# Shrinking Projection Method of Fixed Point Problems for Asymptotically Pseudocontractive Mapping in the Intermediate Sense and Mixed Equilibrium Problems in Hilbert Spaces

Authors: Pattanapong Tianchai
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper is concerned with a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, which extends and improves that of Qin et al. (2010) and many others.


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## Body

## 1. Introduction

Throughout this paper, we always assume thatC be a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by 〈·,·〉 and ∥·∥, respectively. For a sequence {xn} in H, we denote the strong convergence and the weak convergence of {xn} to x∈H by xn→x and xn⇀x, respectively. The domain of the function φ:C→ℝ∪{+∞} is the set
(1.1)dom⁡φ={x∈C:φ(x)<+∞}.Let φ:C→ℝ∪{+∞} be a proper extended real-valued function, and let Φ be a bifunction from C×C into ℝ such that C∩domφ≠∅, where ℝ is the set of real numbers. The so-called the mixed equilibrium problem is to find x∈C such that(1.2)Φ(x,y)+φ(y)-φ(x)≥0,∀y∈C.
The set of solution of problem (1.2) is denoted by MEP(Φ,φ), that is, (1.3)MEP⁡(Φ,φ)={x∈C:Φ(x,y)+φ(y)-φ(x)≥0,∀y∈C}.
It is obvious that if x is a solution of problem (1.2) then x∈domφ. As special cases of problem (1.2), we have the following.  (i)
Ifφ=0, then problem (1.2) is reduced to find x∈C such that
(1.4)Φ(x,y)≥0,∀y∈C.
We denote by EP(Φ) the set of solutions of equilibrium problem, which problem (1.4) was studied by Blum and Oettli [1].(ii)
IfΦ(x,y)=〈Bx,y-x〉 for all x,y∈C where a mapping B:C→H, then problem (1.4) is reduced to find x∈C such that
(1.5)〈Bx,y-x〉≥0,∀y∈C.
We denote by VI(C,B) the set of solutions of variational inequality problem, which problem (1.5) was studied by Hartman and Stampacchia [2].(iii)
IfΦ=0, then problem (1.2) is reduced to find x∈C such that
(1.6)φ(y)-φ(x)≥0,∀y∈C.
We denote by Argmin(φ) the set of solutions of minimize problem.Recall thatPC is the metric projection of H onto C; that is, for each x∈H there exists the unique point in PCx∈C such that ∥x-PCx∥=miny∈C∥x-y∥. A mapping T:C→C is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C, and uniformly L-Lipschitzian if there exists a constant L>0 such that for each n∈ℕ, ∥Tnx-Tny∥≤L∥x-y∥ for all x,y∈C, and a mapping f:C→C is called a contraction if there exists a constant α∈(0,1) such that ∥f(x)-f(y)∥≤α∥x-y∥ for all x,y∈C. A point x∈C is a fixed point of T provided Tx=x. We denote by F(T) the set of fixed points of T; that is, F(T)={x∈C:Tx=x}. If C is a nonempty bounded closed convex subset of H and T is a nonexpansive mapping of C into itself, then F(T) is nonempty (see [3]).Iterative methods are often used to solve the fixed point equationTx=x. The most well-known method is perhaps the Picard successive iteration method when T is a contraction. Picard's method generates a sequence {xn} successively as xn+1=Txn for all n∈ℕ with x1=x chosen arbitrarily, and this sequence converges in norm to the unique fixed point of T. However, if T is not a contraction (for instance, if T is a nonexpansive), then Picard's successive iteration fails, in general, to converge. Instead, Mann's iteration method for a nonexpansive mapping T (see [4]) prevails, generates a sequence {xn} recursively by(1.7)xn+1=αnxn+(1-αn)Txn,∀n∈N,
where x1=x∈C chosen arbitrarily and the sequence {αn} lies in the interval [0,1]. Recall that a mapping T:C→C is said to be  (i)
asymptotically pseudocontractive [5, 6] if there exists a sequence kn⊂[1,∞) with limn→∞kn=1 such that
(1.8)〈Tnx-Tny,x-y〉≤kn‖x-y‖2,∀x,y∈C,∀n∈N;
it is easy to see that (1.8) is equivalent to
(1.9)‖Tnx-Tny‖2≤(2kn-1)‖x-y‖2+‖(I-Tn)x-(I-Tn)y‖2,∀x,y∈C,
for all n∈ℕ,(ii)
asymptotically pseudocontractive in the intermediate sense [7] if there exists a sequence kn⊂[1,∞) with limn→∞kn=1 such that
(1.10)limsup⁡n→∞sup⁡x,y∈C(〈Tnx-Tny,x-y〉-kn‖x-y‖2)≤0;
if we define
(1.11)τn=max⁡{0,sup⁡x,y∈C(〈Tnx-Tny,x-y〉-kn‖x-y‖2)},
then limn→∞τn=0 and it follows that (1.10) is reduced to
(1.12)〈Tnx-Tny,x-y〉≤kn‖x-y‖2+τn,∀x,y∈C,∀n∈N;
it is easy to see that (1.12) is equivalent to
(1.13)‖Tnx-Tny‖2≤(2kn-1)‖x-y‖2+‖(I-Tn)x-(I-Tn)y‖2+2τn,∀x,y∈C,
