# Strong Convergence Theorems for Equilibrium Problems andk-Strict Pseudocontractions in Hilbert Spaces

Authors: Dao-Jun Wen
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We introduce a new iterative scheme for finding a common
element of the set of solutions of an equilibrium problem and the set of common fixed point
of a finite family ofk-strictly pseudo-contractive nonself-mappings. Strong convergence
theorems are established in a real Hilbert space under some suitable conditions. Our
theorems presented in this paper improve and extend the corresponding results announced
by many others.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively.  Let K be a nonempty closed convex subset of H. Let F be a bifunction from K×K into ℝ, where ℝ denotes the set of real numbers. We consider the following problem: Find x∈K such that
(1.1)F(x,y)≥0,∀y∈K,
which is called equilibrium problem. We use EP(F) to denote the set of solution of the problem (1.1). Given a mapping T:K→H, let F(x,y)=〈Tx,y-x〉 for all x,y∈K. Then, z∈EP(F) if and only if 〈Tz,y-z〉≥0 for all y∈K; that is, z is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem (see, e.g., [1–3]).Recall that a nonself-mappingT:K→H is called a k-strict pseudocontraction if there exists a constant k∈[0,1) such that
(1.2)‖Tx-Ty‖2≤‖x-y‖2+k‖(I-T)x-(I-T)y‖2,∀x,y∈K.
We use F(T) to denote the fixed point set of the mapping T, that is, F(T):={x∈K:Tx=x}. As k=0, T is said to be nonexpansive, that is, ∥Tx-Ty∥≤∥x-y∥,for allx,y∈K. T is said to be pseudocontractive if k=1 and is also said to be strongly pseudocontractive if there exists a positive constant λ∈(0,1) such that T+λI is pseudocontractive. Clearly, the class of k-strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. We remark also that the class of strongly pseudocontractive mappings is independent of the class of k-strict pseudocontractions (see, e.g., [4, 5]).Iterative methods for equilibrium problem and nonexpansive mappings have been extensively investigated; see, for example, [1–18] and the references therein. However, iterative methods for strict pseudocontractions are far less developed than those for nonexpansive mappings though Browder and Petryshyn [5] initiated their work in 1967; the reason is probably that the second term appearing in the right-hand side of (1.2) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strict pseudocontraction T. On the other hand, strict pseudocontractions have more powerful applications than nonexpansive mappings do in solving inverse problems (see, e.g., [6]). Therefore, it is interesting to develop the theory of iterative methods for equilibrium problem and strict pseudocontractions.In 2007, Acedo and Xu [12] proposed the following parallel algorithm for a finite family of ki-strict pseudocontractions {Ti}i=1N in Hilbert space H:
(1.3)∀x0∈K,xn+1=(1-αn)xn+αn∑i=1NλiTixn,
where {αn}⊂(0,1), and {λi}i=1N is a finite sequence of positive numbers such that ∑i=1Nλi=1. They proved that the sequence {xn} defined by (1.3) converges weakly to a common fixed point of {Ti}i=1N under some appropriate conditions. Moreover, by applying additional projections, they further proved that algorithm can be modified to have strong convergence.Recently, S. Takahashi and W. Takahashi [13] studied the equilibrium problem and fixed point of nonexpansive self-mappings T in Hilbert spaces by a viscosity approximation methods for finding an element of EP(F)⋂F(T). Very recently, by using the general approximation method, Qin et al. [14] obtained a strong convergence theorem for finding an element of F(T). On the other hand, Ceng et al. [16] proposed an iterative scheme for finding an element of EP(F)⋂F(T) and then obtained some weak and strong convergence theorems.In this paper, inspired and motivated by research going in this area, we introduce a modified parallel iteration, which is defined in the following way:(1.4)F(un,y)+1r〈y-un,un-xn〉≥0,∀y∈K,yn=αnun+(1-αn)∑i=1Nηi(n)Tiun,xn+1=βnu+γnxn+(1-βn-γn)yn,n≥0,
where u∈K is a given point, {Ti}i=iN:K→H is a finite family of ki-strictly pseudocontractive nonself-mappings, {ηi(n)}i=1N is a finite sequences of positive numbers, {αn}, {βn}, and {γn} are some sequences in (0,1).Our purpose is not only to modify the parallel algorithm (1.3) to the case of equilibrium problems and common fixed point for a finite family of ki-strictly pseudocontractive nonself-mappings, but also to establish strong convergence theorems in a real Hilbert space under some different conditions. Our theorems presented in this paper improve and extend the main results of [9, 12–14, 16].


