International Journal of Analysis and Applications ISSN 2291-8639 Volume 13, Number 1 (2017), 54-63 http://www.etamaths.com A GENERALIZED ITERATIVE ALGORITHM FOR HIERARCHICAL FIXED POINTS PROBLEMS AND VARIATIONAL INEQUALITIES VAHID DADASHI∗ AND SOMAYEH AMJADI Abstract. In this paper we propose a method for approximating of the common fixed point in ∞⋂ n=1 F(Tn) where {Tn} is a countable family of nonexpansive mappings on a closed convex subset C of a real Hilbert space H. Then, we prove strong convergence theorems with less control conditions for {Tn} which solves some variational inequality. The main results improve and extend the corre- sponding results of ”F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, On a two-step algorithm for hierarchical fixed point problems and variational inequalities, J. Inequal. Appl., 2009 (2009), Article ID 208692” and ”Y. Yao, Y.J. Cho, and Y.C. Liou, Iterative algorithms for hierarchical fixed points problems and variational inequalities, Mathematical and Computer Modelling, 52(9) (2010), 1697–1705”. 1. Introduction Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product 〈., .〉 and norm ‖.‖, respectively. Recall that a mapping T : C → C is called nonexpansive if ‖Tx−Ty‖≤‖x−y‖ for all x,y ∈ C and a nonself-mapping f : C → H is called a ρ−contraction on C if there exists a constant ρ ∈ [0, 1) such that ‖f(x) − f(y)‖ ≤ ρ‖x − y‖ for all x,y ∈ C. The set of all fixed points of T is denoted by F(T), that is F(T) = {x ∈ C | x = Tx}. Note that each ρ−contraction f has a unique fixed point in C, and for any fixed element x0 ∈ C, Picard’s iteration xn+1 = fn(x0) converges strongly to a unique fixed point of f. However, a simple example shows that Picard’s iteration cannot be used in the case of nonexpansive mappings. One method in [6] used for nonexpansive mappings is to employ a Halpern-type iterative scheme which produces a sequence {xn} as follows:{ x1 = x ∈ C xn+1 = βnu + (1 −βn)Txn, n ≥ 1, (1.1) where u ∈ C is arbitrary and {βn}⊂ [0, 1]. In this paper, we consider the following variational inequalities problem: Find x∗ ∈ F(T) such that 〈(I −S)x∗,x−x∗〉≥ 0, ∀x ∈ F(T), (1.2) where T and S are nonexpansive mappings such that F(T) is nonempty. It is easy to see that x∗ is a solution of the variational inequalities (1.2) if and only if it is a fixed point of the nonexpansive mapping PF(T)S, where PF(T) stands for the metric projection on the closed convex set F(T). In 2000, Moudafi [8] introduced a viscosity approximation method for a nonexpansive mapping as follows: { x1 = x ∈ C xn+1 = βnf(xn) + (1 −βn)Txn, n ≥ 1, (1.3) where f is a contractive mapping and {βn}⊂ [0, 1]. In a real Hilbert space and under certain control conditions, he proved the sequence {xn} defined by (1.3) converges strongly to a fixed point of T which is the unique solution to the variational inequality 〈(I −f)x∗,x−x∗〉≥ 0 for all x ∈ F(T). Received 19th July, 2016; accepted 20th September, 2016; published 3rd January, 2017. 