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International Scholarly Research NetworkISRN Mathematical AnalysisVolume 2011, Article ID 529407, 17 pagesdoi:10.5402/2011/529407

Research ArticleShrinking Projection Method of Common FixedPoint Problems for Discrete Asymptotically StrictlyPseudocontractive Semigroups and MixedEquilibrium Problems in Hilbert Spaces

Pattanapong Tianchai

Faculty of Science, Maejo University, Chiangmai 50290, Thailand

Correspondence should be addressed to Pattanapong Tianchai, [email protected]

Received 13 July 2011; Accepted 18 August 2011

Academic Editor: A. Levy

Copyright q 2011 Pattanapong Tianchai. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

This paper is concerned with a common element of the set of common fixed point for adiscrete asymptotically strictly pseudocontractive semigroup and the set of solutions of the mixedequilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets isobtained by a general iterative scheme based on the shrinking projection method which extendsand improves the corresponding ones due to Kim [Proceedings of the Asian Conference onNonlinear Analysis and Optimization (Matsue, Japan, 2008), 139–162].

1. Introduction

Throughout this paper, we always assume that C is a nonempty closed convex subset of areal Hilbert spaceH with inner product and norm denoted by 〈·, ·〉 and ‖ · ‖, respectively. Thedomain of the function ϕ : C → R ∪ {+∞} is the set

domϕ ={x ∈ C : ϕ(x) < +∞}

. (1.1)

Let ϕ : C → R ∪ {+∞} be a proper extended real-valued function, and let Φ be a bifunctionfrom C × C into R such that C ∩ domϕ/= ∅, where R is the set of real numbers. The so-calledmixed equilibrium problem is to find x ∈ C such that

Φ(x, y

)+ ϕ

(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.2)

2 ISRN Mathematical Analysis

The set of solutions of problem (1.2) is denoted by MEP(Φ, ϕ); that is,

MEP(Φ, ϕ

)={x ∈ C : Φ

(x, y

)+ ϕ

(y) − ϕ(x) ≥, ∀y ∈ C

}. (1.3)

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) If ϕ = 0, then problem (1.2) is reduced to find x ∈ C such that

Φ(x, y

) ≥ 0, ∀y ∈ C. (1.4)

We denote by EP(Φ) the set of solutions of equilibrium problem, problem (1.4)whichwas studied by Blum and Oettli [1].

(ii) If Φ(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

⟨Bx, y − x

⟩ ≥ 0, ∀y ∈ C. (1.5)

We denote by V I(C,B) the set of solutions of variational inequality problem, problem(1.5) which was studied by Hartman and Stampacchia [2].

(iii) If Φ = 0, then problem (1.2) is reduced to find x ∈ C such that

ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C. (1.6)

We denote by Argmin(ϕ) the set of solutions of minimize problem.

Recall that PC is the metric projection of H onto C; that is, for each x ∈ H, there existsthe unique point in PCx ∈ C such that ‖x − PCx‖ = miny∈C‖x − y‖. A mapping T : C → Cis called nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ C, and a mapping f : C → C iscalled a contraction if there exists a constant α ∈ (0, 1) such that ‖f(x) − f(y)‖ ≤ α‖x − y‖ forall x, y ∈ C. A point x ∈ C is a fixed point of T provided Tx = x. We denote by F(T) the set offixed points of T ; that is, F(T) = {x ∈ C : Tx = x}. If C is a nonempty bounded closed convexsubset 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 equation Tx = x. Themost well-known method is perhaps the Picard successive iteration method when T is acontraction. Picard’s method generates a sequence {xn} successively as xn+1 = Txn for alln ∈ N with x1 = x chosen arbitrarily, and this sequence converges in norm to the uniquefixed point of T . However, if T is not a contraction (for instance, if T is a nonexpansive), thenPicard’s successive iteration fails, in general, to converge. Instead, Mann’s iteration methodfor a nonexpansive mapping T (see [4]) prevails and generates a sequence {xn} recursivelyby

xn+1 = αnxn + (1 − αn)Txn, ∀n ∈ N, (1.7)

where x1 = x ∈ C chosen arbitrarily and the sequence {αn} lies in the interval [0, 1]. Recallthat a mapping T : C → C is said to be as follows.

