# A Strong Convergence Theorem for a Common Fixed Point of Two Sequences of Strictly Pseudocontractive Mappings in Hilbert Spaces and Applications

Authors: Kasamsuk Ungchittrakool
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We prove a strong convergence theorem for a common fixed point of two sequences of strictly
pseudocontractive mappings in Hilbert spaces. We also provide some applications of the main theorem to find a common
element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem in
Hilbert spaces. The results extend and improve the recent ones announced by Marino and Xu (2007) and others.


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## Body

## 1. Introduction

LetH be a real Hilbert space and C a nonempty closed convex subset of H. Let T:C→C be a self-mapping of C. Recall that T is said to be a strict pseudocontraction if there exists a constant 0⩽k<1 such that(1.1)‖Tx-Ty‖2⩽‖x-y‖+k‖(I-T)x-(I-T)y‖2
for all x,y∈C. (We also say that T is a k-strict pseudocontraction if T satisfies (1.1)). We use F(T) to denote the set of fixed points of T (i.e., F(T)={x∈C:Tx=x}). Note that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings which are mappings T on C such that (1.2)‖Tx-Ty‖⩽‖x-y‖
for all x,y∈C. That is, T is nonexpansive if and only if T is a 0-strict pseudocontraction.In 1953, Mann [1] introduced the following iterative scheme:(1.3)x0∈Cchosenarbitrarily,xn+1=αnxn+(1-αn)Txn,n=0,1,2,…,
where the sequence {αn} is chosen in [0,1]. Mann's iteration process (1.3) has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [2]. In an infinite-dimensional Hilbert space, the Mann's iteration (1.3) can conclude only weak convergence [3, 4]. In 1967, Browder and Petryshyn [5] established the first convergence result for a k-strict pseudocontraction in a real Hilbert space. They proved weak and strong convergence theorems by using (1.3) with a constant control sequence {αn}≡α for all n. However, this scheme has only weak convergence even in a Hilbert space. Therefore, many authors try to modify the normal Mann's iteration process to have strong convergence; see, for example, [6–10] and the references therein.Attempts to modify (1.3) so that strong convergence is guaranteed have been made. In 2003, Nakajo and Takahashi [9] proposed the following modification of (1.3) for a single nonexpansive mapping T by using the hybrid projection method in a Hilbert space H(1.4)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)Txn,Cn={z∈C:‖z-yn‖⩽‖z-xn‖},Qn={z∈C:〈xn-z,x-xn〉⩾0},xn+1=PCn∩Qn(x),n=0,1,2,…,
where PC denotes the metric projection from H onto a closed convex subset C of H. They proved that if the sequence {αn} is bounded above from one, then {xn} defined by (1.4) converges strongly to PF(T)(x).In 2007, Marino and Xu [11] proved the following strong convergence theorem by using the hybrid projection method for a strict pseudocontraction. They defined a sequence as follows:(1.5)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)Txn,Cn={z∈C:‖yn-z‖2⩽‖xn-z‖2+(1-αn)(k-αn)‖xn-Txn‖2},Qn={z∈C:〈xn-z,x0-xn〉⩾0},xn+1=PCn∩Qn(x0),n=0,1,2,…,
They proved that if 0⩽αn<1, then {xn} defined by (1.5) converges strongly to PF(T)(x0).Motivated and inspired by the above-mentioned results, it is the purpose of this paper to improve and generalize the algorithm (1.5) to the new general process of two sequences of strictly pseudocontractive mappings in Hilbert spaces. Let C be a closed convex subset of a Hilbert space H and Tn,Sn:C→C two sequences of strictly pseudocontractive mappings such that ⋂n=0∞F(Tn)∩⋂n=0∞F(Sn)≠∅. Define {xn} in the following ways:(1.6)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)zn,zn=βnTnxn+(1-βn)Snxn,Ĉn={z∈C:‖yn-z‖2⩽‖xn-z‖2+(1-αn)βn(kTn-αn)‖xn-Tnxn‖2+(1-αn)(1-βn)(kSn-αn)‖xn-Snxn‖2-(1-αn)2βn(1-βn)‖Tnxn-Snxn‖2},Qn={z∈C:〈xn-z,x0-xn〉⩾0},xn+1=PĈn∩Qn(x0),
where {αn}, {βn} are sequences in [0,1].We prove that the algorithm (1.6) converges strongly to a common fixed point of two sequences of strictly pseudocontractive mappings {Tn} and {Sn} provided that {Tn}, {Sn}, {αn}, and {βn} satisfy some appropriate conditions, and then we apply the result for finding a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem in Hilbert spaces. Our results extend and improve the corresponding ones announced by Marino and Xu [11] and others.Throughout the paper, we will use the following notation:(i)
→ for strong convergence and ⇀ for weak convergence,(ii)
ωw(xn)={x:∃xnr⇀x} denotes the weak ω-limit set of {xn}.