2010 Mathematics Subject Classification. 47H09, 47H10. Key words and phrases. fixed points; iterative algorithms; nonexpansive mappings; variational inequalities; ρ−contraction. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 54 A GENERALIZED ITERATIVE ALGORITHM FOR HIERARCHICAL FIXED POINTS PROBLEMS 55 Mainge and Moudafi [7] introduced an iterative scheme for approximating a specific solution of a fixed point problem as follows:   x1 = x ∈ C yn = αnSxn + (1 −αn)Txn, n ≥ 1, xn+1 = βnf(xn) + (1 −βn)yn, n ≥ 1, (1.4) where f is a contractive mapping, {αn},{βn}⊂ [0, 1] and S and T are nonexpansive mappings. They proved that if the sequence {xn} given by scheme (1.4) is bounded, then {xn} strongly convergence to the fixed point of a nonexpansive mapping T with respect to a nonexpansive mapping S under some control conditions on {αn} and {βn}. Recently, Alimohammady and Dadashi [1] studied the iterative scheme (1.5) for a countable family of nonexpansive mappings {Tn} as follows:  x1 = x ∈ C yn = αnSxn + (1 −αn)Tnxn, n ≥ 1, xn+1 = βnf(xn) + (1 −βn)yn, n ≥ 1, (1.5) where f is a contractive mapping, {αn},{βn}⊂ [0, 1] and {Tn} is a sequence of nonexpansive mappings. They proved that the iterative scheme (1.5) strongly convergence to a common fixed point of {Tn} with respect to a nonexpansive mapping S. On the other hand, Cianciaruso et al. in [2] studied the sequence generated by the algorithm  x1 = x ∈ C yn = βnSxn + (1 −βn)xn, n ≥ 1, xn+1 = αnf(xn) + (1 −αn)Tyn, n ≥ 1, (1.6) where f is a contractive mapping, {αn},{βn}⊂ [0, 1] and S and T are nonexpansive mappings. They proved the sequence {xn} generated by (1.6) strongly converges to the fixed point of a nonexpansive mapping T with respect to a nonexpansive mapping S under some control conditions on {αn} and {βn}. Also, they show that this fixed point is a unique solution of a variational inequality. Another results about fixed point and variational inequality problems can be found in [3, 4, 9] and the references therein. Very recently, Yao et al. in [10] introduced another iterative algorithm and proved some strong convergence results for solving the hierarchical fixed point problem (1.2). In this paper, inspired and motivated by the above iterative schemes, we introduced and studied a new composite iterative scheme for countable family of nonexpansive mappings Tk (k ∈ N) with respect to a finite family of nonexpansive mapping Sk(k ∈{1, 2, ...,N} for some N ∈ N) as follows:{ yn = βnSnxn + (1 −βn)xn, xn+1 = PC(αnf(xn) + (1 −αn)Tnyn), ∀n ≥ 1, (1.7) where f is a contractive mapping, {αn},{βn} ⊂ [0, 1] and Sn = Sn mod N . In particular, if we take f ≡ 0, then it is reduced to the iterative scheme:{ yn = βnSnxn + (1 −βn)xn, xn+1 = PC((1 −αn)Tnyn), ∀n ≥ 1, (1.8) The main results improve and extend the corresponding results of [2, 10]. In particular, It should be noticed that we prove strong convergence theorems with less control conditions for {Tn} which solves some variational inequality. 2. Preliminaries In this section, we recall the well known results and give some useful lemmas that will be used in the next section. Let C be a nonempty closed convex subset of a real Hilbert space H. For every point x ∈H, there exists a unique nearest point in C, denoted by PC(x), such that ‖x−PC(x)‖≤‖x−y‖, ∀y ∈ C. PC is called the metric projection of H onto C. Recall that, PC is characterized by the following Lemma 56 DADASHI AND AMJADI Lemma 2.1. Let x ∈ H and z ∈ C be any points. Then z = PC(x) if and only if 〈x− z,y − z〉 ≤ 0, ∀y ∈ C. Lemma 2.2. [5] Let C be a nonempty closed convex subset of a real Hilbert space H and let T : C → C be a nonexpansive mapping with F(T) 6= ∅. If {xn} is a sequence in C weakly converging to x and if {(I −T)xn} converges strongly to y, then (I −T)x = y; in particular, if y = 0, then x ∈ F(T). Lemma 2.3. Let f : C → H be a contraction with coefficient ρ ∈ [0, 1) and T : C → C be a nonexpansive mapping. Then, (i) the mapping (I −f) is strongly monotone with coefficient (1 −ρ) i.e. 〈x−y, (I −f)x− (I −f)y〉≥ (1 −ρ)‖x−y‖2, ∀x,y ∈ C; (ii) the mapping (I −T) is monotone that is 〈x−y, (I −T)x− (I −T)y〉≥ 0, ∀x,y ∈ C. Lemma 2.4. [11] Assume that {αn} is a sequence of nonnegative numbers such that αn+1 ≤ (1 −γn)αn + δn, ∀n ≥ 0, where {γn} is a subsequence in (0, 1) and {δn} is a sequence in R such that (i) ∞∑ n=1 γn = ∞, (ii) lim sup n→∞ δn γn ≤ 0 or ∞∑ n=1 |δn| < ∞. Then lim n→∞ αn = 0. 3. Main results In this section, we prove several strong convergence theorems of the iterative scheme (1.7). Through- out this section, C is a nonempty closed convex subset of a real Hilbert space H, Tn for each n ∈ N and Sn for each n = 1, 2, ...,N are nonexpansive mappings of C into itself such that F := ∞⋂ n=1 F(Tn) is nonempty and f : C →H be a ρ−contraction (possibly nonself) with ρ ∈ [0, 1). Theorem 3.1. Suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (C1) lim n→∞ αn = 0, ∞∑ n=1 αn = ∞, (C2) lim n→∞ βn αn = 0. Then the sequence {xn} generated by (1.7) converges strongly to a point z ∈ F , which is the unique solution of the variational inequality: 〈(I −f)z,x−z〉≥ 0, ∀x ∈ F. (3.1) In particular, if f = 0, then {xn} generated by (1.8) converges in norm to the minimum norm common fixed point z of