# Viscosity-Projection Method for a Family of General Equilibrium Problems and Asymptotically Strict Pseudocontractions in the Intermediate Sense

Authors: Dao-Jun Wen
Journal: The Scientific World Journal
 (2013)

---

## Abstract

In this paper, a Meir-Keeler contraction is introduced to propose a viscosity-projection approximation method for finding a common element of the set of solutions of a family of general equilibrium problems and the set of fixed points of asymptotically strict pseudocontractions in the intermediate sense. Strong convergence of the viscosity iterative sequences is obtained under some suitable conditions. Results presented in this paper extend and unify the previously known results announced by many other authors.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H. Let A:C→H be a nonlinear mapping and F:C×C→ℝ be a bifunction, where ℝ denotes the set of real numbers. We consider the following generalized equilibrium problem: Find x∈C such that
(1)F(x,y)+〈Ax,y-x〉≥0,∀y∈C.
We use EP(F,A) to denote the set of solution of problem (1). If A≡0, the zero mapping, then the problem (1) reduces to the normal equilibrium problem: Find x∈C such that
(2)F(x,y)≥0,∀y∈C.
We use EP(F) to denote the set of solution of problem (2). If F≡0, then the problem (1) reduces to the classical variational inequality problem: Find x∈C such that
(3)〈Ax,y-x〉≥0,∀y∈C.
We use VI(C,A) to denote the set of solution of problem (3). The generalized equilibrium problem (1) is very general in the sense that it includes, as special cases, saddle point problems, variational inequalities, optimization problems, mini-max problems, the Nash equilibrium problem in noncooperative games, and others (see, e.g., [1–4]).Recall that a nonlinear mappingT:C→C is said to be nonexpansive if
(4)∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.T is said to be uniformly L-Lipschitz continuous if there exists a constant L>0 such that
(5)∥Tnx-Tny∥≤L∥x-y∥,n≥1,∀x,y∈C.T is said to be asymptotically nonexpansive if there exists a sequence kn∈[1,∞) with kn→1 as n→∞ such that
(6)∥Tnx-Tny∥≤kn∥x-y∥,n≥1,∀x,y∈C.T is said to be asymptotically nonexpansive in the intermediate sense [5] if it is continuous and the following inequality holds:
(7)limsupn→∞supx,y∈C(∥Tnx-Tny∥-∥x-y∥)≤0.
Putting ξn=max{0,supx,y∈C(∥Tnx-Tny∥-∥x-y∥)}, we see that ξn→0 as n→∞. Then scheme (7) is reduced to the following:
(8)∥Tnx-Tny∥≤∥x-y∥+ξn,∀x,y∈C.
The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk [5] as a generalization of the class of asymptotically nonexpansive mappings. It is known that, if C is a nonempty bounded closed convex subset of a real Hilbert space H, then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point (see, e.g., [6]).Recall also thatT is said to be a λ-strict pseudocontraction [7, 8] if there exists a coefficient λ∈[0,1) such that
(9)∥Tx-Ty∥2≤∥x-y∥2+λ∥(I-T)x-(I-T)y∥2,∀x,y∈C.T is said to be an asymptotically λ-strict pseudocontraction [9, 10] if there exists a sequence kn∈[1,∞) with kn→1 as n→∞ and a constant λ∈[0,1) such that
(10)∥Tnx-Tny∥2≤kn∥x-y∥2+λ∥(x-Tnx)-(y-Tny)∥2,n≥1,∀x,y∈C.T is said to be an asymptotically λ-strict pseudocontraction in the intermediate sense [11, 12] if there exists a sequence kn∈[1,∞) with kn→1 as n→∞ and a constant λ∈[0,1) such that
(11)limsupn→∞supx,y∈C(∥Tnx-Tny∥2-kn∥x-y∥2-λ∥(x-Tnx)-(y-Tny)∥2)≤0,∀x,y∈C.
Putting en=max{0,supx,y∈C(∥Tnx-Tny∥2-kn∥x-y∥2-λ∥(x-Tnx)-(y-Tny)∥2)}, we see that en→0 as n→∞. Then scheme (11) is reduced to the following:
(12)∥Tnx-Tny∥2≤kn∥x-y∥2+λ∥(x-Tnx)-(y-Tny)∥2+en,n≥1,∀x,y∈C.
We use Fix(T) to denote the set of fixed point of T, that is, Fix(T)={x∈C:Tx=x}. The class of asymptotically strict pseudocontractions in the intermediate sense was introduced as a generalization of the asymptotically strict pseudocontractions and asymptotically nonexpansive in the intermediate sense. Clearly, a nonexpansive mapping is a 0-strict pseudocontraction, and an asymptotically nonexpansive mapping is an asymptotically 0-strict pseudocontraction. (see, e.g., [7–12]).Fixed point technique represent an important tool for finding the approximate solution of equilibrium problem and its variant forms, which have been studied extensively in recent years due to their applications in physics, economics, optimization, and pure and applied sciences. Some numerical methods have been proposed for finding a common element of the set of fixed point of various types of nonexpansive mappings and the set of solution of equilibrium problems with bifunctions satisfying certain conditions; see [8–20] and references therein.In 2009, Qin et al. [10] introduced the following explicit iterative algorithm for finding a common fixed point of a finite family of asymptotically λi-strictly pseudocontractions Ti for each i=1,2,…,N(13)xn=αn-1xn-1+(1-αn-1)Ti(n)h(n)xn-1,n≥1,
where x0∈C, {αn}n=0∞ is a sequence in (0,1) and n=[h(n)-1]N+i(n),  i=i(n)=1,2,…,N. They also obtain weak and strong convergence theorems based on the cyclic scheme above.Recently, Sahu et al. [11] considered a new iterative scheme for asymptotically strictly pseudocontractive mappings in the intermediate sense. To be more precise, they proved the following theorem.Theorem SXY.
