# Some Results on Strictly Pseudocontractive Nonself-Mappings and Equilibrium Problems in Hilbert Spaces

Authors: Yan Hao; Sun Young Cho
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

An equilibrium problem and a strictly pseudocontractive nonself-mapping are investigated. Strong convergence theorems of common elements are established based on hybrid projection algorithms in the framework of real Hilbert spaces.


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## Body

## 1. Introduction

Bifunction equilibrium problems which were considered by Blum and Oettli [1] have intensively been studied. It has been shown that the bifunction equilibrium problem covers fixed point problems, variational inequalities, inclusion problems, saddle problems, complementarity problem, minimization problem, and the Nash equilibrium problem; see [1–4] and the references therein. Iterative methods have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization; see [5–16] and the references therein. In this paper, we investigate an equilibrium problem and a strictly pseudocontractive nonself-mapping based on hybrid projection algorithms. Strong convergence theorems of common elements lie in the solution set of the equilibrium problem and the fixed point set of the strictly pseudocontractive nonself-mapping.Throughout this paper, we always assume thatH is a real Hilbert space with the inner product 〈·,·〉 and the norm ∥·∥. Let C be a nonempty closed convex subset of H and F a bifunction of C×C into ℝ, where ℝ denotes the set of real numbers. In this paper, we consider the following equilibrium problem:
(1.1)Findx∈CsuchthatF(x,y)≥0,∀y∈C.
The set of such an x∈C is denoted by EP(F), that is,
(1.2)EP(F)={x∈C:F(x,y)≥0,∀y∈C}.Given a mappingT:C→H, let F(x,y)=〈Tx,y-x〉 for all x,y∈C. Then z∈EP(F) if and only if 〈Tz,y-z〉≥0 for all y∈C, that is, z is a solution of the variational inequality.To study the equilibrium problem (1.1), we may 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 eachx,y,z∈C,
(1.3)limsupt↓0F(tz+(1-t)x,y)≤F(x,y);(A4)
for eachx∈C, y↦F(x,y) is convex and lower semicontinuous.IfH is an Euclidean space, then we see that F(x,y)=x-y is a simple example satisfying the above assumptions. See [2] for more details. Let S:D(S)→R(S), where D(S) and R(S) denote the domain and the range of the mapping S. If D(S)=R(S), then the mapping S is said to be a self-mapping. If D(S)≠R(S), then the mapping S is said to be a nonself-mapping. Let S:C→H be a nonself-mapping. In this paper, we use F(S) to denote the fixed point set of S. Recall the following definitions. S is said to be nonexpansive if
(1.4)∥Sx-Sy∥≤∥x-y∥,∀x,y∈C.S is said to be strictly pseudocontractive if there exists a constant κ∈[0,1) such that
(1.5)∥Sx-Sy∥2≤∥x-y∥2+κ∥(I-S)x-(I-S)y∥2,∀x,y∈C.For such a case,S is also said to be κ-strict pseudocontraction. It is clear that (1.5) is equivalent to
(1.6)〈Sx-Sy,x-y〉≤∥x-y∥2-1-κ2∥(x-Sx)-(y-Sy)∥2,∀x,y∈C.S is said to be pseudocontractive if
(1.7)∥Sx-Sy∥2≤∥x-y∥2+∥(I-S)x-(I-S)y∥2,∀x,y∈C.It is clear that (1.7) is equivalent to
(1.8)〈Sx-Sy,x-y〉≤∥x-y∥2,∀x,y∈C.The class ofκ-strict pseudocontractions which was introduced by Browder and Petryshyn [17] in 1967 has been considered by many authors. It is easy to see that the class of strict pseudocontractions falls into the one between the class of nonexpansive mappings and the class of pseudocontractions. For studying the class of strict pseudocontractions, Zhou [18] proposed the following convex combination method: define a mapping St:C→H by
(1.9)Stx=tx+(1-t)Sx,∀x∈C.He showed thatSt is nonexpansive if t∈[κ,1); see [18] for more details.Recently, many authors considered the problem of finding a common element in the fixed point set of a nonexpansive mapping and in the solution set of the equilibrium problem (1.1) based on iterative methods; see, for instance, [19–27].In 2007, Tada and Takahashi [23] considered an iterative method for the equilibrium problem (1.1) and a nonexpansive nonself-mapping. To be more precise, they obtained the following results.Theorem TT.
LetC be a closed convex subset of a real Hilbert space H, let f:C×C→ℝ be a bifunction satisfying (A1)–(A4), and let S be a nonexpansive mapping of C into H such that F(S)∩EP(f)≠∅. Let {xn} and {un} be sequences generated by x1=x∈H, and let
(1.10)un∈Csuchthatf(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,wn=(1-αn)xn+αnSun,Cn={z∈H:∥wn-z∥≤∥xn-z∥},Qn={z∈H:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,
for everyn∈ℕ, where {αn}⊂[a,1], for some a∈(0,1) and {rn}⊂(0,∞) satisfies liminfn→∞rn>0. Then the sequence {xn} converges strongly to PF(S)∩EP(f)(x).We remark that the iterative process (1.10) is called the hybrid projection iterative process. Recently, the hybrid projection iterative process which was first considered by Haugazeau [28] in 1968 has been studied for fixed point problem of nonlinear mappings and equilibrium problems by many authors. Since the sequence generated in the hybrid projection iterative process depends on the sets Cn and Qn, the hybrid projection iterative process is also known as “CQ” iterative process; see [29] and the reference therein.Recently, Takahashi et al. [30] considered the shrinking projection process for the fixed point problem of nonexpansive self-mapping. More precisely, they obtain the iterative sequence monontonely without the help of the set Qn; see [30] for more details.In this paper, we reconsider the same shrinking projection process for the equilibrium problem (1.1) and a strictly pseudocontractive nonself-mapping. We show that the sequence generated in the proposed iterative process converges strongly to some common element in the fixed point set of a strictly pseudocontractive nonself-mapping and in the solution set of the equilibrium problem (1.1). The main results presented in this paper mainly improved the corresponding results in Tada and Takahashi [23].


