# Strong and Weak Convergence Theorems for Equilibrium Problems and Weak Relatively Uniformly Nonexpansive Multivalued Mappings in Banach Spaces

Authors: Zi-Ming Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Equilibrium problem and fixed point problem are considered. A general iterative algorithm is introduced for finding a common element of the set of solutions to the equilibrium problem and the common set of fixed points of two weak relatively uniformly nonexpansive multivalued mappings. Furthermore, strong and weak convergence results for the common element in the two sets mentioned above are established in some Banach space.


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## Body

## 1. Introduction

LetE be a smooth Banach space, and let C be a nonempty closed convex subset of E. In the sequel, we denote by 2C the family of all nonempty subsets of C. We use ϕ:E×E→ℝ to denote the Lyapunov functional defined by
(1.1)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E.
We know several fundamental properties of ϕ as follows: ϕ(x,y)≥0 for all x,y∈E. For a sequence {yn}⊂E and x∈E, {yn} is bounded if and only if {ϕ(x,yn)} is bounded.LetT:C→2C be a multivalued mapping. We denote by F(T) the set of fixed points of T, that is,
(1.2)F(T)={x∈C:x∈Tx}.For a multivalued mappingT, we define an asymptotic fixed point and a strong asymptotic fixed point of T as follows.Definition 1.1 (see [1]).
LetT:C→2C be a multivalued mapping.(1)
A pointp in C is said to be an asymptotic fixed point of T if C contains a sequence {xn} which converges weakly to p and there exists a sequence {yn} such that yn∈Txn,limn→∞∥yn-xn∥=0. The set of asymptotic fixed point of T will be denoted by F^(T).(2)
A pointp in C is said to be a strong asymptotic fixed point of T if C contains a sequence {xn} which converges strongly to p and there exists a sequence {yn} such that yn∈Txn,limn→∞∥yn-xn∥=0. The set of strong asymptotic fixed point of T will be denoted by F~(T).Definition 1.2 (see [1]).
A multivalued mappingT:C→2C is called relatively nonexpansive multivalued mapping (weak relatively nonexpansive multivalued mapping) if the following conditions are satisfied:(1)
F(T) is nonempty;(2)
ϕ(u,v)≤ϕ(u,x),forallu∈F(T),forallx∈C,∃v∈Tx;(3)
F^(T)=F(T)(F~(T)=F(T)).Definition 1.3 (see [1]).
A multivalued mappingT:C→2C is called relatively uniformly nonexpansive multivalued mapping (weak relatively uniformly nonexpansive multivalued mapping) if the following conditions are satisfied:(1)
F(T) is nonempty;(2)
ϕ(u,v)≤ϕ(u,x),forallu∈F(T),forallx∈C,forallv∈Tx;(3)
F^(T)=F(T)(F~(T)=F(T)).Remark 1.4.
By comparing condition (2) of Definitions1.2 and 1.3, one easily draws the following conclusions:(1)
the class of relatively nonexpansive multivalued mappings contains the class of relatively uniformly nonexpansive multivalued mappings as a subclass, but the converse may be not true;(2)
the class of weak relatively nonexpansive multivalued mappings contains the class of weak relatively uniformly nonexpansive multivalued mappings as a subclass, but the converse may be not true.For any operatorT, F(T)⊂F~(T)⊂F^(T) is held. So we have the following remark.Remark 1.5.
From Definitions1.2 and 1.3, the following conclusions can easily be drawn: (1)
the class of weak relatively nonexpansive multivalued mappings contains the class of relatively nonexpansive multivalued mappings as a subclass, but the converse may be not true;(2)
the class of weak relatively uniformly nonexpansive multivalued mappings contains the class of relatively uniformly nonexpansive multivalued mappings as a subclass, but the converse may be not true.Remark 1.6.
The examples of weak relatively uniformly nonexpansive multivalued mapping can be found in Su [1] and Homaeipour and Razani [2].
LetE be a real Banach space, and let E* be the dual space of E. Let f be a bifunction from C×C to ℝ. The equilibrium problem is to find
(1.3)x^∈Csuchthatf(x^,y)≥0,∀y∈C.