for all n∈ℕ; it is obvious that if τn=0 for all n∈ℕ, then the class of asymptotically pseudocontractive mappings in the intermediate sense is reduced to the class of asymptotically pseudocontractive mappings.The Mann’s algorithm for nonexpansive mappings has been extensively investigated (see [8–10] and the references therein). One of the well-known results is proven by Reich [10] for a nonexpansive mapping T on C, which asserts the weak convergence of the sequence {xn} generated by (1.7) in a uniformly convex Banach space with a Frechet differentiable norm under the control condition ∑n=1∞αn(1-αn)=∞. It is known that the Mann's iteration (1.7) is in general not strongly convergent (see [11]). The strong convergence guaranteed has been proposed by Nakajo and Takahashi [12], they modified the Mann's iteration method (1.7) which is to find a fixed point of a nonexpansive mapping by a hybrid method, which called that the shrinking projection method (or the CQ method) as the following theorem.Theorem NT.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let  T be  a nonexpansive mapping of C into itself such that F(T)≠∅.  Suppose that x1=x∈C chosen arbitrarily and {xn} the sequence defined by
(1.14)yn=αnxn+(1-αn)Txn,Cn={z∈C:‖yn-z‖≤‖xn-z‖},Qn={z∈C:〈xn-z,x1-xn〉≥0},xn+1=PCn∩Qn(x1),∀n∈N,
where 0≤αn≤α<1. Then {xn} converges strongly to PF(T)(x1).Subsequently, Schu [5] modified Ishikawa's iteration method (see [13]) for the class of asymptotically pseudocontractive mappings as the following theorem.Theorem S.
LetC be a nonempty bounded closed convex subset of a real Hilbert space H.  Let T:C→C be a completely continuous uniformly L-Lipschitzian such that L>0 and asymptotically pseudocontractive mapping defined as in (1.9) with the sequence {kn}⊂[1,∞) such that limn→∞kn=1.  Let qn=2kn-1 for all n∈ℕ.  Suppose that x1=x∈C chosen arbitrarily and {xn} the sequence defined by
(1.15)yn=βnTnxn+(1-βn)xn,xn+1=αnTnyn+(1-αn)xn,∀n∈N,
where {αn},{βn}⊂(0,1) such that ϵ≤αn≤βn≤b for some ϵ>0 and b∈(0,L-2(1+L2-1)) and ∑n=1∞(qn2-1)<∞. Then {xn} converges strongly to some fixed point of T.Quite recently, Zhou [14] showed that every uniformly L-Lipschitzian and asymptotically pseudocontractions which are also uniformly asymptotically regular has a fixed point and the fixed point set is closed and convex, and he also introduced iterative scheme to find a fixed point of a uniformly L-Lipschitzian and asymptotically pseudocontractive mapping as the following theorem.Theorem Z.
LetC be a nonempty bounded closed convex subset of a real Hilbert space  H. Let  T:C→C be a uniformly L-Lipschitzian such that  L>0 and asymptotically pseudocontractive mapping with a fixed point defined as in (1.8) with the sequence  {kn}⊂[1,∞) such that  limn→∞kn=1. Suppose that  x1=x∈C chosen arbitrarily and  {xn} the sequence defined by
(1.16)yn=(1-αn)xn+αnTnxn,Cn={z∈C:αn(1-(1+L)αn)‖xn-Tnxn‖2≤〈xn-z,yn-Tnyn〉+(kn-1)(diam⁡C)2},Qn={z∈C:〈xn-z,x1-xn〉≥0},xn+1=PCn∩Qn(x1),∀n∈N,
where  {αn}⊂[a,b] such that  0<a<b<1/(1+L). Then  {xn} converges strongly to  PF(T)(x1).To be more precisely, Qin et al. [7] showed in the framework of a real Hilbert spaces H for the uniformly L-Lipschitzian and asymptotically pseudocontractive mapping in the intermediate sense that the fixed point set is closed and convex (see Lemma 1.4 in [7]) and the demiclosedness principle holds (see Lemma 1.5 in [7]), and they also introduced an iterative scheme to find a fixed point of a uniformly L-Lipschitzian such that L>0 and asymptotically pseudocontractive mapping in the intermediate sense on a nonempty bounded closed convex C⊂H defined as in (1.13) with the sequences {kn}⊂[1,∞) and {τn}⊂[0,∞) such that limn→∞kn=1 and limn→∞τn=0, and let qn=2kn-1 for all n∈ℕ as follows:(1.17)x1=x∈Cchosenarbitrarily,zn=(1-βn)xn+βnTnxn,yn=(1-αn)xn+αnTnzn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖2+αnθn+αnβn(βnqn+βn2L2+βn-1)‖Tnxn-xn‖2},Qn={z∈C:〈xn-z,x1-xn〉≥0},xn+1=PCn∩Qn(x1),∀n∈N,
where θn=(qn[1+βn(qn-1)]-1)·(diamC)2+2(βnqn+1)τn<∞. They proved that under the sequences {αn},{βn}⊂(0,1) such that a≤αn≤βn≤b for some a>0 and b∈(0,L-2(1+L2-1)), if F(T) is nonempty, then the sequence {xn} generated by (1.17) converges strongly to a fixed point of T.Inspired and motivated by the works mentioned above, in this paper, we introduce a general iterative scheme (3.1) below to find a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method which extends and improves Qin et al. [7] and many others.


