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.

1. Introduction

Let H 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‖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 ‖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: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 Hx0∈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: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: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:

→ for strong convergence and ⇀ for weak convergence,

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.

Let H be a real Hilbert space. There holds the following identity:

∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉 for all x,y∈H.

∥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.

Let H 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
{v∈C:‖y-v‖2⩽‖x-v‖2+〈z,v〉+a}
is convex (and closed).

Recall that given a closed convex subset C 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 ‖x-PCx‖⩽‖x-z‖∀z∈C.

Lemma 2.3.

Let C 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
〈x-z,z-y〉⩾0∀y∈C.

Lemma 2.4 (Martinez-Yanes and Xu [<xref ref-type="bibr" rid="B10">8</xref>]).

Let C 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
‖xn-u‖⩽‖u-q‖∀n.
Then, xn→q.

Given a closed convex subset C 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 ‖Tx-p‖2⩽‖x-p‖2+k‖x-Tx‖2
for all x∈C and p∈F(T).

Proposition 2.5 (Marino and Xu [<xref ref-type="bibr" rid="B9">11</xref>, Proposition 2.1]).

Assume C is a closed convex subset of a Hilbert space H, and let T:C→C be a self-mapping of C.

If T is a k-strict pseudocontraction, then T satisfies Lipschitz condition
‖Tx-Ty‖⩽1+k1-k‖x-y‖∀x,y∈C.

If T 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.

If T 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 [<xref ref-type="bibr" rid="B12">12</xref>, Lemma 3.1]).

Let C 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
limk,l→∞ρlk=0,
where
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
Tx=limn→∞Tnx∀x∈C.
Then, limn→∞sup{hB(∥Tz-Tnz∥):z∈B}=0.

Lemma 2.7.

Let H 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
Tx=limn→∞Tnx∀x∈C.
Then, F(T) is closed and convex so that the projection PF(T) is well defined.

Proof.

To see that F(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,
∥Tp̂-pn∥2=‖Tp̂-Tpn‖2=‖limm→∞Tmp̂-limm→∞Tmpn‖2=limm→∞‖Tmp̂-Tmpn‖2⩽limsupm→∞(‖p̂-pn‖2+km‖(p̂-Tmp̂)-(pn-Tmpn)‖2)⩽‖p̂-pn‖2+(limsupm→∞km)(limsupm→∞‖(p̂-Tmp̂)-(pn-Tmpn)‖2)=‖p̂-pn‖2+(limsupm→∞km)‖(p̂-limm→∞Tmp̂)-(pn-limm→∞Tmpn)‖2=‖p̂-pn‖2+(limsupm→∞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 the CQ 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.

Let C 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
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
limk,l→∞sup{hB(‖Tkz-Tlz‖):z∈B}=0=limk,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 that Ĉn 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
‖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,‖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
‖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
〈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 of Qn actually implies xn=PQn(x0). This together with the fact F(T)∩F(S)⊂Qn further implies
‖xn-x0‖⩽‖p-x0‖∀p∈F(T)∩F(S).
In particular, {xn} is bounded and
‖xn-x0‖⩽‖q-x0‖∀n,
where q=PF(T)∩F(S)(x0).

The fact xn+1∈Qn asserts that 〈xn+1-xn,xn-x0〉⩾0. This together with Lemma 2.1(i) and Lemma 2.3 implies
‖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
‖xn+1-xn‖⟶0.
By the fact xn+1∈Ĉn, we get
‖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
‖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
‖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
(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
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
‖xn+1-Tnxn‖2=‖(xn+1-xn)+(xn-Tnxn)‖2=‖xn+1-xn‖2+2〈xn+1-xn,xn-Tnxn〉+‖xn-Tnxn‖2,‖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
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
limn→∞‖xn-Tnxn‖=0=limn→∞‖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
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
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
limn→∞‖xn-Txn‖=0.
By the same argument, we have
limn→∞‖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).

If Tn=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.

Let C 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
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, if T=S, then zn=Txn and 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 [<xref ref-type="bibr" rid="B9">11</xref>, Theorem 4.1]).

Let C 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
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).

If Tn=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.

Let C 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
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
limk,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.

Let H 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 thatφ(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 bifunction φ:C×C→ℝ, let us assume that φ satisfies the following conditions:

φ(x,x)=0 for all x∈C;

φ is monotone, that is, φ(x,y)+φ(y,x)⩽0 for all x,y∈C;

for each x,y,z∈C,
limt↓0φ(tz+(1-t)x,y)⩽φ(x,y);

for each x∈C, y↦φ(x,y) is convex and lower semicontinuous.

The following lemma appears implicitly in [13].

Lemma 4.1 (Blum and Oettli [<xref ref-type="bibr" rid="B2">13</xref>]).

Let C 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
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 [<xref ref-type="bibr" rid="B4">14</xref>]).

Assume that φ:C×C→ℝ satisfies (A1)–(A4). For r>0 and x∈H, define a mapping Sr:H→C as follows:
Sr(x)={z∈C:φ(z,y)+1r〈y-z,z-x〉⩾0,∀y∈C}
for all z∈H. Then, the following hold:

Sr is single-valued;

Sr is firmly nonexpansive, that is, for any x,y∈H, ∥Srx-Sry∥2⩽〈Srx-Sry,x-y〉;

F(Sr)=EP(φ);

EP(φ) is closed and convex.

The following corollary is an application of Corollary 3.4 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.

Let C 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
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
limk,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
φ(vl,y)+1rl〈y-vl,vl-v〉⩾0∀y∈C,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
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,
〈vk-vl,vl-vk〉+〈vk-vl,(1-rlrk)(vk-v)〉⩾0.
Let b:=infnrn. Thus, we have
(‖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
(‖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
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
Sx=limn→∞Srnx∀x∈C.
Next, we will show that
F(S)=⋂n=0∞F(Srn)=EP(φ).
Since {rn}⊂(0,∞), infnrn>0 and (4.14), it is easy to see that
EP(φ)=⋂n=0∞F(Srn)⊂F(S).
Let p∈F(S). By the definition of Sr, we have
φ(Srnp,y)+1rn〈y-Srnp,Srnp-p〉⩾0∀y∈C.
By (A2), we have
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
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).

Acknowledgments

This research is supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand. The author would like to thank the referees for reading this paper carefully and providing valuable suggestions and comments of this paper.

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