# A Hybrid Subgradient Algorithm for Finding a Common Solution of an Equilibrium Problem and a Family of Strict Pseudocontraction Mappings

Authors: Ekkarath Thailert; Rabian Wangkeeree; Chanoksuda Khantree
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We propose a new strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of common fixed points of a family of strict pseudocontraction mappings in a real Hilbert space. The strong convergence theorem of proposed algorithms is investigated without the Lipschitz condition for the bifunctions. Our results complement many known recent results in the literature.


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## Body

## 1. Introduction

LetH be a real Hilbert space endowed with an inner product 〈·〉 and a norm ∥·∥ associated with this inner product, respectively. Let C be a nonempty closed convex subset of H, and let f be a bifunction from C×C to R such that f(x,x)=0 for all x∈C. An equilibrium problem in the sense of Blum and Oettli [1] is stated as follows:
(1)Findx*∈Csuchthatf(x*,y)≥0∀y∈C.
Problem of the form (1) on one hand covers many important problems in optimization as well as in nonlinear analysis such as (generalized) variational inequality, nonlinear complementary problem, and nonlinear optimization problem, just to name a few. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. On the other hand, it is rather convenient for reformulating many practical problems in economics, transportation, and engineering (see [1, 2] and the references quoted therein). We denote the set of solutions of the problem (1) by Sol(f,C).The existence of solution and its characterizations can be found, for example, in [3], while the methods for solving problem (1) have been developed by many researchers [4, 5]. On the other hand, iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems [6]. The problem P(C,f,S) of finding a common point in the solution set of problem EP(C,f) and the set of fixed points of a nonexpansive mapping S recently becomes an attractive subject, and various methods have been developed for solving this problem. Most of the existing algorithms for this problem are based on the proximal point method applying to equilibrium problem EP(C,f) combining with a Mann’s iteration to the problem of finding a fixed point of S.In 2006, S. Takahashi and W. Takahashi [7] proposed an iterative scheme under the nameviscosity approximation methods for finding a common element of set of solutions of (1) and the set of fixed points of nonexpansive mapping S in a real Hilbert space H. This method generated an iteration sequence {xk} starting from a given initial point x0∈H and computed xk+1 as
(2)Finduk∈Csuchthatf(uk,y)+1rk〈y-uk,uk-xk〉≥0,∀y∈C,Computexk+1=αkg(xk)+(1-αk)S(uk),k≥0,
where g is a contraction of H into itself and the sequences of parameters {rk} and {αk} were chosen appropriately. Under certain choice of {αk} and {rk}, the authors showed that two iterative sequences {xk} and {uk} converge strongly to z=PFix⁡(S,C)∩Sol(f,C)(g(z)), where PC denotes the projection onto C.Recently, Anh in [8] proposed to use the extragradient-type iteration instead of the proximal point iteration given in [9] for solving problem P(C,f,T). More precisely, given zk∈C, the proximal point iteration given in [9] is replaced by the two following mathematical programs, which seems numerically easier than previous ones. More precisely, the following algorithm is suggested in [8]:
(3)Foraninitialpointx0∈C,yk=argmin{f(xk,y)+12λk∥y-xk∥2:y∈C},zk=argmin{f(yk,z)+12λk∥z-xk∥2:z∈C},k≥0.
