# Strong Convergence Results for Equilibrium Problems and Fixed Point Problems for Multivalued Mappings

Authors: J. Vahidi; A. Latif; M. Eslamian
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Using viscosity approximation method, we study
strong convergence to a common element of the set of solutions of
an equilibrium problem and the set of common fixed points of a finite
family of multivalued mappings satisfying the condition (E) in the
setting of Hilbert space. Our results improve and extend some recent
results in the literature.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let C be a nonempty closed convex subset H. A subset C⊂H is called proximal if, for each x∈H, there exists an element y∈C such that
(1)∥x-y∥=dist(x,C)=inf{∥x-z∥:z∈C}.A single-valued mappingT:C→C is said to be nonexpansive, if
(2)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.LetPC be a nearest point projection of H into C; that is, for x∈H, PCx is a unique nearest point in C with the property
(3)∥x-PCx∥∶=inf{∥x-y∥:y∈C}.We denote byCB(C), K(C), and P(C) the collection of all nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of C respectively. The Hausdorff metric H on CB(H) is defined by
(4)H(A,B)∶=max{supx∈Adist(x,B),supy∈Bdist(y,A)},
for all A,B∈CB(H).LetT:H→2H be a multivalued mapping. An element x∈H is said to be a fixed point of T, if x∈Tx and the set of fixed points of T is denoted by F(T).A multivalued mappingT:H→CB(H) is called (i)
nonexpansive if(5)H(Tx,Ty)≤∥x-y∥,x,y∈H;(ii)
quasi-nonexpansive ifF(T)≠∅ and H(Tx,Tp)≤∥x-p∥ for all x∈H and all p∈F(T).Recently, García-Falset et al. [1] introduced a new condition on single-valued mappings, called condition (E), which is weaker than nonexpansiveness.Definition 1.
A mappingT:H→H is said to satisfy condition (Eμ) provided that
(6)∥x-Ty∥≤μ∥x-Tx∥+∥x-y∥,x,y∈H.
We say thatT satisfies condition (E) whenever T satisfies (Eμ) for some μ≥1.Recently, Abkar and Eslamian [2, 3] generalized this condition for multivalued mappings as follows.Definition 2.
A multivalued mappingT:H→CB(H) is said to satisfy condition (E) provided that
(7)H(Tx,Ty)≤μdist(x,Tx)+∥x-y∥,x,y∈H,
for some μ≥1.It is obvious that every nonexpansive multivalued mappingT:H→CB(H) satisfies the condition (E), and every mapping T:H→CB(H) which satisfies the condition (E) with nonempty fixed point set F(T) is quasi-nonexpansive.Example 3.
Let us define a mappingT on [0,3] by
(8)T(x)={[0,x3],x≠3[1,2]x=3.
It is easy to see that T satisfies the condition (E) but is not nonexpansive. Indeed, for x,y∈[0,3), H(Tx,Ty)=|(x-y)/3|≤|x-y|. Let x=0 and y=3. Then H(Tx,Ty)=2≤3=|x-y|. If x∈(0,3) and y=3, then, we have dist(x,Tx)=2x/3 and dist(y,Ty)=1; hence
(9)H(Tx,Ty)=2-x3≤3-x+4x3=|x-y|+2dist(x,Tx).
Thus, T satisfies the condition (E). However, T is not nonexpansive; indeed for x=3 and y=7/3, H(Tx,Ty)=11/9>2/3=|x-y|.
LetΨ:C×C→ℝ be a bifunction. The equilibrium problem associated with the bifunction Ψ and the set C is:
(10)findx∈CsuchthatΨ(x,y)≥0,∀y∈C.
Such a point x∈C is called the solution of the equilibrium problem. The set of solutions is denoted by EP(Ψ).
A broad class of problems in optimization theory, such as variational inequality, convex minimization, and fixed point problems, can be formulated as an equilibrium problem; see [4, 5]. In the literature, many techniques and algorithms have been proposed to analyze the existence and approximation of a solution to equilibrium problem; see [6]. Many researchers have studied various iteration processes for finding a common element of the set of solutions of the equilibrium problems and the set of fixed points of a class of nonlinear mappings. For example, see [7–22].