LetC be a nonempty closed and convex subset of a real Hilbert space H and T:C→C be a uniformly continuous asymptotically λ-strictly pseudocontractive mapping in the intermediate sense with a sequence {kn} such that Fix(T) is nonempty and bounded. Let {αn} be a sequence in [0,1] such that 0<δ≤αn≤1-λ for all n∈ℕ. Let {xn}⊂C be a sequence generated by the following (CQ) algorithm:
(14)u=x1∈C,yn=(1-αn)un+αnTnxn,Cn={w∈C:∥yn-w∥2≤∥xn-w∥2+θn},Qn={w∈C:〈xn-w,u-xn〉≥0},xn+1=PCn∩Qn(u),
where θn=(kn-1)ρn2+en and ρn=sup{∥xn-p∥:p∈Fix(T)}<∞. Then, {xn} converges strongly to PFix(T)(u), where PFix(T) is metric projection from H onto Fix(T).
In 2011, Hu and Cai [12] modified schemes (13) and (14) to the case of asymptotically strictly pseudocontractive mappings in the intermediate sense concerning the equilibrium problem and proposed the following modified hybrid method:
(15)x0∈C,u0∈C,F(un,y)+〈Axn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,zn=(1-βn)un+βnTi(n)h(n)un,yn=(1-αn)un+αnzn,Cn={w∈C:∥yn-w∥2≤∥xn-w∥2+θn},Qn={w∈C:〈xn-w,x0-xn〉≥0},xn+1=PCn∩Qn(x0),
where θn=(kh(n)-1)ρn2+eh(n)→0 as n→∞ and ρn=sup{∥xn-p∥:p∈Ω}<∞. Moreover, they obtained convergence theorems under some suitable conditions.
On the other hand, Moudafi [13] introduced the following viscosity approximation method for fixed point problem of nonexpansive mapping
(16)xn+1=αnf(xn)+(1-αn)Txn,
where f is a contractive mapping. He proved that the viscosity iterative sequence {xn} convergence strongly to a fixed point of T, which is the unique solution of the variational inequality:
(17)〈(I-f)x,z-x〉≥0,∀z∈Fix(T).
Furthermore, S. Takahashi and W. Takahashi [14] and Inchan [15] modified the viscosity approximation methods for finding a common element of the set of fixed point problems and equilibrium problems.
In 2012, Kimura and Nakajo [16] introduced a Meir-Keeler contraction and proposed a modified viscosity approximations by the shrinking projection method in Hilbert spaces, the so-called viscosity-projection method. To be more precise, they proved the following theorem.Theorem KN.
LetC be a nonempty closed convex subset of H, and let {Tn} be a sequence of mappings of C into itself with Ω=⋂n=1∞‍Fix(Tn)≠Ø which satisfies the following condition: there exists {an}⊂ℝ with liminfn→∞an>-1 such that ∥Tnx-z∥2≤∥x-z∥2-an∥x-Tnx∥2 for every n∈ℕ,x∈C, and z∈Ω. Let f be a Meir-Keeler contraction of C into itself, and let {xn} be a sequence generated by
(18)x1=x∈C,C1=C,yn=Tnxn,Cn+1={w∈Cn:∥yn-w∥2≤∥xn-w∥2-an∥xn-yn∥2},xn+1=PCn+1f(xn),
for each n∈ℕ. For every sequence {zn}⊂C and zn→z∈C and Tnzn→z imply that z∈Ω. Then, {xn} converges strongly to q∈Ω, which satisfies q=PΩf(q).In this paper, inspired and motivated by research going on in this area, we introduce a new viscosity-projection method for a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, which is defined in the following way:(19)x1∈C,C1=C,un=FrM,nAMFrM-1,nAM-1⋯Fr2,nA2Fr1,nA1xn,zn=(1-βn)un+βnTi(n)h(n)un,yn=(1-αn)un+αnzn,Cn+1={w∈Cn:∥yn-w∥2≤∥xn-w∥2+θn},xn+1=PCn+1f(xn),
where θn=(kh(n)-1)ρn2+eh(n)→0 as n→∞ and ρn=sup{∥xn-p∥:p∈Ω}<∞.Our purpose is not only to extend the viscosity-projection method with a Meir-Keeler contraction to the case of a family of general equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense, but also to obtain a strong convergence theorem by using the proposed schemes under some appropriate conditions. Results presented in this paper extend and unify the corresponding ones of [10–13, 16].