It was proved that if f is pseudomonotone and satisfies the Lipschitz-type condition, there are Lipschitz constants c1>0 and c2>0 if
(4)f(x,y)+f(y,z)≥f(x,z)-c1∥x-y∥2-c2∥y-z∥2,111111111111111111∀x,y,z∈C,
then the sequence {zk} strongly converges to a solution of problem P(C,f,T). Recently, Anh and Muu [10] emphasized that the Lipschitz-type condition (4), in general, is not satisfied, and if yes, finding the constants c1 and c2 is not an easy task. Furthermore, solving the strongly convex programs (3) is expensive except special cases when C has a simple structure. They suggested and studied a new algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space. The proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions. More precisely, they introduced the following algorithm:
(5)Pickanyx0∈C;yk∈∂εkf(xk,·)(xk);γk≔max⁡{λk,∥yk∥},αk≔βkγk;wk=PC(xk-αkyk);xk+1=δkwk+(1-δk)Twk,foreachk=0,1,…,
where ∂ɛf(x,·)(x) stands for ɛ-subdifferential of the convex function f(x,·) at x and {εk}, {λk}, {βk}, and {δk} were chosen appropriately. Under the certain conditions, {xk} converse strongly to a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space.On the other hand, the problem of finding a common fixed point element of a finite family of self-mappingsS≔{Si}i=1p (p≥1) is expressed as follows:
(6)Findx*∈Csuchthatx*∈⋂i=1p‍Fix⁡(Si,C),
where Fix⁡(Si,C) is the set of the fixed points of the mapping Si (i=1,…,p). Let us denote by
(7)Fix⁡(S,C)≔⋂i=1p‍Fix⁡(Si,C)
the solution sets of the fixed-point problem (6). Problem of finding a fixed point of a mapping or a family of mappings is a classical problem in nonlinear analysis. The theory and solution methods of this problem can be found in many research papers and monographs (see [11]). The problem of finding a common fixed point of a finite sequence of mappings has been studied by many researchers. For instance, in 2005, Blum and Oettli [1] proposed an iterative algorithm for finding a common fixed point of p strict pseudocontraction mapping Si (i=1,…,p). The method computed a sequence {xk} starting from x0∈H and taking
(8)xk+1=αkxk+(1-αk)∑i=1p‍λk,iSi(xk),k≥1,
where the sequence of parameters {λk} was chosen in a specific way to ensure the convergence of the iterative sequence {xk}. The authors proved that the sequence {xk} converges weakly to a point x¯∈⋂i=1p‍Fix⁡(Si,C). Very recently, Anh et al. [12] suggested and analyzed an algorithm for finding a common solution of two problems (1) and (6). Typically, this problem can be written as follows:
(9)Findx*∈Fix⁡(S,C)∩Sol(f,C).
They presented an algorithm for finding a solution of problem (9). More precisely, they suggested the following algorithm:
(10)Foraninitialpointx0∈C,yk=argmin{λkf(xk,y)+12∥y-xk∥2:y∈C},tk=argmin{λkf(yk,y)+12∥y-xk∥2:y∈C},xk+1=αktk+(1-αk)∑i=1p‍λk,iSi(tk),k≥0,
where f is pseudomonotone and continuous on C and Lipschitz-type continuous on C, {λk},{λk,i}, for all i=1,2,…,p, and {αk} are the sequences of parameters which were chosen appropriately. Under certain conditions on the parameters and f, they proved that a sequence {xk} converges weakly to a solution x* of (9).In this paper, motivated by the idea of Anh and Muu [10] and Anh et al. [12], we propose a new algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of common fixed points of a family of strict pseudocontraction mappings in a real Hilbert space. The strong convergence of proposed algorithms is investigated under certain assumptions. Our results complement many known recent results in the literature.


## 2. Preliminaries

LetC be a nonempty convex subset of a Hilbert space H. We write xk⇀x to indicate that the sequence {xk} converges weakly to x as k→∞ and xk→x to indicate that the sequence {xk} converges strongly to x as k→∞. Since C is closed, convex, for any x∈H, there exists a unique point in C, denoted by PC(x) satisfying
(11)∥x-PC(x)∥≤∥x-y∥,∀y∈C.PC is called the metric projection of H to C. It is well known that PC satisfies the following properties:
(12)〈x-y,PC(x)-PC(y)〉≥∥PC(x)-PC(y)∥2,∀x,y∈H,(13)〈x-PC(x),PC(x)-y〉≥0,∀x∈H,y∈C,(14)∥x-y∥2≥∥x-PC(x)∥2+∥y-PC(x)∥2,1111111111111∀x∈H,y∈C.
The concept of strict pseudocontraction is considered in [13], defined as follows.Definition 1.
LetC be a nonempty closed convex subset of a real Hilbert space H. A mapping S:C→C is said to be a strict pseudocontraction if there exists a constant 0≤L<1 such that
(15)∥S(x)-S(y)∥2≤∥x-y∥2+L∥(I-S)(x)-(I-S)(y)∥2,11111111111111111111111111111∀x,y∈C,
where I is the identity mapping on H. If L=0, then S is called nonexpansive on C.The following proposition lists some useful properties of a strict pseudocontraction mapping.Proposition 2 (see [13]).