Fixed points and fixed point iteration process for nonexpansive mappings have been studied extensively by many authors to solve nonlinear operator equations, as well as variational inequalities; see, for example, [23–28]. In the recent years, fixed point theory for multivalued mappings has been studied by many authors; see [29–40] and the references therein.
In this paper, using viscosity approximation method, we study strong convergence to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of multivalued mappings satisfying the condition(E) in the setting of Hilbert space. Our results improve and extend some recent results in the literature.


## 2. Preliminaries

For solving the equilibrium problem, we assume that the bifunctionΨ satisfies the following conditions: (A1)
Ψ(x,x)=0 for any x∈C;(A2)
Ψ is monotone; that is, Ψ(x,y)+Ψ(y,x)≤0 for any x,y∈C;(A3)
Ψ is upper-hemicontinuous; that is, for each x,y,z∈C,
(11)limsupt→0+Ψ(tz+(1-t)x,y)≤Ψ(x,y);(A4)
Ψ(x,.) is convex and lower semicontinuous for each x∈C.Lemma 4 (see [4]).
LetC be a nonempty closed convex subset of H and let Ψ be a bifunction of C×C into ℝ satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(12)Ψ(z,y)+1r〈y-z,z-x〉≥0∀y∈C.Lemma 5 (see [6]).
Assume thatΨ:C×C→ℝ satisfies (A1)–(A4). For r>0 and x∈H, define a mapping Sr:H→C as follows:(13)Srx={z∈C:Ψ(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}.
Then, the following hold: (i)
Sr is single valued;(ii)
Sr is firmly nonexpansive; that is, for any x,y∈H,
(14)∥Srx-Sry∥2≤〈Srx-Sry,x-y〉;(iii)
F(Sr)=EP(Ψ);(iv)
EP(Ψ) is closed and convex.Lemma 6 (see [41]).
LetH be a real Hilbert space. Then, for all x,y,z∈H and α,β,γ∈[0,1] with α+β+γ=1 one has
(15)∥αx+βy+γz∥2=α∥x∥2+β∥y∥2+γ∥z∥2-αβ∥x-y∥2-αγ∥x-z∥2-βγ∥z-y∥2.Lemma 7.
For everyx and y in a Hilbert space H, the following inequality holds:
(16)∥x+y∥2≤∥x∥2+2〈y,x+y〉.Lemma 8 (see [42]).
Let{an} be a sequence of nonnegative real numbers, {αn} a sequence in (0,1) with ∑n=1∞‍αn=∞, {γn} a sequence of nonnegative real numbers with ∑n=1∞‍γn<∞, and {βn} a sequence of real numbers with limsupn→∞βn≤0. Suppose that the following inequality holds:
(17)an+1≤(1-αn)an+αnβn+γn,n≥0.
Then, limn→∞an=0.Lemma 9 (see [43]).
Let{un} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {uni} of {un} such that uni<uni+1 for all i≥0. For every sufficiently large number n≥n0, define an integer sequence {τ(n)} as
(18)τ(n)=max{k≤n:uk<uk+1}.
Then, τ(n)→∞ as n→∞ and for all n≥n0,
(19)max{uτ(n),un}≤uτ(n)+1.Lemma 10 (see [20]).
LetC be a closed convex subset of a real Hilbert space H. Let T:C→CB(C) be a quasi-nonexpansive multivalued mapping. If F(T)≠∅ and T(p)={p} for all p∈F(T). Then F(T) is closed and convex.Lemma 11 (see [20]).
LetC be a closed convex subset of a real Hilbert space H. Let T:C→P(C) be a multivalued mapping such that PT is quasi-nonexpansive and F(T)≠∅, where PT(x)={y∈Tx:∥x-y∥=dist(x,Tx)}. Then, F(T) is closed and convex.Lemma 12 (see [16, 20]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→K(C) be a multivalued mapping satisfying the condition (E). If xn converges weakly to v and limn→∞dist(xn,Txn)=0, then v∈Tv.


## 3. A Strong Convergence Theorem

Theorem 13.
