Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
RESEARCH
Open Access
Superimposed algorithms for the split equilibrium problems and fixed point problems Li-Jun Zhu1 , Zhangsong Yao2 , Yeong-Cheng Liou3,4* and Yonghong Yao5 * Correspondence:
[email protected] 3 Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan 4 Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 807, Taiwan Full list of author information is available at the end of the article
Abstract The split common problems for finding the equilibrium points and fixed points have been studied. A parallel superimposed algorithm is introduced to solve this split common problem. Strong convergence theorems are shown with some analysis techniques. MSC: 49J30; 47H09; 65K10 Keywords: split equilibrium problem; split fixed point; nonexpansive mapping; parallel superimposed algorithms
1 Introduction In the present article, our main purpose is to study the split problem. First, we recall some relevant background in the literature. Problem 1: the split feasibility problem Let C and Q be two nonempty closed convex subsets of Hilbert spaces H and H , respectively, and let A : H → H be a bounded linear operator. The problem of finding a point x∗ such that x∗ ∈ C
and
Ax∗ ∈ Q
(.)
is called the split feasibility; it was first introduced by Censor and Elfving [] in finite dimensional Hilbert spaces. Such problems arise in the field of intensity-modulated radiation therapy when one attempts to describe physical dose constraints and equivalent uniform dose constraints within a single model. When C ∈ RN and Q ∈ RM are a single pair of sets, Censor and Elfving [] introduced the simultaneous multi-projections algorithm: – n+ xn+ = A– λ I + λ AA∗ λ v + λ AA∗ vn+ ,
n ≥ ,
(.)
f
= PCi i (bn ) (i = , ), and bn is the solution of the equawhere λ > , λ > , λ + λ = , vn+ i tion i=
λi ∇fi (bn ) =
∇fi vni .
i=
©2014 Zhu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 2 of 15
Note that the simultaneous multi-projections algorithm (.) involve a matrix inversion A– at each iterative step. This is very time-consuming, particularly if the dimensions are large. In order to solve this problem, Byrne [] derived a new algorithm, called the CQalgorithm: xn+ = PC xn – τ A∗ (I – PQ )Axn ,
n ≥ ,
where τ ∈ (, L ) with L being the largest eigenvalue of the matrix A∗ A, I is the unit matrix or operator, and PC and PQ denote the orthogonal projections onto C and Q, respectively. The CQ-algorithm and its variant forms have now been studied for the split feasibility problem; see, for instance [–].
Problem 2: the split common fixed point problem If every closed convex subset of a Hilbert space is the fixed point set of its associating projection, then the split feasibility problem becomes a special case of the split common fixed point problem of finding a point x∗ with the property: x∗ ∈ Fix(U) and Ax∗ ∈ Fix(T).
(.)
This problem was first introduced by Censor and Segal [] who invented an algorithm which generates a sequence {xn } according to the iterative procedure: xn+ = U xn – γ A∗ (I – T)Axn ,
n ∈ N.
(.)
Moudafi [] extended (.) to the following relaxed algorithm: xn+ = Uαn xn + γ A∗ (Tβ – I)Axn ,
n ∈ N,
where β ∈ (, ), αn ∈ (, ) are relaxation parameters. Consequently, Wang and Xu [] considered a general cyclic algorithm. Very recently, the split problem has also been extended to solve other problems, such as the split monotone variational inclusions and the split variational inequalities, please refer to [, –] and [–].
Problem 3: the equilibrium problem Consider the following equilibrium problem: Finding x∗ ∈ C such that F x∗ , x ≥ ,
∀x ∈ C,
(.)
where F : C × C → R is a bifunction. We will denote by EP(F) the set of solutions of (.). The equilibrium problems, in its various forms, found application in optimization problems, fixed point problems, and convex minimization problems; in other words, equilibrium problems are a unified model for problems arising in physics, engineering, economics, and so on (see [–]). Motivated by the split common fixed point problem and the equilibrium problem, He and Du [] presented the following split equilibrium problem and fixed point problem: Find a point x∗ ∈ Fix(T) ∩ EP(F) such that Ax∗ ∈ Fix(S) ∩ EP(G),
(.)