LetC be a nonempty closed convex subset of a real Hilbert space H; let S:C→C be a L-strict pseudocontraction, and, for each i=1,…,p, Si:C→C is a Li -strict pseudocontraction for some 0≤Li<1. Then, (i)
Ssatisfies the following Lipschitz condition:
(16)∥S(x)-S(y)∥≤1+L1-L∥x-y∥,∀x,y∈C;(ii)
I-S is demiclosed at 0. That is, if the sequence {xk} contains in C such that xk⇀x¯ and (I-S)(xk)→0, then (I-S)(x¯)=0;(iii)
the set of fixed pointsFix⁡(S) is closed and convex;(iv)
ifηi>0 (i=1,…,p) and ∑i=1p‍ηi=1, then ∑i=1p‍ηiSi is a L¯-strict pseudocontraction with L¯:=max⁡{Li:1≤i≤p};(v)
ifηi is chosen as in (iv) and {Si:i=1,…,p} has a common fixed point, then
(17)Fix⁡(∑i=1p‍ηiSi)=⋂i=1p‍Fix⁡(Si,C).Lemma 3 (see [14]).
Suppose that{αk} and {βk} are two sequences of nonnegative real numbers such that
(18)αk+1≤αk+βk,k≥0,
where ∑k=0∞‍βk<∞. Then, the sequence {αk} is convergent.The following idea of theɛ-subdifferential of convex functions can be found in the work of Brøndsted and Rockafellar [15] but the theory of ɛ-subdifferential calculus was given by Hiriart-Urruty [16].Definition 4.
Consider a proper convex functionϕ:C→R¯. For a given ɛ>0, the ɛ-subdifferential of ϕ at x0∈dom⁡ϕ is given by
(19)∂ɛϕ(x0)={x∈C:ϕ(y)-ϕ(x0)≥〈x,y-x0〉-ɛ,∀y∈C}.Remark 5.
It is known that if the functionϕ is proper lower semicontinuous convex, then, for every x∈dom⁡ϕ, the ɛ-subdifferential ∂ɛϕ(x) is a nonempty closed convex set (see [17]).


## 3. Main Results

Now, we are in a position to state and prove the main strong convergence theorem for the given iterative scheme.Assumption 6.
Let the bifunctionf:C×C→R be satisfied by the following conditions: (i)
for eachx, f(x,x)=0 and f(x,·) is lower semicontinuous convex on C;(ii)
if{xk}⊆C is bounded and εk↓0 as k→∞, then the sequence {yk} with yk∈∂εkf(xk,·)(xk) is bounded;(iii)
f is pseudomonotone on C with respect to every solution of EP(C,f) and satisfies the following condition, called strict paramonotonicity property:
(20)x∈Sol(C,f),y∈C,f(y,x)=0⟹y∈Sol(C,f);(iv)
for eachx∈C, f(·,x) is weakly upper semicontinuous on the open set C.Assumption 7.
Consider the following:
(i) for eachi=1,2,…,p, Si is Li-strict pseudocontraction for some 0≤Li<1;
(ii) the solution setΩ of the problem P(C,f,Si) is nonempty; that is,
(21)Ω≔Sol(f,C)⋂‍⋂i=1p‍Fix⁡(Si,C)≠∅.Assumption 8.
Suppose that the sequences{λk}, {βk}, {εk}, {ηk,i}, and {δk} of nonnegative numbers satisfy the following conditions: (i)
0<λk<λ, 0<L¯<δk<1, and δk→1/2, where L¯≔max⁡{Li:1≤i≤p};(ii)
βk>0, ∑k=0∞‍βk=+∞, and ∑k=0∞‍βk2<+∞;(iii)
∑k=0∞‍βkεk<+∞;(iv)
∑i=1p‍ηk,i=1 for all k≥1.Algorithm 9.