## 2. Preliminaries

LetK be a nonempty closed and convex subset of a Hilbert space H. We use PK to denote the metric or nearest point projection of H onto K; that is, for x∈H, PKx is the only point in K such that ∥x-PKx∥=inf{∥x-z∥:z∈K}. we write xn⇀x and xn→x indicate that the sequence {xn} convergence weakly and strongly to x, respectively.It is well known that Hilbert spaceH satisfies Opial's condition [8], that is, for any sequence {xn} with xn⇀x and every y∈H with y≠x, we have
(2.1)lim⁡inf⁡n→∞‖xn-x‖<lim⁡inf⁡n→∞‖xn-y‖.To study the equilibrium problem (1.1), we may assume that the bifunction F of K×K into ℝ satisfies the following conditions. (A1)
F(x,x)=0 for all x∈K.(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0 for all x,y∈K.(A3)
For eachx,y,z∈K, limt→0F(tz+(1-t)x,y)≤F(x,y).(A4)
For eachx∈K,y↦F(x,y) is convex and lower semi-continuous.In order to prove our main results, we need the following Lemmas and Propositions.Lemma 2.1 (see [1, 3]).
LetF be a bifunction from K×K into ℝ satisfying (A1)–(A4). Then, for any r>0 and x∈H, there exists z∈K such that
(2.2)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈K.
Further, if Trx={z∈K:F(z,y)+(1/r)〈y-z,z-x〉≥0,forally∈K}, then the following holds.(1)
Tr is single-valued.(2)
Tr is firmly nonexpansive, that is, ∥Trx-Try∥2≤〈Trx-Try,x-y〉, forallx,y∈H.(3)
F(Tr)=EP(F).(4)
EP(F) is closed and convex.Lemma 2.2 (see [7]).
Let(E,〈·,·〉) be an inner product space. Then, for all x,y,z∈E and α,β,γ∈[0,1] with α+β+γ=1, we have
(2.3)‖αx+βy+γz‖2=α‖x‖2+β‖y‖2+γ‖z‖2-αβ‖x-y‖2-αγ‖x-z‖2-βγ‖y-z‖2.Lemma 2.3 (see [19]).
Let{xn} and {zn} be bounded sequence in Banach space E, and let {λn} be a sequence in [0,1] such that 0<liminfn→∞λn≤limsupn→∞λn<1. Suppose xn+1=λnxn+(1-λn)zn and
(2.4)limsup⁡n→∞(‖zn+1-zn‖-‖xn+1-xn‖)≤0,∀n≥0.
Then limn→∞∥zn-xn∥=0.Lemma 2.4 (see [2, 10]).
LetT:K→H be a k-strict pseudocontraction. For λ∈[k,1), define S:K→H by Sx=λx+(1-λ)Tx for each x∈K. Then, as λ∈[k,1), S is a nonexpansive mapping such that F(S)=F(T).Lemma 2.5 (see [10]).
IfT:K→H is a k-strict pseudocontraction, then the fixed point set F(T) is closed convex so that the projection PF(T) is well defined.Lemma 2.6 (see [9]).
LetK be a nonempty bounded closed convex subset of H. Given x∈H and z∈K, then z=PKx if and only if there holds the relation:
(2.5)〈x-z,z-y〉≥0,∀y∈K.Lemma 2.7 (see [20]).
Assume{an} is a sequence of nonnegative real numbers such that
(2.6)an+1≤(1-γn)an+γnδn,n≥0,
where {γn} is a sequence in (0,1) and {δn} is a real sequence such that (i)
∑n=0∞γn=∞,(ii)
limsupn→∞δn≤0 or ∑n=0∞|γnδn|<∞.Then limn→∞an=0.Proposition 2.8 (see, e.g., Acedo and Xu [12]).
LetK be a nonempty closed  convex subset of Hilbert space H. Given an integer N≥1, assume that {Ti}i=1N:K→H is a finite family of ki-strict pseudocontractions. Suppose that {λi}i=1N is a positive sequence such that ∑i=1Nλi=1. Then ∑i=1NλiTi is a k-strict pseudocontraction with k=max{ki:1≤i≤N}.Proposition 2.9 (see, e.g., Acedo and Xu [12]).
Let{Ti}i=1N and {λi}i=1N be given as in Proposition 2.8 above. Then F(∑i=1NλiTi)=⋂i=1NF(Ti).


## 3. Main Results

Theorem 3.1.
