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Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 RESEARCH Open Access The strong convergence theorems for split common fixed point problem of asymptotically nonexpansive mappings in Hilbert spaces Xin-Fang Zhang1 , Lin Wang1* , Zhao Li Ma2 and Li Juan Qin3 Dedicated to Professor SS Chang on the occasion of his 80th birthday. * Correspondence: [email protected] 1 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Long Quan Road, Kunming, China Full list of author information is available at the end of the article Abstract In this paper, an iterative algorithm is introduced to solve the split common fixed point problem for asymptotically nonexpansive mappings in Hilbert spaces. The iterative algorithm presented in this paper is shown to possess strong convergence for the split common fixed point problem of asymptotically nonexpansive mappings although the mappings do not have semi-compactness. Our results improve and develop previous methods for solving the split common fixed point problem. MSC: 47H09; 47J25 Keywords: split common fixed point problem; asymptotically nonexpansive mapping; strong convergence; Hilbert space; algorithm 1 Introduction and preliminaries Throughout this paper, let H and H be real Hilbert spaces whose inner product and norm are denoted by ·, · and · , respectively; let C and Q be nonempty closed convex subsets of H and H , respectively. A mapping T : C → C is said to be nonexpansive if Tx – Ty ≤ x – y for any x, y ∈ C. A mapping T : C → C is said to be quasi-nonexpansive if Tx – p ≤ x – p for any x ∈ C and p ∈ F(T), where F(T) is the set of fixed points of T. A mapping T : C → C is called asymptotically nonexpansive if there exists a sequence {kn } ⊂ [, ∞) satisfying limn→∞ kn = such that T n x – T n y ≤ kn x – y for any x, y ∈ C. A mapping T : C → C is semi-compact if, for any bounded sequence {xn } ⊂ C with limn→∞ xn – Txn = , there exists a subsequence {xnj } ⊂ {xn } such that {xnj } converges strongly to some point x∗ ∈ C. The split feasibility problem (SFP) is to find a point q ∈ H with the property q∈C and Aq ∈ Q, (.) where A : H → H is a bounded linear operator. Assuming that SFP (.) is consistent (i.e., (.) has a solution), it is not hard to see that x ∈ C solves (.) if and only if it solves the following fixed point equation: x = PC I – γ A∗ (I – PQ )A x, x ∈ C, (.) ©2015 Zhang 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. Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 Page 2 of 11 where PC and PQ are the (orthogonal) projections onto C and Q, respectively, γ > is any positive constant, and A∗ denotes the adjoint of A. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction []. