(1.14)x0=x∈C,yn=αnxn+(1-αn)Sxn,Cn={z∈C:‖z-yn‖≤‖z-xn‖},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,n≥0,
where {αn}⊂[0,α],α∈[0,1], and PCn∩Qn is the metric projection from a Hilbert space H onto Cn∩Qn. They proved that {xn} generated by (1.14) converges strongly to a fixed point of S. Xu [30] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection.Matsushita and Takahashi [17] introduced the following iterative algorithm in the framework of Banach spaces:(1.15)x0=x∈C,Cn=co⁡¯{z∈C:‖z-Sz‖≤tn‖xn-Sxn‖},Dn={z∈C:〈xn-z,J(x-xn)〉≥0},xn+1=PCn∩Dnx,n≥0,
where co¯D denoted the convex closure of the set D,{tn} is a sequence in (0,1) with tn→0, and PCn∩Dn is the metric projection from E onto Cn∩Dn.Very recently, Dehghan [24] introduced the following iterative algorithm for finding fixed points of an asymptotically nonexpansive mapping S in a uniformly convex and smooth Banach space:(1.16)x0=x∈C,C0=D0=C,Cn=co⁡¯{z∈Cn-1:‖z-Snz‖≤tn‖xn-Snxn‖},Dn={z∈Dn-1:〈xn-z,J(x-xn)〉≥0},xn+1=PCn∩Dnx,n≥0,
where co¯D denotes the convex closure of the set D,J is the normalized duality mapping, {tn} is a sequence in (0,1) with tn→0, and PCn∩Dn is the metric projection from E onto Cn∩Dn. The strong convergence theorem of the iterative sequence {xn} defined by (1.16) is obtained in a uniformly convex and smooth Banach space.In this paper, motivated and inspired by the above results, we first suggest and analyze the new generalized mixed equilibrium problem with respect to relaxedη-ξ semi-monotone mapping. Using the KKM technique, we obtain the existence of solutions for such problem in a Banach space. Next, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the fixed point set of an asymptotically nonexpansive mapping. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main results extend various results existing in the current literature.


## 2. Preliminaries

LetE be a real Banach space, and let U={x∈E:∥x∥=1} be the unit sphere of E. A Banach space E is said to be strictly convex if for any x,y∈U,(2.1)x≠yimplies‖x+y‖<2.It is also said to beuniformly convex if for each ε∈(0,2], there exists δ>0 such that for any x,y∈U,(2.2)‖x-y‖≥εimplies‖x+y‖<2(1-δ).It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a functionδ:[0,2]→[0,1] called the modulus of convexity of E as follows:(2.3)δ(ε)=inf⁡{1-‖x+y2‖:x,y∈E,‖x‖=‖y‖=1,‖x-y‖≥ε}.ThenE is uniformly convex if and only if δ(ε)>0 for all ε∈(0,2]. A Banach space E is said to be smooth if the limit(2.4)lim⁡t→0‖x+ty‖-‖x‖t
exists for all x,y∈U. Let C be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space E. Then for any x∈E, there exists a unique point x0∈C such that(2.5)‖x0-x‖≤min⁡y∈C‖y-x‖.The mappingPC:E→C defined by PCx=x0 is called the metric projection from E onto C. The following theorem is wellknown.Theorem 2.1 (see [31]).
LetC be a nonempty, closed convex subset of a smooth Banach space E and let x∈E, and y∈C. Then the following are equivalent:(a)
y is a best approximation to x:y=PCx.(b)
y is a solution of the variational inequality:(2.6)〈y-z,J(x-y)〉≥0,∀z∈C,
where J is a duality mapping and PC is the metric projection from E onto C.It is wellknown that ifPC is a metric projection from a real Hilbert space H onto a nonempty, closed, and convex subset C, then PC is nonexpansive. But, in a general Banach space, this fact is not true.In the sequel, we will need the following lemmas.Lemma 2.2 (see [32]).
LetE be a uniformly convex Banach space, let {αn} be a sequence of real numbers such that 0<b≤αn≤c<1 for all n≥1, and let {xn} and {yn} be sequences in E such that limsupn→∞∥xn∥≤d,limsupn→∞∥yn∥≤d, and limn→∞∥αnxn+(1-αn)yn∥=d. Then limn→∞∥xn-yn∥=0.Theorem 2.3 (see [33]).
LetC be a bounded, closed, and convex subset of a uniformly convex Banach space E. Then there exists a strictly increasing, convex, and continuous function γ:[0,∞)→[0,∞) such that γ(0)=0 and
(2.7)γ(‖S(∑i=1nλixi)-∑i=1nλiSxi‖)≤max⁡1≤j≤k≤n(‖xj-xk‖-‖Sxj-Sxk‖),
for all n∈𝒩, {x1,x2,…,xn}⊂C, {λ1,λ2,…,λn}⊂[0,1] with ∑i=1nλi=1 and nonexpansive mapping S of C into E.Theorem 2.4 (see [24]).
LetC be a bounded, closed, and convex subset of a uniformly convex Banach space E. Then there exists a strictly increasing, convex, and continuous function γ:[0,∞)→[0,∞) such that γ(0)=0 and
(2.8)γ(1km‖Sm(∑i=1nλixi)-∑i=1nλiSmxi‖)≤max⁡1≤j≤k≤n(‖xj-xk‖-1km‖Smxj-Smxk‖),
for all n∈𝒩, {x1,x2,…,xn}⊂C; {λ1,λ2,…,λn}⊂[0,1] with ∑i=1nλi=1 and an asymptotically nonexpansive mapping S of C into E with the sequence {km}.Now, let us recall the following well-known concepts and results.Definition 2.5.