LetK be a nonempty closed convex subset of Hilbert space H, and let F be a bifunction from K×K into ℝ satisfying (A1)–(A4). Let {Ti}i=1N:K→H be a finite family of ki-strict pseudocontractions such that k=max{ki:1≤i≤N} and ℱ=⋂i=1NF(Ti)⋂EP(F)≠ϕ. Assume {ηi(n)}i=1N is a finite sequences of positive numbers such that ∑i=1Nηi(n)=1 for all n≥0. Given u∈K and x0∈K, {αn},{βn}, and {γn} are some sequences in (0,1); the following control conditions are satisfied.(i)
k≤αn≤λ<1 for all n≥0 and limn→∞αn=α,(ii)
limn→∞βn=0 and ∑n=0∞βn=∞,(iii)
0<liminfn→∞γn≤limsupn→∞γn<1,(iv)
limn→∞|ηi(n+1)-ηi(n)|=0.Then the sequence {xn} generated by (1.4) converges strongly to q∈ℱ, where q=Pℱu.Proof.
From Lemma2.1, we see that EP(F)=F(Tr), and note that un can be rewritten as un=Trxn. Putting An=∑i=1Nηi(n)Ti, we have An:K→H is a k-strict pseudocontraction and F(An)=⋂i=1NF(Ti) by Propositions 2.8 and 2.9, where k=max{ki:1≤i≤N}.
From (1.4), condition (i), and Lemma 2.2, taking a point p∈ℱ, we have
(3.1)‖yn-p‖2=‖αn(un-p)+(1-αn)(Anun-p)‖2=αn‖un-p‖2+(1-αn)‖Anun-p‖2-αn(1-αn)‖un-Anun‖2≤αn‖un-p‖2+(1-αn)[‖un-p‖2+k‖un-Anun‖2]-αn(1-αn)‖un-Anun‖2=‖un-p‖2-(1-αn)(αn-k)‖un-Anun‖2≤‖Trxn-Trp‖2≤‖xn-p‖2.
Furthermore, we have
(3.2)‖yn-p‖≤‖un-p‖≤‖xn-p‖.
It follows from (1.4) and (3.2) that
(3.3)‖xn+1-p‖=‖βnu+γnxn+(1-βn-γn)yn-p‖≤βn‖u-p‖+γn‖xn-p‖+(1-βn-γn)‖yn-p‖≤βn‖u-p‖+(1-βn)‖xn-p‖≤max⁡⁡{‖u-p‖,‖x0-p‖}.
Consequently, sequence {xn} is bounded and so are {un} and {yn}.
Define a mappingTnx:=αnx+(1-αn)Anx for each x∈K. Then Tn:K→H is nonexpansive. Indeed, by using (1.1), condition (i), and Lemma 2.2, we have for all x,y∈K that
(3.4)‖Tnx-Tny‖2=‖αn(x-y)+(1-αn)(Anx-Any)‖2=αn‖x-y‖2+(1-αn)‖Anx-Any‖2-αn(1-αn)‖x-Anx-(y-Any)‖2≤αn‖x-y‖2+(1-αn)[‖x-y‖2+k‖x-Anx-(y-Any)‖2]-αn(1-αn)‖x-Anx-(y-Any)‖2=‖x-y‖2-(1-αn)(αn-k)‖x-Anx-(y-Any)‖2≤‖x-y‖2,
which shows that Tn:K→H is nonexpansive.
Next we show thatlimn→∞∥xn+1-xn∥=0. Setting xn+1=γnxn+(1-γn)zn, we have
(3.5)zn+1-zn=xn+2-γn+1xn+11-γn+1-xn+1-γnxn1-γn=βn+1u+(1-βn+1-γn+1)yn+11-γn+1-βnu+(1-βn-γn)yn1-γn=βn+11-γn+1(u-yn+1)+(yn+1-yn)-βn1-γn(u-yn).
It follows that
(3.6)‖zn+1-zn‖≤βn+11-γn+1‖u-yn+1‖+‖yn+1-yn‖+βn1-γn‖u-yn‖.
From (1.4), we have yn=Tnun and
(3.7)‖yn+1-yn‖≤‖Tn+1un+1-Tn+1un‖+‖Tn+1un-Tnun‖≤‖un+1-un‖+‖αn+1un+(1-αn+1)An+1un-[αnun+(1-αn)Anun]‖≤‖un+1-un‖+|αn+1-αn|‖un-Anun‖+(1-αn+1)‖An+1un-Anun‖≤‖un+1-un‖+|αn+1-αn|‖un-Anun‖+(1-αn+1)∑i=1N|ηi(n+1)-ηi(n)|‖Tiun‖.