## 2.  Preliminaries

LetC be a nonempty closed and convex subset of a real Hilbert space H. Let PC be the metric projection from H onto C. That is, for x∈H,PCx is the only point in C such that ∥x-PCx∥=inf{∥x-z∥:z∈C}. We know that the mapping PC is firmly nonexpansive, that is,
(2.1)∥PCx-PCy∥2≤〈PCx-PCy,x-y〉,∀x,y∈H.The following lemma can be found in [1, 2].Lemma 2.1.
LetC be a nonempty closed convex subset of H, and let F:C×C→ℝ be a bifunction satisfying  (A1)–(A4). Then, for any r>0 and x∈H, there exists z∈C such that (2.2)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Further, define(2.3)Trx={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
for allr>0 and x∈H. Then, the following hold:  (a)
Tr is single valued;(b)
Tr is firmly nonexpansive, that is,
(2.4)∥Trx-Try∥2≤〈Trx-Try,x-y〉,∀x,y∈H;(c)
F(Tr)=EP(F);(d)
EP(F) is closed and convex.Lemma 2.2 (see [18]).
LetH be a real Hilbert space, C a nonempty closed convex subset of H, and S:C→H a strict pseudocontraction. Then the mapping I-S is demiclosed at zero, that is, if {xn} is a sequence in C such that xn⇀x- and xn-Sxn→0, then x-∈F(S).Lemma 2.3 (see [18]).
LetC be a nonempty closed convex subset of a real Hilbert space H and S:C→H a κ-strict pseudocontraction. Define a mapping Sαx=αx+(1-α)Sx for all x∈C. If α∈[κ,1), then the mapping Sα is a nonexpansive mapping such that F(Sα)=F(S).