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real Hilbert space H with inner product 〈·,·〉 and norm ∥·∥, respectively. We use notation ⇀ for weak convergence and → for strong convergence of a sequence. For every point x∈H, there exists a unique nearest point in C, denoted by PCx, such that
(20)∥x-PCx∥≤∥x-y∥,∀y∈C.PC is called the metric projection of H onto C defined by PC(x)=argminy∈C∥x-y∥. It is well known that PC is nonexpansive mapping, and u=PCx is equivalent to (see, e.g., [21]) the following:
(21)〈x-u,u-y〉≥0,∀y∈C.Recall that a mappingA:C→H is said to be monotone if
(22)〈Ax-Ay,x-y〉≥0,∀x,y∈C.A is said to be r-strongly monotone if there exists a constant r>0 such that
(23)〈Ax-Ay,x-y〉≥r∥x-y∥2,∀x,y∈C.A is said to be α-inverse strongly monotone if there exists a constant α>0 such that
(24)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.
It is easy to see that if A is an α-inverse strongly monotone mapping from C into H, then A is 1/α-Lipschitz continuous.To study the generalized equilibrium problem (1), we may assume that the bifunction F:C×C→ℝ satisfies the following conditions: (A1)
F(x,x)=0for all x∈C;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0for all x,y∈C;(A3)
for eachx,y,z∈C, limt→0F(tz+(1-t)x,y)≤F(x,y);(A4)
for eachx∈C,y↦F(x,y) is convex and lower semi-continuous.Lemma 1 (see [1, 3]).
LetF:C×C→ℝ be a bifunction satisfying (A1)–(A4). Then, for any r>0 and x∈H, there exists z∈C such that
(25)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Further, if Frx={z∈C:F(z,y)+(1/r)〈y-z,z-x〉≥0,∀y∈C}, then the following hold: (1)
Fr is single-valued;(2)
Fr is firmly nonexpansive, that is, ∥Frx-Fry∥2≤〈Frx-Fry,x-y〉 for all x,y∈H;(3)
Fix(Fr)=EP(F);(4)
EP(F) is closed and convex.Lemma 2 (see [8]).
In a Hilbert spaceH, there hold the following identities: (i)
∥x+y∥2≤∥x∥2+2〈y,(x+y)〉, for all x,y∈H;(ii)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2, for all t∈[0,1], for all x,y∈H.Lemma 3 (see [8]).
LetC be a nonempty closed convex subset of a real Hilbert space H. For any x,y,z∈H and given also a real number a∈ℝ, the set
(26){v∈C:∥y-v∥2≤∥x-v∥2+〈z,v〉+a}
is closed and convex.Lemma 4 (see [11]).
LetC be a nonempty closed convex subset of a real Hilbert space H and T:C→C be a uniformly L-Lipschitz continuous and asymptotically λ-strict pseudocontraction in the intermediate sense. Then Fix(T) is closed and convex.Lemma 5 (see [11]).
LetC be a nonempty closed convex subset of a real Hilbert space H and T:C→C be a uniformly L-Lipschitz continuous and asymptotically λ-strict pseudocontraction in the intermediate sense. Then I-T is demiclosed at zero, that is, if the sequence {xn}⊂C such that xn⇀x and xn-Txn→0 as n→∞, then x∈Fix(T).Lemma 6 (see [11]).
LetC be a nonempty closed convex subset of a real Hilbert space H and T:C→C be an asymptotically λ-strict pseudocontraction in the intermediate sense with γn=kn-1. Then
(27)∥Tnx-Tny∥≤11-λ{[1+(1-λ)γn]∥x-y∥2+(1-λ)enλ∥x-y∥+[1+(1-λ)γn]∥x-y∥2+(1-λ)en},n≥1,∀x,y∈C.
Recall also that a mappingf of a complete metric space (X,d) into itself is called a contraction with coefficient r∈(0,1) if ∥f(x)-f(y)∥≤r∥x-y∥, for all x,y∈X. It is known that f has a unique fixed point (see, e.g., [22]). On the other hand, Meir and Keeler [23] defined the following mapping, called the Meir-Keeler contraction. A mapping f:X→X is called a Meir-Keeler contraction if, for every ϵ>0, there exists δ>0 such that d(x,y)<ϵ+δ implies that
(28)d(f(x),f(y))<ϵ,∀x,y∈X.
We know that Meir-Keeler contraction is a generalization of contraction, and the following result, which extends the Banach contraction principle, is proved in [23].Lemma 7 (see [23]).
A Meir-Keeler contraction defined on a complete metric space has a unique fixed point.Lemma 8 (see [24]).
Letf be a Meir-Keeler contraction on a convex subset C of a Banach space E. Then, for every ϵ>0, there exists r∈(0,1) such that ∥x-y∥≥ϵ implies that
(29)∥f(x)-f(y)∥≤r∥x-y∥,∀x,y∈C.
Let{Cn} be a sequence of nonempty closed convex subsets of H. We define a subset s-LinCn of H as follows: x∈s-LinCn if and only if there exists {xn}⊂H such that xn→x and xn∈Cn for all n∈ℕ. Similarly, a subset w-LsnCn of H is defined by y∈w-LsnCn if and only if there exists a subsequence {Cni} of {Cn} and a sequence {yi}⊂H such that yi⇀y and yi∈Cni for all i∈ℕ. If C0⊂H satisfies
(30)C0=s-LinCn=w-LsnCn,
it is said that {Cn} converges to C0 in the sense of Mosco [25], and we write C0=M-limnCn. One of the simplest examples of Mosco convergence is a decreasing sequence {Cn} with respect to inclusion. The Mosco limit of such a sequence is ⋂n=1∞‍Cn. For more details, see [26].Lemma 9 (see [27]).
