# Shrinking Projection Method of Common Fixed Point Problems for Discrete Asymptotically Strictly Pseudocontractive Semigroups and Mixed Equilibrium Problems in Hilbert Spaces

Authors: Pattanapong Tianchai
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

This paper is concerned with a common element of the set of common fixed point for a discrete asymptotically strictly pseudocontractive semigroup 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 the corresponding ones due to Kim [Proceedings of the Asian Conference on Nonlinear Analysis and Optimization (Matsue, Japan, 2008), 139–162].


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## Body

## 1. Introduction

Throughout this paper, we always assume thatC is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by 〈·,·〉 and ∥·∥, 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 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 solutions 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)≥,∀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, problem (1.4) which 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, problem (1.5) which 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 a mapping f:C→C is called acontraction 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 and 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 as follows.(i)
κ-strictly pseudocontractive (see [5]) if there exists a constant κ∈[0,1) such that
(1.8)‖Tx-Ty‖2≤‖x-y‖2+κ‖(I-T)x-(I-T)y‖2,∀x,y∈C,
in brief, we use κ-SPC to denote the κ-strictly pseudocontractive, it is obvious that T is a nonexpansive if and only if T is a 0-SPC.(ii)
Asymptoticallyκ-SPC (see [6]) if there exists a constant κ∈[0,1) and a sequence {γn} of nonnegative real numbers with limn→∞γn=0 such that
(1.9)‖Tnx-Tny‖2≤(1+γn)‖x-y‖2+κ‖(I-Tn)x-(I-Tn)y‖2,∀x,y∈C,
for all n∈ℕ, if κ=0 then T is an asymptotically nonexpansive with kn=1+γn for all n∈ℕ; that is, T is an asymptotically nonexpansive (see [7]) if there exists a sequence {kn}⊂[1,∞) with limn→∞kn=1 such that
(1.10)‖Tnx-Tny‖≤kn‖x-y‖,∀x,y∈C,
for all n∈ℕ, it is known that the class of κ-SPC mappings, and the classes of asymptotically κ-SPC mappings are independent (see [8]).The Mann's algorithm for nonexpansive mappings has been extensively investigated (see [5, 9, 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)=∞. Recently, Marino and Xu [11] devoloped and extended Reich's result to SPC mapping in Hilbert space setting. More precisely, they proved the weak convergence of the Mann's iteration process (1.7) for a κ-SPC mapping T on C, and, subsequently, this result was improved and carried over the class of asymptotically κ-SPC mappings by Kim and Xu [12].It is known that the Mann's iteration (1.7) is in general not strongly convergent (see [13]). The strong convergence is guaranteed and has been proposed by Nakajo and Takahashi [14], 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 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 let  {xn} be the sequence defined by
(1.11)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, Marino and Xu [15] introduced an iterative scheme for finding a fixed point of a κ-SPC mapping as the following theorem.Theorem MX.
LetC be a closed convex subset of a Hilbert space  H, and,  T:C→C be a  κ-SPC mapping for some  0≤κ<1. Assume that  F(T)≠∅. Suppose that  x1=x∈C chosen arbitrarily, and let  {xn} be the sequence defined by
(1.12)yn=αnxn+(1-αn)Txn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖2+(1-αn)(κ-αn)‖xn-Txn‖2},Qn={z∈C:〈xn-z,x1-xn〉≥0},xn+1=PCn∩Qn(x1),∀n∈N,
where 0≤αn<1. Then the sequence {xn} converges strongly to PF(T)(x1).
Quite recently, Kim and Xu [12] has improved and carried Theorem MX over the more wider class of asymptotically κ-SPC mappings as the following theorem.Theorem KX.
LetC be a closed convex subset of a Hilbert space  H, and,  T:C→C be an asymptotically  κ-SPC mapping for some  0≤κ<1 and a bounded sequence  {γn}⊂[0,∞) such that  limn→∞γn=0. Assume that  F(T) is a nonempty bounded subset of  C. Suppose that  x1=x∈C chosen arbitrarily, and let  {xn} be the sequence defined by