## 3. Main Results

Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F1 and F2 be two bifunctions from C×C to ℝ which satisfy (A1)–(A4), respectively. Let S:C→H be a κ-strict pseudocontraction. Assume that ℱ:=EP(F1)∩EP(F2)∩F(S)≠∅. Let {αn}, {βn}, and {γn} be sequences in [0,1]. Let {xn} be a sequence generated in the following manner:
(Δ)x1∈H,C1=H,zn=γnun+(1-γn)vn,yn=αnxn+(1-αn)(βnzn+(1-βn)Szn),Cn+1={w∈Cn:∥yn-w∥≤∥xn-w∥},xn+1=PCn+1x1,n≥0,
whereun is chosen such that
(3.1)F1(un,u)+1rn〈u-un,un-xn〉≥0,∀u∈C,
andvn is chosen such that
(3.2)F2(vn,v)+1sn〈v-vn,vn-xn〉≥0,∀v∈C.
Assume that the control sequences{αn}, {βn}, {γn}, {rn}, and {sn} satisfy the following restrictions:(a)
0≤αn≤a<1,κ≤βn≤b<1,0<c≤γn≤d<1;(b)
0<e≤rn,0<f≤sn
for somea,b,c,d,e,f∈ℝ. Then the sequence {xn} generated in the (Δ) converges strongly to some point x-, where x-=Pℱx1.Proof.
First, we show thatCn is closed and convex for each n≥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. Note that C1=H is convex. Suppose that Cm is convex for some positive integer m. Next, we show that Cm+1 is convex for the same m. Note that ∥yn-w∥≤∥xn-w∥ is equivalent to
(3.3)2〈xn-yn,w〉≤∥xn∥2-∥yn∥2.
Takew1 and w2 in Cm+1, and put w-=tw1+(1-t)w2. It follows that w1∈Cm,w2∈Cm,
(3.4)2〈xn-yn,w1〉≤∥xn∥2-∥yn∥2,2〈xn-yn,w2〉≤∥xn∥2-∥yn∥2.
Combining (3.4), we can obtain that 2〈xn-yn,w-〉≤∥xn∥2-∥yn∥2, that is, ∥yn-w-∥≤∥xn-w-∥. In view of the convexity of Cm, we see that w-∈Cm. This shows that w-∈Cm+1. This concludes that Cn is closed and convex for each n≥1.
Define a mappingSn:C→H by Snx=βnx+(1-βn)Sx for all x∈C. It follows from Lemma 2.3 that Sn is nonexpansive and F(Sn)=F(S) for all n≥1. Next, we show that ℱ⊂Cn for all n≥1. It is easy to see that ℱ⊂C1=H. Suppose that ℱ⊂Ck for some integer k≥1. We intend to claim that ℱ⊂Ck+1 for the same k. For any p∈ℱ⊂Ck, we have
(3.5)∥yk-p∥≤αk∥xk-p∥+(1-αk)∥Skzk-p∥∥yk-p∥≤αk∥xk-p∥+(1-αk)∥zk-p∥≤αk∥xk-p∥+(1-αk)(γk∥uk-p∥+(1-γk)∥vk-p∥)=αk∥xk-p∥+(1-αk)(γk∥Trkxk-p∥+(1-γk)∥Tskxk-p∥)≤αk∥xk-p∥+(1-αk)(γk∥xk-p∥+(1-γk)∥xk-p∥)=∥xk-p∥.
This shows thatp∈Ck+1. This proves that ℱ⊂Cn for all n≥1.
Sincexn=PCnx1 and xn+1=PCn+1x1∈Cn+1⊂Cn, we have that
(3.6)0≤〈x1-xn,xn-xn+1〉=〈x1-xn,xn-x1+x1-xn+1〉≤-∥x1-xn∥2+∥x1-xn∥∥x1-xn+1∥.
It follows that(3.7)∥x1-xn∥≤∥x1-xn+1∥.
On the other hand, for anyp∈ℱ⊂Cn, we see that ∥x1-xn∥≤∥x1-p∥. In particular, we have
(3.8)∥x1-xn∥≤∥x1-Pℱx1∥.
This shows that the sequence{xn} is bounded. In view of (3.7), we see that limn→∞∥xn-x1∥ exists. It follows from (3.6) that
(3.9)∥xn-xn+1∥2=∥xn-x1+x1-xn+1∥2=∥xn-x1∥2+2〈xn-x1,x1-xn+1〉+∥x1-xn+1∥2=∥xn-x1∥2+2〈xn-x1,x1-xn+xn-xn+1〉+∥x1-xn+1∥2=∥xn-x1∥2-2∥xn-x1∥2+2〈xn-x1,xn-xn+1〉+∥x1-xn+1∥2≤∥x1-xn+1∥2-∥xn-x1∥2.
This yields that(3.10)limn→∞∥xn-xn+1∥=0.
Sincexn+1=PCn+1x1∈Cn+1, we see that
(3.11)∥yn-xn+1∥≤∥xn-xn+1∥.
It follows that(3.12)∥yn-xn∥≤∥yn-xn+1∥+∥xn+1-xn∥≤2∥xn-xn+1∥.
From (3.10), we obtain that
(3.13)limn→∞∥xn-yn∥=0.
On the other hand, we have(3.14)∥xn-yn∥=∥xn-αnxn-(1-αn)Snzn∥=(1-αn)∥xn-Snzn∥.
From the restriction (a), we obtain from (3.13) that