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [–]. Let S : H → H and T : H → H be two mappings satisfying F(S) = {x ∈ H : Sx = x} = φ and F(T) = {x ∈ H : Tx = x} = φ, respectively; let A : H → H be a bounded linear operator. The split common fixed point problem (SCFP) for mappings S and T is to find a point q ∈ H with the property q ∈ F(S) and Aq ∈ F(T). (.) We use to denote the set of solutions of SCFP (.), that is, = {q ∈ F(S) : Aq ∈ F(T)}. Since each closed and convex subset may be considered as a fixed point set of a projection on the subset, hence the split common fixed point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP) []. Split feasibility problems and split common fixed point problems have been studied by some authors [–]. In , Moudafi [] proposed the following iteration method to approximate a split common fixed point of demi-contractive mappings: for arbitrarily chosen x ∈ H , un = xn + γβA∗ (T – I)Axn , xn+ = ( – αn )un + αn Uun , n ∈ N, and he proved that {xn } converges weakly to a split common fixed point x∗ ∈ , where U : H → H and T : H → H are two demi-contractive mappings, A : H → H is a bounded linear operator. Using the iterative algorithm above, in , Moudafi [] also obtained a weak convergence theorem for the split common fixed point problem of quasi-nonexpansive mappings in Hilbert spaces. After that, some authors also proposed some iterative algorithms to approximate a split common fixed point of other nonlinear mappings, such as nonspreading type mappings [], asymptotically quasi-nonexpansive mappings [], κ-asymptotically strictly pseudononspreading mappings [], asymptotically strictly pseudocontraction mappings [] etc., but they just obtained weak convergence theorems when those mappings do not have semi-compactness. This naturally brings us to the following question. Can we construct an iterative scheme which can guarantee the strong convergence for split common fixed point problems without assumption of semi-compactness? In this paper, we introduce the following iterative scheme. Let x ∈ H , C = H , the sequence {xn } is defined as follows: ⎧ yn = αn zn + ( – αn )Tn zn , ⎪ ⎪ ⎪ ⎨ z = x + λA∗ (T n – I)Ax , n n n ⎪ Cn+ = {v ∈ Cn : yn – v ≤ kn zn – v, zn – v ≤ kn xn – v}, ⎪ ⎪ ⎩ xn+ = PCn+ (x ), n ≥ , (.) Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 Page 3 of 11 where T : H → H and T : H → H are two asymptotically nonexpansive mappings, A : H → H is a bounded linear operator, A∗ denotes the adjoint of A. Under some suitable conditions on parameters, the iterative scheme {xn } is shown to converge strongly to a split common fixed point of asymptotically nonexpansive mappings T and T without the assumption of semi-compactness on T and T . The following lemma and results are useful for our proofs. Lemma . [] Let E be a real uniformly convex Banach space, K be a nonempty closed subset of E, and let T : K → K be an asymptotically nonexpansive mapping. Then I – T is demiclosed at zero, that is, if {xn } ⊂ K converges weakly to a point p ∈ K and limn→∞ xn – Txn = , then p = Tp. Let C be a closed convex subset of a real Hilbert space H. PC denotes the metric projection of H onto C. It is well known that PC is characterized by the properties: for x ∈ H and z ∈ C, z = PC (x) ⇔ x – z, z – y ≥ , ∀y ∈ C (.) and y – PC (x) + x – PC (x) ≤ x – y , ∀y ∈ C, ∀x ∈ H. (.) In a real Hilbert space H, it is also well known that λx + ( – λ)y = λx + ( – λ)y – λ( – λ)x – y , ∀x, y ∈ H, ∀λ ∈ R (.) and x, y = x + y – x – y , ∀x, y ∈ H. (.) 