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 3 of 15
where Fix(S) and Fix(T) are the sets of fixed points of two nonlinear mappings S and T, respectively, EP(F) and EP(G) are the solution sets of two equilibrium problems with bifunctions F and G, respectively, and A is a bounded linear mapping. Denote the solution set of (.) by = x ∈ Fix(T) ∩ EP(F) : Ax ∈ Fix(S) ∩ EP(G) .
Special cases . If F = and G = , then (.) is reduced to the following split common fixed point problem, which has been considered by many authors, for example, [, , ] and []: Find a point x∗ ∈ Fix(T) such that Ax∗ ∈ Fix(S).
(.)
. If S = PQ and T = PC , then (.) is reduced to the split feasibility problem (.). . If S and T are all identity operators, then (.) is reduced to the split equilibrium problem which has been considered in []: Find a point x∗ ∈ EP(F) such that Ax∗ ∈ EP(G). Based on the work in this direction, in this paper we will develop new algorithms to solve the split equilibrium problem and the fixed point problem (.). We first introduce a parallel superimposed algorithm. Consequently, strong convergence theorems are shown with some analysis techniques.
2 Preliminaries Let H be a real Hilbert space with inner product ·, · and norm · , respectively. Let C be a nonempty closed convex subset of H. Definition . A mapping T : C → C is called nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ C. We will use Fix(T) to denote the set of fixed points of T, that is, Fix(T) = {x ∈ C : x = Tx}. Definition . A mapping f : C → C is called contractive if f (x) – f (y) ≤ ρ x – y for all x, y ∈ C and for some constant ρ ∈ (, ). In this case, we call f is a ρ-contraction. Definition . A linear bounded operator B : H → H is called strongly positive if there exists a constant γ > such that Bx, x ≥ γ x for all x, y ∈ H.
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 4 of 15
Definition . We call PC : H → C the metric projection if for each x ∈ H x – PC (x) = inf x – y : y ∈ C . It is well known that the metric projection PC : H → C is characterized by x – PC (x), y – PC (x) ≤ for all x ∈ H, y ∈ C. From this, we can deduce that PC is firmly nonexpansive, that is, PC (x) – PC (y) ≤ x – y, PC (x) – PC (y)
(.)
for all x, y ∈ H. Hence PC is also nonexpansive. It is well known that in a real Hilbert space H, the following two equalities hold: tx + ( – t)y = t x + ( – t) y – t( – t) x – y
(.)
for all x, y ∈ H and t ∈ [, ], and x + y = x + x, y + y
(.)
for all x, y ∈ H. It follows that x + y ≤ x + y, x + y
(.)
for all x, y ∈ H. Throughout this paper, we assume that a bifunction F : C ×C → R satisfies the following conditions: (H) F(x, x) = for all x ∈ C; (H) F is monotone, i.e., F(x, y) + F(y, x) ≤ for all x, y ∈ C; (H) for each x, y, z ∈ C, limt↓ F(tz + ( – t)x, y) ≤ F(x, y); (H) for each x ∈ C, y → F(x, y) is convex and lower semicontinuous. Lemma . ([]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C × C → R be a bifunction which satisfies conditions (H)-(H). Let r > and x ∈ C. Then there exists z ∈ C such that F(z, y) + y – z, z – x ≥ , r
∀y ∈ C.