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real Hilbert space H. For solving the mixed equilibrium problem, let us assume that the bifunction Φ:C×C→ℝ, the function φ:C→ℝ∪{+∞} and the set C satisfy 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,
(2.1)lim⁡t↓0Φ(tz+(1-t)x,y)≤Φ(x,y);(A4)
for eachx∈C,y↦Φ(x,y) is convex and lower semicontinuous;(A5)
for eachy∈C,x↦Φ(x,y) is weakly upper semicontinuous;(B1)
for eachx∈C and r>0, there exists a bounded subset Dx⊂C and yx∈C such that for any z∈C∖Dx,
(2.2)Φ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0;(B2)
C is a bounded set.Lemma 2.1 (see [15]).
LetH be a Hilbert space. For any x,y∈H and λ∈ℝ, we have
(2.3)‖λx+(1-λ)y‖2=λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)‖x-y‖2.Lemma 2.2 (see [3]).
LetC be a nonempty closed convex subset of a Hilbert space H. Then the following inequality holds:
(2.4)〈x-PCx,PCx-y〉≥0,∀x∈H,y∈C.Lemma 2.3 (see [16]).
LetC be a nonempty closed convex subset of a Hilbert space H, Φ:C×C→ℝ satisfying the conditions (A1)–(A5), and let φ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0, define a mapping Sr:C→C as follows:
(2.5)Sr(x)={z∈C:Φ(z,y)+φ(y)-φ(z)+1r〈y-z,z-x〉≥0,∀y∈C},
for all x∈C. Then, the following statement hold:  (1)
for eachx∈C, Sr(x)≠∅;(2)
Sr is single-valued;(3)
Sr is firmly nonexpansive; that is, for any x,y∈C,
(2.6)‖Srx-Sry‖2≤〈Srx-Sry,x-y〉;(4)
F(Sr)=MEP(Φ,φ);(5)
MEP(Φ,φ) is closed and convex.Lemma 2.4 (see [3]).
Every Hilbert spaceH has Radon-Riesz property or Kadec-Klee property, that is, for a sequence {xn}⊂H with xn⇀x and ∥xn∥→∥x∥ then xn→x.Lemma 2.5 (see [7]).
LetC be a nonempty closed convex of a real Hilbert space H, and let T:C→C be a uniformly L-Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense such that F(T) is nonempty. Then I-T is demiclosed at zero. That is, whenever {xn} is a sequence in C weakly converging to some x∈C and the sequence {(I-T)xn} strongly converges to some y, it follows that (I-T)x=y.


## 3. Main Results

Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H, Φ a bifunction from C×C into ℝ satisfying the conditions (A1)–(A5), and φ:C→ℝ∪{+∞}  a proper lower semicontinuous and convex function with either (B1) or (B2) holds. Let T:C→C be a uniformly L -Lipschitzian such that L>0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {kn}⊂[1,∞) and {τn}⊂[0,∞) such that limn→∞kn=1 and limn→∞τn=0. Let qn=2kn-1 for all n∈ℕ. Assume that Ω∶=F(T)∩MEP(Φ,φ) be a nonempty bounded subset of C. For x1=x∈C chosen arbitrarily, suppose that {xn},{yn},{zn}, and {un} are generated iteratively by
(3.1)un∈CsuchthatΦ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,zn=(1-βn)un+βnTnun,yn=(1-αn)un+αnTnzn,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2+αnθn+αnβn(βnqn+βn2L2+βn-1)‖Tnun-un‖2},Qn+1={z∈Cn∩Qn:〈xn-z,x1-xn〉≥0},C1=Q1=C,xn+1=PCn+1∩Qn+1(x1),∀n∈N,
where θn=(qn[1+βn(qn-1)]-1)·Δn2+2(βnqn+1)τn and Δn=sup{∥xn-z∥:z∈Ω}<∞ satisfying the following conditions:  (C1)
{αn},{βn}⊂(0,1) such that a≤αn≤βn≤b for some a>0 and b∈(0,L-2(1+L2-1));(C2)
{rn}⊂[r,∞) for some r>0;(C3)
∑n=1∞|rn+1-rn|<∞.Then the sequences {xn},{yn},{zn}, and {un} converge strongly to w=PΩ(x1).Proof.