ISRN Mathematical Analysis 3

(i) κ-strictly pseudocontractive (see [5]) if there exists a constant κ ∈ [0, 1) such that

∥∥Tx − Ty

∥∥2 ≤ ∥

∥x − y∥∥2 + κ

∥∥(I − T)x − (I − T)y

∥∥2, ∀x, y ∈ C, (1.8)

in brief, we use κ-SPC to denote the κ-strictly pseudocontractive, it is obvious thatT is a nonexpansive if and only if T is a 0-SPC.

(ii) Asymptotically κ-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

∥∥Tnx − Tny

∥∥2 ≤ (

1 + γn)∥∥x − y

∥∥2 + κ

∥∥(I − Tn)x − (I − Tn)y

∥∥2

, ∀x, y ∈ C, (1.9)

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 asequence {kn} ⊂ [1,∞) with limn→∞kn = 1 such that

∥∥Tnx − Tny∥∥ ≤ kn

∥∥x − y∥∥, ∀x, y ∈ C, (1.10)

for all n ∈ N, it is known that the class of κ-SPC mappings, and the classes ofasymptotically 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 byReich [10] for a nonexpansive mapping T on C, which asserts the weak convergence ofthe sequence {xn} generated by (1.7) in a uniformly convex Banach space with a Frechetdifferentiable norm under the control condition

∑∞n=1 αn(1 − αn) = ∞. Recently, Marino and

Xu [11] devoloped and extended Reich’s result to SPCmapping inHilbert space setting. Moreprecisely, they proved the weak convergence of the Mann’s iteration process (1.7) for a κ-SPCmapping T on C, and, subsequently, this result was improved and carried over the class ofasymptotically 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 byNakajo and Takahashi[14], they modified the Mann’s iteration method (1.7) which is to find a fixed point of anonexpansive mapping by a hybrid method, which called the shrinking projection method(or the CQ method) as the following theorem.

Theorem NT. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be anonexpansive mapping of C into itself such that F(T)/= ∅. Suppose that x1 = x ∈ C chosen arbitrarily,and let {xn} be the sequence defined by

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,

(1.11)

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 ofa κ-SPC mapping as the following theorem.

Theorem MX. Let C be a closed convex subset of a Hilbert space H, and, T : C → C be a κ-SPCmapping for some 0 ≤ κ < 1. Assume that F(T)/= ∅. Suppose that x1 = x ∈ C chosen arbitrarily, andlet {xn} be the sequence defined by

yn = αnxn + (1 − αn)Txn,

Cn ={z ∈ C :

∥∥yn − z

∥∥2 ≤ ‖xn − z‖2 + (1 − αn)(κ − αn)‖xn − Txn‖2

},


(1.12)

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. Let C be a closed convex subset of a Hilbert space H, and, T : C → C be an as-ymptotically κ-SPC mapping for some 0 ≤ κ < 1 and a bounded sequence {γn} ⊂ [0,∞) such thatlimn→∞γn = 0. Assume that F(T) is a nonempty bounded subset of C. Suppose that x1 = x ∈ Cchosen arbitrarily, and let {xn} be the sequence defined by

yn = αnxn + (1 − αn)Tnxn,

Cn ={z ∈ C :

∥∥yn − z∥∥2 ≤ ‖xn − z‖2 + (κ − αn(1 − αn))‖xn − Tnxn‖2 + θn

},


(1.13)

where θn = Δ2n(1 − αn)γn → 0 as n → ∞, Δn = sup{‖xn − z‖ : z ∈ F(T)} < ∞, and 0 ≤ αn < 1

such that lim supn→∞αn < 1 − κ. Then the sequence {xn} converges strongly to PF(T)(x1).