(3.15)limn→∞∥xn-Snzn∥=0.
For anyp∈ℱ, we have that
(3.16)∥un-p∥2=∥Trnxn-Trnp∥2≤〈Trnxn-Trnp,xn-p〉=〈un-p,xn-p〉=12(∥un-p∥2+∥xn-p∥2-∥un-xn∥2).
This implies that(3.17)∥un-p∥2≤∥xn-p∥2-∥un-xn∥2.
In a similar way, we get that(3.18)∥vn-p∥2≤∥xn-p∥2-∥vn-xn∥2.
It follows from (3.17) and (3.18) that
(3.19)∥yn-p∥2≤αn∥xn-p∥2+(1-αn)∥Snzn-p∥2≤αn∥xn-p∥2+(1-αn)∥zn-p∥2≤αn∥xn-p∥2+(1-αn)(γn∥un-p∥2+(1-γn)∥vn-p∥2)≤∥xn-p∥2-(1-αn)γn∥un-xn∥2-(1-αn)(1-γn)∥vn-xn∥2.
This implies that(3.20)(1-αn)γn∥un-xn∥2≤∥xn-p∥2-∥yn-p∥2≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
In view of the restriction (a), we obtain from (3.13) that
(3.21)limn→∞∥un-xn∥=0.
It also follows from (3.19) that
(3.22)(1-αn)(1-γn)∥vn-xn∥2≤∥xn-p∥2-∥yn-p∥2≤(∥xn-p∥+∥yn-p∥)∥xn-yn∥.
In view of the restriction (a), we obtain from (3.13) that
(3.23)limn→∞∥vn-xn∥=0.
Since{xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀q. From (3.21) and (3.23), we see that uni⇀q and vni⇀q, respectively. From (3.21) and the restriction (b), we see that
(3.24)limn→∞∥un-xn∥rn=0.
Now, we are in a position to show thatq∈EP(F1). Note that
(3.25)F1(un,u)+1rn〈u-un,un-xn〉≥0,∀u∈C.
From (A2), we see that(3.26)1rn〈u-un,un-xn〉≥F1(u,un).
Replacingn by ni, we arrive at
(3.27)〈u-uni,uni-xnirni〉≥F1(u,uni).
In view of (3.24) and (A4), we get that
(3.28)F1(u,q)≤0,∀u∈C.
For any t with 0<t≤1 and u∈C, let ut=tu+(1-t)q. Since u∈C and q∈C, we have ut∈C and hence F1(ut,q)≤0. It follows that
(3.29)0=F1(ut,ut)≤tF1(ut,u)+(1-t)F1(ut,q)≤tF1(ut,u),
which yields that(3.30)F1(ut,u)≥0,∀u∈C.
Lettingt↓0, we obtain from (A3) that
(3.31)F1(q,u)≥0,∀u∈C.
This means that q∈EP(F1). In the same way, we can obtain that q∈EP(F2). Next, we show that q∈F(S). Note that
(3.32)∥zn-xn∥≤γn∥un-xn∥+(1-γn)∥vn-xn∥.
It follows from (3.21) and (3.23) that
(3.33)limn→∞∥zn-xn∥=0.
On the other hand, we have(3.34)∥xn-Snxn∥≤∥Snxn-Snzn∥+∥Snzn-xn∥≤∥xn-zn∥+∥Snzn-xn∥.
It follows from (3.15) and (3.33) that
(3.35)limn→∞∥xn-Snxn∥=0.
Note that(3.36)∥Sxn-xn∥≤∥Sxn-Snxn∥+∥Snxn-xn∥≤βn∥Sxn-xn∥+∥Snxn-xn∥,
which yields that(3.37)(1-βn)∥Sxn-xn∥≤∥Snxn-xn∥.
This implies from the restriction (a) and (3.35) that
(3.38)limn→∞∥Sxn-xn∥=0.
It follows from Lemma2.2 that q∈F(S). This shows that q∈ℱ. Since x-=Pℱx1, we obtain that
(3.39)∥x1-x-∥≤∥x1-q∥≤liminfi→∞∥x1-xni∥≤limsupi→∞∥x1-xni∥≤∥x1-x-∥,
which yields that(3.40)limi→∞∥x1-xni∥=∥x1-q∥=∥x1-x-∥.
It follows that {xni} converges strongly to x-. Therefore, we can conclude that the sequence {xn} converges strongly to x-=Pℱx1. This completes the proof.From Theorem3.1, we have the following results.Corollary 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from C×C to ℝ which satisfies (A1)–(A4). Let S:C→H be a κ-strict pseudocontraction. Assume that ℱ:=EP(F)∩F(S)≠∅. Let {αn} and {βn} be sequences in [0,1]. Let {xn} be a sequence generated in the following manner:
(3.41)x1∈H,C1=H,yn=αnxn+(1-αn)(βnun+(1-βn)Sun),Cn+1={w∈Cn:∥yn-w∥≤∥xn-w∥},xn+1=PCn+1x1,n≥0,
whereun is chosen such that
(3.42)F(un,u)+1rn〈u-un,un-xn〉≥0,∀u∈C.