Pickp∈Ω. Therefore, by (3.1) and the definition of Srn in Lemma 2.3, we have
(3.2)un=Srnxn∈dom⁡φ,
and by Tp=p, and Lemma 2.3 (4), we have
(3.3)Tnp=p=Srnp.
By (3.2), (3.3), and the nonexpansiveness of Srn, we have
(3.4)‖un-p‖=‖Srnxn-Srnp‖≤‖xn-p‖.
By (3.3), Lemma 2.1, the uniformly L-Lipschitzian of T, and the asymptotically pseudocontractive mapping in the intermediate sense of T, we have
(3.5)‖yn-p‖2=‖(1-αn)(un-p)+αn(Tnzn-p)‖2=(1-αn)‖un-p‖2+αn‖Tnzn-p‖2-αn(1-αn)‖Tnzn-un‖2≤(1-αn)‖un-p‖2+αn(qn‖zn-p‖2+‖zn-Tnzn‖2+2τn)-αn(1-αn)‖Tnzn-un‖2,(3.6)‖zn-Tnzn‖2=‖(1-βn)(un-Tnzn)+βn(Tnun-Tnzn)‖2=(1-βn)‖un-Tnzn‖2+βn‖Tnun-Tnzn‖2-βn(1-βn)‖Tnun-un‖2≤(1-βn)‖un-Tnzn‖2+βnL2‖un-zn‖2-βn(1-βn)‖Tnun-un‖2=(1-βn)‖un-Tnzn‖2+βn(βn2L2+βn-1)‖Tnun-un‖2,(3.7)‖zn-p‖2=‖(1-βn)(un-p)+βn(Tnun-p)‖2=(1-βn)‖un-p‖2+βn‖Tnun-p‖2-βn(1-βn)‖Tnun-un‖2≤(1-βn)‖un-p‖2+βn(qn‖un-p‖2+‖un-Tnun‖2+2τn)-βn(1-βn)‖Tnun-un‖2=[1+βn(qn-1)]‖un-p‖2+βn2‖Tnun-un‖2+2βnτn.
Substituting (3.6) and (3.7) into (3.5), and by the condition (C1) and (3.4), we have
(3.8)‖yn-p‖2≤‖un-p‖2+αn(qn[1+βn(qn-1)]-1)‖un-p‖2+2αn(βnqn+1)τn+αnβn(βnqn+βn2L2+βn-1)‖Tnun-un‖2-αn(βn-αn)‖un-Tnzn‖2≤‖un-p‖2+αn((qn[1+βn(qn-1)]-1)‖un-p‖2+2(βnqn+1)τn‖un-Tnzn‖2)+αnβn(βnqn+βn2L2+βn-1)‖Tnun-un‖2≤‖un-p‖2+αnθn+αnβn(βnqn+βn2L2+βn-1)‖Tnun-un‖2≤‖xn-p‖2+αnθn+αnβn(βnqn+βn2L2+βn-1)‖Tnun-un‖2,
where θn∶=(qn[1+βn(qn-1)]-1)·Δn2+2(βnqn+1)τn and Δn∶=sup{∥xn-z∥:z∈Ω}.
Firstly, we prove thatCn∩Qn is closed and convex for all n∈ℕ. It is obvious that C1∩Q1 is closed, and by mathematical induction that Cn∩Qn is closed for all n≥2, that is Cn∩Qn is closed for all n∈ℕ. Let ϵn=αnθn+αnβn(βnqn+βn2L2+βn-1)∥Tnun-un∥2. Since for any z∈C, ∥yn-z∥2≤∥xn-z∥2+ϵn is equivalent to
(3.9)‖yn-xn‖2+2〈yn-xn,xn-z〉-ϵn≤0,
for all n∈ℕ. Therefore, for any z1,z2∈Cn+1∩Qn+1⊂Cn∩Qn and ϵ∈(0,1), we have
(3.10)∥yn-xn∥2+2〈yn-xn,xn-(ϵz1+(1-ϵ)z2)〉-ϵn=ϵ(‖yn-xn‖2+2〈yn-xn,xn-z1〉-ϵn)+(1-ϵ)(‖yn-xn‖2+2〈yn-xn,xn-z2〉-ϵn)≤0,
for all n∈ℕ, and we have
(3.11)〈xn-(ϵz1+(1-ϵ)z2),x1-xn〉=ϵ〈xn-z1,x1-xn〉+(1-ϵ)〈xn-z2,x1-xn〉≥0,
for all n∈ℕ. Since C1∩Q1 is convex, and by putting n=1 in (3.9), (3.10), and (3.11), we have C2∩Q2 is convex. Suppose that xk is given and Ck∩Qk is convex for some k≥2. It follows by putting n=k in (3.9), (3.10), and (3.11) that Ck+1∩Qk+1 is convex. Therefore, by mathematical induction, we have Cn∩Qn is convex for all n≥2, that is, Cn∩Qn is convex for all n∈ℕ. Hence, we obtain that  Cn∩Qn is closed and convex for all n∈ℕ.