LetC be a nonempty closed convex subset of a real Hilbert space H and Ψ a bifunction of C×C into ℝ satisfying (A1)–(A4). Let Ti:C→CB(C) (i=1,2,…,m) be a finite family of multivalued mappings, each satisfying condition (E). Assume further that ℱ=⋂i=1m‍F(Ti)⋂‍EP(Ψ)≠∅ and Ti(p)={p},(i=1,2,…,m) for each p∈ℱ. Let f be a k-contraction of C into itself. Let {xn} and {un} be sequences generated the following algorithm:
(20)x0∈C,un∈CsuchthatΨ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈Cyn,1=an,1un+bn,1xn+cn,1zn,1,yn,2=an,2un+bn,2zn,1+cn,2zn,2,yn,3=an,3un+bn,3zn,2+cn,3zn,3⋮yn,m=an,mun+bn,mzn,m-1+cn,mzn,m,xn+1=ϑnf(xn)+(1-ϑn)yn,m,∀n≥0,
where zn,1∈T1(un), zn,k∈Tk(yn,k-1) for k=2,…,m, and {an,i}, {bn,i}, {cn,i}, {ϑn}, and {rn} satisfy the following conditions: (i)
{an,i},{bn,i},{cn,i}⊂[a,b]⊂(0,1),an,i+bn,i+cn,i=1,(i=1,2,…,m),(ii)
{ϑn}⊂(0,1), limn→∞ϑn=0, ∑n=1∞‍ϑn=∞,(iii)
{rn}⊂(0,∞), and liminfn→∞rn>0. Then, the sequences {xn} and {un} converge strongly to q∈ℱ, where q=Pℱf(q).Proof.
LetQ=Pℱ. It is easy to see that Qf is a contraction. By Banach contraction principle, there exists a q∈ℱ such that q=Pℱf(q). From Lemma 5 for all n≥0, we have (21)∥un-q∥=∥Srnxn-Srnq∥≤∥xn-q∥.
We show that {xn} is bounded. Since, for each i=1,2,…,m, Ti satisfies the condition (E) and we have
(22)∥yn,1-q∥=∥an,1un+bn,1xn+cn,1zn,1-q∥≤an,1∥un-q∥+bn,1∥xn-q∥+cn,1∥zn,1-q∥=an,1∥un-q∥+bn,1∥xn-q∥+cn,1dist(zn,1,T1q)≤an,1∥un-q∥+bn,1∥xn-q∥+cn,1H(T1un,T1q)≤an,1∥un-q∥+bn,1∥xn-q∥+cn,1∥un-q∥≤∥xn-q∥,(23)∥yn,2-q∥=∥an,2un+bn,2zn,1+cn,2zn,2-q∥≤an,2∥un-q∥+bn,2∥zn,1-q∥+cn,2∥zn,2-q∥=an,2∥un-q∥+bn,2dist(zn,1,T1q)+cn,2dist(zn,2,T2q)≤an,2∥un-q∥+bn,2H(T1un,T1q)+cn,2H(T2yn,1,T2q)≤an,2∥un-q∥+bn,2∥un-q∥+cn,2∥yn,1-q∥≤∥xn-q∥.
By continuing this process, we obtain(24)∥yn,m-q∥≤∥xn-q∥.
This implies that(25)∥xn+1-q∥=∥ϑnfxn+(1-ϑn)yn-q∥≤ϑn∥fxn-q∥+(1-ϑn)∥yn-q∥≤ϑn(∥fxn-fq∥+∥fq-q∥)+(1-ϑn)∥xn-q∥≤ϑnk∥xn-q∥+ϑn∥fq-q∥+(1-ϑn)∥xn-q∥=(1-ϑn(1-k))∥xn-q∥+ϑn∥fq-q∥≤max{∥xn-q∥,∥fq-q∥1-k}.
By induction, we get
(26)∥xn-q∥≤max{∥x0-q∥,∥fq-q∥1-k},
for all n∈ℕ. This implies that {xn} is bounded and we also obtain that {un},{yn},{fxn}, and {zn,i} are bounded. Next, we show that limn→∞dist(un,Tiun)=0 for each i∈ℕ. By Lemma 6, we have
(27)∥yn,1-q∥2=∥an,1un+bn,1xn+cn,1zn,1-q∥2≤an,1∥un-q∥2+bn,1∥xn-q∥2+cn,1∥zn,1-q∥2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2=an,1∥un-q∥2+bn,1∥xn-q∥2+cn,1dist(zn,1,T1q)2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2≤an,1∥un-q∥2+bn,1∥xn-q∥2+cn,1H(T1un,T1q)2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2≤an,1∥un-q∥2+bn,1∥xn-q∥2+cn,1∥un-q∥2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2≤∥xn-q∥2-an,1bn,1∥xn-un∥2-an,1cn,1∥un-zn,1∥2.