2 Main results Theorem . Let H and H be two Hilbert spaces, A : H → H be a bounded linear operator, T : H → H be an asymptotically nonexpansive mapping with the sequence {kn() } ⊂ [, ∞) satisfying limn→∞ kn() = , and T : H → H be an asymptotically nonexpansive mapping with the sequence {kn() } ⊂ [, ∞) satisfying limn→∞ kn() = , F(T ) = ∅ and F(T ) = ∅, respectively. Let x ∈ H , C = H , and let the sequence {xn } be defined as follows: ⎧ zn = xn + λA∗ (Tn – I)Axn , ⎪ ⎪ ⎪ ⎨ y = α z + ( – α )T n z , n n n n n ⎪ Cn+ = {v ∈ Cn : yn – v ≤ kn zn – v, zn – v ≤ kn xn – v}, ⎪ ⎪ ⎩ xn+ = PCn+ (x ), n ≥ , (.) where A∗ denotes the adjoint of A, λ ∈ (, A∗ ) and {αn } ⊂ (, η] ⊂ (, ) satisfies lim infn→∞ αn ( – αn ) > , kn = max{kn() , kn() }, n ≥ . If = {p ∈ F(T ) : Ap ∈ F(T )} = ∅, then {xn } converges strongly to x∗ ∈ . Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 Page 4 of 11 Proof We will divide the proof into five steps. Step . We first show that Cn is closed and convex for any n ≥ . Since C = H , so C is closed and convex. Assume that Cn is closed and convex. For any v ∈ Cn , since yn – v ≤ kn zn – v ⇔ zn – v ≤ kn xn – v ⇔ kn zn – yn – kn v + v, v ≤ kn zn – yn , kn xn – zn – kn v + v, v ≤ kn xn – zn , we know that Cn+ is closed and convex. Therefore Cn is closed and convex for any n ≥ . Step . We prove ⊂ Cn for any n ≥ . Let p ∈ , then from (.) we have zn – p = xn – p + λA∗ Tn – I Axn = xn – p + λA∗ Tn – I Axn + λ xn – p, A∗ Tn – I Axn , (.) where λ xn – p, A∗ Tn – I Axn = λ Axn – Ap, Tn – I Axn = λ A(xn – p) + Tn – I Axn – Tn – I Axn , Tn – I Axn = λ Tn Axn – Ap, Tn – I Axn – Tn – I Axn = λ Tn Axn – Ap + Tn – I Axn n – Axn – Ap – T – I Axn ≤ λ kn Axn – Ap – Tn – I Axn – Axn – Ap n = –λ T – I Axn + λ kn – Axn – Ap . (.) Substituting (.) into (.), we can obtain that zn – p ≤ xn – p + λ A∗ Tn – I Axn – λ Tn – I Axn + λ kn – Axn – Ap = xn – p – λ – λA∗ Tn – I Axn + λA kn – xn – p ≤ kn xn – p – λ – λA∗ Tn – I Axn . (.) In addition, it follows from (.) that yn – p = αn (zn – p) + ( – αn ) Tn zn – p ≤ kn zn – p. Therefore, from (.) and (.), we know that p ∈ Cn and ⊂ Cn for any n ≥ . (.) Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 Page 5 of 11 Step . We will show that {xn } is a Cauchy sequence. Since ⊂ Cn+ ⊂ Cn and xn+ = PCn+ (x ) ⊂ Cn , then xn+ – x ≤ p – x for n ≥ and p ∈ . (.) It means that {xn } is bounded. For any n ≥ , by using (.), we have xn+ – xn + x – xn = xn+ – PCn (x ) + x – PCn (x ) ≤ xn+ – x , which implies that ≤ xn – xn+ ≤ xn+ – x – xn – x . Thus {xn – x } is nondecreasing. Therefore, by the boundedness of {xn }, limn→∞ xn –x exists. For some positive integers m, n with m ≤ n, from xn = PCn (x ) ⊂ Cm and (.), we have xm – xn + x – xn = xm – PCn (x ) + x – PCn (x ) ≤ xm – x . (.) Since limn→∞ xn – x exists, it follows from (.) that limn→∞ xn – xm = . Therefore {xn } is a Cauchy sequence. Step . We will show that limn→∞ zn – T zn = limn→∞ Axn – T Axn = . Since xn+ = PCn+ (x ) ∈ Cn+ ⊂ Cn , we have zn – xn ≤ zn – xn+ + xn+ – xn ≤ ( + kn )xn+ – xn → , yn – xn ≤ yn – xn+ + xn+ – xn ≤ + kn xn+ – xn → , yn – zn ≤ yn – xn + xn – zn → . (.) (.) (.) Notice that λ( – λA∗ ) > , it follows from (.) that n T – I Axn ≤ kn xn – p – zn – p ∗ λ( – λA ) ≤ (kn – )xn – p + (xn – p + zn – p)(xn – p – zn – p) λ( – λA∗ ) ≤ (kn – )xn – p + xn – zn (xn – p + zn – p) , λ( – λA∗ ) thus, since {xn } is bounded and limn→∞ kn = , from (.) we have lim Tn – I Axn = . n→∞ On the other hand, since yn – p = αn (zn – p) + ( – αn ) Tn zn – p = αn zn – p + ( – αn )Tn zn – p – αn ( – αn )Tn zn – zn ≤ + kn – zn – p – αn ( – αn )Tn zn – zn , (.) Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 Page 6 of 11 we have αn ( – αn )Tn zn – zn ≤ zn – p – yn – p + kn – zn – p ≤ zn – p + yn – p zn – yn + kn – zn – p . Since lim infn→∞ αn ( – αn ) > and limn→∞ kn = , we know that lim Tn – I zn = . (.) n→∞ In addition, since zn+ – zn ≤ zn+ – xn+ + xn+ – xn + xn – zn , we know that limn→∞ zn+ – zn = . So from zn – T zn = zn – zn+ + zn+ – Tn+ zn+ + Tn+ zn+ – Tn+ zn + Tn+ zn – T zn ≤ ( + kn+ )zn – zn+ + zn+ – T n+ zn+ + k T n zn – zn , we can obtain that lim zn – T zn = . n→∞ (.) Similarly, we have lim Axn – T Axn = . n→∞ (.) Step . We will show that {xn } converges strongly to an element of . Since {xn } is a Cauchy sequence, we may assume that xn → x∗ , from (.) we have zn → ∗ x , which implies that zn x∗ . So it follows from (.) and Lemma . that x∗ ∈ F(T ). In addition, since A is a bounded linear operator, we have that limn→∞ Axn – Ax∗ = . Hence, it follows from (.) and Lemma . that Ax∗ ∈ F(T ). This means that x∗ ∈ and {xn } converges strongly to x∗ ∈ . The proof is completed. In Theorem ., as T = T and H = H , we have the following result. Corollary . Let H be a Hilbert space, T : H → H be an asymptotically nonexpansive mapping with a sequence {kn } ⊂ [, ∞) satisfying limn→∞ kn = . The sequence {xn } is defined as follows: x ∈ H , C = H ⎧ zn = xn + λ(T n – I)xn , ⎪ ⎪ ⎪ ⎨ yn = αn zn + ( – αn )T n zn , ⎪ Cn+ = {v ∈ Cn : yn – v ≤ kn zn – v, zn – v ≤ kn xn – v}, ⎪ ⎪ ⎩ xn+ = PCn+ (x ), n ≥ , (.) where λ ∈ (, ) and {αn } ⊂ (, η] ⊂ (, ) satisfies lim infn→∞ αn ( – αn ) > . If F(T) = {p ∈ H : p = Tp} = ∅, then {xn } converges strongly to a fixed point x∗ of T. Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 Page 7 of 11 In Theorem ., when T and T are two nonexpansive mappings, the following result holds. Corollary . Let H and H be two Hilbert spaces, A : H → H be a bounded linear operator, T : H → H and T : H → H be two nonexpansive mappings such that F(T ) = ∅ and F(T ) = ∅, respectively. Let x ∈ H , C = H , and let the sequence {xn } be defined as follows: ⎧ zn = xn + λA∗ (T – I)Axn , ⎪ ⎪ ⎪ ⎨ yn = αn zn + ( – αn )T zn , ⎪ C n+ = {v ∈ Cn : yn – v ≤ zn – v ≤ xn – v}, ⎪ ⎪ ⎩ xn+ = PCn+ (x ), n ≥ , (.) where A∗ denotes the adjoint of A, λ ∈ (, A∗ ) and {αn } ⊂ (, η] ⊂ (, ) satisfies lim infn→∞ αn ( – αn ) > . If = {p ∈ F(T ) : Ap ∈ F(T )} = ∅, then {xn } converges strongly ∗ to x ∈ . Remark . When T and T are two quasi-nonexpansive mappings and I – T and I – T are demiclosed at zero, Corollary . also holds. Example . Let C be a unit ball in a real Hilbert space l , and let T : C → C be a mapping defined by T : (x , x , . . .) → , x , a x , a x , . . . . It is proved in Goebel and Kirk [] that (i) T x – T y ≤ x – y, ∀x, y ∈ C; (ii) Tn x – Tn y ≤ nj= aj x – y, ∀x, y ∈ C, ∀n ≥ . – Taking aj = j– , j ≥ , it is easy to see that ∞ j= aj = . So we can take k = , and n kn = j= aj , n ≥ , then lim kn = lim n→∞ n→∞ n – j– = . j= Therefore T is an asymptotically nonexpansive mapping from C into itself with F(T ) = {(, , . . . , , . . .)