## 2. Preliminaries

This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.Lemma 2.1.
LetH be a real Hilbert space. There holds the following identity:  (i)
∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,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 2.2.
LetH be a real Hilbert space. Given a closed convex subset C⊂H and x,y,z∈H. Given also a real number a∈ℝ. The set
(2.1){v∈C:‖y-v‖2⩽‖x-v‖2+〈z,v〉+a}
is convex (and closed).Recall that given a closed convex subsetC of a real Hilbert space H, the nearest point projection PC from H onto C assigns to each x∈H its nearest point denoted by PCx which is a unique point in C with the property (2.2)‖x-PCx‖⩽‖x-z‖∀z∈C.Lemma 2.3.
LetC be a closed convex subset of real Hilbert space H. Given x∈H and z∈C. Then, z=PCx if and only if there holds the relation
(2.3)〈x-z,z-y〉⩾0∀y∈C.Lemma 2.4 (Martinez-Yanes and Xu [8]).
LetC be a closed convex subset of real Hilbert space H. Let {xn} be a sequence in H and u∈H. Let q=PCu. If {xn} is such that ωw(xn)⊂C and satisfies the condition
(2.4)‖xn-u‖⩽‖u-q‖∀n.
Then, xn→q.Given a closed convex subsetC of a real Hilbert space H and a mapping T:C→C. Recall that T is said to be a quasistrict pseudocontraction if F(T) is nonempty and there exists a constant 0⩽k<1 such that (2.5)‖Tx-p‖2⩽‖x-p‖2+k‖x-Tx‖2
for all x∈C and p∈F(T).Proposition 2.5 (Marino and Xu [11, Proposition 2.1]).
AssumeC is a closed convex subset of a Hilbert space H, and let T:C→C be a self-mapping of C. (i)
IfT is a k-strict pseudocontraction, then T satisfies Lipschitz condition
(2.6)‖Tx-Ty‖⩽1+k1-k‖x-y‖∀x,y∈C.(ii)
IfT is a k-strict pseudocontraction, then the mapping I-T is demiclosed (at 0). That is, if {xn} is a sequence in C such that xn⇀x̂ and (I-T)xn→0, then (I-T)x̂=0.(iii)
IfT is a k-quasistrict pseudocontraction, then the fixed-point set F(T) of T is closed and convex so that the projection PF(T) is well defined.Lemma 2.6 (Plubtieng and Ungchittrakool [12, Lemma 3.1]).
LetC be a nonempty subset of a Banach space E and {Tn} a sequence of mappings from C into E. Suppose that for any bounded subset B of C there exists continuous increasing function hB from ℝ+ into ℝ+ such that hB(0)=0 and
(2.7)lim⁡k,l→∞ρlk=0,
where
(2.8)ρlk:=sup⁡{hB(‖Tkz-Tlz‖):z∈B}<∞,
for all k,l∈ℕ. Then, for each x∈C, {Tnx} converges strongly to some point of E. Moreover, let T be a mapping from C into E defined by
(2.9)Tx=lim⁡n→∞Tnx∀x∈C.
Then, limn→∞sup{hB(∥Tz-Tnz∥):z∈B}=0.Lemma 2.7.
LetH be a real Hilbert space, let C be a nonempty closed convex subset of H, and let {Tn} be a sequence such that for each n, Tn is kn-strict pseudo contraction from C into H with lim supn→∞kn<1 and
(2.10)Tx=lim⁡n→∞Tnx∀x∈C.
Then, F(T) is closed and convex so that the projection PF(T) is well defined.Proof.