(1.13)yn=αnxn+(1-αn)Tnxn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖2+(κ-αn(1-αn))‖xn-Tnxn‖2+θn},Qn={z∈C:〈xn-z,x1-xn〉≥0},xn+1=PCn∩Qn(x1),∀n∈N,
where θn=Δn2(1-αn)γn→0 as n→∞, Δn=sup{∥xn-z∥:z∈F(T)}<∞, and 0≤αn<1 such that limsupn→∞αn<1-κ. Then the sequence {xn} converges strongly to PF(T)(x1).Recall that a discrete family𝒮={Tn:n≥0} of self-mappings of C is said to be a Lipschitzian semigroup on C if the following conditions are satisfied.  (1)
T0=I where I denotes the identity operator on C.(2)
Tn+mx=TnTmx,∀n,m≥0,∀x∈C.(3)
There exists a sequence{Ln} of nonnegative real numbers such that(1.14)‖Tnx-Tny‖≤Ln‖x-y‖,∀x,y∈C,∀n≥0.
A discrete Lipschitzian semigroup 𝒮 is called nonexpansive semigroup if Ln=1 for all n≥0, contraction semigroup if 0<Ln<1 for all n≥0 and, asymptotically nonexpansive semigroup if limsupn→∞Ln≤1, respectively. We use F(𝒮) to denote the common fixed point set of the semigroup 𝒮; that is, F(𝒮)={x∈C:Tnx=x,∀n≥0}.Very recently, Kim [16] introduced asymptoticallyκ-SPC semigroup, a discrete family 𝒮={Tn:n≥0} of self-mappings of C which is said to be asymptotically κ-SPC semigroup on C if, in addition to (1), (2) and the following condition (3′) are satisfied.  (3′)
There exists a constantκ∈[0,1) and a bounded sequence {Ln} of nonnegative real numbers with limsupn→∞Ln≤1 such that(1.15)‖Tnx-Tny‖2≤Ln‖x-y‖2+κ‖(I-Tn)x-(I-Tn)y‖2,∀x,y∈C,∀n≥0.Note that for both discrete asymptotically nonexpansive semigroups and discrete asymptoticallyκ-SPC semigroups, we can always assume that the Lipschitzian constants {Ln}n≥0 are such that Ln≥1 for all n≥0 and limn→∞Ln=1; otherwise, we replace Ln for all n≥0 with Ln′=max{supm≥nLm,1}. Therefore, for a single asymptotically κ-SPC mapping T:C→C note that (1.15) immediately reduces to (1.9) by taking Tn≡Tn and γn=Ln-1 such that Ln≥1 for all n≥0 and limn→∞Ln=1.To be more precise, Kim also showed in the framework of Hilbert spaces for the asymptoticallyκ-SPC semigroups that Tn is continuous on C for all n≥0 and that F(𝒮) is closed and convex (see Lemma  3.2 in [16]), and the demiclosedness principle (see Theorem  3.3 in [16]) holds in the sense that if {xn} is a sequence in C such that xn⇀z and limsupm→∞limsupn→∞∥xn-Tmxn∥=0 then, z∈F(𝒮)=⋂n=0∞F(Tn), and he also introduced an iterative scheme to find a common fixed point of a discrete asymptotically κ-SPC semigroup and a bounded sequence {Ln}⊂[1,∞) such that limn→∞Ln=1 as follows:(1.16)x0=x∈Cchosenarbitrarily,yn=αnxn+(1-αn)Tnxn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖2+(1-αn)(θn+(κ-αn)‖xn-Tnxn‖2)},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qn(x0),∀n∈N∪{0},
where θn=(Ln-1)·sup{∥xn-z∥2:z∈F(𝒮)}<∞. He proved that under the parameter 0≤αn<1 for all n∈ℕ∪{0}, if F(𝒮) is a nonempty bounded subset of C, then the sequence {xn} generated by (1.16) converges strongly to PF(𝒮)(x0).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 common fixed point for a discrete asymptotically κ-SPC semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained based on the shrinking projection method which extend and improve the corresponding ones due to Kim [16].


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real Hilbert space H. 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, and the weak ω-limit set of {xn} by ωw(xn)={x:∃xnj⇀x}.For solving the mixed equilibrium problem, let us assume that the bifunctionΦ: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 [17]).
LetH be a Hilbert space. For any x,y∈H and λ∈ℝ, one has
(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 [18]).
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 [16]).
LetC be a nonempty closed convex subset of a Hilbert space H, and let 𝒮={Tn:n≥0} be an asymptotically κ-strictly pseudocontractive semigroup on C. Let {xn} be a sequence in C such that limn→∞∥xn-xn+1∥=0 and limn→∞∥xn-Tnxn∥=0. Then ωw(xn)⊂F(𝒮).