Next, we prove thatΩ⊂Cn∩Qn for all n∈ℕ. It is obvious that p∈Ω⊂C=C1∩Q1. Therefore, by (3.1) and (3.8), we have p∈C2, and note that p∈C=Q2, and so p∈C2∩Q2. Hence, we have Ω⊂C2∩Q2. Since C2∩Q2 is a nonempty closed convex subset of C, there exists a unique element x2∈C2∩Q2 such that x2=PC2∩Q2(x1). Suppose that xk∈Ck∩Qk is given such that xk=PCk∩Qk(x1), and p∈Ω⊂Ck∩Qk for some k≥2. Therefore, by (3.1) and (3.8), we have p∈Ck+1. Since xk=PCk∩Qk(x1), therefore, by Lemma 2.2, we have
(3.12)〈xk-z,x1-xk〉≥0,
for all z∈Ck∩Qk. Thus, by (3.1), we have p∈Qk+1, and so p∈Ck+1∩Qk+1. Hence, we have Ω⊂Ck+1∩Qk+1. Since Ck+1∩Qk+1 is a nonempty closed convex subset of C, there exists a unique element xk+1∈Ck+1∩Qk+1 such that xk+1=PCk+1∩Qk+1(x1). Therefore, by mathematical induction, we obtain Ω⊂Cn∩Qn for all n≥2, and so Ω⊂Cn∩Qn for all n∈ℕ, and we can define xn+1=PCn+1∩Qn+1(x1) for all n∈ℕ. Hence, we obtain that the iteration (3.1) is well defined.
Next, we prove that{xn} is bounded. Since xn=PCn∩Qn(x1) for all n∈ℕ, we have
(3.13)∥xn-x1∥≤∥z-x1∥,
for all z∈Cn∩Qn. It follows by p∈Ω⊂Cn∩Qn that ∥xn-x1∥≤∥p-x1∥ for all n∈ℕ. This implies that {xn} is bounded, and so are {yn},{zn}, and {un}.
Next, we prove that∥xn-xn+1∥→0 and ∥un-un+1∥→0 as n→∞. Since xn+1=PCn+1∩Qn+1(x1)∈Cn+1∩Qn+1⊂Cn∩Qn, therefore, by (3.13), we have ∥xn-x1∥≤∥xn+1-x1∥ for all n∈ℕ. This implies that {∥xn-x1∥} is a bounded nondecreasing sequence; there exists the limit of ∥xn-x1∥, that is
(3.14)lim⁡n→∞‖xn-x1‖=m,
for some m≥0. Since xn+1∈Qn+1, therefore, by (3.1), we have
(3.15)〈xn-xn+1,x1-xn〉≥0.
It follows by (3.15) that
(3.16)‖xn-xn+1‖2=‖(xn-x1)+(x1-xn+1)‖2=‖xn-x1‖2+2〈xn-x1,x1-xn〉+2〈xn-x1,xn-xn+1〉+‖xn+1-x1‖2≤‖xn+1-x1‖2-‖xn-x1‖2.
Therefore, by (3.14), we obtain
(3.17)‖xn-xn+1‖⟶0asn⟶∞.
Indeed, from (3.1) we have
(3.18)Φ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,(3.19)Φ(un+1,y)+φ(y)-φ(un+1)+1rn+1〈y-un+1,un+1-xn+1〉≥0,∀y∈C;
substituting y=un+1 into (3.18) and y=un into (3.19), we have
(3.20)Φ(un,un+1)+φ(un+1)-φ(un)+1rn〈un+1-un,un-xn〉≥0,Φ(un+1,un)+φ(un)-φ(un+1)+1rn+1〈un-un+1,un+1-xn+1〉≥0.
Therefore, by the condition (A2), we get
(3.21)0≤Φ(un,un+1)+Φ(un+1,un)+〈un+1-un,un-xnrn-un+1-xn+1rn+1〉≤〈un+1-un,un-xnrn-un+1-xn+1rn+1〉.