LetB be a subset of topological vector space X. A mapping G:B→2X is called a KKM mapping if co{x1,x2,…,xm}⊂⋃i=1mG(xi) for xi∈B and i=1,2,…,m, where coA denotes the convex hull of the set A.Lemma 2.6 (see [34]).
LetB be a nonempty 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)≠∅.Theorem 2.7 (see [35] (Kakutani-Fan-Glicksberg Fixed Point Theorem)).
Let E be a locally convex Hausdorff topological vector space andC a nonempty, convex, and compact subset of E. Suppose T:C→2C is a upper semi-continuous mapping with nonempty, closed, and convex values. Then T has a fixed point in C.Definition 2.8 (see [36]).
LetC be a nonempty, closed convex of a Banach space E. Let T:C→E* and let η:C×C→ℛ be two mappings. T is said to be η-hemicontinuousif, for any fixed x,y∈C, the mapping f:[0,1]→(-∞,∞) defined by f(t)=〈T(x+t(y-x)),η(y,x)〉 is continuous at 0+.
For solving the mixed equilibrium problem, let us assume the following conditions for a bifunctionf:C×C→ℛ:
(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)
for ally∈C, f(·,y) is weakly upper semicontinuous;(A4)
for allx∈C, f(x,·) is convex. The following lemmas can be found in [37].Lemma 2.9 (see [37]).
LetC be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex and reflexive Banach space E, let T:C→E* be an η-hemicontinuous and relaxed η-ξ monotone mapping. Let f be a bifunction from C×C to ℛ satisfying (A1) and (A4), and let φ be a lower semicontinuous and convex function from C to ℛ. Let r>0 and z∈C. Assume that(i)
η(x,x)=0,forallx∈C;(ii)
for any fixedu,v∈C, the mapping x↦〈Tv,η(x,u)〉 is convex. Then the following problems (2.9) and (2.10) are equivalent. Find x∈C such that:
(2.9)f(x,y)+φ(y)+〈Tx,η(y,x)〉+1r〈y-x,J(x-z)〉≥φ(x),∀y∈C.
Find x∈C such that
(2.10)f(x,y)+〈Ty,η(y,x)〉+φ(y)+1r〈y-x,J(x-z)〉≥φ(x)+ξ(y-x),∀y∈C.Lemma 2.10 (see [37]).
LetC be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E, let T:C→E* be an η-hemicontinuous and relaxed η-ξ monotone mapping. Let f be a bifunction from C×C to ℛ satisfying (A1), (A3), and (A4), and let φ be a lower semicontinuous and convex function from C to ℛ. Let r>0 and z∈C. Assume that(i)
η(x,y)+η(y,x)=0 for all x,y∈C;(ii)
for any fixedu,v∈C, the mapping x↦〈Tv,η(x,u)〉 is convex and lower semicontinuous;(iii)
ξ:E→ℛ is weakly lower semicontinuous; that is,  for any net {xβ},{xβ} converges to x in σ(E,E*) implies that ξ(x)≤liminfξ(xβ). Then, the solution set of the problem (2.9) is nonempty, that is, there exists x0∈C such that
(2.11)f(x0,y)+〈Tx0,η(y,x0)〉+φ(y)+1r〈y-x0,J(x0-z)〉≥φ(x0),∀y∈C.


## 3. Existence Results of Generalized Mixed Equilibrium Problem

In this section, we prove the following crucial lemma concerning the generalized mixed equilibrium problem with respect to relaxedη-ξ semi-monotone mapping (GMEP(f,A,η,φ)) in a real Banach space with the smooth and strictly convex second dual space.Lemma 3.1.
LetE be a real Banach space with the smooth and strictly convex second dual space E**, let C be a nonempty bounded closed convex subset of E**, let A:C×C→E* be a relaxed η-ξ semi-monotone mapping. Let f:C×C→ℛ be a bifunction satisfying (A1), (A3), and (A4), and let φ:C→ℛ∪{+∞} be a proper lower semicontinuous and convex function. Let r>0 and z∈C. Assume that(i)
η(x,y)+η(y,x)=0 for all x,y∈C;(ii)
for any fixedu,v,w∈C, the mapping x↦〈A(v,w),η(x,u)〉 is convex and lower semicontinuous;(iii)
for eachx∈C,A(x,·):C→E* is finite-dimensional continuous: that is, for any finite-dimensional subspace F⊂E**,A(x,·):C∩F→E* is continuous;(iv)
ξ:E**→ℛ is convex lower semicontinuous. Then there exists u0∈C such that
(3.1)f(u0,v)+〈A(u0,u0),η(v,u0)〉+φ(v)+1r〈v-u0,J(u0-z)〉≥φ(u0),∀v∈C.Proof.
LetF⊆E** be a finite-dimensional subspace with CF∶=F∩C≠∅. For each w∈C, consider the following problem: find u0∈CF such that
(3.2)f(u0,v)+〈A(w,u0),η(v,u0)〉+φ(v)+1r〈v-u0,J(u0-z)〉-φ(u0)≥0,∀v∈CF.
SinceCF⊆F is bounded closed and convex, A(w,·) is continuous on CF and relaxed η-ξ monotone for each fixed w∈C, from Lemma 2.10, we know that problem (3.2) has a solution u0∈CF.