The set of solutions of (1.3) is denoted by EP(f). Given a mapping T:C→E*, let f(x,y)=〈Tx,y-x〉 for all x,y∈C. Then x^∈EP(f) if and only if 〈Tx^,y-x^〉≥0 for all y∈C, that is, x^ is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.3). Some methods have been proposed to solve the equilibrium problem in Hilbert spaces, see [3–5] for details.In recent years, iterative methods for approximating fixed points of multivalued mappings in Banach spaces have been studied by many authors, see [2, 6–9] for details. In 2011, Homaeipour and Razani [2] introduced the concept of relatively nonexpansive multivalued mappings and proved some weak and strong convergence theorems to approximate a fixed point for a single relatively nonexpansive multivalued mapping in a uniformly convex and uniformly smooth Banach space E which improved and extended the corresponding results of Matsushita and Takahashi [10]. Very recently, Su [1] not only redefined relatively nonexpansive multivalued mappings, which was different from Homaeipour and Razani [2]'s definition, but also introduced some interesting examples about the multivalued mappings. On the other hand, in 2009, Qin et al. [11] introduced an iterative algorithm for the equilibrium problem (1.3) and relatively nonexpansive mappings. Moreover, they proved a weak convergence theorem for finding a common element of the set of solutions to the equilibrium problem (1.3) and the common set of fixed points of two relatively nonexpansive mappings, which improved and extended the corresponding results of Takahashi and Zembayashi [12].Motivated and inspired by the above facts, the purpose of this paper will introduce an iterative algorithm for the equilibrium problem (1.3) and two weak relatively uniformly nonexpansive multivalued mappings. Furthermore, a weak convergence theorem will given for finding a common element of the set of solutions to the equilibrium problem (1.3) and the common set of fixed points of two weak relatively uniformly nonexpansive multivalued mappings in some Banach space. Our results improve and extend the corresponding results of Qin et al. [11] and Takahashi and Zembayashi [12].


## 2.    Preliminaries

LetE be a real Banach space with norm ∥·∥, and let J be the normalized duality mapping from E into 2E* given by
(2.1)Jx={x*∈E*:〈x,x*〉=∥x∥∥x*∥,∥x∥=∥x*∥},
for all x∈E, where E* denotes the dual space of E and 〈·,·〉 the generalized duality pairing between E and E*. It is well known that if E* is uniformly convex, then J is uniformly continuous on bounded subsets of E.As we all know that ifC is a nonempty closed convex subset of a Hilbert space H, and PC:H→C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces, and, consequently, it is not available in more general Banach spaces. In this connection, Alber [13] introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces. The generalized projection ΠC:E→C is a map that assigns to an arbitrary point x∈E the minimum point of the Lyapunov functional ϕ(x,y), that is, ΠCx=x¯, where x¯ is the solution to the minimization problem;
(2.2)ϕ(x¯,x)=infy∈Cϕ(y,x).
The existence and uniqueness of the operator ΠC follow from the properties of the Lyapunov functional ϕ(x,y) and strict monotonicity of the mapping J, see, for example, [13, 14]. In Hilbert spaces, ΠC=PC. It is obvious from the definition of function ϕ that
(2.3)(∥y∥-∥x∥)2≤ϕ(y,x)≤(∥y∥-∥x∥)2,∀x,y∈E.A Banach spaceE is said to be strictly convex if ∥(x+y)/2∥<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. It is said to be uniformly convex if limn→∞∥xn-yn∥=0 for any two sequences {xn} and {yn} in E such that ∥xn∥=∥yn∥=1 and limn→∞∥(xn+yn)/2∥=1. Let U={x∈E:∥x∥=1} be the unit sphere of E. Then the Banach space E is said to be smooth provided by
(2.4)limt→0∥x+ty∥-∥x∥t,
which exists for each x,y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈E. It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.We also need the following lemmas for the proof of our main results.Lemma 2.1 (see [2]).
LetE be a strictly convex and smooth Banach space, then ϕ(x,y)=0 if and only if x=y.Lemma 2.2 (see [2]).