It follows that
(3.22)0≤〈un+1-un,(un-un+1)+(un+1-xn)-rnrn+1(un+1-xn+1)〉=〈un+1-un,un-un+1〉+〈un+1-un,(un+1-xn+1)+(xn+1-xn)-rnrn+1(un+1-xn+1)〉.
Thus, we have
(3.23)‖un+1-un‖2≤〈un+1-un,(xn+1-xn)+(1-rnrn+1)(un+1-xn+1)〉≤‖un+1-un‖(‖xn+1-xn‖+|1-rnrn+1|‖un+1-xn+1‖).
It follows by the condition (C2) that
(3.24)‖un+1-un‖≤‖xn+1-xn‖+|rn+1-rn|rn+1‖un+1-xn+1‖≤‖xn+1-xn‖+(Mr)|rn+1-rn|,
where M=supn≥1∥un-xn∥<∞. Therefore, by the condition (C3) and (3.17), we obtain
(3.25)‖un-un+1‖⟶0asn⟶∞.
Next, we prove that∥Tnun-un∥→0 and ∥Tun-un∥→0 as n→∞. Since xn+1∈Cn+1, by (3.1), we have
(3.26)‖yn-xn+1‖2≤‖xn-xn+1‖2+αnθn+αnβn(βnqn+βn2L2+βn-1)‖Tnun-un‖2,
it follows by the condition (C1) that
(3.27)a2(1-bqn-b2L2-b)∥Tnun-un∥2≤αnβn(1-βnqn-βn2L2-βn)‖Tnun-un‖2≤‖xn-xn+1‖2+αnθn-‖yn-xn+1‖2≤‖xn-xn+1‖2+bθn.
Since limn→∞qn=1 and the condition (C1), we have limn→∞(1-bqn-b2L2-b)=1-2b-b2L2>0. Therefore, from (3.27) by (3.17) and limn→∞θn=0, we obtain
(3.28)‖Tnun-un‖⟶0asn⟶∞.
By the uniformly L-Lipschitzian of T, we have
(3.29)‖un-Tun‖≤‖un-un+1‖+‖un+1-Tn+1un+1‖+‖Tn+1un+1-Tn+1un‖+‖Tn+1un-Tun‖≤(1+L)‖un-un+1‖+‖un+1-Tn+1un+1‖+L‖Tnun-un‖.
Therefore, by (3.25) and (3.28), we obtain
(3.30)‖un-Tun‖⟶0asn⟶∞.
Next, we prove that∥yn-xn∥→0, ∥un-zn∥→0 and ∥un-xn∥→0 as n→∞. From (3.26), by the condition (C1), we have
(3.31)‖yn-xn+1‖2≤‖xn-xn+1‖2+bθn+b2(bqn+b2L2+b-1)‖Tnun-un‖2;
it follows that
(3.32)‖yn-xn‖2=‖(yn-xn+1)+(xn+1-xn)‖2=‖yn-xn+1‖2+2〈yn-xn+1,xn+1-xn〉+‖xn+1-xn‖2≤‖yn-xn+1‖2+2‖yn-xn+1‖‖xn+1-xn‖+‖xn+1-xn‖2≤2‖xn+1-xn‖(‖xn+1-xn‖+‖yn-xn+1‖)+bθn-b2(1-bqn-b2L2-b)‖Tnun-un‖2.
Therefore, by (3.17), limn→∞θn=0, and limn→∞(1-bqn-b2L2-b)=1-2b-b2L2>0, we obtain
(3.33)‖yn-xn‖⟶0asn⟶∞.
From (3.1), we have ∥un-zn∥=βn∥un-Tnun∥; therefore, by (3.28), we obtain
(3.34)‖un-zn‖⟶0asn⟶∞.
By (3.2), (3.3), and the firmly nonexpansiveness of Srn, we have
(3.35)‖un-p‖2≤〈Srnxn-Srnp,xn-p〉=〈un-p,xn-p〉=12(‖un-p‖2+‖xn-p‖2-‖un-xn‖2),
it follows that
(3.36)∥un-p∥2≤∥xn-p∥2-∥un-xn∥2.
Therefore, from (3.8), we have
(3.37)∥yn-p∥2≤∥un-p∥2+αnθn+αnβn(βnqn+βn2L2+βn-1)∥Tnun-un∥2≤∥xn-p∥2-∥un-xn∥2+αnθn+αnβn(βnqn+βn2L2+βn-1)∥Tnun-un∥2;
it follows by the condition (C1) that
(3.38)‖un-xn‖2≤‖xn-p‖2-‖yn-p‖2+αnθn+αnβn(βnqn+βn2L2+βn-1)‖Tnun-un‖2≤‖xn-yn‖(‖xn-p‖+‖yn-p‖)+bθn-b2(1-bqn-b2L2-b)‖Tnun-un‖2.