Now, define a set-valued mappingG:CF→2CF as follows:
(3.3)Gw={u∈CF:f(u,v)+〈A(w,u),η(v,u)〉+φ(v)+1r〈v-u,J(u-z)〉-φ(u)≥0,∀v∈CF}.
It follows from Lemma 2.9 that, for each fixed w∈CF:
(3.4){u∈CF:f(u,v)+〈A(w,u),η(v,u)〉+φ(v)+1r〈v-u,J(u-z)〉-φ(u)≥0,∀v∈KF}={u∈CF:f(u,v)+〈A(w,v),η(v,u)〉+φ(v)+1r〈v-u,J(u-z)〉-φ(u)≥ξ(v-u),∀v∈KF}.
Since every convex lower semicontinuous function in Banach spaces is weakly lower semicontinuous, the proper convex lower semicontinuity of φ and ξ, condition (ii), (A3) and (A4) implies that G:CF→2CF has nonempty bounded closed and convex values. Using (A3) and the complete continuity of A(·,u), we can conclude that G is upper semicontinuous. It follows from Theorem 2.7 that G has a fixed point w*∈CF, that is,
(3.5)〈f(w*,v)+〈A(w*,w*),η(v,w*)〉+φ(v)+1r〈v-w*,J(w*-z)〉-φ(w*)≥0,∀v∈CF.
Let
(3.6)F={F⊂E**:FisfinitedimensionalwithF∩C≠∅},
and let
(3.7)WF={1ru∈C:f(u,v)+〈A(u,v),η(v,u)〉+φ(v)+1r〈v-u,J(u-z)〉-φ(u)≥ξ(v-u),∀v∈CF},∀F∈F.
By (3.5) and Lemma 2.9, we know that WF is nonempty and bounded. Denote by W¯F* the weak*-closure of WF in E**. Then, W¯F* is weak* compact in E**.
For anyFi∈ℱ, i=1,2,…,N, we know that W⋂i=1NFi⊂⋂i=1NWFi, so {W¯F*:F∈ℱ} has the finite intersection property. Therefore, it follows that(3.8)⋂F∈FW¯F*≠∅.
Let u0∈⋂F∈ℱW¯F*. We claim that
(3.9)f(u0,v)+〈A(u0,u0),η(v,u0)〉+φ(v)+1r〈v-u0,J(u0-z)〉-φ(u0)≥0,∀v∈C.
Indeed, for each v∈C, let F∈ℱ be such that v∈CF and u0∈CF. Then, there exists uj∈WF such that uj⇀u0. The definition of WF implies that
(3.10)f(uj,v)+〈A(uj,v),η(v,uj)〉+φ(v)+1r〈v-uj,J(uj-z)〉-φ(uj)≥ξ(v-uj),
that is
(3.11)f(uj,v)+〈A(uj,v),η(v,uj)〉+φ(v)+1r〈v-z,J(uj-z)〉-1r‖z-uj‖2-φ(uj)≥ξ(v-uj),
for all j=1,2,…. Using the complete continuity of A(·,u), (A3), (ii), the continuity of J, the convex lower semicontinuity of φ, ξ, and ∥·∥2, and letting j→∞, we get
(3.12)f(u0,v)+〈A(u0,v),η(v,u0)〉+φ(v)+1r〈v-u0,J(u0-z)〉-φ(u0)≥ξ(v-u0),∀v∈C.
From Lemma 2.9, we have
(3.13)f(u0,v)+〈A(u0,u0),η(v,u0)〉+φ(v)+1r〈v-u0,J(u0-z)〉-φ(u0)≥0,∀v∈C.
Hence, we complete the proof.SettingA≡0 and φ≡0 in Lemma 3.1, we have the following result.Corollary 3.2.
LetE be a real Banach space with the smooth and strictly convex second dual space E**, let C be a nonempty bounded closed convex subset of E**. Let f:C×C→ℛ be a bifunction satisfying (A1), (A3), and (A4). Let r>0 and z∈C. Then there exists u0∈C such that
(3.14)f(u0,v)+1r〈v-u0,J(u0-z)〉≥0,∀v∈C.IfE is reflexive (i.e., E=E**) smooth and strictly convex real Banach space, then we have the following result.Corollary 3.3.
LetE be a reflexive smooth and strictly convex Banach space, let C be a nonempty bounded closed convex subset of E, let A:C×C→E* be a relaxed η-ξ semi-monotone mapping. Let f:C×C→ℛ be a bifunction satisfying (A1), (A3), and (A4), and let φ:C→ℛ∪{+∞} be a proper lower semicontinuous and convex function. Let r>0 and z∈C. Assume that(i)
η(x,y)+η(y,x)=0 for all x,y∈C;(ii)
for any fixedu,v,w∈C, the mapping x↦〈A(v,w),η(x,u)〉 is convex and lower semicontinuous;(iii)
for eachx∈C,A(x,·):C→E* is finite-dimensional continuous.(iv)
ξ:E→ℛ is convex lower semicontinuous.Then, there exists u0∈C such that
(3.15)f(u0,v)+〈A(u0,u0),η(v,u0)〉+φ(v)+1r〈v-u0,J(u0-z)〉≥φ(u0),∀v∈C.IfE is reflexive (i.e., E=E**) smooth and strictly convex, A is semi-monotone, then we obtain the following result.Corollary 3.4.