To see thatF(T) is closed, assume that {pn} is a sequence in F(T) such that pn→p̂. Since Tn is a kn-quasistrict pseudocontraction, we get, for each n,
(2.11)∥Tp̂-pn∥2=‖Tp̂-Tpn‖2=‖lim⁡m→∞Tmp̂-lim⁡m→∞Tmpn‖2=lim⁡m→∞‖Tmp̂-Tmpn‖2⩽limsup⁡m→∞(‖p̂-pn‖2+km‖(p̂-Tmp̂)-(pn-Tmpn)‖2)⩽‖p̂-pn‖2+(limsup⁡m→∞km)(limsup⁡m→∞‖(p̂-Tmp̂)-(pn-Tmpn)‖2)=‖p̂-pn‖2+(limsup⁡m→∞km)‖(p̂-lim⁡m→∞Tmp̂)-(pn-lim⁡m→∞Tmpn)‖2=‖p̂-pn‖2+(limsup⁡m→∞km)‖p̂-Tp̂‖2.
Taking the limit as n→∞ yields ∥Tp̂-p̂∥2⩽κ∥p̂-Tp̂∥2, where κ:=lim supm→∞km. Since 0⩽κ<1, we have Tp̂=p̂.


## 3. Main Result

In this section, we prove a strong convergence theorem by using the hybrid projection method (some authors call this theCQ method) for finding a common element of the set of fixed points of two sequences of strictly pseudocontractive mappings in Hilbert spaces.Theorem 3.1.
LetC be a closed convex subset of a Hilbert space H. For each n, let Tn,Sn:C→C be kTn,kSn-strict pseudocontractions for some 0⩽kTn, kSn<1 with lim supn→∞kTn,lim supn→∞kSn<1, respectively, and assume that ⋂n=0∞F(Tn)∩⋂n=0∞F(Sn)≠∅. Let {xn}n=0∞ be the sequence generated by
(3.1)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)zn,zn=βnTnxn+(1-βn)Snxn,Ĉn={z∈C:‖yn-z‖2⩽‖xn-z‖2+(1-αn)βn(kTn-αn)‖xn-Tnxn‖2+(1-αn)(1-βn)(kSn-αn)‖xn-Snxn‖2-(1-αn)2βn(1-βn)‖Tnxn-Snxn‖2},Qn={z∈C:〈xn-z,x0-xn〉⩾0},xn+1=PĈn∩Qn(x0).
Assume that {αn} and {βn} are chosen so that 0⩽αn<1 and 0<a⩽βn⩽b<1 for all n. Suppose that for any bounded subset B of C there exists an increasing, continuous, and convex function hB from ℝ+ into ℝ+ such that hB(0)=0, and
(3.2)lim⁡k,l→∞sup⁡{hB(‖Tkz-Tlz‖):z∈B}=0=lim⁡k,l→∞sup⁡{hB(‖Skz-Slz‖):z∈B}.
Let T,S:C→C such that Tx=limn→∞Tnx and Sx=limn→∞Snx for all x∈C, respectively, and suppose that F(T)=⋂n=0∞F(Tn) and F(S)=⋂n=0∞F(Sn). Then, {xn} converges strongly to a common fixed point q=PF(T)∩F(S)(x0).Proof.
It is not hard to check thatĈn and Qn are closed and convex for all n (via Lemma 2.2 and the properties of the inner product). Then, if Ĉn∩Qn is nonempty for all n, the sequence {xn} is well defined. Now, we will show that ⋂n=0∞F(Tn)∩⋂n=0∞F(Sn)⊂Ĉn for all n. Let p∈⋂n=0∞F(Tn)∩⋂n=0∞F(Sn), we observe that
(3.3)‖zn-p‖2=‖βn(Tnxn-p)+(1-βn)(Snxn-p)‖2=βn‖Tnxn-p‖2+(1-βn)‖Snxn-p‖2-βn(1-βn)‖Tnxn-Snxn‖2⩽βn(‖xn-p‖2+kTn‖xn-Tnxn‖2)+(1-βn)(‖xn-p‖2+kSn‖xn-Snxn‖2)-βn(1-βn)‖Tnxn-Snxn‖2=‖xn-p‖2+βnkTn‖xn-Tnxn‖2+(1-βn)kSn‖xn-Snxn‖2-βn(1-βn)‖Tnxn-Snxn‖2,(3.4)‖xn-zn‖2=‖βn(xn-Tnxn)+(1-βn)(xn-Snxn)‖2=βn‖xn-Tnxn‖2+(1-βn)‖xn-Snxn‖2-βn(1-βn)‖Tnxn-Snxn‖2.