Let{Cn} be a sequence of nonempty closed convex subsets of H. If C0=M-limnCn exists and is nonempty, then, for each x∈H, {PCnx} converges strongly to PC0x.
For the rest of this paper, letFm:C×C→ℝ be a bifunction satisfying (A1)–(A4) and Am:C→H be an αm-inverse strongly monotone mapping, for some m=1,2,…,M. For each rm>0 and x∈H, define a mapping FrmAm:H→C as follows:
(31)FrmAm(x)={1rmz∈C:Fm(z,y)+〈Amx,y-z〉+1rm〈y-z,z-x〉≥0,∀y∈C}.
It follows from Lemma 1 that FrmAm=Frm(I-rmAm) for each m=1,2,…,M.
LetTi:C→C be a uniformly Li-Lipschitz continuous and asymptotically λi-strict pseudocontractive mapping in the intermediate sense with the sequences {kn,i}⊂[1,∞) such that limn→∞kn,i=1 and {en,i}⊂[0,∞) such that limn→∞en,i=0, for some i=1,2,…,N, that is,
(32)∥Tinx-Tiny∥2≤kn,i∥x-y∥2+λi∥(x-Tinx)-(y-Tiny)∥2+en,i,∀x,y∈C.
Remark that L=max{Li:1≤i≤N}, λ=max{λi:1≤i≤N}, kn=max{kn,i:1≤i≤N,n∈ℕ} and en=max{en,i:1≤i≤N,n∈ℕ}.


## 3. Main Results

Theorem 10.
LetC be a nonempty closed convex subset of Hilbert space H. Let Fm:C×C→ℝ be a bifunction satisfying (A1)–(A4), and let Am:C→H be an αm-inverse strongly monotone mapping, for each m=1,2,…,M. Let Ti:C→C be a uniformly Li-Lipschitz continuous and asymptotically λi-strict pseudocontractive mapping in the intermediate sense with the sequences {kn,i} and {en,i} for each i=1,2,…,N. If f is a Meir-Keeler contraction of C into itself and Ω=(⋂i=1N‍Fix(Ti))∩(⋂m=1M‍EP(Fm,Am)) is nonempty and bounded. Assume that {αn},{βn} are sequences in [0,1] such that 0<a≤αn≤1, 0<b≤βn≤1-λ and {rm,n}⊂(0,∞) such that rm,n∈[c,d]⊂(0,2αm), for each m=1,2,…,M and n∈ℕ. Then the sequence {xn} generated by (19) converges strongly to q=PΩf(q).Proof.
We split the proof into six steps.
Step  1. We prove that PΩf exists a unique fixed point. To do this, we first show that (I-rm,nAm) is nonexpansive for each m=1,2,…,M. Indeed,
(33)∥(I-rm,nAm)x-(I-rm,nAm)y∥2=∥(x-y)-rm,n(Amx-Amy)∥2=∥x-y∥2-2rm,n〈Amx-Amy,x-y〉+rm,n2∥Amx-Amy∥2≤∥x-y∥2-rm,n(2αm-rm,n)∥Amx-Amy∥2≤∥x-y∥2.
It follows that (I-rm,nAm) is nonexpansive. By Lemma 1, we know that ⋂m=1M‍EP(Fm,Am) is closed and convex. We also know from Lemma 4 that ⋂i=1N‍Fix(Ti) is closed, and convex. Hence, Ω=(⋂i=1N‍Fix(Ti))∩(⋂m=1M‍EP(Fm,Am)) is a nonempty, closed and convex subset of C. Consequently, PΩ is well-defined. Since PΩ is nonexpansive, the composed mapping PΩf of C into itself is a Meir-Keeler contraction on C; see [24, Proposition 3]. By Lemma 7, there exists a unique fixed point z∈Ω of PΩf.
Step  2. We show that Cn is closed convex subset of C for each n≥1. By the assumption of Cn+1, it is easy to see that Cn is closed for each n≥1. We only show that Cn is convex for each n≥1. It is obvious that C1=C is closed and convex. Suppose that Ck is closed and convex for some k≥1. For any w∈Ck, we see that
(34)∥yk-w∥2≤∥xk-w∥2+θk,k≥1,
is equivalent to
(35)2〈xk-yk,w〉≤∥xk∥2-∥yk∥2+θk.
Taking w1 and w2 in Ck+1 and putting w¯=tw1+(1-t)w2, it follows that w1,w2∈Ck, and so
(36)2〈xk-yk,w1〉≤∥xk∥2-∥yk∥2+θk,(37)2〈xk-yk,w2〉≤∥xk∥2-∥yk∥2+θk.
Combing (36) and (37), we obtain that
(38)2〈xk-yk,w¯〉≤∥xk∥2-∥yk∥2+θk.
That is,
(39)∥yk-w¯∥2≤∥xk-w¯∥2+θk.
In view of the convexity of Ck, we see that w¯∈Ck. This implies that w¯∈Ck+1. Therefore, Ck+1 is convex. Consequently, Cn is closed and convex for each n≥1.