}. n Let D be an orthogonal subspace of Rn with the norm x = i= xi and the inner n product x, y = i= xi yi for x = (x , . . . , xn ) and y = (y , . . . , yn ). For each x = (x , x , . . . , xn ) ∈ D, we define a mapping T : D → D by T x = if ni= xi < ; (x , x , . . . , xn ) (–x , –x , . . . , –xn ) if ni= xi ≥ . It is easy to show that Tn x – Tn y = x – (–)n y = x + y = x – y or Tn x – Tn y = (–)n x – y = x + y = x – y for any x, y ∈ D. Therefore T is an Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 Page 8 of 11 asymptotically nonexpansive mapping from D into itself with F(T ) = {(, , . . . , )} ∪ {(x , x , . . . , xn ) : ni= xi < } since Tn x – Tn y ≤ kn x – y for any sequence {kn } ⊂ [, ∞) with limn→∞ kn = . Obviously, C and D are closed convex subsets of l and RN , respectively. Let A : C → D be defined by Ax = (x , x , . . . , xn ) for x = (x , x , . . .) ∈ C. Then A is a bounded linear operator with adjoint operator A∗ z = (x , x , . . . , xn , , , . . .) for z = (x , x , . . . , xn ) ∈ D. Clearly, = {(, , . . . , , . . .)}, A = A∗ = . – j– , n ≥ , γ = and αn = . – n , n ≥ . It follows Taking C = C, k = , kn+ = n+ j= from Theorem . that {xn } converges strongly to (, , . . .) ∈ . 3 Applications and examples Application to the equilibrium problem Let H be a real Hilbert space, C be a nonempty closed and convex subset of H, and let the bifunction F : C × C → R satisfy the following conditions: (A) F(x, x) = , ∀x ∈ C; (A) F(x, y) + F(y, x) ≤ , ∀x, y ∈ C; (A) For all x, y, z ∈ C, limt↓ F(tz + ( – t)x, y) ≤ F(x, y); (A) For each x ∈ C, the function y −→ F(x, y) is convex and lower semi-continuous. The so-called equilibrium problem for F is to find a point x∗ ∈ C such that F(x∗ , x) ≥ for all y ∈ C. The set of its solutions is denoted by EP(F). Lemma . [] Let C be a nonempty closed convex subset of a Hilbert space H, and let F : C × C → R be a bifunction satisfying (A)-(A). Let r > and x ∈ H. Then there exists z ∈ K such that F(z, y) + y – z, z – x ≥ , r ∀y ∈ C. Lemma . [] Assume that F : C × C → R satisfies (A)-(A). For r > and x ∈ H, define a mapping Tr : H → H as follows: Tr (x) = z ∈ C : F(z, y) + y – z, z – x ≥ , ∀y ∈ C , r ∀x ∈ H. Then () Tr is single-valued; () Tr is firmly nonexpansive, that is, for all x, y ∈ H, Tr x – Tr y ≤ Tr x – Tr y, x – y; () F(Tr ) = EP(F); () EP(F) is nonempty, closed and convex. Theorem . Let H and H be two Hilbert spaces, A : H → H be a bounded linear operator, T : H → H be a nonexpansive mapping, F : H × H → R be a bifunction satisfying (A)-(A). Assume that C := EP(F) = ∅ and Q := F(T) = ∅. Taking C = H , for arbitrarily Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 Page 9 of 11 chosen x ∈ H , the sequence {xn } is defined as follows: ⎧ ⎪ z = xn + λA∗ (T – I)Axn , ⎪ ⎪ n ⎪ ⎪ ⎪ ⎨ F(un , y) + rn y – un , un – zn ≥ , ∀y ∈ H , yn = αn zn + ( – αn )un , ⎪ ⎪ ⎪ Cn+ = {v ∈ Cn : yn – v ≤ zn – v ≤ xn – v}, ⎪ ⎪ ⎪ ⎩ xn+ = PCn+ (x ), n ≥ , (.) where A∗ denotes the adjoint of A, {rn } ⊂ (, ∞), λ ∈ (, A∗ ) and {αn } ⊂ (, η] ⊂ (, ) satisfies lim infn→∞ αn ( – αn ) > . If = {p ∈ C : Ap ∈ Q} = ∅, then the sequence {xn } converges strongly to a point x∗ ∈ . Proof It follows from Lemma . that un = Tr (zn ), F(Tr ) = EP(F) is nonempty, closed and convex and Tr is a firmly nonexpansive mapping. Hence all conditions in Corollary . are satisfied. The conclusion of Theorem . can be directly obtained from Corollary .. Let E and E be two real Hilbert spaces. Let C be a closed convex subset of E , K be a closed convex subset of E , A : E → E be a bounded linear operator. Assume that F is a bi-function from C × C into R and G is a bi-function from K × K into R. The split equilibrium problem (SEP) is to find an element p ∈ C such that F(p, y) ≥ , ∀y ∈ C (.) and such that u := Ap ∈ C solves G(u, v) ≥ , ∀v ∈ K. (.) Let = {p ∈ EP(F) : Ap ∈ EP(G)} denote the solution set of the split equilibrium problem SEP. Example . [] Let E = E = R, C := [, +∞) and K := (–∞, –]. Let A(x) = –x for all R, then A is a bounded linear operator. Let F : C × C → R and G : K × K → R be defined by F(x, y) = y – x and G(u, v) = (u – v), respectively. Clearly, EP(F) = {} and A() = – ∈ EP(G). So = {p ∈ EP(F) : Ap ∈ EP(G)} = ∅. Example . [] Let E = R with the standard norm | · | and E = R with the norm α = (a + a ) for some α = (a , a ) ∈ R . K := [, +∞) and C := {α = (a , a ) ∈ R |a – a ≥ }. Define a bi-function F(w, α) = w – w + a – a , where w = (w , w ), α = (a , a ) ∈ C, then F is a bi-function from C × C into R with EP(F) = {p = (p , p )|p – p = }. For each α = (a , a ) ∈ E , let Aα = a – a , then A is a bounded linear operator from E into E . In fact, it √ is also easy to verify that A(aα + bα ) = aA(α ) + bA(α ) and A = for some α , α ∈ E and a, b ∈ R. Now define another bi-function G as follows: G(u, v) = v – u for all u, v ∈ K . Then G is a bi-function from K × K into R with EP(G) = {}. Clearly, when p ∈ EP(F), we have Ap = ∈ EP(G). So = {p ∈ EP(F) : Ap ∈ EP(G)} = ∅. Corollary . Let H and H be two Hilbert spaces, A : H → H be a bounded linear operator, F : H × H → R be a bifunction satisfying EP(F) = ∅ and G : H × H → R be a Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 Page 10 of 11 bifunction satisfying EP(G) = ∅. Taking C = H , for arbitrarily chosen x ∈ H , the sequence {xn } is defined as follows: ⎧ F(un , y) + rn y – un , un – xn ≥ , ∀y ∈ H , ⎪ ⎪ ⎪ ⎪ ⎪ G(vn , z) + rn z – vn , vn – Aun ≥ , ∀z ∈ H , ⎪ ⎪ ⎪ ⎨ zn = un + λA∗ (TrGn – I)Aun , ⎪ yn = αn zn + ( – αn )TrFn xn , ⎪ ⎪ ⎪ ⎪ ⎪ Cn+ = {v ∈ Cn : yn – v ≤ zn – v ≤ xn – v}, ⎪ ⎪ ⎩ xn+ = PCn+ (x ), n ≥ , (.) where A∗ denotes the adjoint of A, {rn } ⊂ (, ∞), λ ∈ (, A∗ ) and {αn } ⊂ (, η] ⊂ (, ) satisfies lim infn→∞ αn ( – αn ) > . If = {p ∈ EP(F) : Ap ∈ EP(G)} = ∅, then the sequence {xn } converges strongly to a point x∗ ∈ . Remark . Since Example . and Example . satisfy the conditions of Corollary ., the split equilibrium problems in Example . and Example . can