By (3.3) and (3.4) we obtain
(3.5)‖yn-p‖2=‖αn(xn-p)+(1-αn)(zn-p)‖2=αn‖xn-p‖2+(1-αn)‖zn-p‖2-αn(1-αn)‖xn-zn‖2⩽‖xn-p‖2+(1-αn)βn(kTn-αn)‖xn-Tnxn‖2+(1-αn)(1-βn)(kSn-αn)‖xn-Snxn‖2-(1-αn)2βn(1-βn)‖Tnxn-Snxn‖2.
Thus, we have F(T)∩F(S)⊂Ĉn for all n. Next, we will show that F(T)∩F(S)⊂Qn for all n. If n=0, then F(T)∩F(S)⊂C=Q0. Assume that F(T)∩F(S)⊂Qn. Since xn+1 is the projection of x0 onto Ĉn∩Qn, by Lemma 2.3 we have
(3.6)〈xn+1-z,x0-xn+1〉⩾0∀z∈Ĉn∩Qn.
Noting that F(T)∩F(S)⊂Ĉn∩Qn by the induction assumption, it implies that F(T)∩F(S)⊂Qn+1, thus by induction F(T)∩F(S)⊂Qn for all n. Hence, F(T)∩F(S)⊂Ĉn∩Qn for all n. So, {xn} is well defined.
Notice that the definition ofQn actually implies xn=PQn(x0). This together with the fact F(T)∩F(S)⊂Qn further implies
(3.7)‖xn-x0‖⩽‖p-x0‖∀p∈F(T)∩F(S).
In particular, {xn} is bounded and
(3.8)‖xn-x0‖⩽‖q-x0‖∀n,
where q=PF(T)∩F(S)(x0).
The factxn+1∈Qn asserts that 〈xn+1-xn,xn-x0〉⩾0. This together with Lemma 2.1(i) and Lemma 2.3 implies
(3.9)‖xn+1-xn‖2=‖(xn+1-x0)-(xn-x0)‖2=‖xn+1-x0‖2-‖xn-x0‖2-2〈xn+1-xn,xn-x0〉⩽‖xn+1-x0‖2-‖xn-x0‖2.
It turns out that
(3.10)‖xn+1-xn‖⟶0.
By the fact xn+1∈Ĉn, we get
(3.11)‖xn+1-yn‖2⩽‖xn+1-xn‖2+(1-αn)βn(kTn-αn)‖xn-Tnxn‖2+(1-αn)(1-βn)(kSn-αn)‖xn-Snxn‖2-(1-αn)2βn(1-βn)‖Tnxn-Snxn‖2.
Observe that
(3.12)‖xn+1-yn‖2=‖αn(xn+1-xn)+(1-αn)(xn-zn)‖2=αn‖xn+1-xn‖2+(1-αn)‖xn+1-zn‖2-αn(1-αn)‖xn-zn‖2,‖xn+1-zn‖2=‖βn(xn+1-Tnxn)+(1-βn)(xn+1-Snxn)‖2=βn‖xn+1-Tnxn‖2+(1-βn)‖xn+1-Snxn‖2-βn(1-βn)‖Tnxn-Snxn‖2.
With simple calculation by using (3.12) and (3.4), we have
(3.13)‖xn+1-yn‖2=αn‖xn+1-xn‖2+(1-αn)βn‖xn+1-Tnxn‖2+(1-αn)(1-βn)‖xn+1-Snxn‖2-αn(1-αn)βn‖xn-Tnxn‖2-αn(1-αn)(1-βn)‖xn-Snxn‖2-(1-αn)2βn(1-βn)‖Tnxn-Snxn‖2.
So, when we combine (3.11) and (3.13) and compute, we obtain
(3.14)(1-αn)βn‖xn+1-Tnxn‖2+(1-αn)(1-βn)‖xn+1-Snxn‖2⩽(1-αn)‖xn+1-xn‖2+(1-αn)βnkTn‖xn-Tnxn‖2+(1-αn)(1-βn)kSn‖xn-Snxn‖2.
Since αn<1 for all n, we have
(3.15)βn‖xn+1-Tnxn‖2+(1-βn)‖xn+1-Snxn‖2⩽‖xn+1-xn‖2+βnkTn‖xn-Tnxn‖2+(1-βn)kSn‖xn-Snxn‖2.