Tn, n ∈ N, namely, the point z is the unique solution to the quadratic minimization problem: z = arg min x∈ ∞⋂ n=1 F(Tn) ‖x‖2. (3.2) A GENERALIZED ITERATIVE ALGORITHM FOR HIERARCHICAL FIXED POINTS PROBLEMS 57 Proof. First, we claim that {xn} is bounded. Indeed, take an arbitrary fixed u ∈ F = ∞⋂ n=1 F(Tn) and using (C2), we can assume, without loss of generality, that βn ≤ αn for all n ≥ 1. From (1.7), we have ‖xn+1 −u‖ = ‖PC(αnf(xn) + (1 −αn)Tnyn) −PC(u)‖ ≤ αn‖f(xn) −f(u)‖ + αn‖f(u) −u‖ + (1 −αn)‖Tnyn −u‖ ≤ αnρ‖xn −u‖ + αn‖f(u) −u‖ + (1 −αn)‖yn −u‖ ≤ αnρ‖xn −u‖ + αn‖f(u) −u‖ + (1 −αn)βn‖Snxn −u‖ + (1 −αn)(1 −βn)‖xn −u‖ ≤ (1 − (1 −ρ)αn)‖xn −u‖ + (1 −ρ)αn { ‖f(u) −u‖ + ‖Snu−u‖ 1 −ρ } ≤ max { ‖xn −u‖, ‖f(u) −u‖ + ‖Snu−u‖ 1 −ρ } ≤ max 1≤K≤N { ‖x1 −u‖, ‖f(u) −u‖ + ‖Sku−u‖ 1 −ρ } , which implies that the sequence {xn} is bounded and so are the sequences {f(xn)}, {yn}, {Tnxn}, {Tnyn} and {Snxn}. Now, we prove that xn → z where, z = PFf(z). From Lemma 2.1 and set un := αnf(xn) + (1 −αn)Tnyn, we get ‖xn+1 −z‖2 = 〈PC(un) −un,PC(un) −z〉 + 〈un −z,xn+1 −z〉 ≤ 〈un −z,xn+1 −z〉 = αn〈f(xn) −f(z),xn+1 −z〉 + (1 −αn)〈Tnyn −z,xn+1 −z〉 + αn〈f(z) −z,xn+1 −z〉 ≤ αnρ‖xn −z‖‖xn+1 −z‖ + (1 −αn)‖Tnyn −z‖‖xn+1 −z‖ +αn〈f(z) −z,xn+1 −z〉, (3.3) and hence by the definition of {yn}, we have ‖Tnyn −z‖ ≤ ‖Tnyn −Tnxn‖ + ‖Tnxn −xn‖ + ‖xn −z‖ ≤‖yn −xn‖ + ‖Tnxn −xn‖ + ‖xn −z‖ ≤ βn‖Snxn −xn‖ + ‖Tnxn −xn‖ + ‖xn −z‖. (3.4) Also, we have ‖Tnxn −xn‖≤‖Tnxn −Tnz‖ + ‖z −xn‖≤ 2‖xn −z‖. (3.5) Substituting (3.4) and (3.5) into (3.3) to obtain ‖xn+1 −z‖2 ≤ (αnρ + 3(1 −αn))‖xn −z‖‖xn+1 −z‖ + (1 −αn)βn‖Snxn −xn‖‖xn+1 −z‖ +αn〈f(z) −z,xn+1 −z〉 ≤ αnρ + 3(1 −αn) 2 ( ‖xn −z‖2 + ‖xn+1 −z‖2 ) + (1 −αn)βn‖Snxn −xn‖‖xn+1 −z‖ +αn〈f(z) −z,xn+1 −z〉 So ‖xn+1 −z‖2 ≤ ( 1 − 2αn(ρ− 3) + 4 αn(ρ− 3) + 1 ) ‖xn −z‖2 + 2(1 −αn)βn αn(3 −ρ) − 1 ‖Snxn −xn‖‖xn+1 −z‖ + 2αn αn(3 −ρ) − 1 〈f(z) −z,xn+1 −z〉 = (1 −γn)‖xn −z‖2 + δn which γn = 2αn(ρ−3)+4 αn(ρ−3)+1 and δn = 2(1−αn)βn αn(3−ρ)−1 ‖Snxn −xn‖‖xn+1 − z‖ + 2αnαn(3−ρ)−1〈f(z) − z,xn+1 − z〉. Then, Lemma 2.4 implies that xn → z as n →∞. In particular, if f = 0, then {xn} generated by (1.8) converges strongly to z ∈ ∞⋂ n=1 F(Tn) such that z is the unique solution of the variational inequality 〈z,x−z〉≥ 0, ∀x ∈ F, 58 DADASHI AND AMJADI and hence, for each x ∈ ∞⋂ n=1 F(Tn) ‖z‖2 ≤〈z,x〉≤ ‖z‖‖x‖. Then for each x ∈ ∞⋂ n=1 F(Tn), ‖z‖2 ≤‖x‖2, that is, z is the unique solution to the quadratic minimiza- tion problem (3.2). � Corollary 3.2. Let S , T be nonexpansive mapping of C with F(T) 6= ∅. Suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (C1) and (C2). Then the sequence {xn} generated by { yn = βnSxn + (1 −βn)xn, xn+1 = PC(αnf(xn) + (1 −αn)Tyn), ∀n ≥ 1, (3.6) converges strongly to a point z ∈ F(T), which is the unique solution