Further, if UrF (x) = {z ∈ C : F(z, y) + r y – z, z – x ≥ , ∀y ∈ C}, then the following hold: (i) UrF is single-valued and UrF is firmly nonexpansive, i.e., for any x, y ∈ H, UrF x – UrF y ≤ UrF x – UrF y, x – y ; (ii) EP(F) is closed and convex and EP(F) = Fix(UrF ). Lemma . ([]) Let the mapping UrF be defined as in Lemma .. Then, for r, s > and x, y ∈ H, F U (x) – U F (y) ≤ x – y + |s – r| U F (y) – y. r s s s
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 5 of 15
Lemma . ([]) Let {xn } and {yn } be two bounded sequences in a Banach space X and let {βn } be a sequence in [, ] with < lim infn→∞ βn ≤ lim supn→∞ βn < . Suppose that xn+ = ( – βn )yn + βn xn for all n ≥ and lim sup yn+ – yn – xn+ – xn ≤ . n→∞
Then limn→∞ yn – xn = . Lemma . ([]) Let C be a closed convex subset of a real Hilbert space H and let S : C → C be a nonexpansive mapping. Then the mapping I – S is demiclosed. That is, if {xn } is a sequence in C such that xn → x∗ weakly and (I – S)xn → y strongly, then (I – S)x∗ = y. Lemma . ([]) Assume that {an } is a sequence of nonnegative real numbers such that an+ ≤ ( – γn )an + δn ,
n ∈ N,
where {γn } is a sequence in (, ) and {δn } is a sequence such that
∞ () n= γn = ∞;
() lim supn→∞ γδnn ≤ or ∞ n= |δn | < ∞. Then limn→∞ an = .
3 Main results In this section, we introduce our algorithm and prove our main results. Let H and H be two real Hilbert spaces and let C and D be two nonempty closed convex subsets of H and H , respectively. Let A : H → H be a bounded linear operator with its adjoint A∗ , B be a strongly positive bounded linear operator on H with coefficient γ > . Let f : C → C be a ρ-contraction and F : C ×C → R and G : D×D → R be two bifunctions satisfying the conditions (H)-(H). Let S : D → D and T : C → C be two nonexpansive mappings. Algorithm . Taking x ∈ H arbitrarily, we define a sequence {xn } by the following: xn+ = αn σ f (xn ) + βn xn + ( – βn )I – αn B TUλFn xn + δA∗ SUγGn – I Axn
(.)
for all n ∈ N, where {λn } and {γn } are two real number sequences in (, ∞), δ ∈ (, A ) and σ > are two constants and {αn } and {βn } are two real number sequences in (, ).
Theorem . Suppose = ∅ and suppose the following conditions hold:
(C): limn→∞ αn = and ∞ n= αn = ∞; (C): < lim infn→∞ βn ≤ lim supn→∞ βn < ; = ; (C): lim infn→∞ λn > and limn→∞ λλn+ n = ; (C): lim infn→∞ γn > and limn→∞ γγn+ n (C): σρ < γ .
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 6 of 15
Then the sequence {xn } generated by algorithm (.) converges strongly to p = Proj (σ f + I – B)p, which solves the following VI: (σ f – B)x, y – x ≤ ,
∀y ∈ .
(.)
Proof First, we know that the solution of (.) is unique. We denote the unique solution by p. That is, p = Proj (σ f + I – B)p. Then we have p ∈ Fix(T) ∩ EP(F) and Ap ∈ Fix(S) ∩ EP(G). Set zn = UγGn Axn , yn = xn + δA∗ (SUγGn – I)Axn and un = UλFn (xn + δA∗ (SUγGn – I)Axn ) for all n ∈ N. Then un = UλFn yn . From Lemma ., we know that UλFn and UγGn are firmly nonexpansive. By these facts, we have the following conclusions: zn – Ap = UγGn Axn – Ap ≤ Axn – Ap , un – p = UλFn yn – p ≤ yn – p
(.)
G SU Axn – Ap ≤ U G Axn – Ap γn γn ≤ Axn – Ap – UγGn Axn – Axn .
(.)
(.)
and
Applying Lemma ., we deduce un+ – un = UλFn+ yn+ – UλFn yn λn+ – λn un+ – yn+ ≤ yn+ – yn + λn+
(.)
and zn+ – zn = UγGn+ Axn+ – UγGn Axn γn+ – γn zn+ – Axn . ≤ Axn+ – Axn + γn+
(.)