Applying Lemma 6 once more, we have
(28)∥yn,2-q∥2=∥an,2un+bn,2zn,1+cn,2zn,2-q∥2≤an,2∥un-q∥2+bn,2∥zn,1-q∥2+cn,2∥zn,2-q∥2-an,2cn,2∥un-zn,2∥2=an,2∥un-q∥2+bn,2dist(zn,1,T1q)2+cn,2dist(zn,2,T2q)2-an,2cn,2∥un-zn,2∥2≤an,2∥un-q∥2+bn,2H(T1un,T1q)2+cn,2H(T1yn,1,T2q)2-an,2cn,2∥un-zn,2∥2≤an,2∥un-q∥2+bn,2∥un-q∥2+cn,2∥yn,1-q∥2-an,2cn,2∥un-zn,2∥2≤∥xn-q∥2-an,2cn,2∥un-zn,2∥2-an,1cn,1cn,2∥un-zn,1∥2-an,1bn,1cn,2∥xn-un∥2.
By continuing this process we have
(29)∥yn,m-q∥2=∥an,mun+bn,mzn,m-1+cn,mzn,m-q∥2≤an,m∥un-q∥2+bn,m∥zn,m-1-q∥2+cn,m∥zn,m-q∥2-an,mcn,m∥un-zn,m∥2=an,m∥un-q∥2+bn,mdist(zn,m-1,Tm-1q)2+cn,mdist(zn,m,Tmq)2-an,mcn,m∥un-zn,m∥2≤an,m∥un-q∥2+bn,mH(Tm-1yn,m-2,Tm-1q)2+cn,mH(Tmyn,m-1,Tmq)2-an,mcn,m∥un-zn,m∥2≤an,m∥un-q∥2+bn,m∥yn,m-2-q∥2+cn,m∥yn,m-1-q∥2-an,mcn,m∥un-zn,m∥2≤∥un-q∥2-an,mcn,m∥un-zn,m∥2-an,m-1cn,m-1cn,m∥un-zn,m-1∥2-⋯-an,1cn,1cn,2…cn,m∥un-zn,1∥2-an,1bn,1cn,2…cn,m∥un-xn∥2,
which implies that
(30)∥xn+1-q∥2=∥ϑnfxn+(1-ϑn)yn,m-q∥2≤ϑn∥fxn-q∥2+(1-ϑn)∥yn,m-q∥2≤ϑn∥fxn-q∥2+(1-ϑn)∥un-q∥2-(1-ϑn)an,mcn,m∥un-zn,m∥2-(1-ϑn)an,m-1cn,m-1cn,m∥un-zn,m-1∥2-⋯-(1-ϑn)an,1cn,1cn,2…cn,m∥un-zn,1∥2-(1-ϑn)an,1bn,1cn,2…cn,m∥un-xn∥2.
Therefore, we have that
(31)(1-ϑn)an,1bn,1cn,2…cn,m∥un-xn∥2≤∥xn-q∥2-∥xn+1-q∥2+ϑn∥γfxn-q∥.
In order to prove that xn→q as n→∞, we consider the following two cases.
Case 1. Suppose that there exists n0 such that {∥xn-q∥} is nonincreasing, for all n≥n0. Boundedness of {∥xn-q∥} implies that ∥xn-q∥ is convergent. From (31) and conditions (i), (ii) we have that
(32)limn→∞∥un-xn∥=0.
By a similar argument, for k=1,2,…,m, we obtain that
(33)limn→∞∥un-zn,k∥=0.
Hence,
(34)limn→∞dist(un,T1un)≤limn→∞∥un-zn,1∥=0,limn→∞dist(un,Tkyn,k-1)≤limn→∞∥un-zn,k∥=0,(k=2,…,m).
Therefore, we have
(35)limn→∞∥un-yn,1∥≤limn→∞bn,1∥un-xn∥+limn→∞cn,1∥un-zn,1∥=0.