## 3. Main Results

Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space H, let Φ be a bifunction from C×C into ℝ satisfying the conditions (A1)–(A5), and let φ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function with either (B1) or (B2) holds. Let 𝒮={Tn:n≥0} be an asymptotically κ-SPC semigroup on C for some κ∈[0,1) and a bounded sequence {Ln}⊂[1,∞) such that limn→∞Ln=1. Assume that Ω:=F(𝒮)∩MEP(Φ,φ) is a nonempty bounded subset of C. For x0=x∈C chosen arbitrarily, suppose that {xn},{yn}, and {un} are generated iteratively by
(3.1)un∈CsuchthatΦ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=αnun+(1-αn)Tnun,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2+(1-αn)(θn+(κ-αn)‖un-Tnun‖2)},Qn+1={z∈Cn∩Qn:〈xn-z,x0-xn〉≥0},C0=Q0=C,xn+1=PCn+1∩Qn+1(x0),∀n∈N∪{0},
where θn=(Ln-1)·sup{∥xn-z∥2:z∈Ω}<∞ satisfying the following conditions:  (C1)
{αn}⊂[a,b] such that κ<a<b<1,(C2)
{rn}⊂[r,∞) for some r>0,(C3)
∑n=0∞|rn+1-rn|<∞.Then the sequences {xn},{yn}, and {un} converge strongly to w=PΩ(x0).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 F(𝒮)=⋂n=0∞F(Tn) 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), (3.4), Lemma 2.1, and the asymptotically κ-SPC semigroupness of 𝒮, we have
(3.5)‖yn-p‖2=‖αn(un-p)+(1-αn)(Tnun-p)‖2=αn‖un-p‖2+(1-αn)‖Tnun-p‖2-αn(1-αn)‖un-Tnun‖2≤αn‖un-p‖2+(1-αn)(Ln‖un-p‖2+κ‖un-Tnun‖2)-αn(1-αn)‖un-Tnun‖2=(1+(1-αn)(Ln-1))‖un-p‖2+(1-αn)(κ-αn)‖un-Tnun‖2≤‖xn-p‖2+(1-αn)(θn+(κ-αn)‖un-Tnun‖2),
where θn:=(Ln-1)·sup{∥xn-z∥2:z∈Ω} for all n∈ℕ∪{0}.
Firstly, we prove thatCn∩Qn is closed and convex for all n∈ℕ∪{0}. It is obvious that C0∩Q0 is closed and by mathematical induction that Cn∩Qn is closed for all n≥1; that is Cn∩Qn is closed for all n∈ℕ∪{0}. Let ϵn=(1-αn)(θn+(k-αn)∥un-Tnun∥2) since, for any z∈C, ∥yn-z∥2≤∥xn-z∥2+ϵn is equivalent to
(3.6)‖yn-xn‖2+2〈yn-xn,xn-z〉-ϵn≤0,
for all n∈ℕ∪{0}. Therefore, for any z1,z2∈Cn+1∩Qn+1⊂Cn∩Qn and ϵ∈(0,1), we have
(3.7)‖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∪{0}, and we have
(3.8)〈xn-(ϵz1+(1-ϵ)z2),x0-xn〉=ϵ〈xn-z1,x0-xn〉+(1-ϵ)〈xn-z2,x0-xn〉≥0,
for all n∈ℕ∪{0}. Since C0∩Q0 is convex and by putting n=0 in (3.6), (3.7), and (3.8), we have that C1∩Q1 is convex. Suppose that xk is given and Ck∩Qk is convex for some k≥1. It follows by putting n=k in (3.6), (3.7), and (3.8) that Ck+1∩Qk+1 is convex. Therefore, by mathematical induction, we have that Cn∩Qn is convex for all n≥1; that is, Cn∩Qn is convex for all n∈ℕ∪{0}. Hence, we obtain that Cn∩Qn is closed and convex for all n∈ℕ∪{0}.
Next, we prove thatΩ⊂Cn∩Qn for all n∈ℕ∪{0}. It is obvious that p∈Ω⊂C=C0∩Q0. Therefore, by (3.1) and (3.5), we have p∈C1 and note that p∈C=Q1, and so p∈C1∩Q1. Hence, we have Ω⊂C1∩Q1. Since C1∩Q1 is a nonempty closed convex subset of C, there exists a unique element x1∈C1∩Q1 such that x1=PC1∩Q1(x0). Suppose that xk∈Ck∩Qk is given such that xk=PCk∩Qk(x0) and p∈Ω⊂Ck∩Qk for some k≥1. Therefore, by (3.1) and (3.5), we have p∈Ck+1. Since xk=PCk∩Qk(x0); therefore, by Lemma 2.2, we have
(3.9)〈xk-z,x0-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(x0). Therefore, by mathematical induction, we obtain Ω⊂Cn∩Qn for all n≥1, and so Ω⊂Cn∩Qn for all n∈ℕ∪{0}, and we can define xn+1=PCn+1∩Qn+1(x0) for all n∈ℕ∪{0}. Hence, we obtain that the iteration (3.1) is well defined.