From (.), we have xn+ – p = αn σ f (xn ) – Bp + βn (xn – p) + ( – βn )I – αn B (Tun – p) ≤ αn σ f (xn ) – f (p) + αn σ f (p) – Bp + βn xn – p + ( – βn – αn γ ) un – p .
(.)
Using (.), we get yn – p = xn – p + δA∗ (Szn – Axn ) = xn – p + δ A∗ (Szn – Axn ) + δ xn – p, A∗ (Szn – Axn ) .
(.)
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 7 of 15
Since A is a linear operator with its adjoint A∗ , we have xn – p, A∗ (Szn – Axn ) = A(xn – p), Szn – Axn = Axn – Ap + Szn – Axn – (Szn – Axn ), Szn – Axn = Szn – Ap, Szn – Axn – zn – Axn .
(.)
Again from (.), we obtain Szn – Ap, Szn – Axn =
Szn – Ap + Szn – Axn – Axn – Ap .
(.)
From (.), (.), and (.), we have xn – p, A∗ (Szn – Axn ) = Szn – Ap + Szn – Axn – Axn – Ap – Szn – Axn ≤ Axn – Ap – zn – Axn + Szn – Axn – Axn – Ap – Szn – Axn = – zn – Axn – Szn – Axn .
(.)
Substituting (.) into (.) to deduce
yn – p ≤ xn – p + δ A Szn – Axn + δ – zn – Axn – Szn – Axn = xn – p + δ A – δ Szn – Axn – δ zn – Axn ≤ xn – p . It follows that yn – p ≤ xn – p . Thus, from (.), we get xn+ – p ≤ αn σρ xn – p + αn σ f (p) – Bp + βn xn – p + ( – βn – αn γ ) xn – p = – (γ – σρ)αn xn – p + αn σ f (p) – Bp σ f (p) – Bp . ≤ max xn – p , γ – σρ The boundedness of the sequence {xn } follows. Next, we estimate un+ – un . Observe that yn+ – yn = xn+ – xn + δ A∗ (Szn+ – Axn+ ) – A∗ (Szn – Axn )
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 8 of 15
= xn+ – xn + δ A∗ (Szn+ – Axn+ ) – A∗ (Szn – Axn ) + δ xn+ – xn , A∗ (Szn+ – Axn+ ) – (Szn – Axn ) ≤ xn+ – xn + δ A Szn+ – Szn – (Axn+ – Axn ) + δ Axn+ – Axn , Szn+ – Szn – (Axn+ – Axn ) = xn+ – xn + δ A Szn+ – Szn – (Axn+ – Axn ) + δ Szn+ – Szn , Szn+ – Szn – (Axn+ – Axn ) – δ Szn+ – Szn – (Axn+ – Axn ) = xn+ – xn + δ A Szn+ – Szn – (Axn+ – Axn )
+ δ Szn+ – Szn + Szn+ – Szn – (Axn+ – Axn ) – Axn+ – Axn – δ Szn+ – Szn – (Axn+ – Axn ) = xn+ – xn + δ A – δ Szn+ – Szn – (Axn+ – Axn ) + δ Szn+ – Szn – Axn+ – Axn ≤ xn+ – xn + δ A – δ Szn+ – Szn – (Axn+ – Axn ) + δ zn+ – zn – Axn+ – Axn .
(.)
Since δ ∈ (, A ), we derive by virtue of (.) and (.) that
yn+ – yn ≤ xn+ – xn γn+ – γn zn+ – zn + Axn+ – Axn . + δ γn+
(.)
According to (.) and (.), we have un+ – un = UλFn+ yn+ – UλFn yn
λn+ – λn ≤ yn+ – yn + un+ – yn+ λn+ λn+ – λn yn+ – yn un+ – yn+ ≤ yn+ – yn + λn+
λn+ – λn un+ – yn+ + λn+ λn+ – λn yn+ – yn un+ – yn+ ≤ xn+ – xn + λn+
λn+ – λn un+ – yn+ + λn+ γn+ – γn zn+ – zn + Axn+ – Axn + δ γn+ λn+ – λn γn+ – γn + δ M, ≤ xn+ – xn + λn+ γn+
(.)