LetE be a uniformly convex and smooth Banach space and r>0. Then,
(2.5)g(∥y-z∥)≤ϕ(y,z),
for all y,z∈Br(0)={x∈E:∥x∥≤r}, where g:[0,∞)→[0,∞) is a continuous, strictly increasing, and convex function with g(0)=0.Lemma 2.3 (see [11]).
LetC be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E. Then
(2.6)ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y),∀x∈Candy∈E.Lemma 2.4 (see [11]).
LetE be a uniformly convex Banach space and Br(0) be a closed ball of E. Then there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(2.7)∥λx+μy+γz∥2≤λ∥x∥2+μ∥y∥2+γ∥z∥2-λμg(∥x-y∥),
for all x,y,z∈Br(0)={x∈E:∥x∥≤r} and λ,μ,γ∈[0,1] with λ+μ+γ=1.Lemma 2.5.
LetE be a strictly convex and smooth Banach space, and let C be a closed convex subset of E. Suppose T:C→2C is a weak relatively uniformly nonexpansive multivalued mapping. Then, F(T) is closed and convex.Proof.
First, we show thatF(T) is closed. Let {pn} be a sequence in F(T) such that pn→p as n→∞. Since the multivalued operator T is uniformly weak relatively nonexpansive, one has
(2.8)ϕ(pn,p~)≤ϕ(pn,p),
for all p~∈Tp and for all n∈ℕ. Therefore,
(2.9)ϕ(p,p~)=limn→∞ϕ(pn,p~)≤limn→∞ϕ(pn,p)=ϕ(p,p).
Applying Lemma 2.1, one gets p=p~. Hence Tp={p}. Therefore, p∈F(T).
Next, we show thatF(T) is convex. To this end, for arbitrary p1,p2∈F(T),t∈(0,1). Putting p=tp1+(1-t)p2, we prove that Tp={p}. Let q∈Tp, we have
(2.10)ϕ(p,q)=∥p∥2-2〈p,Jq〉+∥q∥2=∥p∥2-2〈tp1+(1-t)p2,Jq〉+∥q∥2=∥p∥2-2t〈p1,Jq〉-2(1-t)〈p2,Jq〉+∥q∥2=∥p∥2+tϕ(p1,q)+(1-t)ϕ(p2,q)-t∥p1∥2-(1-t)∥p2∥2≤∥p∥2+tϕ(p1,p)+(1-t)ϕ(p2,p)-t∥p1∥2-(1-t)∥p2∥2=∥p∥2-2〈tp1+(1-t)p2,Jp〉+∥p∥2=∥p∥2-2〈p,Jp〉+∥p∥2=0.
Using Lemma 2.1 again, we also obtain p=q. Hence, T(p)={p}, that is, p∈F(T). Therefore, F(T) is convex.For solving the equilibrium problem for a bifunctionf:C×C→ℝ, let us assume that f 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) for each x,y,z∈C,
(2.11)limt↓0f(tz+(1-t)x,y)≤f(x,y);(A4) for each x∈C,y↦f(x,y) is convex and lower semicontinuous.Lemma 2.6 (see [12]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C×C to R satisfying (A1)–(A4), and let r>0 and x∈E. Then, there exists z∈C such that
(2.12)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.7 (see [12]).
LetC be a closed subset of a strictly convex, uniformly smooth, and reflexive Banach space E, and let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). For all r>0 and x∈E, define a mapping Tr:E→C as follows:
(2.13)Trx={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},
for all x∈E. Then, the following hold:   (1)
Tr is single-valued;(2)
Tr is firmly nonexpansive mapping, that is, for all x,y∈E,
(2.14)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;(3)
F(Tr)=EP(f)(4)
EP(f) is closed and convex.Lemma 2.8 (see [12]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0 and x∈E and q∈F(Tr),
(2.15)ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).Lemma 2.9 (see [12]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let x∈E, and let z∈C. Then
(2.16)z=ΠCx⟺〈y-z,Jx-Jz〉≤0,∀y∈C.


## 3.   Main Results

In this section, we prove a weak convergence theorem for finding a common element of the set of solutions for an equilibrium problem and the set of fixed points of two weak relatively uniformly nonexpansive multivalued mappings in a Banach space. Before proving the result, we need the following theorem.Theorem 3.1.