Now, the iteration scheme for finding a common point in the set of solutions of problemEP(C,f) and the set of common fixed points of Li-strict pseudocontraction, for each i=1,2,…,p, can be written as follows:
(22)x0∈C;yk∈∂εkf(xk,·)(xk);γk≔max⁡{λk,∥yk∥},αk≔βkγk;wk=PC(xk-αkyk);(23)xk+1=δkwk+(1-δk)∑i=1p‍ηk,iSi(wk),1111111111foreachk=0,1,….Remark 10 (see [10], Remark 2.1).
Consider the following:(i)
iff is pseudomonotone on C with respect to the solution set Sol(C,f) of the problem EP(C,f), then, under Assumptions 6 (i) and (iv), the set Sol(C,f) is convex;(ii)
Assumption6 (ii) is true if whenever Assumption 6 (i) is satisfied and the bifunction f is continuous on C×C;(iii)
Assumption6 (iii) is true if f is pseudomonotone on C and satisfies the paramonotone property as
(24)x∈Sol(C,f),y∈C,f(y,x)=f(y,x)=0⟹y∈Sol(C,f);(iv)
sincef(x,·) is lower semicontinuous convex on C, applying Remark 5, we conclude that ∂εkf(xk,·)(xk)≠∅. Thus, Algorithm 9 is well defined.Theorem 11.
Suppose that Assumptions6–8 are satisfied. Then, the sequences {xk} and {wk} generated by Algorithm 9 converge strongly to the same point x¯∈Ω, where x¯=lim⁡k→∞PΩ(xk).Proof.
The proof is divided into several steps as follows.
Step 1. For every x*∈C and every k, we show that
(25)∥xk+1-x*∥2≤∥xk-x*∥2+2αkf(xk,x*)+2αkϵk+2βk2,
and there exists the limit
(26)c≔lim⁡k→∞∥xk-x*∥.
For each k≥1, let
(27)S¯k≔∑i=1p‍ηk,iSi.
By Proposition 2, we see that S¯k is a L¯-strict pseudocontraction on C and the sequence {xk} generated by Algorithm 9 can be rewritten as
(28)xk+1≔δkwk+(1-δk)S¯k(wk),∀k≥1.
Then, for all k≥1, we have
(29)∥xk+1-x*∥2=∥δkwk+(1-δk)S¯k(wk)-x*∥2=∥δk(wk-x*)+(1-δk)(S¯k(wk)-x*)∥2=δk∥wk-x*∥2+(1-δk)∥S¯k(wk)-S¯k(x*)∥2-δk(1-δk)∥wk-S¯k(wk)∥2≤δk∥wk-x*∥2+(1-δk)×(∥wk-x*∥2+L¯∥(I-S¯k)(wk)-(I-S¯k)(x*)∥2)-δk(1-δk)∥wk-S¯k(wk)∥2=∥wk-x*∥2+(1-δk)(L¯∥S¯k(wk)-wk∥2)-δk(1-δk)∥wk-S¯k(wk)∥2=∥wk-x*∥2+(1-αk)(L¯-δk)∥S¯k(wk)-wk∥2≤∥xk-x*∥2-∥xk-wk∥2+2〈xk-wk,x*-wk〉+(1-δk)(L¯-δk)∥S¯k(wk)-wk∥2(30)≤∥xk-x*∥2-∥xk-wk∥2+2〈xk-wk,x*-wk〉≤∥xk-x*∥2+2〈xk-wk,x*-wk〉.
Since wk=PC(xk-αkyk) and x*∈C, we have
(31)〈xk-wk,x*-wk〉≤αk〈yk,x*-wk〉.
Combining this inequality with (30) yields
(32)∥xk+1-x*∥2≤∥xk-x*∥2+2〈xk-wk,x*-wk〉≤∥xk-x*∥2+2αk〈yk,x*-wk〉=∥xk-x*∥2+2αk〈yk,x*-xk〉+2αk〈yk,xk-wk〉≤∥xk-x*∥2+2αk〈yk,x*-xk〉+2αk∥yk∥∥xk-wk∥=∥xk-x*∥2+2αk〈yk,x*-xk〉+2βkmax⁡{λk,∥yk∥}∥yk∥∥xk-wk∥≤∥xk-x*∥2+2αk〈yk,x*-xk〉+2βk∥xk-wk∥.