Next, we prove that{xn} is bounded. Since xn=PCn∩Qn(x0) for all n∈ℕ∪{0}, we have
(3.10)‖xn-x0‖≤‖z-x0‖,
for all z∈Cn∩Qn. It follows by p∈Ω⊂Cn∩Qn that ∥xn-x0∥≤∥p-x0∥ for all n∈ℕ∪{0}. This implies that {xn} is bounded and so are {yn} and {un}.
Next, we prove that∥xn-xn+1∥→0 and ∥un-un+1∥→0 as n→∞. Since xn+1=PCn+1∩Qn+1(x0)∈Cn+1∩Qn+1⊂Cn∩Qn; therefore, by (3.10), we have ∥xn-x0∥≤∥xn+1-x0∥ for all n∈ℕ∪{0}. This implies that {∥xn-x0∥} is a bounded nondecreasing sequence; there exists the limit of ∥xn-x0∥; that is,
(3.11)lim⁡n→∞‖xn-x0‖=m,
for some m≥0. Since xn+1∈Qn+1; therefore, by (3.1), we have
(3.12)〈xn-xn+1,x0-xn〉≥0.
It follows by (3.12) that
(3.13)‖xn-xn+1‖2=‖(xn-x0)+(x0-xn+1)‖2=‖xn-x0‖2+2〈xn-x0,x0-xn〉+2〈xn-x0,xn-xn+1〉+‖xn+1-x0‖2≤‖xn+1-x0‖2-‖xn-x0‖2.
Therefore, by (3.11), we obtain
(3.14)‖xn-xn+1‖⟶0asn⟶∞.
Indeed, from (3.1), we have
(3.15)Φ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,(3.16)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 (3.15) and y=un into (3.16), we have
(3.17)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
(3.18)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
(3.19)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
(3.20)‖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
(3.21)‖un+1-un‖≤‖xn+1-xn‖+|rn+1-rn|rn+1‖un+1-xn+1‖≤‖xn+1-xn‖+(Lr)|rn+1-rn|,
where L=supn≥0∥un-xn∥<∞. Therefore, by the condition (C3) and (3.14), we obtain
(3.22)‖un-un+1‖⟶0asn⟶∞.
Next, we prove that∥un-Tnun∥→0, ∥yn-xn∥→0, and ∥un-xn∥→0 as n→∞. Since xn+1∈Cn+1, by (3.1), we have
(3.23)‖yn-xn+1‖2≤‖xn-xn+1‖2+(1-αn)(θn+(κ-αn)‖un-Tnun‖2).
It follows that
(3.24)(1-αn)(αn-κ)‖un-Tnun‖2≤‖xn-xn+1‖2+(1-αn)θn-‖yn-xn+1‖2≤‖xn-xn+1‖2+θn.
Therefore, by the condition (C1), (3.14), and limn→∞θn=0, we obtain
(3.25)‖un-Tnun‖⟶0asn⟶∞.
From (3.23) and the condition (C1), we have
(3.26)‖yn-xn+1‖2≤‖xn-xn+1‖2+θn.
Therefore,
(3.27)‖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‖)+θn.
Hence, by (3.14) and limn→∞θn=0, we obtain
(3.28)‖yn-xn‖⟶0asn⟶∞.
By (3.2), (3.3), and the firmly nonexpansiveness of Srn, we have
(3.29)‖un-p‖2≤〈Srnxn-Srnp,xn-p〉=〈un-p,xn-p〉=12(‖un-p‖2+‖xn-p‖2-‖un-xn‖2).
It follows that
(3.30)‖un-p‖2≤‖xn-p‖2-‖un-xn‖2.
Therefore, by the condition (C1) and (3.5), we have
(3.31)‖yn-p‖2≤(1+(1-αn)(Ln-1))‖un-p‖2+(1-αn)(κ-αn)‖un-Tnun‖2≤(Ln-αn(Ln-1))‖un-p‖2≤Ln‖un-p‖2≤Ln(‖xn-p‖2-‖un-xn‖2).
It follows that
(3.32)Ln‖un-xn‖2≤Ln‖xn-p‖2-‖yn-p‖2=(Ln-1)‖xn-p‖2+‖xn-p‖2-‖yn-p‖2≤θn+‖xn-yn‖(‖xn-p‖+‖yn-p‖).
Hence, by (3.28) and limn→∞θn=0, we obtain
(3.33)‖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 (3.22) and (3.25), we have ∥uni-uni+1∥→0, and ∥uni-Tniuni∥→0 as i→∞; therefore, by Lemma 2.5, we obtain w¯∈F(𝒮). From (3.1), we have
(3.34)0≤Φ(un,y)+φ(y)-φ(un)+1rn〈y-un,un-xn〉,∀y∈C.
It follows by the condition (A2) that
(3.35)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,
(3.36)φ(y)-φ(uni)+〈y-uni,uni-xnirni〉≥Φ(y,uni),∀y∈C.
Therefore, from (3.33) and by uni⇀w¯ as i→∞, we obtain
(3.37)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 (3.37), we have
(3.38)Φ(yt,w¯)+φ(w¯)-φ(yt)≤0.
By (3.38), the conditions (A1) and (A4), and the convexity of φ, we have