LetC be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to R satisfying (A1)–(A4) and let T,S:C→C be two weak relatively uniformly nonexpansive multivalued mappings such that ℱ=F(T)∩F(S)∩EP(f)≠∅. Let {xn} be a sequence generated by the following manner:
(3.1)xn∈Csuchthatf(xn,y)+1rn〈y-xn,Jxn-Jun〉≥0,∀y∈C,un+1=J-1(αnJxn+βnJyn+γnJzn),
where yn∈Txn, zn∈Sxn, and J are the duality mapping on E. Assume that {αn}, {βn}, and {γn} are three sequences in [0,1] satisfying the following conditions:  (a)
αn+βn+γn=1;(b)
liminfn→∞αnβn>0,liminfn→∞αnγn>0;(c)
{rn}⊂[a,∞) for some a>0.Then {Πℱxn} converges strongly to z∈ℱ, where Πℱ is the generalized projection of E onto ℱ.Proof.
Letp∈ℱ. Putting xn=Trnun for all n∈ℕ, it is well known that Trn is relatively nonexpansive, one has
(3.2)ϕ(p,xn+1)=ϕ(p,Trnun+1)≤ϕ(p,un+1)=ϕ(p,J-1(αnJxn+βnJyn+γnJzn))=∥p∥2-2αn〈p,Jxn〉-2βn〈p,Jyn〉-2γn〈p,Jzn〉+∥αnJxn+βnJyn+γnJzn∥2≤∥p∥2-2αn〈p,Jxn〉-2βn〈p,Jyn〉-2γn〈p,Jzn〉+αn∥Jxn∥2+βn∥Jyn∥2+γn∥Jzn∥2=ϕ(p,xn)+βnϕ(p,yn)+γnϕ(p,zn)≤ϕ(p,xn).
Therefore, limn→∞ϕ(p,xn) exists. Since ϕ(p,xn) is bounded, {xn},{yn},and{zn} are bounded.
Definevn=Πℱxn for all n∈ℕ. Then, from vn∈ℱ and (3.2), one gets
(3.3)ϕ(vn,xn+1)≤ϕ(vn,xn).
Since Πℱ is the generalized projection, from Lemma 2.3, one sees
(3.4)ϕ(vn+1,xn+1)=ϕ(Πℱxn+1,xn+1)≤ϕ(vn,xn+1)-ϕ(vn,Πℱxn+1)=ϕ(vn,xn+1)-ϕ(vn,vn+1)≤ϕ(vn,xn+1).
Hence, from (3.3), one has
(3.5)ϕ(vn+1,xn+1)≤ϕ(vn,xn).
Therefore, {ϕ(vn,xn)} is a convergent sequence. Applying (3.3) again, one also obtains that, for all m∈ℕ,
(3.6)ϕ(vn,xn+m)≤ϕ(vn,xn).
From vn+m=Πℱxn+m and Lemma 2.3, one has
(3.7)ϕ(vn,vn+m)+ϕ(vn+m,xn+m)≤ϕ(vn,xn+m)≤ϕ(vn,xn),
and hence
(3.8)ϕ(vn,vn+m)≤ϕ(vn,xn)-ϕ(vn+m,xn+m).
Let r=supn∈ℕ∥vn∥. From Lemma 2.2, there exists a continuous, strictly increasing, and convex function g with g(0)=0 such that
(3.9)g(∥x-y∥)≤ϕ(x,y)forx,y∈Br.
Therefore, one has
(3.10)g(∥vn-vn+m∥)≤ϕ(vn,vn+m)≤ϕ(vn,xn)-ϕ(vn+m,xn+m).
Since {ϕ(vn,xn)} is a convergent sequence, from the property of g, one obtains that {vn} is a Cauchy sequence. Since ℱ is closed, {vn} converges strongly to z∈ℱ. This completes the proof of Theorem 3.1.In the following, we give our weak convergence result in this paper.Theorem 3.2.