Step  3. We show that Ω⊂Cn for each n≥1. Put Θnm=Frm,nAmFrm-1,nAm-1⋯Fr2,nA2Fr1,nA1 for every m=1,2,…,M and Θn0=I for all n∈ℕ. Note that Frm,nAm=Frm,n(I-rm,nAm) is nonexpansive. Therefore, un=ΘnMxn. It is obvious that Ω⊂C1=C. Suppose that Ω⊂Ck for some k≥1. Taking p∈Ω, it follows from Lemma 1 that
(40)∥un-p∥=∥ΘnMxn-ΘnMp∥=∥FrM,n(I-rM,nAM)ΘnM-1xn-FrM,n(I-rM,nAM)ΘnM-1p∥≤∥(I-rM,nAM)ΘnM-1xn-(I-rM,nAM)ΘnM-1p∥≤∥ΘnM-1xn-ΘnM-1p∥⋮≤∥Θn0xn-Θn0p∥=∥xn-p∥.
From (19), we observe that
(41)∥zn-p∥2=∥(1-βn)un+βnTi(n)h(n)un-p∥2=(1-βn)∥un-p∥2+βn∥Ti(n)h(n)un-p∥2-βn(1-βn)∥un-Ti(n)h(n)un∥2≤(1-βn)∥un-p∥2+βn[kh(n)∥un-p∥2+λ∥un-Ti(n)h(n)un∥2+eh(n)]-βn(1-βn)∥un-Ti(n)h(n)un∥2≤kh(n)∥un-p∥2-βn(1-βn-λ)×∥un-Ti(n)h(n)un∥2+βneh(n)≤kh(n)∥un-p∥2+βneh(n).
By virtue of convexity of ∥·∥2, combining (40) and (41), we have
(42)∥yn-p∥2=∥(1-αn)un+αnzn-p∥2≤(1-αn)∥un-p∥2+αn∥zn-p∥2≤(1-αn)∥un-p∥2+αn[kh(n)∥un-p∥2+βneh(n)]≤∥un-p∥2+(kh(n)-1)∥un-p∥2+eh(n)=∥un-p∥2+θn(43)≤∥xn-p∥2+θn,
where θn=(kh(n)-1)ρn2+eh(n)→0 as n→∞ and ρn=sup{∥xn-p∥:p∈Ω}<∞. Setting n=k in (40)–(43), it follows that p∈Ck+1. Therefore, Ω∈Cn for each n≥1.
Step  4. Next, we prove that limn→∞xn=q, where q=P⋂n=1∞‍Cnf(q). Note that Cn is a closed convex subset of H and ∅≠Ω⊂Cn+1⊂Cn for all n∈ℕ. Thus, {xn} is well-defined. Since the composed mapping P⋂n=1∞‍Cnf is a Meir-Keeler contraction on C, there exists a unique fixed point q=P⋂n=1∞‍Cnf(q)∈⋂n=1∞‍Cn by Lemma 7. Let wn=PCnf(q) for each n∈ℕ. We get ⋂n=1∞‍Cn=M-limnCn, since Ω⊂Cn+1⊂Cn for every n∈ℕ. Thus, from Lemma 9, we get
(44)wn⟶P⋂n=1∞Cnf(q)=q.
We prove thatxn→q in the following. If it were so, it would hold that limsupn→∞∥xn-q∥>0. Let ϵ>0, such that limsupn→∞∥xn-q∥>ϵ. By the definition of Meir-Keeler contraction, there exists δ>0 with limsupn→∞∥xn-q∥>ϵ+δ such that ∥x-y∥<ϵ+δ implies that
(45)∥f(x)-f(y)∥<ϵ,∀x,y∈C.
From Lemma 8, there exists r∈(0,1) such that ∥x-y∥≥ϵ+δ implies that
(46)∥f(x)-f(y)∥<r∥x-y∥,∀x,y∈C.
By (44), there exists n0∈ℕ such that ∥wn-q∥<δ for each n≥n0. As in the proof of [24, Theorem 8], we consider the following two cases: (i)
There existsn1≥n0 such that ∥xn1-q∥<ϵ+δ.(ii)
∥xn-q∥≥ϵ+δ for every n≥n0.
In case (i), it holds that(47)∥xn1+1-wn1+1∥≤∥f(xn1)-f(q)∥<ϵ.
Thus we get
(48)∥xn1+1-q∥≤∥xn1+1-wn1+1∥+∥wn1+1-q∥<ϵ+δ,
which means that
(49)limsupn→∞∥xn-q∥≤ϵ+δ<limsupn→∞∥xn-q∥.
This is a contradiction. In case (ii), we have
(50)∥f(xn)-f(q)∥<r∥xn-q∥,n≥n0.
Thus we get
(51)∥xn1+1-wn1+1∥≤∥f(xn)-f(q)∥<r∥xn-q∥≤r(∥xn-wn∥+∥wn-q∥),
It follows from (44) that
(52)limsupn→∞∥xn-wn∥=limsupn→∞∥xn1+1-wn1+1∥≤rlimsupn→∞∥xn-wn∥<limsupn→∞∥xn-wn∥.
This is a contradiction again. Therefore, we obtain that limn→∞xn=q. Moreover, since xn+1=PCn+1f(xn), we have 〈f(xn)-xn+1,xn+1-y〉≥0 for each y∈Cn+1. By Ω⊂Cn+1, we have
(53)〈f(q)-q,q-y〉≥0,∀y∈Ω,
which is equivalent to q=PΩf(q) from the property of metric projection.
Step  5. Now, we prove that limn→∞∥xn-un∥=0. From xn→q as n→∞, we have
(54)limn→∞∥xn+1-xn∥=0.