Assume that the control sequences{αn}, {βn}, and {rn} satisfy the following restrictions:  (a)
0≤αn≤a<1,κ≤βn≤b<1;(b)
0<e≤rn
for somea,b,e∈ℝ. Then the sequence {xn} converges strongly to some point x-, where x-=Pℱx1.Proof.
PuttingF1≡F2≡F and rn≡sn in Theorem 3.1, we see that zn≡un. From the proof of Theorem 3.1, we can conclude the desired conclusion immediately.Remark 3.3.
Corollary3.2 improves Theorem TT in the following aspects.(1)
From the viewpoint of mappings, the class of nonexpansive mappings is extended to the class of strict pseudocontractions.(2)
From the viewpoint of computation, the setQn is removed.Corollary 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let S:C→H be a κ-strict pseudocontraction with fixed points. Let {αn} and {βn} be sequences in [0,1]. Let {xn} be a sequence generated in the following manner:
(3.43)x1∈H,C1=H,yn=αnxn+(1-αn)(βnPCxn+(1-βn)SPCxn),Cn+1={w∈Cn:∥yn-w∥≤∥xn-w∥},xn+1=PCn+1x1,n≥0.
Assume that the control sequences{αn} and {βn} satisfy the restrictions 0≤αn≤a<1 and κ≤βn≤b<1 for some a,b∈ℝ. Then the sequence {xn} converges strongly to some point x-, where x-=PF(S)x1.Proof.
PuttingF1≡F2≡0 and rn≡sn≡1, we can obtain from Theorem 3.1 the desired conclusion easily.
IfS is nonexpansive and βn≡0, then Corollary 3.4 is reduced to the following.Corollary 3.5.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let S:C→H be a nonexpansive mapping with fixed points. Let {αn} be a sequence in [0,1]. Let {xn} be a sequence generated in the following manner:
(3.44)x1∈H,C1=H,yn=αnxn+(1-αn)SPCxn,Cn+1={w∈Cn:∥yn-w∥≤∥xn-w∥},xn+1=PCn+1x1,n≥0.
Assume that the control sequence{αn} satisfies the restriction 0≤αn≤a<1 for some a∈ℝ. Then the sequence {xn} converges strongly to some point x-, where x-=PF(S)x1.Recently, many authors studied the following convex feasibility problem (CFP):(3.45)findinganx∈⋂m=1N‍Cm,
where N≥1 is an integer and each Cm is a nonempty closed and convex subset of a real Hilbert space H. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning.Next, we consider the case that eachCm is the solution set of an equilibrium problem.Theorem 3.6.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Fm be a bifunction from C×C to ℝ which satisfies (A1)–(A4) for each 1≤m≤N. Assume that ℱ:=∩m=1NEP(Fm)≠∅. Let {αn}, {γ(n,1)},…, and {γ(n,N)} be sequences in [0,1]. Let r(n,1),…, and r(n,N) be sequences in (0,∞). Let {xn} be a sequence generated in the following manner:
(3.46)x1∈H,C1=H,Fm(u(n,m),u)+1r(n,m)〈u-u(n,m),u(n,m)-xn〉≥0,∀u∈C,yn=αnxn+(1-αn)∑m=1N‍γ(n,m)u(n,m),Cn+1={w∈Cn:∥yn-w∥≤∥xn-w∥},xn+1=PCn+1x1,n≥0.
Assume that the control sequences satisfy the following restrictions:(a)
0≤αn≤a<1,∑m=1N‍γ(n,m)=1, 0<cm≤γ(n,m)≤dm<1;(b)
0<em≤r(n,m), where a,c1,…,cm,d1,…,dm,e1,…,em∈ℝ. Then the sequence {xn} generated in the above iterative process converges strongly to some point x-, where x-=Pℱx1.Proof.
LetS be the identity mapping and βn≡0, then we can obtain from Theorem 3.1 the desired conclusion easily.



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