(3.39)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
(3.40)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(x0). Next, we prove that xn→w as n→∞. Since w=PΩ(x0), we have ∥x0-w∥≤∥x0-z∥ for all z∈Ω; it follows that
(3.41)‖x0-w‖≤‖x0-w¯‖.
Since w∈Ω⊂Cn∩Qn; therefore, by (3.10), we have
(3.42)‖x0-xn‖≤‖x0-w‖.
Since ∥xni-uni∥→0 by (3.33) and uni⇀w¯, we have xni⇀w¯ as i→∞. Therefore, by (3.41), (3.42), and the weak lower semicontinuity of norm, we have
(3.43)‖x0-w‖≤‖x0-w¯‖≤lim⁡inf⁡i→∞‖x0-xni‖≤limsup⁡i→∞‖x0-xni‖≤‖x0-w‖.
It follows that
(3.44)‖x0-w‖=lim⁡i→∞‖x0-xni‖=‖x0-w¯‖.
Since xni⇀w¯ as i→∞; therefore, we have
(3.45)(x0-xni)⇀(x0-w¯)asi⟶∞.
Hence, from (3.44), (3.45), the Kadec-Klee property, and the uniqueness of w=PΩ(x0), we obtain
(3.46)xni⟶w¯=wasi⟶∞.
It follows that {xn} converges strongly to w and so are {yn} and {un}. This completes the proof.Remark 3.2.
The iteration (3.1) is the difference with the iterative scheme of Kim [16] as the following.  (1)
The sequence{xn} is a projection sequence of x0 onto Cn∩Qn for all n∈ℕ∪{0} such that(3.47)C0∩Q0⊃C1∩Q1⊃⋯⊃Cn∩Qn⊃⋯⊃Ω.(2)
A solving of a common element of the set of common fixed point for a discrete asymptoticallyκ-SPC semigroup and the set of solutions of the mixed equilibrium problems by iteration is obtained.For solving the equilibrium problem, let us define the condition(B3) as the condition (B1) such that φ=0. We have the following result.Corollary 3.3.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let Φ be a bifunction from C×C into ℝ satisfying the conditions (A1)–(A5) with either (B2) or (B3) holds. Let 𝒮={Tn:n≥0} be an asymptotically κ-SPC semigroup on C for some κ∈[0,1) and a bounded sequence {Ln}⊂[1,∞) such that limn→∞Ln=1. Assume that Ω:=F(𝒮)∩EP(Φ) is a nonempty bounded subset of C. For x0=x∈C chosen arbitrarily, suppose that {xn},{yn}, and {un} are generated iteratively by
(3.48)un∈Csuch thatΦ(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=αnun+(1-αn)Tnun,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2+(1-αn)(θn+(κ-αn)‖un-Tnun‖2)},Qn+1={z∈Cn∩Qn:〈xn-z,x0-xn〉≥0},C0=Q0=C,xn+1=PCn+1∩Qn+1(x0),∀n∈N∪{0},
where θn=(Ln-1)·sup{∥xn-z∥2:z∈Ω}<∞ satisfying the following conditions:  (C1)
{αn}⊂[a,b] such that κ<a<b<1,(C2)
{rn}⊂[r,∞) for some r>0,(C3)
∑n=0∞|rn+1-rn|<∞.Then the sequences {xn},{yn}, and {un} converge strongly to w=PΩ(x0).Proof.
It is concluded from Theorem3.1 immediately, by putting φ=0.Corollary 3.4.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let 𝒮={Tn:n≥0} be an asymptotically κ-SPC semigroup on C for some κ∈[0,1) and a bounded sequence {Ln}⊂[1,∞) such that limn→∞Ln=1. Assume that F(𝒮) is a nonempty bounded subset of C. For x0=x∈C chosen arbitrarily, suppose that {xn} and {yn} are generated iteratively by
(3.49)yn=αnxn+(1-αn)Tnxn,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2+(1-αn)(θn+(κ-αn)‖xn-Tnxn‖2)},Qn+1={z∈Cn∩Qn:〈xn-z,x0-xn〉≥0},C0=Q0=C,xn+1=PCn+1∩Qn+1(x0),∀n∈N∪{0},
where θn=(Ln-1)·sup{∥xn-z∥2:z∈F(𝒮)}<∞ and {αn}⊂[a,b] such that κ<a<b<1. Then the sequences {xn} and {yn} converge strongly to w=PF(𝒮)(x0).Proof.
It is concluded from Corollary3.3 immediately, by putting Φ=0.