By Lemma 2.1, un=Trxn and un+1=Trxn+1, we have
(3.8)F(un,y)+1r〈y-un,un-xn〉≥0,∀y∈K,(3.9)F(un+1,y)+1r〈y-un+1,un+1-xn+1〉≥0,∀y∈K.
Putting y=un+1 in (3.8) and y=un in (3.9), we obtain
(3.10)F(un,un+1)+1r〈un+1-un,un-xn〉≥0,F(un+1,un)+1r〈un-un+1,un+1-xn+1〉≥0.
So, from (A2) and r>0, we have
(3.11)〈un+1-un,un-un+1+un+1-xn-(un+1-xn+1)〉≥0,
and hence
(3.12)‖un+1-un‖2≤〈un+1-un,xn+1-xn〉,
which implies that
(3.13)‖un+1-un‖≤‖xn+1-xn‖.
Combining (3.6), (3.7), and (3.13), we have
(3.14)‖zn+1-zn‖≤βn+11-γn+1‖u-yn+1‖+‖xn+1-xn‖+βn1-γn‖u-yn‖+|αn+1-αn|‖un-Anun‖+(1-αn+1)∑i=1N|ηi(n+1)-ηi(n)|‖Tiun‖.
This together with (i), (ii) and (iv) imply that
(3.15)limsup⁡n→∞(‖zn+1-zn‖-‖xn+1-xn‖)≤0.
Hence, by Lemma 2.3, we obtain
(3.16)lim⁡n→∞‖zn-xn‖=0.
Consequently,
(3.17)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1-γn)‖zn-xn‖=0.
On the other hand, by (1.4) and (iii), we have
(3.18)‖xn+1-yn‖≤βn‖u-yn‖+γn‖xn-xn+1‖+γn‖xn+1-yn‖,
which implies that
(3.19)‖xn+1-yn‖≤βn1-γn‖u-yn‖+γn1-γn‖xn-xn+1‖.
Combining (ii), (3.17), and (3.19), we have
(3.20)lim⁡n→∞‖xn+1-yn‖=0.
Note that
(3.21)‖xn-yn‖≤‖xn-xn+1‖+‖xn+1-yn‖,
which together with (3.17) and (3.20) implies
(3.22)lim⁡n→∞‖xn-yn‖=0.
Moreover, for p∈ℱ=⋂i=1NF(Ti)⋂EP(F), we have
(3.23)‖un-p‖2=‖Trxn-Trp‖2≤〈xn-p,un-p〉=12(‖xn-p‖2+‖un-p‖2-‖xn-un‖2),
and hence
(3.24)‖un-p‖2≤‖xn-p‖2-‖xn-un‖2.
From Lemma 2.2, (3.2) and (3.24), we have
(3.25)‖xn+1-p‖2=‖βnu+γnxn+(1-βn-γn)yn-p‖2≤βn‖u-p‖2+γn‖xn-p‖2+(1-βn-γn)‖yn-p‖2≤βn‖u-p‖2+γn‖xn-p‖2+(1-βn-γn)(‖xn-p‖2-‖xn-un‖2)≤βn‖u-p‖2+‖xn-p‖2-(1-βn-γn)‖xn-un‖2,
and hence
(3.26)(1-βn-γn)‖xn-un‖2≤βn‖u-p‖2+‖xn-p‖2-‖xn+1-p‖2≤βn‖u-p‖2+‖xn-xn+1‖(‖xn-p‖+‖xn+1-p‖).
By (ii) and (3.17), we obtain
(3.27)lim⁡n→∞‖xn-un‖=0.
It follows from (3.22) and (3.27) that
(3.28)lim⁡n→∞‖yn-un‖=0.
Define Sn:K→H by Snx=αx+(1-α)Anx. Then, Sn is a nonexpansive with F(Sn)=F(An) by Lemma 2.4. Note that limn→∞αn=α∈[k,1) by condition (i) and
(3.29)‖un-Snun‖≤‖un-yn‖+‖yn-Snun‖≤‖un-yn‖+‖αnun+(1-αn)Anun-[αun+(1-α)Anun]‖≤‖un-yn‖+|αn-α|‖un-Anun‖,
which combines with condition (i) and (3.28) yielding that
(3.30)lim⁡n→∞‖un-Snun‖=0.