LetC be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to R satisfying (A1)–(A4) and let T,S:C→C be two weak relatively uniformly nonexpansive multivalued mappings such that ℱ=F(T)∩F(S)∩EP(f)≠∅. Let {xn} be a sequence generated by the following manner:
(3.11)xn∈Csuchthatf(xn,y)+1rn〈y-xn,Jxn-Jun〉≥0,∀y∈C,un+1=J-1(αnJxn+βnJyn+γnJzn),
where yn∈Txn, zn∈Sxn, and J are the duality mapping on E. Assume that {αn}, {βn}, and {γn} are three sequences in [0,1] satisfying the following conditions:  (a)
αn+βn+γn=1;(b)
liminfn→∞αnβn>0,liminfn→∞αnγn>0;(c)
{rn}⊂[a,∞) for some a>0.If J is weakly sequentially continuous, then {xn} converges weakly to z∈ℱ, where z=limn→∞Πℱxn.Proof.
In view of the proof Theorem3.1, one has that {xn}, {yn}, and {zn} are bounded. Let
(3.12)r=supn≥0{∥xn∥,∥yn∥,∥zn∥},
from Lemma 2.4, for p∈ℱ, one has
(3.13)ϕ(p,xn+1)=ϕ(p,Trnun+1)≤ϕ(p,un+1)=ϕ(p,J-1(αnJxn+βnJyn+γnJzn))=∥p∥2-2αn〈p,Jxn〉-2βn〈p,Jyn〉-2γn〈p,Jzn〉+∥αnJxn+βnJyn+γnJzn∥2≤∥p∥2-2αn〈p,Jxn〉-2βn〈p,Jyn〉-2γn〈p,Jzn〉+αn∥Jxn∥2+βn∥Jyn∥2+γn∥Jzn∥2-αnβng(∥Jyn-Jxn∥)=ϕ(p,xn)+βnϕ(p,yn)+γnϕ(p,zn)-αnβng(∥Jyn-Jxn∥)≤ϕ(p,xn)-αnβng(∥Jyn-Jxn∥).
It follows that
(3.14)αnβng(∥Jyn-Jxn∥)≤ϕ(p,xn)-ϕ(p,xn+1).
In view of liminfn→∞αnβn>0, by taking the limit in (3.14), one sees
(3.15)g(∥Jyn-Jxn∥)→0,asn→∞.
From the property of g, one has
(3.16)∥Jyn-Jxn∥→0,asn→∞.
Since J-1 is also uniformly norm-to-norm continuous on bounded sets, one obtains
(3.17)∥yn-xn∥→0,asn→∞.
Similarly, one could obtain
(3.18)∥zn-xn∥→0,asn→∞.
Noticing that {xn} is bounded, one gets that there exists a subsequence {xnk} of {xn} such that xnk converges weakly to x*∈C. From (3.17) and (3.18), F~(T)=F(T) and F~(S)=F(S), one has
(3.19)x*∈F~(T)∩F~(S)=F(T)∩F(S).
Next, we show thatx*∈EP(f). Let s=supn≥1{∥xn∥,∥un∥}. From Lemma 2.2, there exists a continuous, strictly increasing, and convex function g1 with g1(0)=0 such that
(3.20)g1(∥x-y∥)≤ϕ(x,y),∀x,y∈Bs.
Noticing that xn=Trnun and from Lemma 2.8 and (3.13), for p∈ℱ, one has
(3.21)g1(∥xn-un∥)≤ϕ(xn,un)≤ϕ(p,un)-ϕ(p,xn)≤ϕ(p,xn-1)-ϕ(p,xn).
Noticing that limn→∞ϕ(p,xn) exists, one gets
(3.22)limn→∞g1(∥xn-un∥)=0.
It follows from the property of g1 that
(3.23)limn→∞∥xn-un∥=0.
Since J is uniformly norm-to-norm continuous on bounded sets, one has
(3.24)limn→∞∥Jxn-Jun∥=0.
From the assumption rn≥a, one obtains
(3.25)limn→∞∥Jxn-Jun∥rn=0.
Since xn=Trnun, one has
(3.26)f(xn,y)+1rn〈y-xn,Jxn-Jun〉≥0,∀y∈C.