Since xn+1=PCn+1f(xn)∈Cn+1, we have
(55)∥yn-xn+1∥2≤∥xn-xn+1∥2+θn.
It follows from (55) and Lemma 2 that
(56)∥yn-xn∥2=∥yn-xn+1+xn+1-xn∥2=∥yn-xn+1∥2+2〈yn-xn+1,xn+1-xn〉+∥xn+1-xn∥2≤2∥xn-xn+1∥2+2∥yn-xn+1∥×∥xn+1-xn∥+θn.
Since θn=(kh(n)-1)ρn2+eh(n)→0 as n→∞ and (54), we obtain
(57)limn→∞∥xn-yn∥=0.
For eachp∈Ω, m=1,2,…,M, it follows from (33) and (40) that
(58)∥Θnmxn-p∥2=∥Frm,n(I-rm,nAm)Θnm-1xn-Frm,n(I-rm,nAm)p∥2≤∥(I-rm,nAm)Θnm-1xn-(I-rm,nAm)p∥2≤∥Θnm-1xn-p∥2-rm,n(2αm-rm,n)×∥AmΘnm-1xn-Amp∥2≤∥xn-p∥2-rm,n(2αm-rm,n)×∥AmΘnm-1xn-Amp∥2.
By (40), (42), and (58), we have
(59)∥yn-p∥2≤∥un-p∥2+θn=∥ΘnMxn-ΘnMp∥+θn≤∥Θnmxn-Θnmp∥+θn≤∥xn-p∥2-rm,n(2αm-rm,n)×∥AmΘnm-1xn-Amp∥2+θn,
which implies that
(60)rm,n(2αm-rm,n)∥AmΘnm-1xn-Amp∥2≤∥xn-p∥2-∥yn-p∥2+θn≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+θn.
Since rm,n∈[c,d]⊂(0,2αm) for each m=1,2,…,M and θn→0 as n→∞. From (57), we have
(61)limn→∞∥AmΘnm-1xn-Amp∥=0,m=1,2,…,M.
On the other hand, it follows from the nonexpansive I-rm,nAm and Lemma 1 that
(62)∥Θnmxn-p∥2=∥Frm,n(I-rm,nAm)Θnm-1xn-Frm,n(I-rm,nAm)Θnm-1p∥2≤〈(I-rm,nAm)Θnm-1xn-(I-rm,nAm)p,Θnmxn-p〉=12[∥(I-rm,nAm)Θnm-1xn-(I-rm,nAm)p∥2+∥Θnmxn-p∥2-∥(I-rm,nAm)Θnm-1xn-(I-rm,nAm)p-(Θnmxn-p)∥2]≤12[∥Θnm-1xn-Θnmxn-rm,n(AmΘnm-1xn-Amp)∥2∥Θnm-1xn-p∥2+∥Θnmxn-p∥2-∥Θnm-1xn-Θnmxn-rm,n(AmΘnm-1xn-Amp)∥2],
which implies that
(63)∥Θnmxn-p∥2≤∥Θnm-1xn-p∥2-∥Θnm-1xn-Θnmxn-rm,n(AmΘnm-1xn-Amp)∥2=∥Θnm-1xn-p∥2-∥Θnm-1xn-Θnmxn∥2-rm,n2∥AmΘnm-1xn-Amp∥2+2rm,n〈Θnm-1xn-Θnmxn,AmΘnm-1xn-Amp〉≤∥xn-p∥2-∥Θnm-1xn-Θnmxn∥2+2rm,n∥Θnm-1xn-Θnmxn∥·∥AmΘnm-1xn-Amp∥.
From (40), (42), and (63), we obtain
(64)∥yn-p∥2≤∥un-p∥2+θn=∥ΘnMxn-p∥+θn≤∥Θnmxn-p∥+θn≤∥xn-p∥2-∥Θnm-1xn-Θnmxn∥2+2rm,n∥Θnm-1xn-Θnmxn∥·∥AmΘnm-1xn-Amp∥+θn,
which implies that
(65)∥Θnm-1xn-Θnmxn∥2≤∥xn-p∥2-∥yn-p∥2+2rm,n∥Θnm-1xn-Θnmxn∥×∥AmΘnm-1xn-Amp∥+θn≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+2rm,n∥Θnm-1xn-Θnmxn∥∥AmΘnm-1xn-Amp∥+θn.
It follows from (57) and (65) that
(66)limn→∞∥Θnm-1xn-Θnmxn∥=0,m=1,2,…,M.
Note that
(67)∥xn-un∥≤∥Θn0xn-Θn1xn∥+∥Θn1xn-Θn2xn∥+⋯+∥ΘnM-1xn-ΘnMxn∥.
Therefore,
(68)limn→∞∥xn-un∥=0.
Step  6. Finally, we prove that q∈(⋂i=1N‍Fix(Ti))∩(⋂m=1M‍EP(Fm,Am)). To do this, we first show that q∈⋂i=1N‍Fix(Ti). Note that
(69)∥un+1-un∥≤∥un+1-xn+1∥+∥xn+1-xn∥+∥xn-un∥.
From (54) and (68), we get
(70)limn→∞∥un+1-un∥=0.
It follows that
(71)limn→∞∥un+i-un∥=0,i=1,2,…,N.