Recall that a discrete family S = {Tn : n ≥ 0} of self-mappings of C is said to be aLipschitzian 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

∥∥Tnx − Tny∥∥ ≤ Ln

∥∥x − y∥∥, ∀x, y ∈ C, ∀n ≥ 0. (1.14)

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 iflim supn→∞Ln ≤ 1, respectively. We use F(S) to denote the common fixed point set of thesemigroup S; that is, F(S) = {x ∈ C : Tnx = x, ∀n ≥ 0}.


Very recently, Kim [16] introduced asymptotically κ-SPC semigroup, a discrete familyS = {Tn : n ≥ 0} of self-mappings of C which is said to be asymptotically κ-SPC semigroupon C if, in addition to (1), (2) and the following condition (3′) are satisfied.

( 3′ ) There exists a constant κ ∈ [0, 1) and a bounded sequence {Ln} of nonnegative realnumbers with lim supn→∞Ln ≤ 1 such that

∥∥Tnx − Tny

∥∥2 ≤ Ln

∥∥x − y

∥∥2 + κ

∥∥(I − Tn)x − (I − Tn)y

∥∥2

, ∀x, y ∈ C, ∀n ≥ 0. (1.15)

Note that for both discrete asymptotically nonexpansive semigroups and discrete as-ymptotically κ-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 forall n ≥ 0 with L′

n = max{supm≥nLm, 1}. Therefore, for a single asymptotically κ-SPC mappingT : C → C note that (1.15) immediately reduces to (1.9) by taking Tn ≡ Tn and γn = Ln − 1such 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 as-ymptotically κ-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 (seeTheorem 3.3 in [16]) holds in the sense that if {xn} is a sequence in C such that xn ⇀ zand lim supm→∞lim supn→∞‖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 asymptoticallyκ-SPC semigroup and a bounded sequence {Ln} ⊂ [1,∞) such that limn→∞Ln = 1 as follows:

x0 = x ∈ C chosen arbitrarily,


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},

(1.16)

where θn = (Ln − 1) · sup{‖xn − z‖2 : z ∈ F(S)} < ∞. He proved that under the parameter0 ≤ 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 ageneral iterative scheme (3.1) below to find a common element of the set of common fixedpoint for a discrete asymptotically κ-SPC semigroup and the set of solutions of the mixedequilibrium problems in Hilbert spaces. The strong convergence theorem for the above twosets is obtained based on the shrinking projection method which extend and improve the cor-responding ones due to Kim [16].

2. Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space H. For a sequence {xn} inH, we denote the strong convergence and the weak convergence of {xn} to x ∈ H by xn → xand 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 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 each x, y, z ∈ C,

limt↓0

Φ(tz + (1 − t)x, y

) ≤ Φ(x, y

). (2.1)

(A4) For each x ∈ C, y �→ Φ(x, y) is convex and lower semicontinuous.

(A5) For each y ∈ C, x �→ Φ(x, y) is weakly upper semicontinuous.

(B1) For each x ∈ C and r > 0, there exists a bounded subset Dx ⊂ C and yx ∈ C suchthat for any z ∈ C \Dx,

Φ(z, yx

)+ ϕ

(yx

) − ϕ(z) +1r

⟨yx − z, z − x

⟩< 0. (2.2)

(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

∥∥λx + (1 − λ)y∥∥2 = λ‖x‖2 + (1 − λ)

∥∥y∥∥2 − λ(1 − λ)

∥∥x − y∥∥2

. (2.3)

Lemma 2.2 (see [3]). Let C be a nonempty closed convex subset of a Hilbert space H. Then thefollowing inequality holds:

⟨x − PCx, PCx − y

⟩ ≥ 0, ∀x ∈ H, y ∈ C. (2.4)

Lemma 2.3 (see [18]). Let C be a nonempty closed convex subset of a Hilbert spaceH,Φ : C×C →R satisfying the conditions (A1)–(A5), and let ϕ : C → R∪{+∞} be a proper lower semicontinuousand convex function. Assume that either (B1) or (B2) holds. For r > 0, define a mapping Sr : C → Cas follows:

Sr(x) ={z ∈ C : Φ

(z, y

)+ ϕ

(y) − ϕ(z) +

1r

⟨y − z, z − x

⟩ ≥ 0, ∀y ∈ C

}, (2.5)

for all x ∈ C. Then, the following statement hold.