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 9 of 15
where M > is a constant such that λn+ – λn un+ – yn+ sup yn+ – yn un+ – yn+ + λn+ n + δ zn+ – zn + Axn+ – Axn ≤ M.
Therefore, λn+ – λn γn+ – γn + δ M. un+ – un ≤ xn+ – xn + γ λ n+
(.)
n+
From (.), we write xn+ = βn xn + ( – βn )wn where wn = Tun + n ∈ N. Then we have
αn (σ f (xn ) – BTun ) –βn
for all
wn+ – wn = Tun+ – Tun +
αn+ αn σ f (xn+ ) – BTun+ – σ f (xn ) – BTun – βn+ – βn αn+ σ f (xn+ ) – BTun+ + αn σ f (xn ) – BTun ≤ Tun+ – Tun + – βn+ – βn αn+ α n σ f (xn+ ) – BTun+ + σ f (xn ) – BTun ≤ un+ – un + – βn+ – βn λn+ – λn γn+ – γn + δ M ≤ xn+ – xn + γ λ n+ n+ αn+ σ f (xn+ ) – BTun+ + αn σ f (xn ) – BTun . + – βn+ – βn
Noting the condition (C) and the boundedness of the sequences {un+ }, {yn+ }, {zn+ }, {Axn }, {f (xn )}, and {BTun }, we have lim sup wn+ – wn – xn+ – xn ≤ . n→∞
By Lemma ., we deduce lim xn – wn = .
n→∞
Hence, lim xn+ – xn = lim ( – βn ) xn – wn = .
n→∞
n→∞
(.)
Since xn+ – xn = αn (σ f (xn ) – BTun ) + ( – βn )(Tun – xn ), we obtain Tun – xn ≤
αn σ f (xn ) – BTun + xn+ – xn . βn
Thus, lim xn – Tun = .
n→∞
(.)
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 10 of 15
Using the firmly nonexpansiveness of UλFn , we have un – p = UλFn yn – p ≤ yn – p – UλFn yn – yn = yn – p – un – yn = yn – p – un – xn – δA∗ SUγGn – I Axn = yn – p – un – xn – δ A∗ SUγGn – I Axn + δ un – xn , A∗ SUγGn – I Axn .
(.)
Applying (.) to (.) to deduce xn+ – p = αn σ f (xn ) – Bp + βn (xn – Tun ) + (I – αn B)(Tun – p) ≤ (I – αn B)(Tun – p) + βn (xn – Tun ) + αn σ f (xn ) – Bp, xn+ – p ≤ I – αn B Tun – p + βn xn – Tun + αn σ f (xn ) – Bp xn+ – p ≤ ( – αn γ ) un – p + βn xn – Tun + αn σ f (xn ) – Bp xn+ – p = ( – αn γ ) un – p + βn xn – Tun + ( – αn γ )βn un – p xn – Tun + αn σ f (xn ) – Bp xn+ – p .
(.)
It follows from (.) that xn+ – p ≤ xn – p – un – yn + βn xn – Tun + ( – αn γ )βn un – p xn – Tun + αn σ f (xn ) – Bp xn+ – p .
(.)
Then un – yn ≤ xn – p – xn+ – p + βn xn – Tun
+ ( – αn γ )βn un – p xn – Tun + αn σ f (xn ) – Bp xn+ – p ≤ xn – p + xn+ – p xn+ – xn + βn xn – Tun + ( – αn γ )βn un – p xn – Tun + αn σ f (xn ) – Bp xn+ – p .
This together with (C), (.), and (.) implies that lim un – yn = .
n→∞
(.)