LetE be a reflexive smooth and strictly convex Banach space, let C be a nonempty bounded closed convex subset of E, let A:C×C→E* be a semi-monotone mapping. Let f:C×C→ℛ be a bifunction satisfying (A1), (A3), and (A4), and let φ:C→ℛ∪{+∞} be a proper lower semicontinuous and convex function. Assume that, for any r>0 and z∈C,(i)
for any fixedu,v,w∈C, the mapping x↦〈A(v,w),x-u)〉 is convex and lower semicontinuous;(ii)
for eachx∈C,A(x,·):C→E* is finite-dimensional continuous.Then, there exists u0∈C such that
(3.16)f(u0,v)+〈A(u0,u0),v-u0〉+φ(v)+1r〈v-u0,J(u0-z)〉≥φ(u0),∀v∈C.Theorem 3.5.
LetE be a real Banach space with the smooth and strictly convex second dual space E**, let C be a nonempty, bounded, closed, and convex subset of E**, let A:C×C→E* be a relaxed η-ξ semi-monotone mapping. Let f be a bifunction from C×C to ℛ satisfying (A1)–(A4) and let φ be a lower semicontinuous and convex function from C to ℛ. For any r>0, define a mapping Φr:E**→C as follows:
(3.17)Φr(x)={u∈C:f(u,v)+〈A(u,u),η(v,u)〉+φ(v)+1r〈v-u,J(u-x)〉≥φ(u),∀v∈C},
for all x∈E. Assume that(i)
η(x,y)+η(y,x)=0 for all x,y∈C;(ii)
for any fixedu,v,w∈C, the mapping x↦〈A(v,w),η(x,u)〉 is convex and lower semicontinuous;(iii)
for eachx∈C,A(x,·):C→E* is finite-dimensional continuous: that is, for any finite-dimensional subspace F⊂E**,A(x,·):C∩F→E* is continuous;(iv)
ξ:E**→ℛ is convex lower semicontinuous;(v)
for anyx,y∈C, ξ(x-y)+ξ(y-x)≥0;(vi)
for anyx,y∈C, A(x,y)=A(y,x). Then, the following holds:(1)
Φr is single-valued;(2)
〈Φrx-Φry,J(Φrx-x)〉≤〈Φrx-Φry,J(Φry-y)〉 for all x,y∈E;(3)
F(Φr)=GMEP(f,A,η,φ);(4)
GMEP(f,A,η,φ) is nonempty, closed, and convex.Proof.
For eachx∈E**, by Lemma 2.10, we conclude that Φr(x) is nonempty.(1)
We prove thatΦr is single-valued. Indeed, for x∈E** and r>0, let z1,z2∈Φr(x). Then,(3.18)f(z1,v)+〈A(z1,z1),η(v,z1)〉+φ(v)+1r〈v-z1,J(z1-x)〉≥φ(z1),∀v∈C,f(z2,v)+〈A(z2,z2),η(v,z2)〉+φ(v)+1r〈v-z2,J(z2-x)〉≥φ(z2),∀v∈C.
Hence,
(3.19)f(z1,z2)+〈A(z1,z1),η(z2,z1)〉+φ(z2)+1r〈z2-z1,J(z1-x)〉≥φ(z1),f(z2,z1)+〈A(z2,z2),η(z1,z2)〉+φ(z1)+1r〈z1-z2,J(z2-x)〉≥φ(z2).
Adding the two inequalities, from (i) we have
(3.20)f(z2,z1)+f(z1,z2)+〈A(z1,z1)-A(z2,z2),η(z2,z1)〉+1r〈z2-z1,J(z1-x)-J(z2-x)〉≥0.
From (A2), we have
(3.21)〈A(z1,z1)-A(z2,z2),η(z2,z1)〉+1r〈z2-z1,J(z1-x)-J(z2-x)〉≥0.
That is,
(3.22)1r〈z2-z1,J(z1-x)-J(z2-x)〉≥〈A(z2,z2)-A(z1,z1),η(z2,z1)〉.
Calculating the right-hand side of (3.22), we have
(3.23)〈A(z2,z2)-A(z1,z1),η(z2,z1)〉=〈A(z2,z2)-A(z2,z1)+A(z2,z1)-A(z1,z2)+A(z1,z2)-A(z1,z1),η(z2,z1)〉=〈A(z2,z2)-A(z2,z1),η(z2,z1)〉+〈A(z2,z1)-A(z1,z2),η(z2,z1)〉+〈A(z1,z2)-A(z1,z1),η(z2,z1)〉≥2ξ(z2-z1)+〈A(z2,z1)-A(z1,z2),η(z2,z1)〉,
and so,
(3.24)1r〈z2-z1,J(z1-x)-J(z2-x)〉≥2ξ(z2-z1)+〈A(z2,z1)-A(z1,z2),η(z2,z1)〉.
In (3.24) exchanging the position of z1 and z2, we get
(3.25)1r〈z1-z2,J(z2-x)-J(z1-x)〉≥2ξ(z1-z2)+〈A(z1,z2)-A(z2,z1),η(z1,z2)〉.
Adding the inequalities (3.24) and (3.25) and using (v) and (vi), we have
(3.26)〈z2-z1,J(z1-x)-J(z2-x)〉≥r(ξ(z2-z1)+ξ(z1-z2))≥0.
Hence,
(3.27)0≤〈z2-z1,J(z1-x)-J(z2-x)〉=〈(z2-x)-(z1-x),J(z1-x)-J(z2-x)〉.