be solved by algorithm (.). Application to the hierarchial variational inequality problem Let H be a real Hilbert space, T and T be two nonexpansive mappings from H to H such that F(T ) = ∅ and F(T ) = ∅. The so-called hierarchical variational inequality problem for nonexpansive mapping T with respect to a nonexpansive mapping T : H → H is to find a point x∗ ∈ F(T ) such that ∗ x – T x∗ , x∗ – x ≤ , ∀x ∈ F(T ). (.) It is easy to see that (.) is equivalent to the following fixed point problem: find x∗ ∈ F(T ) such that x∗ = PF(T ) T x∗ , (.) where PF(T ) is the metric projection from H onto F(T ). Letting C := F(T ) and Q := F(PF(T ) T ) (the fixed point set of the mapping PF(T ) T ) and A = I (the identity mapping on H), then problem (.) is equivalent to the following split feasibility problem: find x∗ ∈ C such that Ax∗ ∈ Q. (.) Hence from Theorem . we have the following theorem. Theorem . Let H, T , T , C and Q be the same as above. Let x ∈ H and C = H , and let the sequence {xn } be defined as follows: ⎧ zn = xn + λ(T – I)xn , ⎪ ⎪ ⎪ ⎨ y = α z + ( – α )T z , n n n n n ⎪ Cn+ = {v ∈ Cn : yn – v ≤ zn – v ≤ xn – v}, ⎪ ⎪ ⎩ xn+ = PCn+ (x ), n ≥ , (.) Zhang et al. Journal of Inequalities and Applications 2015, 2015:1 http://www.journalofinequalitiesandapplications.com/content/2015/1/1 where λ ∈ (, ) and {αn } ⊂ (, η] ⊂ (, ) satisfies lim infn→∞ αn ( – αn ) > . If C ∩ Q = ∅, then the sequence {xn } converges strongly to a solution of the hierarchical variational inequality problem (.). Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to this work. All authors read and approved the final manuscript. Author details 1 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Long Quan Road, Kunming, China. 2 School of Information Engineering, The College of Arts and Sciences, Yunnan Normal University, Long Quan Road, Kunming, 650222, China. 3 Department of Mathematics, Kunming University, Pu Xin Road No. 2, Kunming Economic and Technological Development Zone, Kunming, 650214, China. Acknowledgements The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advice. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070) and the Scientific Research Foundation of Postgraduate of Yunnan University of Finance and Economics. Received: 12 April 2014 Accepted: 11 December 2014 Published: 07 Jan 2015 References 1. Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problems. Inverse Probl. 18, 441-453 (2002) 2. Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projection in a product space. Numer. Algorithms 8, 221-239 (1994) 3. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications. Inverse Probl. 21, 2071-2084 (2005) 4. Censor, Y, Seqal, A: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587-600 (2009) 5. 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Appl. 2012, 162 (2012) 10.1186/1029-242X-2015-1 Cite this article as: Zhang et al.: The strong convergence theorems for split common fixed point problem of asymptotically nonexpansive mappings in Hilbert spaces. Journal of Inequalities and Applications 2015, 2015:1 Page 11 of 11
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