Therefore, by (3.33), limn→∞θn=0, and limn→∞(1-bqn-b2L2-b)=1-2b-b2L2>0, we obtain
(3.39)∥un-xn∥⟶0asn⟶∞.
Since {un} is bounded, there exists a subsequence {uni} of {un} which converges weakly to w¯. Next, we prove that w¯∈Ω. From uni⇀w¯ and ∥uni-Tuni∥→0 as i→∞ by (3.30), therefore, by Lemma 2.5, we obtain w¯∈F(T). From (3.1), we have
(3.40)0≤Φ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉,∀y∈C;
it follows by the condition (A2) that
(3.41)Φ(y,un)≤Φ(y,un)+Φ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉,∀y∈C≤φ(y)-φ(un)+1rn〈y-un,un-xn〉,∀y∈C.
Hence,
(3.42)φ(y)-φ(uni)+〈y-uni,uni-xnirni〉≥Φ(y,uni),∀y∈C.
Therefore, from (3.39) and by uni⇀w¯ as i→∞, we obtain
(3.43)Φ(y,w¯)+φ(w¯)-φ(y)≤0,∀y∈C.
For a constant t with 0<t<1 and y∈C, let yt=ty+(1-t)w¯. Since y,w¯∈C, thus, yt∈C. So, from (3.43), we have
(3.44)Φ(yt,w¯)+φ(w¯)-φ(yt)≤0.
By (3.44), the conditions (A1) and (A4), and the convexity of φ, we have
(3.45)0=Φ(yt,yt)+φ(yt)-φ(yt)≤(tΦ(yt,y)+(1-t)Φ(yt,w¯))+(tφ(y)+(1-t)φ(w¯))-φ(yt)=t(Φ(yt,y)+φ(y)-φ(yt))+(1-t)(Φ(yt,w¯)+φ(w¯)-φ(yt))≤t(Φ(yt,y)+φ(y)-φ(yt));
it follows that
(3.46)Φ(yt,y)+φ(y)-φ(yt)≥0.
Therefore, by the condition (A3) and the weakly lower semicontinuity of φ, we have Φ(w¯,y)+φ(y)-φ(w¯)≥0 as t→0 for all y∈C, and hence, we obtain w¯∈MEP(Φ,φ), and so w¯∈Ω.
SinceΩ is a nonempty closed convex subset of C, there exists a unique w∈Ω such that w=PΩ(x1). Next, we prove that xn→w as n→∞. Since w=PΩ(x1), we have ∥x1-w∥≤∥x1-z∥ for all z∈Ω; it follows that
(3.47)∥x1-w∥≤∥x1-w¯∥.
Since w∈Ω⊂Cn∩Qn, therefore, by (3.13), we have
(3.48)‖x1-xn‖≤‖x1-w‖.
Since ∥xni-uni∥→0 by (3.39) and uni⇀w¯, we have xni⇀w¯ as i→∞. Therefore, by (3.47), (3.48) and the weak lower semicontinuity of norm, we have
(3.49)‖x1-w‖≤‖x1-w¯‖≤liminf⁡i→∞‖x1-xni‖≤limsup⁡i→∞‖x1-xni‖≤‖x1-w‖.
It follows that
(3.50)‖x1-w‖=lim⁡i→∞‖x1-xni‖=‖x1-w¯‖.
Since xni⇀w¯ as i→∞, therefore, we have
(3.51)(x1-xni)⇀(x1-w¯)asi⟶∞.
Hence, from (3.50), (3.51), Kadec-Klee property, and the uniqueness of w=PΩ(x1), we obtain
(3.52)xni⟶w¯=wasi⟶∞.
It follows that {xn} converges strongly to w and so are {yn},{zn}, and {un}. This completes the proof.Remark 3.2.
The iteration (3.1) is the difference with the iterative scheme of Qin et al. [7] as follows.(1)
The sequence{xn} is a projection sequence of x1 onto Cn∩Qn for all n∈ℕ such that
(3.53)C1∩Q1⊃C2∩Q2⊃⋯⊃Cn∩Qn⊃⋯⊃Ω.(2)
A solving of a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of the mixed equilibrium problems by iteration is obtained.