(1) For each x ∈ C, Sr(x)/= ∅.(2) Sr is single-valued.

(3) Sr is firmly nonexpansive; that is, for any x, y ∈ C,


∥∥Srx − Sry

∥∥2 ≤ ⟨

Srx − Sry, x − y⟩. (2.6)

(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; thatis, for a sequence {xn} ⊂ H with xn ⇀ x and ‖xn‖ → ‖x‖ then xn → x.

Lemma 2.5 (see [16]). Let C be a nonempty closed convex subset of a Hilbert space H, and letS = {Tn : n ≥ 0} be an asymptotically κ-strictly pseudocontractive semigroup on C. Let {xn} be asequence inC such that limn→∞‖xn−xn+1‖ = 0 and limn→∞‖xn−Tnxn‖ = 0. Thenωw(xn) ⊂ F(S).

3. Main Results

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, let Φ be abifunction 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 nonemptybounded subset of C. For x0 = x ∈ C chosen arbitrarily, suppose that {xn}, {yn}, and {un} aregenerated iteratively by

un ∈ C such that Φ(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},

(3.1)

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. Pick p ∈ Ω. Therefore, by (3.1) and the definition of Srn in Lemma 2.3, we have

un = Srnxn ∈ domϕ, (3.2)


and, by F(S) = ⋂∞n=0 F(Tn) and Lemma 2.3(4), we have

Tnp = p = Srnp. (3.3)

By (3.2), (3.3), and the nonexpansiveness of Srn , we have

∥∥un − p

∥∥ =

∥∥Srnxn − Srnp

∥∥ ≤ ∥

∥xn − p∥∥. (3.4)

By (3.3), (3.4), Lemma 2.1, and the asymptotically κ-SPC semigroupness of S, we have

∥∥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)


),

(3.5)

where θn := (Ln − 1) · sup{‖xn − z‖2 : z ∈ Ω} for all n ∈ N ∪ {0}.Firstly, we prove that Cn ∩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 isCn ∩Qn is closed for all n ∈ N ∪ {0}. Let εn = (1 − αn)(θn + (k − αn)‖un − Tnun‖2) since, for anyz ∈ C, ‖yn − z‖2 ≤ ‖xn − z‖2 + εn is equivalent to

∥∥yn − xn

∥∥2 + 2⟨yn − xn, xn − z

⟩ − εn ≤ 0, (3.6)

for all n ∈ N ∪ {0}. Therefore, for any z1, z2 ∈ Cn+1 ∩Qn+1 ⊂ Cn ∩Qn and ε ∈ (0, 1), we have

∥∥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,(3.7)

for all n ∈ N ∪ {0}, and we have

〈xn − (εz1 + (1 − ε)z2), x0 − xn〉 = ε〈xn − z1, x0 − xn〉 + (1 − ε)〈xn − z2, x0 − xn〉

≥ 0,(3.8)


for all n ∈ N ∪ {0}. Since C0 ∩Q0 is convex and by putting n = 0 in (3.6), (3.7), and (3.8), wehave 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, bymathematical induction, we have that Cn∩Qn is convex for all n ≥ 1; that is, Cn∩Qn is convexfor all n ∈ N ∪ {0}. Hence, we obtain that Cn ∩Qn is closed and convex for all n ∈ N ∪ {0}.