From (.), we have xn+ – p ≤ ( – αn γ ) un – p + βn xn – Tun
+ ( – αn γ )βn un – p xn – Tun + αn σ f (xn ) – Bp xn+ – p
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 11 of 15
≤ yn – p + βn xn – Tun + ( – αn γ )βn un – p xn – Tun + αn σ f (xn ) – Bp xn+ – p ≤ xn – p + δ A – δ Szn – Axn – δ zn – Axn + βn xn – Tun + ( – αn γ )βn un – p xn – Tun + αn σ f (xn ) – Bp xn+ – p . Hence,
δ – δ A Szn – Axn + δ zn – Axn ≤ xn – p – xn+ – p + βn xn – Tun
+ ( – αn γ )βn un – p xn – Tun + αn σ f (xn ) – Bp xn+ – p ≤ xn – p + xn+ – p xn+ – xn + βn xn – Tun + ( – αn γ )βn un – p xn – Tun + αn σ f (xn ) – Bp xn+ – p , which implies that lim Szn – Axn = lim zn – Axn = .
n→∞
n→∞
So, lim Szn – zn = .
(.)
n→∞
Note that yn – xn = δA∗ SUγGn – I Axn ≤ δ A Szn – Axn . Therefore, lim xn – yn = .
(.)
n→∞
From (.), (.), and (.), we get lim xn – Txn = .
(.)
n→∞
Now, we show that lim supn→∞ (σ f – B)p, xn – p ≤ . Choose a subsequence {xni } of {xn } such that lim sup (σ f – B)p, xn – p = lim (σ f – B)p, xni – p . n→∞
i→∞
(.)
Since the sequence {xni } is bounded, we can choose a subsequence {xnij } of {xni } such that xnij z. For the sake of convenience, we assume (without loss of generality) that xni z.
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 12 of 15
Consequently, we derive from the above conclusions that yni z,
uni z,
and zni Az.
Axni z
(.)
By the demi-closed principle of the nonexpansive mappings S and T (see Lemma .), we deduce z ∈ Fix(T) and Az ∈ Fix(S) (according to (.) and (.), respectively). Next, we show that z ∈ EP(F). Since un = UλFn yn , we have F(un , y) +
y – un , un – yn ≥ , λn
∀y ∈ C.
(.)
It follows from the monotonicity of F that y – un , un – yn ≥ F(y, un ), λn
(.)
and hence uni – yni ≥ F(y, uni ). y – uni , λni
(.) un –yn
Since un – yn → , uni z, and lim infn→∞ λn > , we obtain λi n i → . It follows that i ≥ F(y, z). For t with < t ≤ and y ∈ C, let yt = ty + ( – t)z ∈ C. It follows that F(yt , z) ≤ . So, = F(yt , yt ) ≤ tF(yt , y) + ( – t)F(yt , z) ≤ tF(yt , y).
(.)
Therefore, ≤ F(yt , y). Thus ≤ F(z, y). This implies that z ∈ EP(F). Similarly, we can prove that Az ∈ EP(G). To this end, we deduce z ∈ Fix(T) ∩ EP(F) and Az ∈ Fix(S) ∩ EP(G). That is to say, z ∈ . Therefore, lim sup (σ f – B)p, xn – p = lim (σ f – B)p, xni – p n→∞
i→∞
= lim (σ f – B)p, z – p i→∞
≤ .
(.)
Finally, we prove xn → p. From (.), we have xn+ – p = αn σ f (xn ) – Bp + βn (xn – p) + ( – βn )I – αn B (Tun – p), xn+ – p = αn σ f (xn ) – Bp, xn+ – p + βn xn – p, xn+ – p
+ ( – βn )I – αn B (Tun – p), xn+ – p ≤ αn σ f (xn ) – f (p), xn+ – p + αn σ f (p) – Bp, xn+ – p + βn xn – p xn+ – p + ( – βn – αn γ ) Tun – p xn+ – p ≤ – (γ – σρ)αn xn – p xn+ – p + αn σ f (p) – Bp, xn+ – p ≤
– (γ – σρ)αn xn – p + xn+ – p + αn σ f (p) – Bp, xn+ – p .