We now show thatlimsupn→∞〈u-q,xn-q〉≤0, where q=Pℱu. To see this, we choose a subsequence {xni} of {xn} such that
(3.31)limsup⁡n→∞〈u-q,xn-q〉=lim⁡i→∞〈u-q,xni-q〉.
Since {uni} is bounded, there exists a subsequence {unij} of {uni} converging weakly to u*. Without loss of generality, we assume that uni⇀u* as i→∞. Form (3.27), we obtain xni⇀u* as i→∞. Since K is closed and convex, K is weakly closed. So, we have u*∈K and u*∈F(Sn). Otherwise, from u*≠Snu* and Opial's condition, we obtain
(3.32)lim⁡inf⁡i→∞‖uni-u*‖<lim⁡inf⁡i→∞‖uni-Sniu*‖≤lim⁡inf⁡i→∞(‖uni-Sniuni‖+‖Sniuni-Sniu*‖)≤lim⁡inf⁡i→∞‖uni-u*‖.
This is a contradiction. Hence, we get u*∈F(Sn)=F(An). Moreover, by un=Trxn, we have
(3.33)F(un,y)+1r〈y-un,un-xn〉≥0,∀y∈K.
It follows from (A2) that
(3.34)1r〈y-un,un-xn〉≥F(y,un).
Replacing n by ni, we have
(3.35)1r〈y-uni,uni-xni〉≥F(y,uni).
Since uni-xni→0 and uni⇀u*, it follows from (A4) that F(y,u*)≤0 for all y∈K. Put zt=ty+(1-t)u* for all t∈(0,1] and y∈K. Then, we have zt∈K, and hence, F(zt,u*)≤0. By (A1) and (A4), we have
(3.36)0=F(zt,zt)≤tF(zt,y)+(1-t)F(zt,u*)≤tF(zt,y),
which implies F(zt,y)≥0. From (A3), we have F(u*,y)≥0 for all y∈K, and hence, u*∈EP(F). Therefore, u*∈F(Sn)∩EP(F). From Lemma 2.6, we know that
(3.37)〈u-PFu,u*-PFu〉≤0.
It follows from (3.31) and (3.37) that
(3.38)limsup⁡n→∞〈u-q,xn-q〉=lim⁡i→∞〈u-q,xni-q〉=〈u-q,u*-q〉≤0.
Finally, we prove thatxn→q=Pℱu as n→∞. From (1.4) again, we have
(3.39)‖xn+1-q‖2=〈βnu+γnxn+(1-βn-γn)yn-q,xn+1-q〉=βn〈u-q,xn+1-q〉+γn〈xn-q,xn+1-q〉+(1-βn-γn)〈yn-q,xn+1-q〉≤βn〈u-q,xn+1-q〉+γn‖xn-q‖‖xn+1-q‖+(1-βn-γn)‖yn-q‖‖xn+1-q‖≤(1-βn)‖xn-q‖‖xn+1-q‖+βn〈u-q,xn+1-q〉≤1-βn2(‖xn-q‖2+‖xn+1-q‖2)+βn〈u-q,xn+1-q〉≤1-βn2‖xn-q‖2+12‖xn+1-q‖2+βn〈u-q,xn+1-q〉,
which implies that
(3.40)‖xn+1-q‖2≤(1-βn)‖xn-q‖2+2βn〈u-q,xn+1-q〉.
It follows from (3.38), (3.40), and Lemma 2.7 that limn→∞∥xn-q∥=0. This completes the proof.AsN=1, that is, An=T and ηi(n)≡1 in Theorem 3.1, we have the following results immediately.Theorem 3.2.
LetK be a nonempty closed convex subset of Hilbert space H, and let F be a bifunction from K×K into ℝ satisfying (A1)–(A4). Let T:K→H be a k-strict pseudocontractions such that ℱ=F(T)⋂EP(F)≠ϕ. Let {xn} be a sequence generated in the following manner:
(3.41)F(un,y)+1r〈y-un,un-xn〉≥0,∀y∈K,yn=αnun+(1-αn)Tun,xn+1=βnu+γnxn+(1-βn-γn)yn,n≥0,
where u∈K and x0∈K, {αn},{βn}, and {γn} are some sequences in (0,1). If the following control conditions are satisfied: (i)
k≤αn≤λ<1 for all n≥0 and limn→∞αn=α,(ii)
limn→∞βn=0 and ∑n=0∞βn=∞,(iii)
0<liminfn→∞γn≤limsupn→∞γn<1,then {xn} converges strongly to q∈ℱ, where q=Pℱu.



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