For any positive integer n≥N, note that n=[h(n)-1]N+i(n), where i=i(n)=1,2,…,N. By (19) and the conditions 0<a≤αn≤1 and 0<b≤βn≤1-λ, we have
(72)∥un-Ti(n)h(n)un∥=1βn∥zn-un∥=1αnβn∥yn-un∥≤1ab(∥yn-xn∥+∥xn-un∥).
From (57) and (68), we get
(73)limn→∞∥un-Ti(n)h(n)un∥=0.
By the fact that h(n)=h(n-N)+1 and i(n)=i(n-N), we observe that
(74)∥un-Ti(n)h(n)-1un∥≤∥un-un-N∥+∥un-N-Ti(n-N)h(n-N)un-N∥+∥Ti(n-N)h(n-N)un-N-Ti(n)h(n)-1un∥.
Applying (71), (73), and Lemma 6, we obtain
(75)limn→∞∥un-Ti(n)h(n)-1un∥=0.
By the uniformly L-Lipschitzian of Ti, we have
(76)∥un-Tnun∥≤∥un-Ti(n)h(n)un∥+∥Ti(n)h(n)un-Ti(n)un∥≤∥un-Ti(n)h(n)un∥+L∥un-Ti(n)h(n)-1un∥.
It follows from (73) and (75) that
(77)limn→∞∥un-Tnun∥=0.
Since
(78)∥un-Tn+iun∥≤∥un-un+i∥+∥un+i-Tn+iun+i∥+∥Tn+iun+i-Tn+iun∥≤(1+L)∥un-un+i∥+∥un+i-Tn+iun+i∥.
Combining (71) and (77), we obtain
(79)limn→∞∥un-Tiun∥=0,i=1,2,…,N.
Moreover, for each i=1,2,…,N, we have
(80)∥xn-Tixn∥≤∥xn-un∥+∥un-Tiun∥+∥Tiun-Tixn∥.
This implies that
(81)limn→∞∥xn-Tixn∥=0,i=1,2,…,N.
Note that xn→q as n→∞. It follows from (81) and Lemma 5 that q∈⋂i=1N‍Fix(Ti).
Next, we show thatq∈⋂m=1M‍EP(Fm,Am). From Lemma 1 and since  Θnmxn=Frm,n(I-rm,nAm)Θnm-1xn, m=1,2,…,M, we have
(82)Fm(Θnmxn,y)+〈AmΘnm-1xn,y-Θnmxn〉+1rm,n〈y-Θnmxn,Θnmxn-Θnm-1xn〉≥0,∀y∈C.
By (A2), we have
(83)〈AmΘnm-1xn,y-Θnmxn〉+1rm,n〈y-Θnmxn,Θnmxn-Θnm-1xn〉≥Fm(y,Θnmxn),∀y∈C.
Let zt=ty+(1-t)q for all t∈(0,1] and y∈C. This implies that zt∈C. Then, we have
(84)〈Amzt,zt-Θnmxn〉≥〈Amzt,zt-Θnmxn〉-〈AmΘnm-1xn,zt-Θnmxn〉-〈zt-Θnmxn,Θnmxn-Θnm-1xnrm,n〉+Fm(zt,Θnmxn)≥〈Amzt-AmΘnmxn,zt-Θnmxn〉+〈AmΘnmxn-AmΘnm-1xn,zt-Θnmxn〉-〈zt-Θnmxn,Θnmxn-Θnm-1xnrm,n〉+Fm(zt,Θnmxn).
From (66), we have ∥AmΘnmxn-AmΘnm-1xn∥→0 as n→∞. Moreover, by (A4) and the monotonicity of Am, we obtain
(85)〈Amzt,zt-q〉≥Fm(zt,q),m=1,2,…,M.
Using (A1), (A4), and (85), we obtain
(86)0=Fm(zt,zt)≤tFm(zt,y)+(1-t)Fm(zt,q)≤tFm(zt,y)+(1-t)〈Amzt,zt-q〉≤tFm(zt,y)+(1-t)t〈Amzt,y-q〉.
and hence
(87)Fm(zt,y)+(1-t)〈Amzt,y-q〉≥0.
Let t→0, from (A3) and (87), we have
(88)Fm(q,y)+〈Amq,y-q〉≥0,∀y∈C,m=1,2,…,M.
This implies that q∈EP(Fm,Am), m=1,2,…,M. Therefore, q∈⋂m=1M‍EP(Fm,Am). Consequently, we obtain that q∈Ω. This completes the proof.We also obtain the following results by using the viscosity-hybrid projection methods, which extend and improve the hybrid method (CQ) proposed by Sahu et al. [11] and Hu and Cai [12].Theorem 11.
LetC be a nonempty closed convex subset of Hilbert space H. Let Fm:C×C→ℝ be a bifunction satisfying (A1)–(A4), and let Am:C→H be an αm-inverse strongly monotone mapping, for each m=1,2,…,M. Let Ti:C→C be a uniformly Li-Lipschitz continuous and asymptotically λi-strict pseudocontractive mapping in the intermediate sense with the sequences {kn,i} and {en,i} for each i=1,2,…,N. If f is a Meir-Keeler contraction of C into itself and Ω=(⋂i=1N‍Fix(Ti))∩(⋂m=1M‍EP(Fm,Am)) is nonempty and bounded. Let {xn} be a sequence defined by
(89)x1∈C,Q1=C,un=FrM,nAMFrM-1,nAM-1⋯Fr2,nA2Fr1,nA1xn,zn=(1-βn)un+βnTi(n)h(n)un,yn=(1-αn)un+αnzn,Cn={w∈C:∥yn-w∥2≤∥xn-w∥2+θn},Qn={u∈Qn-1:〈f(xn-1)-xn,xn-u〉≥0},xn+1=PCn⋂‍Qnf(xn),
where θn=(kh(n)-1)ρn2+eh(n)→0 as n→∞ and ρn=sup{∥xn-p∥:p∈Ω}<∞. Assume that {αn} and {βn} are sequences in [0,1] such that 0<a≤αn≤1, 0<b≤βn≤1-λ and {rm,n}⊂(0,∞) such that rm,n∈[c,d]⊂(0,2αm), for each m=1,2,…,M. Then the sequence {xn} generated by (89) converges strongly to q=PΩf(q).Proof.