Notice that
(3.16)‖xn+1-Tnxn‖2=‖(xn+1-xn)+(xn-Tnxn)‖2=‖xn+1-xn‖2+2〈xn+1-xn,xn-Tnxn〉+‖xn-Tnxn‖2,(3.17)‖xn+1-Snxn‖2=‖(xn+1-xn)+(xn-Snxn)‖2=‖xn+1-xn‖2+2〈xn+1-xn,xn-Snxn〉+‖xn-Snxn‖2.
By (3.15), (3.16), and (3.17), we have
(3.18)βn(1-kTn)‖xn-Tnxn‖2+(1-βn)(1-kSn)‖xn-Snxn‖2⩽-2βn〈xn+1-xn,xn-Tnxn〉-2(1-βn)〈xn+1-xn,xn-Snxn〉.
Since {xn}, {Tnxn}, and {Snxn} are bounded, 0<a⩽βn⩽b<1 for all n and lim supn→∞kTn, lim supn→∞kSn<1, it follows from (3.10) and (3.18) that
(3.19)lim⁡n→∞‖xn-Tnxn‖=0=lim⁡n→∞‖xn-Snxn‖.
Since {xn} is bounded, there exists a bounded subset B of C such that {xn}⊂B. From Lemma 2.6, we are able to set Tx=limm→∞Tmx for all x∈C, and then observe that
(3.20)12‖xn-Txn‖⩽12‖xn-Tnxn‖+12‖Tnxn-Txn‖.
Since hB is an increasing, continuous, and convex function from ℝ+ into ℝ+ such that hB(0)=0, we discover that
(3.21)hB(12‖xn-Txn‖)⩽12hB(‖xn-Tnxn‖)+12hB(‖Tnxn-Txn‖)⩽12hB(‖xn-Tnxn‖)+12sup⁡⁡{hB(‖Tnz-Tz‖):z∈B}.
By Lemma 2.6 and the continuity of hB, we have limn→∞hB((1/2)∥xn-Txn∥)=0. And then the properties of hB yield
(3.22)lim⁡n→∞‖xn-Txn‖=0.
By the same argument, we have
(3.23)lim⁡n→∞‖xn-Sxn‖=0.
Now Proposition 2.5 guarantees that ωw(xn)⊂F(T)∩F(S). This fact, the inequality (3.8), and Lemma 2.4 ensure the strong convergence of {xn} to q=PF(T)∩F(S)(x0).IfTn=T and Sn=S for all n, then kTn=kT and kSn=kS for all n. So, Theorem 3.1 reduces to the following corollary.Corollary 3.2.
LetC be a closed convex subset of a Hilbert space H. Let T,S:C→C be kT,kS-strict pseudocontractions for some 0⩽kT, kS<1, respectively, and assume that F(T)∩F(S)≠∅. Let {xn}n=0∞ be the sequence generated by
(3.24)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)zn,zn=βnTxn+(1-βn)Sxn,Ĉn={z∈C:‖yn-z‖2⩽‖xn-z‖2+(1-αn)βn(kT-αn)‖xn-Txn‖2+(1-αn)(1-βn)(kS-αn)‖xn-Sxn‖2-(1-αn)2βn(1-βn)‖Txn-Sxn‖2},Qn={z∈C:〈xn-z,x0-xn〉⩾0},xn+1=PĈn∩Qn(x0),
where {αn} and {βn} be as in Theorem 3.1. Then, {xn} converges strongly to a common fixed point q=PF(T)∩F(S)(x0).In particular, ifT=S, then zn=Txn and (3.25)Ĉn={z∈C:‖yn-p‖2⩽‖xn-p‖2+(1-αn)βn(kT-αn)‖xn-Txn‖2+(1-αn)(1-βn)(kT-αn)‖xn-Txn‖2-(1-αn)2βn(1-βn)‖Txn-Txn‖2}=Cn.
So, Corollary 3.2 reduces to the following corollary.Corollary 3.3 (Marino and Xu [11, Theorem 4.1]).
LetC be a closed convex subset of a Hilbert space H. Let T:C→C be a k-strict pseudocontraction for some 0⩽k<1, and assume that the fixed-point set F(T)≠∅. Let {xn}n=0∞ be the sequence generated by
(3.26)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)Txn,Cn={z∈C:‖yn-z‖2⩽‖xn-z‖2+(1-αn)(k-αn)‖xn-Txn‖2}Qn={z∈C:〈xn-z,x0-xn〉⩾0},xn+1=PCn∩Qn(x0).