## 4. Applications

We introduce the equilibrium problem to the optimization problem:(4.1)min⁡x∈Cζ(x),
where C is a nonempty closed convex subset of a real Hilbert space H and ζ:C→ℝ∪{+∞} is a proper convex and lower semicontinuous. We denote by Argmin(ζ) the set of solutions of problem (4.1).  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 give the interesting result as the following theorem.Theorem 4.1.
LetC be a nonempty closed convex subset of a real Hilbert space H, and let ζ:C→ℝ∪{+∞} be a proper lower semicontinuous and convex function  with either (B2) or (B4) holds. Let 𝒮={Tn:n≥0} be an asymptotically κ-SPC semigroup on C for some κ∈[0,1) and a bounded sequence {Ln}⊂[1,∞) such that limn→∞Ln=1. Assume that Ω:=F(𝒮)∩Argmin(ζ) is a nonempty bounded subset of C. For x0=x∈C chosen arbitrarily, suppose that {xn},{yn} and {un} are generated iteratively by
(4.2)un∈Csuchthatζ(y)-ζ(un)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=αnun+(1-αn)Tnun,Cn+1={z∈Cn∩Qn:‖yn-z‖2≤‖xn-z‖2+(1-αn)(θn+(κ-αn)‖un-Tnun‖2)},Qn+1={z∈Cn∩Qn:〈xn-z,x0-xn〉≥0},C0=Q0=C,xn+1=PCn+1∩Qn+1(x0),∀n∈N∪{0},
where θn=(Ln-1)·sup{∥xn-z∥2:z∈Ω}<∞ satisfying the following conditions:  (C1)
{αn}⊂[a,b] such that κ<a<b<1,(C2)
{rn}⊂[r,∞) for some r>0,(C3)
∑n=0∞|rn+1-rn|<∞.Then the sequences {xn},{yn}, and {un} converge strongly to w=PΩ(x0).Proof.
It is concluded from Corollary3.3 immediately, by defining Φ(x,y)=ζ(y)-ζ(x) for all x,y∈C.



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*Source: 529407-2011-10-20.xml*