By replacing n by nk and from (A2), one sees
(3.27)∥y-xnk∥∥Jxnk-Junk∥rnk≥1rnk〈y-xnk,Jxnk-Junk〉≥-f(xnk,y)≥f(y,xnk),∀y∈C.
Taking k→∞ in the above inequality and from (A4), one has
(3.28)f(y,x*)≤0,∀y∈C.
For 0<t<1 and y∈C, define yt=ty+(1-t)x*. Since y,x*∈C, one gets yt∈C, which yields that f(yt,x*)≤0. It follows from (A1) that
(3.29)0=f(yt,yt)≤tf(yt,y)+(1-t)f(yt,x*)≤tf(yt,y),
that is,
(3.30)f(yt,y)≥0.
Let t↓0; from (A3), we obtain f(x*,y)≥0 for all y∈C. This implies that x*∈EP(f). Therefore, x*∈ℱ.
On the other hand, letvn=Πℱxn; from Lemma 2.9 and x*∈ℱ, we have
(3.31)〈vnk-x*,Jxnk-Jvnk〉≥0.
From Theorem 3.1, one knows that {vn} converges strongly to z∈ℱ. Since J is weakly sequentially continuous, one has
(3.32)〈z-x*,Jx*-Jz〉≥0,
as k→∞. On the other hand, since J is monotone, one has
(3.33)〈z-x*,Jx*-Jz〉≤0,
as k→∞. Also, since E is uniformly convex, one has z=x*. This completes the proof of Theorem 3.2.If we only consider one operatorT, the following corollary can been established by Theorem 3.2.Corollary 3.3.
LetC be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to R satisfying (A1)–(A4),  and let T:C→C be a weak relatively uniformly nonexpansive multivalued mapping such that ℱ=F(T)∩EP(f)≠∅. Let {xn} be a sequence generated by the following manner:
(3.34)xn∈Csuchthatf(xn,y)+1rn〈y-xn,Jxn-Jun〉≥0,∀y∈C,un+1=J-1(αnJxn+(1-αn)Jyn),
where yn∈Txn, and J is the duality mapping on E. Assume that {αn} is a sequence in [0,1]  satisfying the following conditions: (a)
liminfn→∞αn(1-αn)>0;(b)
{rn}⊂[a,∞) for some a>0. If J is weakly sequentially continuous, then {xn} converges weakly to z∈ℱ, where z=limn→∞Πℱxn.IfT and S are two relatively uniformly nonexpansive multivalued mappings, from Definitions 1.1 and 1.3, it is easy to know that the class of weak relatively uniformly nonexpansive multivalued mappings contains the class of relatively uniformly nonexpansive multivalued mappings as a subclass. Therefore, the following corollary can be easily obtained by Theorem 3.2.Corollary 3.4.
LetC be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to R satisfying (A1)–(A4) and let T,S:C→C be two relatively uniformly nonexpansive multivalued mappings such that ℱ=F(T)∩F(S)∩EP(f)≠∅. Let {xn} be a sequence generated by the following manner:
(3.35)xn∈Csuchthatf(xn,y)+1rn〈y-xn,Jxn-Jun〉≥0,∀y∈C,un+1=J-1(αnJxn+βnJyn+γnJzn),
where yn∈Txn, zn∈Sxn, and J are the duality mapping on E. Assume that {αn}, {βn}, and {γn} are three sequences in [0,1] satisfying the following conditions:  (a)
αn+βn+γn=1;(b)
liminfn→∞αnβn>0,liminfn→∞αnγn>0;(c)
{rn}⊂[a,∞) for some a>0.If J is weakly sequentially continuous, then {xn} converges weakly to z∈ℱ, where z=limn→∞Πℱxn.Remark 3.5.
Our results improve Theorem 4.1. of Takahashi and Zembayashi [12] and Theorem 3.5. of Qin et al. [11] in the following senses:  (1)
from single-valued mappings to multivalued ones;(2)
from relatively nonexpansive single-valued mappings (the definition can be found in [1, 11, 12]) to weak relatively uniformly nonexpansive multivalued ones.



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*Source: 489629-2012-11-11.xml*