Using again wk=PC(xk-αkyk) and xk∈C, we have
(33)∥xk-wk∥2≤αk〈yk,x*-wk〉≤αk∥yk∥∥xk-wk∥=βkmax⁡{λk,∥yk∥}∥yk∥∥xk-wk∥≤βk∥xk-wk∥,
which gives that∥xk-wk∥≤βk. Consequently,
(34)lim⁡k→∞∥xk-wk∥=0.
This together with (32) implies that
(35)∥xk+1-x*∥2≤∥xk-x*∥2+2αk〈yk,x*-xk〉+2βk2.
Since yk∈∂ϵkf(xk,·)(xk), x*∈C and f(x,x)=0 for all x∈C, we have
(36)〈yk,x*-xk〉≤f(xk,x*)-f(xk,xk)+ϵk≤f(xk,x*)+ϵk.
Combining (35) and (36), we obtain that
(37)∥xk+1-x*∥2≤∥xk-x*∥2+2αkf(xk,x*)+2αkϵk+2βk2.
On the other hand, since x*∈Sol(C,f), that is, f(x*,x)≥0 for all x∈C, by pseudomonotonicity of f with respect to x*, we have f(x,x*)≤0 for all x∈C. Replacing x by xk∈C, we get f(xk,x*)≤0. Then, from (37), it follows that
(38)∥xk+1-x*∥≤∥xk-x*∥2+2αkϵk+2βk2.
By Assumptions 8 (ii) and (iii), we found that ∑k=1∞‍(2αkϵk+2βk2)<∞. Using Lemma 3 and (38), we arrive at the existence of
(39)c≔lim⁡k→∞∥xk-x*∥.
Step 2. For every x*∈C, we show that
(40)limsup⁡k→∞f(xk,x*)=0.
Since f is pseudomonotone on C and f(x*,xk)≥0, we have -f(xk,x*)≥0. By Step  1,
(41)∥xk+1-x*∥2≤∥xk-x*∥2+2αkf(xk,x*)+2αkϵk+2βk2.
We have
(42)2αk[-f(xk,x*)]≤∥xk-x*∥2-∥xk+1-x*∥2+2αkϵk+2βk2.
Summing up the above inequalities for every k, we obtain that
(43)0≤2∑k=0∞‍αk[-f(xk,x*)]≤∥x0-x*∥2+2∑k=0∞‍αkϵk+2∑k=0∞‍βk2<+∞.
It follows from the boundedness of the sequences {yk} and {λk} that we can assume that
(44)max⁡{λk,∥yk∥}≤M,
for a constant M≥0. Thus,
(45)αk=βkγk=βkmax⁡{λk,∥yk∥}≥βkM,
which together with (43) implies
(46)0≤2M∑k=0∞‍βk[-f(xk,x*)]≤2∑k=0∞‍αk[-f(xk,x*)]<+∞.
Thus,
(47)∑k=0∞‍βk[-f(xk,x*)]<+∞.
Then, by ∑k=0∞‍βk=∞ and -f(xk,x*)≥0, we can deduce that limsup⁡k→∞f(xk,x*)=0 as desired.
Step 3. For any x*∈Ω, suppose that {xkj} is the subsequence of {xk} such that
(48)limsup⁡k→∞f(xk,x*)=lim⁡j→∞f(xkj,x*),
and, without loss of generality, we may assume that xkj⇀x¯ as j→∞ for some x¯∈C. We show that x¯ solves EP(C,f). To this end, since f(·,x*) is weakly upper semicontinuous, we have
(49)f(x¯,x*)≥limsup⁡j→∞f(xkj,x*)=lim⁡j→∞f(xkj,x*)=limsup⁡j→∞f(xk,x*)=0.
On the other hand, since f is pseudomonotone with respect to x* and f(x*,x¯)≥0, we have
(50)f(x¯,x*)≤0.
From (49) and (50), we can conclude that f(x¯,x*)=0. By Assumption 6, we can deduce that x¯ is a solution of EP(f,C) as well.