Next, we prove that Ω ⊂ 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 sop ∈ C1 ∩Q1. Hence, we have Ω ⊂ C1 ∩Q1. Since C1 ∩Q1 is a nonempty closed convex subsetof C, there exists a unique element x1 ∈ C1 ∩ Q1 such that x1 = PC1∩Q1(x0). Suppose thatxk ∈ 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

〈xk − z, x0 − xk〉 ≥ 0, (3.9)

for all z ∈ Ck ∩Qk. Thus, by (3.1), we have p ∈ Qk+1, and so p ∈ Ck+1 ∩Qk+1. Hence, we haveΩ ⊂ Ck+1 ∩ Qk+1. Since Ck+1 ∩ Qk+1 is a nonempty closed convex subset of C, there exists aunique element xk+1 ∈ Ck+1 ∩Qk+1 such that xk+1 = PCk+1∩Qk+1(x0). Therefore, by mathematicalinduction, we obtainΩ ⊂ Cn ∩Qn for all n ≥ 1, and soΩ ⊂ Cn ∩Qn for all n ∈ N ∪ {0}, and wecan define xn+1 = PCn+1∩Qn+1(x0) for all n ∈ N ∪ {0}. Hence, we obtain that the iteration (3.1) iswell defined.

Next, we prove that {xn} is bounded. Since xn = PCn∩Qn(x0) for all n ∈ N∪{0}, we have

‖xn − x0‖ ≤ ‖z − x0‖, (3.10)

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; thereexists the limit of ‖xn − x0‖; that is,

limn→∞

‖xn − x0‖ = m, (3.11)

for some m ≥ 0. Since xn+1 ∈ Qn+1; therefore, by (3.1), we have

〈xn − xn+1, x0 − xn〉 ≥ 0. (3.12)

It follows by (3.12) that

‖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.

(3.13)


Therefore, by (3.11), we obtain

‖xn − xn+1‖ −→ 0 as n −→ ∞. (3.14)

Indeed, from (3.1), we have

Φ(un, y

)+ ϕ

(y) − ϕ(un) +

1rn


⟩ ≥ 0, ∀y ∈ C, (3.15)

Φ(un+1, y

)+ ϕ

(y) − ϕ(un+1) +

1rn+1

⟨y − un+1, un+1 − xn+1

⟩ ≥ 0, ∀y ∈ C. (3.16)

Substituting y = un+1 into (3.15) and y = un into (3.16), we have

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) +1

rn+1〈un − un+1, un+1 − xn+1〉 ≥ 0.

(3.17)

Therefore, by the condition (A2), we get

0 ≤ Φ(un, un+1) + Φ(un+1, un) +⟨un+1 − un,

un − xn

rn− un+1 − xn+1

rn+1

⟩

≤⟨un+1 − un,

un − xn

rn− un+1 − xn+1

rn+1

⟩.

(3.18)

It follows that

0 ≤⟨un+1 − un, (un − un+1) + (un+1 − xn) − rn

rn+1(un+1 − xn+1)

⟩

= 〈un+1 − un, un − un+1〉 +⟨un+1 − un, (un+1 − xn+1) + (xn+1 − xn) − rn

rn+1(un+1 − xn+1)

⟩.

(3.19)

Thus, we have

‖un+1 − un‖2 ≤⟨un+1 − un, (xn+1 − xn) +

(1 − rn

rn+1

)(un+1 − xn+1)

⟩

≤ ‖un+1 − un‖(‖xn+1 − xn‖ +

∣∣∣∣1 −rnrn+1

∣∣∣∣‖un+1 − xn+1‖).

(3.20)


It follows by the condition (C2) that

‖un+1 − un‖ ≤ ‖xn+1 − xn‖ + |rn+1 − rn|rn+1

‖un+1 − xn+1‖

≤ ‖xn+1 − xn‖ +(L

r

)|rn+1 − rn|,

(3.21)

where L = supn≥0‖un − xn‖ < ∞. Therefore, by the condition (C3) and (3.14), we obtain

‖un − un+1‖ −→ 0 as n −→ ∞. (3.22)

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

∥∥yn − xn+1∥∥2 ≤ ‖xn − xn+1‖2 + (1 − αn)


). (3.23)

It follows that

(1 − αn)(αn − κ)‖un − Tnun‖2 ≤ ‖xn − xn+1‖2 + (1 − αn)θn −∥∥yn − xn+1

∥∥2

≤ ‖xn − xn+1‖2 + θn.(3.24)

Therefore, by the condition (C1), (3.14), and limn→∞θn = 0, we obtain

‖un − Tnun‖ −→ 0 as n −→ ∞. (3.25)

From (3.23) and the condition (C1), we have

∥∥yn − xn+1∥∥2 ≤ ‖xn − xn+1‖2 + θn. (3.26)

Therefore,

∥∥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.