Since J is monotone and E** is strictly convex, we obtain that z1-x=z2-x and hence z1=z2. Therefore, Φr is single-valued.(2)
Forx,y∈C, we have(3.28)f(Φrx,Φry)+〈A(Φrx,Φrx),η(Φry,Φrx)〉+φ(Φry)-φ(Φrx)+1r〈Φry-Φrx,J(Φrx-x)〉≥0,f(Φry,Φrx)+〈A(Φry,Φry),η(Φrx,Φry)〉+φ(Φrx)-φ(Φry)+1r〈Φrx-Φry,J(Φry-y)〉≥0.
Adding the above two inequalities and by (i) and (A2), we get
(3.29)〈A(Φrx,Φrx)-A(Φry,Φry),η(Φry,Φrx)〉+1r〈Φry-Φrx,J(Φrx-x)-J(Φry-y)〉≥0,
that is
(3.30)1r〈Φry-Φrx,J(Φrx-x)-J(Φry-y)〉≥〈A(Φry,Φry)-A(Φrx,Φrx),η(Φry,Φrx)〉.
After calculating (3.30), we have
(3.31)1r〈Φry-Φrx,J(Φrx-x)-J(Φry-y)〉≥2ξ(Φry,Φrx)+〈A(Φry,Φrx)-A(Φrx,Φry),η(Φry,Φrx)〉.
In (3.30), exchanging the position of Φrx and Φry, we get
(3.32)1r〈Φrx-Φry,J(Φry-y)-J(Φrx-x)〉≥2ξ(Φrx,Φry)+〈A(Φrx,Φry)-A(Φry,Φrx),η(Φrx,Φry)〉.
Adding the inequalities (3.31) and (3.32), use (i) and (vi), we have
(3.33)〈Φry-Φrx,J(Φrx-x)-J(Φry-y)〉≥r(ξ(Φrx,Φry)+ξ(Φry,Φrx)).
It follows from (iv) that
(3.34)〈Φry-Φrx,J(Φrx-x)-J(Φry-y)〉≥0.
Hence,
(3.35)〈Φrx-Φry,J(Φrx-x)〉≤〈Φrx-Φry,J(Φry-y)〉.(3)
Next, we show thatF(Φr)=GMEP(f,A,η,φ). Indeed, we have the following:
(3.36)u∈F(Φr)⟺u=Φru⟺f(u,v)+〈A(u,u),η(v,u)〉+φ(v)+1r〈v-u,J(u-u)〉≥φ(u),∀v∈C⟺f(u,v)+〈A(u,u),η(v,u)〉+φ(v)≥φ(u),∀v∈C⟺u∈GMEP⁡(f,A,η,φ).Hence, F(Φr)=GMEP(f,A,η,φ).(4)
Finally, we prove thatGMEP(f,A,η,φ) is nonempty, closed, and convex. For each v∈C, we define the multivalued mapping G:C→2E** by(3.37)G(v)={u∈C:f(u,v)+〈A(u,u),η(v,u)〉+φ(v)≥φ(u)}.
Since v∈G(v), we have G(v)≠∅. We prove that G is a KKM mapping on C. Suppose that there exists a finite subset {z1,z2,…,zm} of C, and αi>0 with ∑i=1mαi=1 such that ẑ=∑i=1mαizi∉G(zi) for all i=1,2,…,m. Then
(3.38)f(ẑ,zi)+〈A(ẑ,ẑ),η(zi,ẑ)〉+φ(zi)-φ(ẑ)<0,i=1,2,…,m.
From (A1), (A4), (ii), and the convexity of φ, we have
(3.39)0=f(ẑ,ẑ)+〈A(ẑ,ẑ),η(ẑ,ẑ)〉+φ(ẑ)-φ(ẑ)=f(ẑ,∑i=1mαizi)+〈A(ẑ,ẑ),η(∑i=1mαizi,ẑ)〉+φ(∑i=1mαizi)-φ(ẑ)≤∑i=1mαi(f(ẑ,zi)+〈A(ẑ,ẑ),η(zi,ẑ)〉+φ(zi)-φ(ẑ))<0,
which is a contradiction. Thus, G is a KKM mapping on C.
Next, we prove thatG(y) is closed for each y∈C. For any y∈C, let {xn} be any sequence in G(y) such that xn→x0. We claim that x0∈G(y). Then, for each y∈C, we have(3.40)f(xn,y)+〈A(xn,xn),η(y,xn)〉+φ(y)≥φ(xn).
By monotonicity of A, we obtain that
(3.41)f(xn,y)+〈A(xn,y),η(y,xn)〉+φ(y)≥φ(xn)+ξ(y-xn).
By (A3), (i), (ii), (iv), lower semicontinuity of φ, and the complete continuity of A, we obtain the following
(3.42)φ(x0)+〈A(x0,y),η(x0,y)〉≤liminf⁡n→∞φ(xn)+liminf⁡n→∞〈A(xn,y),η(xn,y)〉≤liminf⁡n→∞(φ(xn)+〈A(xn,y),η(xn,y)〉)=liminf⁡n→∞(φ(xn)-〈A(xn,y),η(y,xn)〉)≤limsup⁡n→∞(φ(xn)-〈A(xn,y),η(y,xn)〉)≤limsup⁡n→∞(f(xn,y)+φ(y)-ξ(y-xn))≤f(x0,y)+φ(y)-ξ(y-x0).
Hence,
(3.43)f(x0,y)+〈A(x0,y),η(y,x0)〉+φ(y)≥φ(x0)+ξ(y-x0),∀y∈C.