For k=2,…,m, we have
(36)limn→∞∥un-yn,k∥≤limn→∞bn,k∥un-zn,k-1∥+limn→∞cn,k∥un-zn,k∥=0.
Using the previous inequality for k=2,…,m, we have
(37)dist(un,Tkun)≤dist(un,Tkyn,k-1)+H(Tkyn,k-1,Tkun)≤dist(un,Tkyn,k-1)+μdist(yn,k-1,Tkyn,k-1)+∥yn,k-1-un∥≤(μ+1)dist(un,Tkyn,k-1)+(μ+1)∥yn,k-1-un∥≤(μ+1)∥un-zn,k∥+(μ+1)∥yn,k-1-un∥⟶0,n⟶∞.
Next, we show that
(38)limsupn→∞〈q-fq,q-xn〉≤0,
where q=Pℱf(q). To show this inequality, we choose a subsequence {xni} of {xn} such that
(39)limi→∞〈q-fq,q-xni〉=limsupn→∞〈q-fq,q-xn〉.
Since {xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly to v. Without loss of generality, we can assume that xni converges weakly to v. Since limn→∞∥xn-un∥=0, we have uni converges weakly to v. We show that v∈ℱ. Let us show v∈EP(Ψ). Since un=Srnxn, we have
(40)Ψ(un,y)+1rn〈y-un,un-xn〉≥0∀y∈C.
From (A2), we have
(41)1rn〈y-un,un-xn〉≥Ψ(y,un).
Replacing n with ni, we have
(42)〈y-uni,uni-xnirni〉≥Ψ(y,uni).
From (A4), we have
(43)0≥Ψ(y,v),∀y∈C.
For t∈(0,1] and y∈C, let yt=ty+(1-t)v. Since y,v∈C, and C is convex, we have yt∈C and hence Ψ(yt,v)≤0. So, from (A1) and (A4) we have
(44)0=Ψ(yt,yt)≤tΨ(yt,y)+(1-t)Ψ(yt,v)≤tΨ(yt,y),
which gives 0≤Ψ(yt,y). Letting t→0, we have, for each y∈C, 0≤Ψ(v,y) Also, since uni⇀v and limn→∞dist(un,Tiun)=0, by Lemma 12 we have v∈⋂i=1m‍F(Ti). Hence, v∈ℱ. Since q=Pℱf(q) and v∈ℱ, it follows that
(45)limsupn→∞〈q-fq,q-xn〉=limi→∞〈q-fq,q-xni〉=〈q-fq,q-v〉≤0.
By using Lemma 7 and inequality (31) we have
(46)∥xn+1-q∥2≤∥(1-ϑn)(yn,m-q)∥2+2ϑn〈fxn-q,xn+1-q〉≤(1-ϑn)2∥yn,m-q∥2+2ϑn〈fxn-fq,xn+1-q〉+2ϑn〈fq-q,xn+1-q〉≤(1-ϑn)2∥xn-q∥2+2ϑnk∥xn-q∥∥xn+1-q∥+2ϑn〈fq-q,xn+1-q〉≤(1-ϑn)2∥xn-q∥2+ϑnk(∥xn-q∥2+∥xn+1-q∥2)+2ϑn〈fq-q,xn+1-q〉≤((1-ϑn)2+ϑnk)∥xn-q∥2+ϑnk∥xn+1-q∥2+2ϑn〈fq-q,xn+1-q〉.
This implies that(47)∥xn+1-q∥2≤(1-2(1-k)ϑn1-ϑnk)∥xn-q∥2+ϑn21-ϑnk∥xn-q∥2+2ϑn1-ϑnk〈fq-q,xn+1-q〉.
From Lemma 8, we conclude that the sequence {xn} converges strongly to q.
Case 2. Assume that there exists a subsequence {xnj} of {xn} such that
(48)∥xnj-q∥<∥xnj+1-q∥,
for all j∈ℕ. In this case, from Lemma 9, there exists a nondecreasing sequence {τ(n)} of ℕ for all n≥n0 (for some n0 large enough) such that τ(n)→∞ as n→∞ and the following inequalities hold for all n≥n0:(49)∥xτ(n)-q∥≤∥xτ(n)+1-q∥,∥xn-q∥≤∥xτ(n)+1-q∥.