Assume that the control sequence {αn}n=0∞ is chosen so that 0⩽αn<1 for all n. Then, {xn} converges strongly to a fixed-point q=PF(T)(x0).IfTn=T for all n and {Sn} is a sequences of nonexpansive mappings, then kTn=k and kSn=0 for all n. So, Theorem 3.1 reduces to the following corollary.Corollary 3.4.
LetC be a closed convex subset of a Hilbert space H. Let T:C→C be a k-strict pseudocontraction for some 0⩽k<1, for each n and Sn:C→C a nonexpansive mapping, and assume that F(T)∩⋂n=0∞F(Sn)≠∅. Let {xn}n=0∞ be the sequence generated by
(3.27)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)zn,zn=βnTxn+(1-βn)Snxn,Ĉn={z∈C:‖yn-z‖2⩽‖xn-z‖2+(1-αn)βn(k-αn)‖xn-Txn‖2-αn(1-αn)(1-βn)‖xn-Snxn‖2-(1-αn)2βn(1-βn)‖Txn-Snxn‖2},Qn={z∈C:〈xn-z,x0-xn〉⩾0},xn+1=PĈn∩Qn(x0),
where {αn} and {βn} be as in Theorem 3.1. Suppose that for any bounded subset B of C there exists an increasing, continuous, and convex function hB from ℝ+ into ℝ+ such that hB(0)=0, and
(3.28)lim⁡k,l→∞sup⁡{hB(‖Skz-Slz‖):z∈B}=0.
Let S:C→C be such that Sx=limn→∞Snx for all x∈C, and suppose F(S)=⋂n=0∞F(Sn). Then, {xn} converges strongly to a common fixed point q=PF(T)∩F(S)(x0).


## 4. Equilibrium Problem

In this section, we have an application of the main result for finding a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem.LetH be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let φ be a bifunction of C×C into ℝ, where ℝ is the set of real numbers. The equilibrium problem for φ:C×C→ℝ is to find x∈C such that(4.1)φ(x,y)⩾0∀y∈C.
The set of solution of (4.1) is denoted by EP(φ)(={x∈C:φ(x,y)⩾0forally∈C}). Many problems in physics, optimization, and economics reduce to find some elements of EP(φ).For solving the equilibrium problem for a bifunctionφ:C×C→ℝ, let us assume that φ satisfies the following conditions:  (A1)
φ(x,x)=0 for all x∈C;(A2)
φ is monotone, that is, φ(x,y)+φ(y,x)⩽0 for all x,y∈C;(A3)
for eachx,y,z∈C,
(4.2)lim⁡t↓0φ(tz+(1-t)x,y)⩽φ(x,y);(A4)
for eachx∈C, y↦φ(x,y) is convex and lower semicontinuous.The following lemma appears implicitly in [13].Lemma 4.1 (Blum and Oettli [13]).
LetC be a nonempty closed convex subset of H, and let φ be a bifunction of C×C into ℝ satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(4.3)φ(z,y)+1r〈y-z,z-x〉⩾0∀y∈C.The following lemma was also given in [14].Lemma 4.2 (Combettes and Hirstoaga [14]).
Assume thatφ:C×C→ℝ satisfies (A1)–(A4). For r>0 and x∈H, define a mapping Sr:H→C as follows:
(4.4)Sr(x)={z∈C:φ(z,y)+1r〈y-z,z-x〉⩾0,∀y∈C}
for all z∈H. Then, the following hold:(1)
Sr is single-valued;(2)
Sr is firmly nonexpansive, that is, for any x,y∈H, ∥Srx-Sry∥2⩽〈Srx-Sry,x-y〉;(3)
F(Sr)=EP(φ);(4)
EP(φ) is closed and convex.The following corollary is an application of Corollary3.4 in the case of finding a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of an equilibrium problem.Corollary 4.3.