From Lemma 2.9, we have
(3.44)f(x0,y)+〈A(x0,x0),η(y,x0)〉+φ(y)≥φ(x0),∀y∈C.
This shows that x0∈G(y), and hence G(y) is closed for each y∈C. Thus, GMEP(f,A,η,φ)=⋂y∈CG(y) is also closed.
Next, we observe thatG(y) is weakly compact. In fact, since C is bounded, closed, and convex, we also have G(y), which is weakly compact in the weak topology. By Lemma 2.6, we can conclude that ⋂y∈CG(y)=GMEP(f,A,η,φ)≠∅.
Finally, we prove thatGMEP(f,A,η,φ) is convex. In fact, let u,v∈F(Φr), and zt=tu+(1-t)v for t∈(0,1). From (2), we know that
(3.45)〈Φru-Φrzt,J(Φrzt-zt)-J(Φru-u)〉≥0.
This yields that
(3.46)〈u-Φrzt,J(Φrzt-zt)〉≥0.
Similarly, we also have
(3.47)〈v-Φrzt,J(Φrzt-zt)〉≥0.
It follows from (3.46) and (3.47) that
(3.48)‖zt-Φrzt‖2=〈zt-Φrzt,J(zt-Φrzt)〉=t〈u-Φrzt,J(zt-Φrzt)〉+(1-t)〈v-Φrzt,J(zt-Φrzt)〉≤0.
Hence, zt∈F(Φr)=GMEP(f,A,η,φ) and hence GMEP(f,A,η,φ) is convex. This completes the proof.IfE is reflexive (i.e., E=E**) smooth and strictly convex, then the following result can be derived as a corollary of Theorem 3.5Corollary 3.6.
LetE be a reflexive smooth and strictly convex Banach space, let C be a nonempty, bounded, closed, and convex subset of E, and let A:C×C→E* be a relaxed η-ξ semi-monotone mapping. Let f be a bifunction from C×C to ℛ satisfying (A1)–(A4) and let φ be a lower semicontinuous and convex function from C to ℛ. Let r>0 and z∈C and define a mapping Φr:E→C as follows:
(3.49)Φr(x)={u∈C:f(u,v)+〈A(u,u),η(v,u)〉+φ(v)+1r〈v-u,J(u-x)〉≥φ(u),∀v∈C},for all x∈E. Assume that(i)
η(x,y)+η(y,x)=0 for all x,y∈C;(ii)
for any fixedu,v,w∈C, the mapping x↦〈A(v,w),η(x,u)〉 is convex and lower semicontinuous;(iii)
for eachx∈C,A(x,·):C→E* is finite-dimensional continuous;(iv)
ξ:E→ℛ is convex lower semicontinuous;(v)
for anyx,y∈C, ξ(x-y)+ξ(y-x)≥0;(vi)
for anyx,y∈C, A(x,y)=A(y,x). Then, the following holds:(1)
Φr is single-valued;(2)
〈Φrx-Φry,J(Φrx-x)〉≤〈Φrx-Φry,J(Φry-y)〉 for all x,y∈E;(3)
F(Φr)=GMEP(f,A,η,φ);(4)
GMEP(f,A,η,φ) is nonempty, closed, and convex.


## 4. Strong Convergence Theorems

In this section, we prove a strong convergence theorem by using a hybrid projection algorithm for an asymptotically nonexpansive mapping in a uniformly convex and smooth Banach space.Theorem 4.1.
LetE be a real Banach space with the smooth and uniformly convex second dual space E**, let C be a nonempty, bounded, closed, and convex subset of E**. Let f be a bifunction from C×C to ℛ satisfying (A1)–(A4), and let φ be a lower semicontinuous and convex function from C to ℛ. Let A:C×C→E* be a relaxed η-ξ semi-monotone and let S:C→C be an asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞) such that kn→1 as n→∞. Assume that Ω∶=F(S)∩GMEP(f,A,η,φ)≠∅. Let {xn} be a sequence in C generated by
(4.1)x0∈C,D0=C0=C,Cn=co⁡¯{z∈Cn-1:‖z-Snz‖≤tn‖xn-Snxn‖},n≥1,un∈Csuchthatf(un,y)+φ(y)+〈A(un,un),η(y,un)〉+1rn〈y-un,J(un-xn)〉≥φ(un),∀y∈C,n≥0,Dn={z∈Dn-1:〈un-z,J(xn-un)〉≥0},n≥1,xn+1=PCn∩Dnx0,n≥0,
where {tn} and {rn} are real sequences in (0,1) such that limn→∞tn=0, and liminfn→∞rn>0. Then {xn} converges strongly, as n→∞, to PΩx0.Proof.
Firstly, we rewrite the (4.1) as follows:
(4.2)x0∈C,D0=C0=C,Cn=co⁡¯{z∈Cn-1:‖z-Snz‖≤tn‖xn-Snxn‖},n≥0,Dn={z∈Dn-1:〈Φrnxn-z,J(xn-Φrnxn)〉≥0},n≥1,xn+1=PCn∩Dnx0,n≥0,
where Φr is the mapping defined by
(4.3)Φr(x)={z∈C:f(z,y)+〈A(z,z),η(y,z)〉+φ(y)+1r〈y-z,J(z-x)〉≥φ(z),∀y∈C}.