We define the condition (B3) as the condition (B1) such thatφ=0. If φ=0, then Theorem 3.1 is reduced immediately to the following result.Corollary 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space  H, and let  Φ a bifunction from  C×C into  ℝ satisfying the conditions (A1)–(A5) with either (B2) or (B3) holds. Let  T:C→C be a uniformly  L-Lipschitzian such that  L>0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences  {kn}⊂[1,∞) and  {τn}⊂[0,∞) such that  limn→∞kn=1 and  limn→∞τn=0. Let  qn=2kn-1 for all  n∈ℕ. Assume that  Ω:=F(T)∩EP(Φ) be a nonempty bounded subset of  C. For  x1=x∈C chosen arbitrarily, suppose that  {xn},{yn},{zn}, and  {un} are generated iteratively by
(3.54)un∈CsuchthatΦ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,zn=(1-βn)un+βnTnun,yn=(1-αn)un+αnTnzn,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2+αnθn+αnβn(βnqn+βn2L2+βn-1)‖Tnun-un‖2},Qn+1={z∈Cn∩Qn:〈xn-z,x1-xn〉≥0},C1=Q1=C,xn+1=PCn+1∩Qn+1(x1),∀n∈N,
where θn=(qn[1+βn(qn-1)]-1)·Δn2+2(βnqn+1)τn and Δn=sup{∥xn-z∥:z∈Ω}<∞ satisfying the following conditions:  (C1)
{αn},{βn}⊂(0,1) such that a≤αn≤βn≤b for some a>0 and b∈(0,L-2(1+L2-1));(C2)
{rn}⊂[r,∞) for some r>0;(C3)
∑n=1∞|rn+1-rn|<∞. Then the sequences {xn},{yn},{zn}, and {un} converge strongly to w=PΩ(x1).IfΦ=0, then Corollary 3.3 is reduced immediately to the following result.Corollary 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a uniformly L-Lipschitzian such that L>0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {kn}⊂[1,∞) and {τn}⊂[0,∞) such that limn→∞kn=1 and limn→∞τn=0. Let qn=2kn-1 for all n∈ℕ. Assume that F(T) be a nonempty bounded subset of C. For x1=x∈C chosen arbitrarily, suppose that {xn},{yn}, and {zn} are generated iteratively by
(3.55)zn=(1-βn)xn+βnTnxn,yn=(1-αn)xn+αnTnzn,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2+αnθn+αnβn(βnqn+βn2L2+βn-1)‖Tnxn-xn‖2},Qn+1={z∈Cn∩Qn:〈xn-z,x1-xn〉≥0},C1=Q1=C,xn+1=PCn+1∩Qn+1(x1),∀n∈N,
where θn=(qn[1+βn(qn-1)]-1)·Δn2+2(βnqn+1)τn, Δn=sup{∥xn-z∥:z∈F(T)}<∞ and {αn},{βn}⊂(0,1) such that a≤αn≤βn≤b for some a>0 and b∈(0,L-2(1+L2-1)). Then the sequences {xn},{yn} and {zn} converge strongly to w=PF(T)(x1).We introduce the equilibrium problem to the optimization problem:(3.56)min⁡x∈Cζ(x),
where C is a nonempty closed convex subset of a real Hilbert space H and let ζ:C→ℝ∪{+∞} is a proper convex and lower semicontinuous. We denote by Argmin(ζ) the set of solution of problem (3.56). We define the condition (B4) as the condition (B3) such that Φ:C×C→ℝ is a bifunction defined by Φ(x,y)=ζ(y)-ζ(x) for all x,y∈C. Observe that EP(Φ)=Argmin(ζ). We obtain that Corollary 3.3 is reduced immediately to the following result.Corollary 3.5.
LetC is a nonempty closed convex subset of a real Hilbert space H, and ζ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function with either (B2) or (B4) holds. Let T:C→C be a uniformly L-Lipschitzian such that L>0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {kn}⊂[1,∞) and {τn}⊂[0,∞) such that limn→∞kn=1 and limn→∞τn=0. Let qn=2kn-1 for all n∈ℕ. Assume that Ω:=F(T)∩Argmin(ζ) be a nonempty bounded subset of C. For x1=x∈C chosen arbitrarily, suppose that {xn},{yn},{zn} and {un} are generated iteratively by
(3.57)un∈Csuchthatζ(y)-ζ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,zn=(1-βn)un+βnTnun,yn=(1-αn)un+αnTnzn,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2+αnθn+αnβn(βnqn+βn2L2+βn-1)‖Tnun-un‖2},Qn+1={z∈Cn∩Qn:〈xn-z,x1-xn〉≥0},C1=Q1=C,xn+1=PCn+1∩Qn+1(x1),∀n∈N,
where θn=(qn[1+βn(qn-1)]-1)·Δn2+2(βnqn+1)τn and Δn=sup{∥xn-z∥:z∈Ω}<∞ satisfying the following conditions:  (C1)
{αn},{βn}⊂(0,1) such that a≤αn≤βn≤b for some a>0 and b∈(0,L-2(1+L2-1));(C2)
{rn}⊂[r,∞) for some r>0;(C3)
∑n=1∞|rn+1-rn|<∞.Then the sequences {xn},{yn},{zn}, and {un} converge strongly to w=PΩ(x1).



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*Source: 187421-2011-10-31.xml*