LetC be a closed convex subset of a Hilbert space H. Let T:C→C be a k-strict pseudocontraction for some 0⩽k<1 and φ a bifunction from C×C into ℝ satisfying (A1)–(A4). Suppose that F(T)∩EP(φ)≠∅. Let {xn}n=0∞ be the sequence generated by
(4.5)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)zn,zn=βnTxn+(1-βn)un,un∈Csuchthatφ(un,y)+1rn〈y-un,un-xn〉⩾0,∀y∈C,Ĉn={z∈C:‖yn-z‖2⩽‖xn-z‖2+(1-αn)βn(k-αn)‖xn-Txn‖2-αn(1-αn)(1-βn)‖xn-un‖2-(1-αn)2βn(1-βn)‖Txn-un‖2},Qn={z∈C:〈xn-z,x0-xn〉⩾0},xn+1=PĈn∩Qn(x0),
where {αn} and {βn} be as in Theorem 3.1 and {rn}n=0∞ is chosen so that {rn}⊂(0,∞) with infnrn>0 and ∑n=0∞|rn+1-rn|<∞. Then, {xn} converges strongly to a common fixed point q=PF(T)∩EP(φ)(x0).Proof.
Obviously,un=Srnxn, where Srn are mappings as in Lemma 4.2. Next, we want to show that for any bounded subset B of C there exists an increasing, continuous, and convex function hB(=(·)2) from ℝ+ into ℝ+ such that hB(0)=0, and
(4.6)lim⁡k,l→∞sup⁡{hB(‖Srkv-Srlv‖):v∈B}=0.
Let B be a bounded subset of C. For each v∈B, let vn=Srnv. Then, by Lemma 4.2, we have
(4.7)φ(vl,y)+1rl〈y-vl,vl-v〉⩾0∀y∈C,(4.8)φ(vk,y)+1rk〈y-vk,vk-v〉⩾0∀y∈C.
Put y=vk in (4.7) and y=vl in (4.8), we have
(4.9)φ(vl,vk)+1rl〈vk-vl,vl-v〉⩾0,φ(vk,vl)+1rk〈vl-vk,vk-v〉⩾0.
So, from (A2), we have 〈vk-vl,(vl-v)/rl-(vk-v)/rk〉⩾0 and hence 〈vk-vl,vl-v-(rl/rk)(vk-v)〉⩾0. Thus,
(4.10)〈vk-vl,vl-vk〉+〈vk-vl,(1-rlrk)(vk-v)〉⩾0.
Let b:=infnrn. Thus, we have
(4.11)(‖Srkv-Srlv‖)2=〈vk-vl,vk-vl〉⩽〈vk-vl,(1-rlrk)(vk-v)〉⩽‖vk-vl‖1b|rk-rl|‖vk-v‖=1b‖Srkv-Srlv‖‖Srkv-v‖|rk-rl|⩽4b‖v-p‖|rk-rl|wherep∈EP⁡(φ)⩽4bM|rk-rl|whereM:=sup⁡{‖v-p‖:v∈B}.
Put k>l. Observe that
(4.12)(‖Srkv-Srlv‖)2⩽4bM|rk-rl|⩽4bM∑n=lk-1|rn+1-rn|⩽4bM∑n=l∞|rn+1-rn|<∞,
for all v∈B, and then
(4.13)ρlk=sup⁡{(‖Srkv-Srlv‖)2:v∈B}⩽4bM∑n=l∞|rn+1-rn|<∞.
Let l→∞, we have limk,l→∞ρlk=0. Then, by Lemma 2.6, we can define a mapping S by
(4.14)Sx=lim⁡n→∞Srnx∀x∈C.
Next, we will show that
(4.15)F(S)=⋂n=0∞F(Srn)=EP⁡(φ).
Since {rn}⊂(0,∞), infnrn>0 and (4.14), it is easy to see that
(4.16)EP⁡(φ)=⋂n=0∞F(Srn)⊂F(S).
Let p∈F(S). By the definition of Sr, we have
(4.17)φ(Srnp,y)+1rn〈y-Srnp,Srnp-p〉⩾0∀y∈C.
By (A2), we have
(4.18)1rn〈y-Srnp,Srnp-p〉⩾φ(y,Srnp)∀y∈C.
From (4.14) and the lower semicontinuity of φ(y,·), we have 0⩾φ(y,p)forally∈C. Let y∈C and set xt=ty+(1-t)p, for t∈(0,1]. Then, we have
(4.19)0=φ(xt,xt)⩽tφ(xt,y)+(1-t)φ(xt,p)⩽tφ(xt,y).
So φ(xt,y)⩾0. Letting t↓0 and using (A3), we get φ(p,y)⩾0forally∈C, and hence p∈EP(φ)=⋂n=1∞F(Srn). Hence, we have (4.15). Then, applying Corollary 3.4, xn→q=PF(T)∩EP(φ)(x0).



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*Source: 876819-2010-12-06.xml*