(3.27)

Hence, by (3.14) and limn→∞θn = 0, we obtain

∥∥yn − xn

∥∥ −→ 0 as n −→ ∞. (3.28)


By (3.2), (3.3), and the firmly nonexpansiveness of Srn , we have

∥∥un − p

∥∥2 ≤ ⟨

Srnxn − Srnp, xn − p⟩=⟨un − p, xn − p

⟩

=12

(∥∥un − p

∥∥2 +

∥∥xn − p

∥∥2 − ‖un − xn‖2

).

(3.29)

It follows that

∥∥un − p

∥∥2 ≤ ∥

∥xn − p∥∥2 − ‖un − xn‖2. (3.30)

Therefore, by the condition (C1) and (3.5), we have

∥∥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

).

(3.31)

It follows that

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∥∥).

(3.32)

Hence, by (3.28) and limn→∞θn = 0, we obtain

‖un − xn‖ −→ 0 as n −→ ∞. (3.33)

Since {un} is bounded, there exists a subsequence {uni} of {un} which converges weakly tow. 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), wehave

0 ≤ Φ(un, y

)+ ϕ

(y) − ϕ(un) +

1rn


⟩, ∀y ∈ C. (3.34)

It follows by the condition (A2) that

Φ(y, un

) ≤ Φ(y, un

)+ Φ

(un, y

)+ ϕ

(y) − ϕ(un) +

1rn


⟩, ∀y ∈ C

≤ ϕ(y) − ϕ(un) +

1rn


⟩, ∀y ∈ C.

(3.35)


Hence,

ϕ(y) − ϕ(uni) +

⟨y − uni ,

uni − xni

rni

⟩≥ Φ

(y, uni

), ∀y ∈ C. (3.36)

Therefore, from (3.33) and by uni ⇀ w as i → ∞, we obtain

Φ(y,w

)+ ϕ(w) − ϕ

(y) ≤ 0, ∀y ∈ C. (3.37)

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

Φ(yt,w

)+ ϕ(w) − ϕ

(yt

) ≤ 0. (3.38)

By (3.38), the conditions (A1) and (A4), and the convexity of ϕ, we have

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

)).

(3.39)

It follows that

Φ(yt, y

)+ ϕ

(y) − ϕ

(yt

) ≥ 0. (3.40)

Therefore, by the condition (A3) and the weakly lower semicontinuity of ϕ, we haveΦ(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.

Since Ω is a nonempty closed convex subset of C, there exists a unique w ∈ Ω suchthat 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

‖x0 −w‖ ≤ ‖x0 −w‖. (3.41)

Since w ∈ Ω ⊂ Cn ∩Qn; therefore, by (3.10), we have

‖x0 − xn‖ ≤ ‖x0 −w‖. (3.42)

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

‖x0 −w‖ ≤ ‖x0 −w‖ ≤ lim infi→∞

‖x0 − xni‖ ≤ lim supi→∞

‖x0 − xni‖ ≤ ‖x0 −w‖. (3.43)


It follows that

‖x0 −w‖ = limi→∞

‖x0 − xni‖ = ‖x0 −w‖. (3.44)

Since xni ⇀ w as i → ∞; therefore, we have

(x0 − xni) ⇀ (x0 −w) as i −→ ∞. (3.45)

Hence, from (3.44), (3.45), the Kadec-Klee property, and the uniqueness of w = PΩ(x0), weobtain

xni −→ w = w as i −→ ∞. (3.46)

It follows that {xn} converges strongly to w and so are {yn} and {un}. This completes theproof.