We have thatCn and Qn are closed convex subsets of H and Ω⊂Cn for every n∈ℕ. We only prove that Ω⊂Qn for every n∈ℕ and that a sequence {xn} is well-defined. We have x1∈C and Ω⊂Q1=C. Assume that xk∈C and Ω⊂Qk for some k∈ℕ. Since Ω⊂Ck∩Qk, there exists a unique element xk+1=PCk∩Qkf(xk), and hence
(90)〈f(xk)-xk+1,xk+1-y〉≥0,∀y∈Ck∩Qk.
This implies that
(91)〈f(xk)-xk+1,xk+1-y〉≥0,∀y∈Ω.
That is, Ω⊂Qk+1. Therefore, we prove that Ω⊂Qn.
On the other hand,P⋂n=1∞‍Qnf is a Meir-Keeler contraction on C, there exists a unique element q=P⋂n=1∞‍Qnf(q)∈⋂n=1∞‍Qn by Lemma 7. Let zn=PQnf(q) for each n∈ℕ. Since Ω⊂Qn+1⊂Qn, it follows from Lemma 9 that zn→q=P⋂n=1∞‍Qnf(q). We also have xn=PQnf(xn-1) by the definition of Qn. Therefore, as in the proof of Theorem 10, we get xn→q, and the desired conclusion follows immediately from Theorem 10. This completes the proof.IfM=N=1, we obtain the following corollary for a general equilibrium problem and asymptotically strict pseudocontraction in the intermediate sense as a special cases.Theorem 12.
LetC be a nonempty closed convex subset of Hilbert space H. Let F:C×C→ℝ be a bifunction satisfying (A1)–(A4) and A:C→H be an α-inverse strongly monotone mapping. Let T:C→C be a uniformly L-Lipschitz continuous and asymptotically λ-strict pseudocontractive mapping in the intermediate sense with the sequences {kn} and {en}. If f is a Meir-Keeler contraction of C into itself and Ω=Fix(T)∩EP(F,A) is nonempty and bounded. Let {xn} be a sequence defined by
(92)x1∈C,C1=C,F(un,y)+〈Axn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,zn=(1-βn)un+βnTnun,yn=(1-αn)un+αnzn,Cn+1={w∈Cn:∥yn-w∥2≤∥xn-w∥2+θn},xn+1=PCn+1f(xn),
where θn=(kn-1)ρn2+en→0 as n→∞ and  ρn=sup{∥xn-p∥:p∈Ω}<∞. Assume that {αn} and {βn} are sequences in [0,1] such that 0<a≤αn≤1, 0<b≤βn≤1-λ and {rn}⊂(0,∞) such that rn∈[c,d]⊂(0,2α). Then the sequence {xn} generated by (92) converges strongly to q=PΩf(q).
IfM=N=1 and F=0, the general equilibrium problem (1) reduces into the classical variational inequality problem (3), we obtain the following corollary as a special case of Theorems 10 and 12.Theorem 13.
LetC be a nonempty closed convex subset of Hilbert space H. Let A:C→H be an α-inverse strongly monotone mapping. Let T:C→C be a uniformly L-Lipschitz continuous and asymptotically λ-strict pseudocontractive mapping in the intermediate sense with the sequences {kn} and {en}. If f is a Meir-Keeler contraction of C into itself and Ω=Fix(T)∩VI(C,A) is nonempty and bounded. Let {xn} be a sequence defined by
(93)x1∈C,C1=C,yn=(1-αn)xn+αnTnPC(xn-rnAxn),Cn+1={w∈Cn:∥yn-w∥2≤∥xn-w∥2+θn},xn+1=PCn+1f(xn),
where θn=(kn-1)ρn2+en→0 as n→∞ and ρn=sup{∥xn-p∥:p∈Ω}<∞. Assume that {αn} is a sequence in [0,1] such that 0<b≤αn≤1-λ and {rn}⊂(0,∞) such that rn∈[c,d]⊂(0,2α). Then the sequence {xn} generated by (93) converges strongly to q=PΩf(q).Proof.
IfF=0, the general equilibrium problem (1) reduces into the classical variational inequality problem (3), and
(94)〈Axn,y-un〉+1rn〈y-un,un-xn〉≥0,∀y∈C,
which is equivalent to
(95)〈y-un,un-(xn-rnAxn)〉≥0,∀y∈C.
Therefore, we have un=PC(xn-rnAxn). The desired conclusion follows immediately from Theorem 10 (Set αn=1, βn=αn). This completes the proof.



---

*Source: 608241-2013-10-28.xml*