(.)
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Page 13 of 15
It follows that xn+ – p ≤ – (γ – σρ)αn xn – p + αn σ f (p) – Bp, xn+ – p .
(.)
Applying Lemma . and (.) to (.), we deduce xn → p. The proof is completed. Algorithm . Taking x ∈ H arbitrarily, we define a sequence {xn } by the following: xn+ = αn σ f (xn ) + βn xn + ( – βn )I – αn B T xn + δA∗ (S – I)Axn
(.)
for all n ∈ N, where δ ∈ (, A ) and σ > are two constants and {αn } and {βn } are two real number sequences in (, ).
Corollary . Suppose = {x ∈ Fix(T) : Ax ∈ Fix(S)} = ∅ and suppose the following conditions hold:
(C): limn→∞ αn = and ∞ n= αn = ∞; (C): < lim infn→∞ βn ≤ lim supn→∞ βn < ; (C): σρ < γ . Then the sequence {xn } generated by algorithm (.) converges strongly to p = Proj (σ f + I – B)p, which solves the following VI: (σ f – B)x, y – x ≤ ,
∀y ∈ .
Algorithm . Taking x ∈ H arbitrarily, we define a sequence {xn } by the following: xn+ = αn σ f (xn ) + βn xn + ( – βn )I – αn B UλFn xn + δA∗ UγGn – I Axn
(.)
for all n ∈ N, where {λn } and {γn } are two real number sequences in (, ∞), δ ∈ (, A ) and σ > are two constants and {αn } and {βn } are two real number sequences in (, ).
Corollary . Suppose = {x ∈ EP(F) : Ax ∈ EP(G)} = ∅ and suppose the following conditions hold:
(C): limn→∞ αn = and ∞ n= αn = ∞; (C): < lim infn→∞ βn ≤ lim supn→∞ βn < ; = ; (C): lim infn→∞ λn > and limn→∞ λλn+ n γn+ (C): lim infn→∞ γn > and limn→∞ γn = ; (C): σρ < γ . Then the sequence {xn } generated by algorithm (.) converges strongly to p = Proj (σ f + I – B)p, which solves the following VI: (σ f – B)x, y – x ≤ ,
∀y ∈ .
Algorithm . Taking x ∈ H arbitrarily, we define a sequence {xn } by the following: xn+ = αn σ f (xn ) + βn xn + ( – βn )I – αn B PC xn + δA∗ (PQ – I)Axn
(.)
for all n ∈ N, where δ ∈ (, A ) and σ > are two constants and {αn } and {βn } are two real number sequences in (, ).
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
Corollary . Suppose = {x ∈ C : Ax ∈ Q} = ∅ and suppose the following conditions hold:
(C): limn→∞ αn = and ∞ n= αn = ∞; (C): < lim infn→∞ βn ≤ lim supn→∞ βn < ; (C): σρ < γ . Then the sequence {xn } generated by algorithm (.) converges strongly to p = Proj (σ f + I – B)p, which solves the following VI: (σ f – B)x, y – x ≤ ,
∀y ∈ .