We first show that the sequence {xn} is well defined. It is easy to verify that Cn∩Dn is closed and convex and Ω⊂Cn for all n≥0. Next, we prove that Ω⊂Cn∩Dn. Since D0=C, we also have Ω⊂C0∩D0. Suppose that Ω⊂Ck-1∩Dk-1 for k≥2. It follows from Theorem 3.5 (2) that
(4.4)〈Φrkxk-Φrku,J(Φrku-u)-J(Φrkxk-xk)〉≥0,
for all u∈Ω. This implies that
(4.5)〈Φrkxk-u,J(xk-Φrkxk)〉≥0,
for all u∈Ω. Hence Ω⊂Dk. By the mathematical induction, we get that Ω⊂Cn∩Dn for each n≥0, and hence {xn} is welldefined. Put w=PΩx0. Since Ω⊂Cn∩Dn and xn+1=PCn∩Dn, we have
(4.6)‖xn+1-x0‖≤‖w-x0‖,n≥0.
Since xn+2∈Dn+1⊂Dn and xn+1=PCn∩Dnx0, we have
(4.7)‖xn+1-x0‖≤‖xn+2-x0‖.
Since {∥xn-x0∥} is bounded, we have limn→∞∥xn-x0∥=d for some a constant d. Moreover, by the convexity of Dn, we also have (1/2)(xn+1+xn+2)∈Dn and hence
(4.8)‖x0-xn+1‖≤‖x0-xn+1+xn+22‖≤12(‖x0-xn+1‖+‖x0-xn+2‖).
This implies that
(4.9)lim⁡n→∞‖12(x0-xn+1)+12(x0-xn+2)‖=lim⁡n→∞‖x0-xn+1+xn+22‖=d.
By Lemma 2.2, we have
(4.10)lim⁡n→∞‖xn-xn+1‖=0.
Next, we show that
(4.11)lim⁡n→∞‖xn-Sxn‖=0.
To obtain (4.11), we need to show that limn→∞∥xn-Sn-kxn∥=0,forallk∈𝒩.
Fixk∈𝒩 and put m=n-k. Since xn=PCn-1∩Dn-1x, we have xn∈Cn-1⊆⋯⊆Cm. Since tm>0, there exist y1,…,yN∈C and nonnegative numbers λ1,…,λN with λ1+⋯+λN=1 such that
(4.12)‖xn-∑i=1Nλiyi‖<tm,
and ∥yi-Smyi∥≤tm∥xm-Smxm∥ for all i∈{1,…,N}. Put M=supx∈C∥x∥,u=PF(S)x and r0=supn≥1(1+kn)∥xn-u∥. Since C and {km} are bounded, (4.12) implies
(4.13)‖xn-1km∑i=1Nλiyi‖≤(1-1km)‖x‖+1km‖xn-∑i=1Nλiyi‖≤(1-1km)M+tm,
and ∥yi-Smyi∥≤tm∥xm-Smxm∥≤tm(1+km)∥xm-u∥≤r0tm for all i∈{1,…,N}. Therefore,
(4.14)‖yi-1kmSmyi‖≤(1-1km)M+r0tm,
for all i∈{1,…,N}. Moreover, asymptotically nonexpansiveness of S and (4.6) give that
(4.15)‖1kmSm(∑i=1Nλiyi)-Smxn‖≤(1-1km)M+tm.
It follows from Theorem 2.4, (4.13)–(4.15) that
(4.16)‖xn-Smxn‖≤‖xn-1km∑i=1Nλiyi‖+1km‖∑i=1Nλi(yi-Smyi)‖+1km‖∑i=1NλiSmyi-Sm(∑i=1Nλiyi)‖+‖1kmSm(∑i=1Nλiyi)-Smxn‖≤2(1-1km)M+2tm+r0tmkm+γ-1(max⁡1≤i≤j≤N(‖yi-yj∥-1km∥Smyi-Smyj‖))≤2(1-1km)M+2tm+r0tmkm+γ-1(max⁡1≤i≤j≤N(‖yi-1kmSmyi‖+‖yj-1kmSmyj‖))≤2(1-1km)M+2tm+r0tmkm+γ-1(2(1-1km)M+2r0tm).
Since limn→∞kn=1 and limn→∞tn=0, it follows from the last inequality that limn→∞∥xn-Smxn∥=0. We have that
(4.17)‖xn-Sxn‖=‖xn-Sn-1xn‖+‖Sn-1xn-Sxn‖≤‖xn-Sn-1xn‖+k1‖Sn-2xn-xn‖⟶0asn⟶∞.
Since {xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀x̃∈C. Therefore, we obtain x̃∈F(S). Next, we show that x̃∈GMEP(f,A,η,φ). By the construction of Dn, we see from Theorem 2.1 that Φrnxn=PDnxn. Since xn+1∈Dn, we get
(4.18)‖xn-Φrnxn‖≤‖xn-xn+1‖⟶0.
From (C2), we also have
(4.19)1rn‖J(xn-Φrnxn)‖=1rn‖xn-Φrnxn‖⟶0,
as n→∞. By (4.19), we also have Φrnixni⇀x̃. By the definition of Φrni, for each y∈C, we obtain
(4.20)f(Φrnixni,y)+〈A(Φrnixni,Φrnixni),η(y,Φrnixni)〉+φ(y)+1rni〈y-Φrnixni,J(Φrnixni-xni)〉≥φ(Φrnixni).