Step  4. We prove that any weakly cluster point of the sequence {xk} is a common fixed point of Li-strict pseudocontraction, for each i=1,2,…,p. In particular, x¯∈⋂i=1p‍Fix⁡(Si,C). Let y¯ be any weakly cluster point of {xk} and let {xkm} be a subsequence of {xk}⊂C weakly converging to y¯. By convexity and the closedness of C, C is weakly closed. Hence, y¯∈C. We first show that
(51)lim⁡m→∞∥xkm-S(xkm)∥=0.
It follows from (29) that
(52)(1-δk)(δk-L¯)∥S¯k(wk)-wk∥2≤∥xk-x*∥2-∥xk+1-x*∥2+2αk〈yk,x*-xk〉+2βk∥xk-wk∥≤∥xk-x*∥2-∥xk+1-x*∥2+2αk∥yk∥∥x*-xk∥+2βk∥xk-wk∥,
which gives that
(53)lim⁡k→∞∥S¯k(wk)-wk∥=0.
Consequently,
(54)∥S¯k(wk)-xk∥≤∥S¯k(wk)-wk∥+∥xk-wk∥⟶0ask⟶∞.
By Proposition 2 (i), we arrive at the following:
(55)∥S¯k(xk)-xk∥≤∥S¯k(xk)-S¯k(wk)∥+∥S¯k(wk)-xk∥≤1+L¯1-L¯∥xk-wk∥+∥S¯k(wk)-xk∥≤1+L¯1-L¯βk+∥S¯k(wk)-xk∥.
Thus, we obtain
(56)lim⁡k→∞∥S¯k(xk)-xk∥=0.
For each i=1,…,p, we suppose that {ηkm,i} converges to ηi as m→∞ such that ∑i=1p‍ηi=1. Then, for each i=1,2,…,p and x∈C, we have
(57)S¯km(x)≔∑m=1p‍ηkm,iSi(x)⟶∑i=1p‍ηiSi(x)≔S(x)asm⟶∞.
It follows from (56) that
(58)∥xkm-S(xkm)∥≤∥xkm-S¯km(xkm)∥+∥S¯km(xkm)-S(xkm)∥=∥xkm-S¯km(xkm)∥+∥∑i=1p‍ηkm,iSi(xkm)-∑i=1p‍ηiSi(xkm)∥=∥xkm-S¯km(xkm)∥+∥∑i=1p‍(ηkm,i-ηi)Si(xkm)∥≤∥xkm-S¯km(xkm)∥+∑i=1p‍|ηkm,i-ηi|∥Si(xkm)∥.
We obtain that
(59)lim⁡j→∞∥xkm-S(xkm)∥=0.
By Proposition 2 (ii), we have
(60)y¯∈Fix⁡(S)=Fix⁡(∑i=1p‍ηiSi)x.
It then follows from Proposition 2 (v) that we have
(61)y¯∈⋂i=1p‍Fix⁡(Si,C).
In particular, we conclude that x¯∈⋂i=1p‍Fix⁡(Si,C).
Step  5. Finally, we prove that
(62)lim⁡k→∞xk=lim⁡k→∞wk=lim⁡k→∞PΩ(xk)=x¯.
It follows from (38) that, for all x*∈Ω,
(63)∥xk+1-x*∥2≤∥xk-x*∥2+μk,
where μk≔2αkϵk+2βk2>0 for all k≥0 and ∑k=0∞‍μk<+∞. Now, using property (14) of the metric projection, we have
(64)∥xk+1-PΩ(xk+1)∥2=∥δk(wk-PΩ(xk))+(1-δk)(S¯k(wk)-PΩ(xk))∥2≤δk∥wk-PΩ(xk)∥2+(1-δk)∥S¯k(wk)-PΩ(xk)∥2≤δk[∥wk-xk∥+∥xk-PΩ(xk)∥]2+(1-δk)∥S¯k(wk)-xk∥2-(1-δk)∥xk-PΩ(xk)∥2=δk∥wk-xk∥2+2δk∥wk-xk∥∥xk-PΩ(xk)∥+δk∥xk-PΩ(xk)∥2+(1-δk)∥S¯k(wk)-xk∥2-(1-δk)∥xk-PΩ(xk)∥2=δk∥wk-xk∥2+2δk∥wk-xk∥∥xk-PΩ(xk)∥+(2δk-1)∥xk-PΩ(xk)∥2+(1-δk)∥S¯k(wk)-xk∥2.