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Author details 1 School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, 750021, China. 2 School of Mathematics and Information Technology, Nanjing Xiaozhuang University, Nanjing, 211171, China. 3 Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan. 4 Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 807, Taiwan. 5 Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China. Acknowledgements Li-Jun Zhu was supported in part by NSFC 61362033 and NZ13087. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3. Received: 29 March 2014 Accepted: 11 September 2014 Published: 02 Oct 2014 References 1. Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221-239 (1994) 2. Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18, 441-453 (2002) 3. Ceng, LC, Ansari, QH, Yao, JC: An extragradient method for split feasibility and fixed point problems. Comput. Math. Appl. 64, 633-642 (2012) 4. Dang, Y, Gao, Y: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27, 015007 (2011) 5. Wang, F, Xu, HK: Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2010, Article ID 102085 (2010). doi:10.1155/2010/102085 6. Xu, HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010) 7. Yang, Q: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20, 1261-1266 (2004) 8. Yao, Y, Wu, J, Liou, YC: Regularized methods for the split feasibility problem. Abstr. Appl. Anal. 2012, Article ID 140679 (2012). doi:10.1155/2012/140679 9. Yao, Y, Kim, TH, Chebbi, S, Xu, HK: A modified extragradient method for the split feasibility and fixed point problems. J. Nonlinear Convex Anal. 13, 383-396 (2012) 10. Zhao, J, Yang, Q: Several solution methods for the split feasibility problem. Inverse Probl. 21, 1791-1799 (2005) 11. Qu, B, Xiu, N: A note on the CQ algorithm for the split feasibility problem. Inverse Probl. 21, 1655-1665 (2005) 12. Wang, Z, Yang, Q, Yang, Y: The relaxed inexact projection methods for the split feasibility problem. Appl. Math. Comput. (2010). doi:10.1016/j.amc.2010.11.058 13. Ceng, LC, Ansari, QH, Yao, JC: Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. Nonlinear Anal. 75, 2116-2125 (2012) 14. Censor, Y, Segal, A: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587-600 (2009) 15. Moudafi, A: The split common fixed-point problem for demi-contractive mappings. Inverse Probl. 26, 587-600 (2010) 16. Wang, F, Xu, HK: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 74, 4105-4111 (2011) 17. Moudafi, A: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74, 4083-4087 (2011) 18. He, ZH: The split equilibrium problems and its convergence algorithms. J. Inequal. Appl. 2012, 162 (2012) 19. He, ZH, Du, WS: Nonlinear algorithms approach to split common solution problems. Fixed Point Theory Appl. 2012, 130 (2012) 20. Moudafi, A: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275-283 (2011) 21. Byrne, C, Censor, Y, Gibali, A, Reich, S: The split common null point problem. J. Nonlinear Convex Anal. 13, 759-775 (2012)
Page 14 of 15
Zhu et al. Journal of Inequalities and Applications 2014, 2014:380 http://www.journalofinequalitiesandapplications.com/content/2014/1/380
22. Censor, Y, Gibali, A, Reich, S: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301-323 (2012) 23. Yao, Y, Marino, G, Muglia, L: A modified Korpelevich’s method convergent to the minimum norm solution of a variational inequality. Optimization 63, 559-569 (2014) 24. Yao, Y, Marino, G, Xu, HK, Liou, YC: Construction of minimum norm fixed points of pseudocontractions in Hilbert spaces. J. Inequal. Appl. 2014, 206 (2014) 25. Chang, SS, Wang, L, Tang, YK, Yang, L: The split common fixed point problem for total asymptotically strictly pseudocontractive mappings. J. Appl. Math. 2012, Article ID 385638 (2012) 26. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994) 27. Moudafi, A: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9, 37-43 (2008) 28. Yao, Y, Noor, MA, Liou, YC: On iterative methods for equilibrium problems. Nonlinear Anal. 70, 497-507 (2009) 29. Ceng, LC, Al-Homidan, S, Ansari, QH, Yao, JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudocontraction mappings. J. Comput. Appl. Math. 223, 967-974 (2009) 30. He, ZH, Du, WS: On hybrid split problem and its nonlinear algorithms. Fixed Point Theory Appl. 2013, 47 (2013) 31. Combettes, PL, Hirstoaga, A: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117-136 (2005) 32. Cianciaruso, F, Marino, G, Muglia, L, Yao, Y: A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl. 2010, Article ID 383740 (2010) 33. Suzuki, T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 103-123 (2005) 34. Geobel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990) 35. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240-256 (2002)
10.1186/1029-242X-2014-380 Cite this article as: Zhu et al.: Superimposed algorithms for the split equilibrium problems and fixed point problems. Journal of Inequalities and Applications 2014, 2014:380
Page 15 of 15