By (A3), (4.19), (ii), the weakly lower semicontinuity of φ and complete continuity of A we have
(4.21)φ(x̃)≤liminf⁡i→∞φ(Φrnixni)≤liminf⁡i→∞f(Φrnixni,y)+liminf⁡i→∞〈A(Φrnixni,Φrnixni),η(y,Φrnixni)〉+φ(y)+liminf⁡i→∞1rni〈y-Φrnixni,J(Φrnixni-xni)〉≤f(x̃,y)+φ(y)+〈A(x̃,x̃),η(y,x̃)〉.
Hence,
(4.22)f(x̃,y)+φ(y)+〈A(x̃,x̃),η(y,x̃)〉≥φ(x̃).
This shows that x̃∈GMEP(f,A,η,φ), and hence x̃∈Ω∶=F(S)∩GMEP(f,A,η,φ).
Finally, we show thatxn→w as n→∞, where w∶=PΩx0. By the weakly lower semicontinuity of the norm, it follows from (4.6) that
(4.23)‖x0-w‖≤‖x0-x̃‖≤liminf⁡i→∞‖x0-xni‖≤limsup⁡i→∞‖x0-xni‖≤‖x0-w‖.
This shows that
(4.24)lim⁡i→∞‖x0-xni‖=‖x0-w‖=‖x0-x̃‖,
and x̃=w. Since E** is uniformly convex, we obtain that x0-xni→x0-w. It follows that xni→w. So, we have xn→w as n→∞. This completes the proof.IfS is a nonexpansive mapping in Theorem 4.1, then we obtain the following result concerning the problem of finding a common element of GMEP(f,A,η,φ) and the fixed point set of a nonexpansive mapping in a Banach space setting.Theorem 4.2.
LetE be a real Banach space with the smooth and uniformly convex second dual space E**, let C be a nonempty, bounded, closed, and convex subset of E**. Let f be a bifunction from C×C to ℛ satisfying (A1)–(A4) and let φ be a lower semicontinuous and convex function from C to ℛ. Let A:C×C→E* be a relaxed η-ξ semi-monotone and let S:C→C be a nonexpansive mapping such that Ω∶=F(S)∩GMEP(f,A,η,φ)≠∅. Let {xn} be a sequence in C generated by
(4.25)x0∈C,D0=C0=C,Cn=co⁡¯{z∈Cn-1:‖z-Sz‖≤tn‖xn-Sxn‖},n≥1,un∈Csuchthatf(un,y)+φ(y)+〈A(un,un),η(y,un)〉+1rn〈y-un,J(un-xn)〉≥φ(un),∀y∈C,n≥0,Dn={z∈Dn-1:〈un-z,J(xn-un)〉≥0},n≥1,xn+1=PCn∩Dnx0,n≥0,
where {tn} and {rn} are real sequences in (0,1) such that limn→∞tn=0, and liminfn→∞rn>0. Then, {xn} converges strongly, as n→∞, to PΩx0.PuttingA≡0 and φ≡0 in Theorem 4.1, then we have the following result in a Banach space.Theorem 4.3.
LetE be a real Banach space with the smooth and uniformly convex second dual space E** and let C be a nonempty, bounded, closed, and convex subset of E**. Let f be a bifunction from C×C to ℛ satisfying (A1)–(A4). Let S:C→C be an asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞) such that Ω∶=F(S)∩EP(f)≠∅. Let {xn} be a sequence in C generated by
(4.26)x0∈C,D0=C0=C,Cn=co⁡¯{z∈Cn-1:‖z-Snz‖≤tn‖xn-Snxn‖},n≥1,un∈Csuchthatf(un,y)+1rn〈y-un,J(un-xn)〉≥0,∀y∈C,n≥0,Dn={z∈Dn-1:〈un-z,J(xn-un)〉≥0},n≥1,xn+1=PCn∩Dnx0,n≥0,
where {tn} and {rn} are real sequences in (0,1) such that limn→∞tn=0, and liminfn→∞rn>0. Then, {xn} converges strongly, as n→∞, to PΩx0.Puttingf≡0,A≡0,φ≡0, and rn≡1 in Theorem 4.1 and applying Theorem 2.1, we get xn=un. Then, we have the following new approximation method concerning the problem of finding a fixed of an asymptotically nonexpansive mapping in a Banach space.Theorem 4.4.
LetE be a real Banach space with the smooth and uniformly convex second dual space E**, let C be a nonempty, bounded, closed, and convex subset of E**. Let S:C→C be an asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞) such that F(S)≠∅. Let {xn} be a sequence in C generated by
(4.27)x0∈C,C0=C,Cn=co⁡¯{z∈Cn-1:‖z-Snz‖≤tn‖xn-Snxn‖},n≥1,xn+1=PCnx0,n≥0,
where {tn} and {rn} is a real sequence in (0,1) such that limn→∞tn=0. Then {xn} converges strongly, as n→∞, to PF(S)x0.IfE is reflexive (i.e., E=E**) smooth and uniformly convex, then the following results can be derived as a corollary of Theorem 4.4.Corollary 4.5.
LetE be a reflexive smooth and uniformly convex real Banach space, let C be a nonempty, bounded, closed, and convex subset of E. Let S:C→C be an asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞] such that F(S)≠∅. Let {xn} be a sequence in C generated by
(4.28)x0∈C,C0=C,Cn=co⁡¯{z∈Cn-1:‖z-Snz‖≤tn‖xn-Snxn‖},n≥1,xn+1=PCnx0,n≥0,
where {tn} and {rn} is a real sequence in (0,1) such that limn→∞tn=0. Then, {xn} converges strongly, as n→∞, to PF(S)x0.



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*Source: 790592-2012-04-10.xml*