Remark 3.2. The iteration (3.1) is the difference with the iterative scheme of Kim [16] as thefollowing.

(1) The sequence {xn} is a projection sequence of x0 onto Cn ∩ Qn for all n ∈ N ∪ {0}such that

C0 ∩Q0 ⊃ C1 ∩Q1 ⊃ · · · ⊃ Cn ∩Qn ⊃ · · · ⊃ Ω. (3.47)

(2) A solving of a common element of the set of common fixed point for a discreteasymptotically κ-SPC semigroup and the set of solutions of the mixed equilibriumproblems 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. Let C be a nonempty closed convex subset of a real Hilbert space H, and let Φ be abifunction from C × C into R satisfying the conditions (A1)–(A5) with either (B2) or (B3) holds. LetS = {Tn : n ≥ 0} be an asymptotically κ-SPC semigroup on C for some κ ∈ [0, 1) and a boundedsequence {Ln} ⊂ [1,∞) such that limn→∞Ln = 1. Assume that Ω := F(S) ∩ EP(Φ) is a nonemptybounded subset of C. For x0 = x ∈ C chosen arbitrarily, suppose that {xn}, {yn}, and {un} aregenerated iteratively by

un ∈ C such that Φ(un, y

)+

1rn


⟩ ≥ 0, ∀y ∈ C,


Cn+1 ={z ∈ Cn ∩Qn :

∥∥yn − z∥∥2 ≤ ‖xn − z‖2 + (1 − αn)


)},

Qn+1 = {z ∈ Cn ∩Qn : 〈xn − z, x0 − xn〉 ≥ 0},


C0 = Q0 = C,

xn+1 = PCn+1∩Qn+1(x0), ∀n ∈ N ∪ {0},(3.48)



(C2) {rn} ⊂ [r,∞) for some r > 0,

(C3)∑∞

n=0 |rn+1 − rn| < ∞.


Proof. It is concluded from Theorem 3.1 immediately, by putting ϕ = 0.

Corollary 3.4. Let C 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. Forx0 = x ∈ C chosen arbitrarily, suppose that {xn} and {yn} are generated iteratively by


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},

(3.49)

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 Corollary 3.3 immediately, by putting Φ = 0.

4. Applications

We introduce the equilibrium problem to the optimization problem:

minx∈C

ζ(x), (4.1)

where C is a nonempty closed convex subset of a real Hilbert spaceH and ζ : C → R∪ {+∞}is a proper convex and lower semicontinuous. We denote by Argmin(ζ) the set of solutions ofproblem (4.1). We define the condition (B4) as the condition (B3) such thatΦ : C×C → R is abifunction defined byΦ(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. Let C 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. LetS = {Tn : n ≥ 0} be an asymptotically κ-SPC semigroup on C for some κ ∈ [0, 1) and a boundedsequence {Ln} ⊂ [1,∞) such that limn→∞Ln = 1. Assume that Ω := F(S) ∩ Argmin(ζ) is anonempty bounded subset of C. For x0 = x ∈ C chosen arbitrarily, suppose that {xn}, {yn} and {un}are generated iteratively by

un ∈ C such that ζ(y) − ζ(un) +

1rn


⟩ ≥ 0, ∀y ∈ C,


Cn+1 ={z ∈ Cn ∩Qn :

∥∥yn − z

∥∥2 ≤ ‖xn − z‖2 + (1 − αn)


)},

Qn+1 = {z ∈ Cn ∩Qn : 〈xn − z, x0 − xn〉 ≥ 0},C0 = Q0 = C,

xn+1 = PCn+1∩Qn+1(x0), ∀n ∈ N ∪ {0},

(4.2)



(C2) {rn} ⊂ [r,∞) for some r > 0,

(C3)∑∞

n=0 |rn+1 − rn| < ∞.


Proof. It is concluded from Corollary 3.3 immediately, by definingΦ(x, y) = ζ(y)− ζ(x) for allx, y ∈ C.

References

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