Since δk→1/2, ∥wk-xk∥→0, and ∥S¯k(wk)-xk∥→0 as k→∞, we obtain that
(65)lim⁡k→∞∥xk+1-PΩ(xk+1)∥=0.
For the simplicity of notation, let zk≔PΩ(xk) for each k≥1. Then, for all m>k, since Ω is convex, we have (1/2)(zm+zk)∈Ω, and therefore
(66)∥zm-zk∥2=2∥xm-zm∥2+2∥xm-zk∥2-4∥xm-12(zm+zk)∥2≤2∥xm-zm∥2+2∥xm-zk∥2-4∥xm-zm∥2=2∥xm-zk∥2-2∥xm-zm∥2.
Replacing x* with zk in (63), we can obtain the following:
(67)∥xm-zk∥2≤∥xm-1-zk∥2+μm-1≤∥xm-2-zk∥2+μm-1+μm-2≤⋯≤∥xk-zk∥2+∑j=km-1‍μj,
Combining this inequality with (66), we have
(68)∥zm-zk∥2≤2∥xk-zk∥2+2∑j=km-1‍μj-2∥xm-zm∥2,
which gives that
(69)lim⁡m→∞,k→∞∥zm-zk∥=0,
which implies also that {zk} is a Cauchy sequence. Hence, {zk} strongly converges to some point z¯∈Ω. However, since zki≔PΩ(xki), letting i→∞, we obtain in the limit that
(70)z¯=lim⁡i→∞PΩ(xki)=PΩ(x¯)=x¯∈Ω.
Therefore, zk≔PΩ(xk)→z¯=x¯∈Ω. Then, from (65), we can conclude that xk→x¯. Finally, since lim⁡k→∞∥xk-wk∥=0, we have lim⁡k→∞wk=x¯.Using Theorem11, we can obtain the new strong convergence result for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of a strict pseudocontraction mapping in a real Hilbert space.Corollary 12.
LetC be a nonempty convex subset of a Hilbert space H. Let the bifunction f:C×C→R be satisfied by all conditions in Assumption 6. Let S:C→C be a L-strict pseudocontraction for some 0≤L<1 such that the solution set Ω1 of the problem P(C,f,S) is nonempty. Suppose that the sequences {λk}, {βk}, {εk}, and {δk} of nonnegative numbers satisfy the following conditions: (i)
0<λk<λ, 0<L<δk<1, and δk→1/2;(ii)
βk>0, ∑k=0∞‍βk=+∞, and ∑k=0∞‍βk2<+∞;(iii)
∑k=0∞‍βkεk<+∞. Then, the sequence {xk} generated by
(71)x0∈C;yk∈∂εkf(xk,·)(xk);γk≔max⁡{λk,∥yk∥},αk≔βkγk;wk=PC(xk-αkyk);xk+1=δkwk+(1-δk)S(wk),k≥0
converge strongly to a point x¯∈Ω1, where x¯=lim⁡k→∞PΩ1(xk).Corollary 13.
LetC be a nonempty convex subset of a Hilbert space H. Let the bifunction f:C×C→R be satisfied by all conditions in Assumption 6. Let S:C→C be a nonexpansive mapping such that the solution set Ω2 of the problem P(C,f,S) is nonempty. Suppose that the sequences {λk}, {βk}, {εk}, and {δk} of nonnegative numbers satisfy the following conditions: (i)
0<λk<λ, 0<a<δk<b<1, and δk→1/2;(ii)
βk>0, ∑k=0∞‍βk=+∞, and ∑k=0∞‍βk2<+∞;(iii)
∑k=0∞‍βkεk<+∞. Then, the sequences {xk} and {wk} generated by
(72)x0∈C;yk∈∂εkf(xk,·)(xk);γk≔max⁡{λk,∥yk∥},αk≔βkγk;wk=PC(xk-αkyk);xk+1=δkwk+(1-δk)S(wk),k≥0
converge strongly to a point x¯∈Ω2, where x¯=lim⁡k→∞PΩ2(xk).



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*Source: 142671-2014-08-20.xml*