From (31) we obtain limn→∞∥uτ(n)-Tiuτ(n)∥=0, and limn→∞∥uτ(n)-xτ(n)∥=0. Following an argument similar to that in Case 1, we have
(50)limn→∞∥xτ(n)-q∥=0,limn→∞∥xτ(n)+1-q∥=0.
Thus, by Lemma 9 we have
(51)0≤∥xn-q∥≤max{∥xτ(n)-q∥,∥xn-q∥}≤∥xτ(n)+1-q∥.
Therefore, {xn} converges strongly to q=Pℱf(q). This completes the proof.Now, we remove the condition thatT(p)={p} for all p∈ℱ, and state the following theorem.Theorem 14.
LetC be a nonempty closed convex subset of a real Hilbert space H and Ψ a bifunction of C×C into ℝ satisfying (A1)–(A4). Let, for each 1≤i≤m, Ti:C→P(C) be multivalued mappings such that PTi satisfies the condition (E). Assume that ℱ=⋂i=1m‍F(Ti)⋂‍EP(Ψ)≠∅. Let f be a k-contraction of C into itself. Let {xn} and {un} be sequences generated the following algorithm:
(52)x0∈C,un∈CsuchthatΨ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈Cyn,1=an,1un+bn,1xn+cn,1zn,1,yn,2=an,2un+bn,2zn,1+cn,2zn,2,yn,3=an,3un+bn,3zn,2+cn,3zn,3⋮yn,m=an,mun+bn,mzn,m-1+cn,mzn,m,xn+1=ϑnfxn+(1-ϑn)yn,m,∀n≥0,
where zn,1∈PT1(un), zn,k∈PTk(yn,k-1) for k=2,…,m, and {an,i}, {bn,i}, {cn,i}, {ϑn} and, {rn} satisfy the following conditions: (i)
{an,i},{bn,i},{cn,i}⊂[a,b]⊂(0,1),an,i+bn,i+cn,i=1,(i=1,2,…,m),(ii)
{ϑn}⊂(0,1), limn→∞ϑn=0, ∑n=1∞‍ϑn=∞,(iii)
{rn}⊂(0,∞), and liminfn→∞rn>0. Then, the sequences {xn} and {un} converge strongly to q∈ℱ, where q=Pℱf(q).Proof.
Letp∈ℱ; then PTi(p)={p},(i=1,2,…,m). Now by substituting PTi instead of Ti, and using a similar argument as in the proof of Theorem 13, the desired result follows.As a corollary for single-valued mappings, we obtain the following result.Corollary 15.
LetC be a nonempty closed convex subset of a real Hilbert space H and Ψ a bifunction of C×C into ℝ satisfying (A1)–(A4). Let, for each 1≤i≤m, Ti:C→C be a finite family of mappings satisfying condition (E). Assume that ℱ=⋂i=1m‍F(Ti)⋂‍EP(Ψ)≠∅. Let f be a k-contraction of C into itself. Let {xn} and {un} be sequences generated the following algorithm:
(53)x0∈C,un∈CsuchthatΨ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈Cyn,1=an,1un+bn,1xn+cn,1T1un,yn,2=an,2un+bn,2T1un+cn,2T2yn,1⋮yn,m=an,mun+bn,mTm-1yn,m-2+Tmyn,m-1,xn+1=ϑnfxn+(1-ϑn)yn,m,∀n≥0,
where {an,i}, {bn,i},{cn,i}, {ϑn}, and {rn} satisfy the following conditions: (i)
{an,i},{bn,i},{cn,i}⊂[a,b]⊂(0,1),an,i+bn,i+cn,i=1, (i=1,2,…,m),(ii)
{ϑn}⊂(0,1), limn→∞ϑn=0, ∑n=1∞‍ϑn=∞(iii)
{rn}⊂(0,∞), and liminfn→∞rn>0. Then, the sequences {xn} and {un} converge strongly to q∈ℱ, where q=Pℱf(q).Remark 16.
Our results generalize the corresponding results of S. Takahashi and W. Takahashi [9] from a single valued nonexpansive mapping to a finite family of multivalued mappings satisfying the condition (E). Our results also improve the recent results of Eslamian [16].



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*Source: 825130-2013-11-13.xml*