of the variational inequality: 〈(I −f)z,x−z〉≥ 0, ∀x ∈ F(T). (3.7) In particular, if f = 0, then {xn} generated by (1.8) converges in norm to the minimum norm fixed point z of T , namely, the point z is the unique solution to the quadratic minimization problem: z = arg min x∈F(T) ‖x‖2. Proof. It is sufficient that assume Sn = S and Tn = T in Theorem 3.1. � Remark 3.3. It is worth to mention that Yao et al. in [10] proved that the sequence {xn} generated by (3.6) converges strongly to a point z ∈ F(T), which is the unique solution of the variational inequality (3.7) under control conditions (C1), (C2) and the following conditions lim n→∞ |αn −αn−1| αn = 0, lim n→∞ |βn −βn−1| βn = 0 ∈ (0,∞); (3.8) or ∞∑ n=1 |αn −αn−1| < ∞, ∞∑ n=1 |βn −βn−1| < ∞; (3.9) But Corollary 3.2 proves that the sequence {xn} converges strongly under control conditions (C1) and (C2) and it does not require conditions (3.8) and (3.9) for convergence. Theorem 3.4. Suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (C1), (C2′) lim n→∞ βn αn = τ ∈ (0,∞); (C3) lim n→∞ |βn −βn−1| + |αn −αn−1| αnβn = 0; (C4) there exist a constant K > 0 such that 1 αn | 1 βn − 1 βn−1 | ≤ K; (C5) ∞∑ n=1 sup { ‖Tnz − Tn−1z‖‖ αnβn , z ∈ B } < ∞ for any bounded subset B of C. Let T be a mapping of C into itself defined by Tz = lim n→∞ Tnz for all z ∈ C and suppose that F(T) = ∞⋂ n=1 F(Tn). Then the sequence {xn} generated by{ yn = βnSxn + (1 −βn)xn, xn+1 = PC(αnf(xn) + (1 −αn)Tnyn), ∀n ≥ 1, (3.10) converges strongly to a point x∗ ∈ F , which is the unique solution of the variational inequality 〈 1 τ (I −f)x∗ + (I −S)x∗,y −x∗〉≥ 0, ∀y ∈ F. (3.11) A GENERALIZED ITERATIVE ALGORITHM FOR HIERARCHICAL FIXED POINTS PROBLEMS 59 Proof. At first, we show that uniqueness of the solution to the variational inequality (3.11) in F(T). In fact, suppose that x∗ and x̃ satisfy in (3.11).Then, since x̃ satisfy in (3.11), for y = x∗, it follows that 〈(I −f)x̃, x̃−x∗〉≤ τ〈(I −S)x̃,x∗ − x̃〉. (3.12) Similarly, we have 〈(I −f)x∗,x∗ − x̃〉≤ τ〈(I −S)x∗, x̃−x∗〉. (3.13) By (3.12), (3.13) and Lemma 2.3, we get (1 −ρ)‖x̃−x∗‖2 ≤ 〈(I −f)x̃− (I −f)x∗, x̃−x∗〉 = 〈(I −f)x̃, x̃−x∗〉−〈(I −f)x∗, x̃−x∗〉 ≤ τ〈(I −S)x̃,x∗ − x̃〉 + τ〈(I −S)x∗, x̃−x∗〉 = −τ〈(I −S)x̃− (I −S)x∗, x̃−x∗〉 ≤ 0. Hence, x∗ = x̃. We can assume from (C2′), without loss of generality, that βn ≤ (τ + 1)αn for all n ≥ 1. By a similar argument as that of Theorem 3.1, we have ‖xn+1 −u‖ ≤ (1 − (1 −ρ)αn)‖xn −u‖ + αn‖f(u) −u‖ + (1 −αn)βn‖Su−u‖ ≤ (1 − (1 −ρ)αn)‖xn −u‖ + αn‖f(u) −u‖ + αn(τ + 1)‖Su−u‖ = (1 − (1 −ρ)αn)‖xn −u‖ + (1 −ρ)αn [ ‖f(u) −u‖ 1 −ρ + (τ + 1)‖Su−u‖ 1 −ρ ] ≤ max { ‖xn −u‖, ‖f(u) −u‖ 1 −ρ + (τ + 1)‖Su−u‖ 1 −ρ } , which implies that the sequence {xn} is bounded. Set un = αnf(xn) + (1 −αn)Tnyn, then we have ‖xn+1 −xn‖ = ‖PC(un) −PC(un−1)‖≤‖un −un−1‖ ≤ αn‖f(xn) −f(xn−1)‖ + |αn −αn−1|‖f(xn−1) −Tn−1yn−1‖ +(1 −αn)‖Tnyn −Tn−1yn−1‖ ≤ αnρ‖xn −xn−1‖ + |αn −αn−1|‖f(xn−1) −Tn−1yn−1‖ + (1 −αn)‖yn −yn−1‖ +‖Tnyn−1 −Tn−1yn−1‖ (3.14) Also by definition of {yn}, we get ‖yn −yn−1‖ ≤ βn‖Sxn −Sxn−1‖ + (1 −βn)‖xn −xn−1‖ + |βn −βn−1|‖Sxn−1 −xn−1‖ ≤ ‖xn −xn−1‖ + |βn −βn−1|‖Sxn−1 −xn−1‖. (3.15) Set, M = max{sup‖f(xn−1)−Tn−1yn−1‖,‖Sxn−1−xn−1‖} and substituting (3.15) in (3.14) we have ‖xn+1 −xn‖ ≤ ‖un −un−1‖ ≤ (1 − (1 −ρ)αn)‖xn −xn−1‖ + M [|αn −αn−1| + |βn −βn−1|] (3.16) +‖Tnyn−1 −Tn−1yn−1‖ ≤ (1 − (1 −ρ)αn)‖xn −xn−1‖ + M(τ + 1)αn [ |αn −αn−1| αn + |βn −βn−1| βn ] +αn [ sup { ‖Tnz −Tn−1z‖ αn ,z ∈ B }] . 60 DADASHI AND AMJADI From (C1), (C3), (C5) and Lemma 2.4, we can deduce that ‖xn+1 −xn‖→ 0. By (3.16) and (C4) we have, ‖xn+1 −xn‖ βn ≤ ‖un −un−1‖ βn ≤ (1 − (1 −ρ)αn) ‖xn −xn−1‖ βn + M [ |αn −αn−1| βn + |βn −βn−1| βn ] + [ ‖Tnyn−1 −Tn−1yn−1‖ βn ] = (1 − (1 −ρ)αn) ‖xn −xn−1‖ βn−1 + (1 − (1 −ρ)αn) [ 1 βn − 1 βn−1 ] ‖xn −xn−1‖ +Mαn [ |αn −αn−1| + |βn −βn−1| αnβn ] + αn [ ‖Tnyn−1 −Tn−1yn−1‖ αnβn ] ≤ (1 − (1 −ρ)αn) ‖xn −xn−1‖ βn−1 + αn [ 1 βn − 1 βn−1 ] 1 αn ‖xn −xn−1‖ +Mαn [ |αn −αn−1| + |βn −βn−1| αnβn ] + αn sup z∈B { ‖Tnz −Tn−1z‖ αnβn } ≤ (1 − (1 −ρ)αn) ‖xn −xn−1‖ βn−1 + αnK‖xn −xn−1‖ +Mαn [ |αn −αn−1| + |βn −βn−1| αnβn ] + αn sup z∈B { ‖Tnz −Tn−1z‖ αnβn } . Again, (C1), (C3), (C5) and Lemma 2.4 imply that lim n→∞ ‖xn+1 −xn‖ βn = 0, lim n→∞ ‖un −un−1‖ βn = 0, and hence by (C2′) we get lim n→∞ ‖un −un−1‖ αn = 0. It follows from (C1) and (C2′) that βn → 0 and by (3.10), ‖yn − xn‖ → 0 and ‖xn+1 − Tnyn‖ → 0. Then, ‖xn −Tnxn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −Tnyn‖ + ‖Tnyn −Tnxn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −Tnyn‖ + ‖yn −xn‖→ 0 as n →∞, therefore, we have ‖xn −Txn‖ ≤ ‖xn −Tnxn‖ + ‖Tnxn −Txn‖ ≤ ‖xn −Tnxn‖ + sup{‖Tnz −Tz‖, z ∈{xn}}→ 0 as n →∞. By demiclosedness principle, Lemma 2.2, we obtain ww(xn) ⊆ F(T) = ∞⋂ n=1 F(Tn). Also ‖yn −Tnyn‖≤‖yn −xn‖ + ‖xn −xn+1‖ + ‖xn+1 −Tnyn‖→ 0. From (3.10), we have xn+1 = PC(un) −un + αnf(xn) + (1 −αn)(Tnyn −yn) + (1 −αn)[βnSxn + (1 −βn)xn], and hence xn −xn+1 = xn −PC(un) + un −αnf(xn) − (1 −αn)(Tnyn −yn) − (1 −αn)βnSxn − (1 −αn)(1 −βn)xn = un −PC(un) + αn(I −f)xn + (1 −αn)(I −Tn)yn + (1 −αn)βn(I −S)xn. Set vn = xn−xn+1 (1−αn)βn . Hence, we obtain vn = 1 (1 −αn)βn (un −PC[un]) + αn (1 −αn)βn (I −f)xn + 1 βn (I −Tn)yn + (I −S)xn. A GENERALIZED ITERATIVE ALGORITHM FOR HIERARCHICAL FIXED POINTS PROBLEMS 61 For any z ∈ ∞⋂ n=1 F(Tn) we have 〈vn,xn −z〉 = 1 (1 −αn)βn 〈un −PC(un),xn −z〉 + αn (1 −αn)βn 〈(I −f)xn,xn −z〉 (3.17) + 1 βn 〈(I −Tn)yn,xn −z〉 + 〈(I −S)xn,xn −z〉 = 1 (1 −αn)βn 〈un −PC(un),PC(un−1) −PC(un) + PC(un) −z〉 + αn (1 −αn)βn 〈(I −f)xn − (I −f)z + (I −f)z,xn −z〉 + 1 βn 〈(I −Tn)yn − (I −Tn)z,xn −z〉 + 〈(I −S)xn − (I −S)z + (I −S)z,xn −z〉 = 1 (1 −αn)βn 〈un −PC(un),PC(un) −z〉 + 1 (1 −αn)βn 〈un −PC(un),PC(un−1) −PC(un)〉 + αn (1 −αn)βn 〈(I −f)xn − (I −f)z,xn −z〉 + αn (1 −αn)βn 〈(I −f)z,xn −z〉 + 1 βn 〈(I −Tn)yn − (I −Tn)z,xn −yn〉 + 1 βn 〈(I −Tn)yn − (I −Tn)z,yn −z〉 +〈(I −S)xn − (I −S)z,xn −z〉 + 〈(I −S)z,xn −z〉 By Lemma 2.3, we obtain 〈vn,xn −z〉 ≥ 1 (1 −αn)βn 〈un −PC(un),PC(un−1) −PC(un)〉 + αn(1 −ρ) (1 −αn)βn ‖xn −z‖2 + αn (1 −αn)βn 〈(I −f)z,xn −z〉 + 1 βn 〈(I −Tn)yn − (I −Tn)z,xn −yn〉 + 〈(I −Sn)z,xn −z〉 ≥ 1 (1 −αn)βn 〈un −PC(un),PC(un−1) −PC(un)〉 + αn(1 −ρ) (1 −αn)βn ‖xn −z‖2 + αn (1 −αn)βn 〈(I −f)z,xn −z〉 +〈(I −Tn)yn,xn −Sxn〉 + 〈(I −S)z,xn −z〉 Then it follows that ‖xn −z‖2 ≤ (1 −αn)βn αn(1 −ρ) [〈vn,xn −z〉− 1 (1 −αn)βn 〈un −PC(un),PC(un−1) −PC(un)〉 − αn (1 −αn)βn 〈(I −f)z,xn −z〉−〈(I −Tn)yn,xn −Sxn〉−〈(I −S)z,xn −z〉] ≤ (1 −αn)βn αn(1 −ρ) [〈vn,xn −z〉−〈(I −Tn)yn,xn −Sxn〉−〈(I −S)z,xn −z〉] + ‖un −un−1‖ αn(1 −ρ) ‖un −PC(un)‖− 1 (1 −ρ) 〈(I −f)z,xn −z〉 Since vn → 0, (I − Tn)yn → 0, ‖un−un−1‖ αn → 0 and ωw(xn) ⊆ F(T) = ∞⋂ n=1 F(Tn), then every weak cluster point of {xn} is also a strong cluster point. It follows from the boundedness of the sequence {xn} that there exists a subsequence {xnk} converging to a point x ′ ∈H. For all z ∈ F(T), it follows 62 DADASHI AND AMJADI from (3.17) that 〈(I −f)xnk,xnk −z〉 = (1 −αnk )βnk αnk 〈vnk,xnk −z〉− (1 −αnk )βnk αnk 〈(I −S)xnk,xnk −z〉 − 1 αnk 〈unk −PC(unk ),PC(unk−1 ) −z〉− (1 −αnk ) αnk 〈(I −Tnk )ynk,xnk −z〉 ≤ (1 −αnk )βnk αnk 〈vnk,xnk −z〉− (1 −αnk )βnk αnk 〈(I −S)z,xnk −z〉 − 1 αnk 〈unk −PC(unk ),PC(unk−1 ) −PC(unk )〉 − (1 −αnk ) αnk 〈(I −Tnk )ynk − (I −Tnk )z,xnk −ynk〉 ≤ (1 −αnk )βnk αnk 〈vnk,xnk −z〉− (1 −αnk )βnk αnk 〈(I −S)z,xnk −z〉 + ‖unk −unk−1‖ αnk ‖unk −PC(unk )‖− (1 −αnk )βnk αnk 〈(I −Tnk )ynk,xnk −Sxnk〉. Letting k →∞, we obtain 〈(I −f)x′,x′ −z〉 ≤ −τ〈(I −S)z,x′ −z〉. Thus x′ is a solution of the variational inequality (3.11) and since (3.11)has the unique solution, it follows that ωw(xn) = ωs(xn) = {x∗} and this ensures that xn → x∗ as n →∞. � Corollary 3.5. Suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (C1), (C3), (C4), (C5) and (C2′) lim n→∞ βn αn = 1; Then the sequence {xn} defined by (1.8) converges strongly to a point x∗ ∈ F, which is the unique solution of the variational inequality 〈(2I −S)x∗,y −x∗〉≥ 0, ∀y ∈ F. (3.18) Proof. It is sufficient that assume f = 0 and τ = 1 in Theorem 3.4. � Corollary 3.6. Let S,T : C → C be two nonexpansive mappings with F(T) 6= ∅. Suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (C1), (C2′′), (C3) and (C4). Then the sequence {xn} defined by (3.6) converges strongly to a point x∗ ∈ F(T), which is the unique solution of the variational inequality 〈 1 τ (I −f)x∗ + (I −S)x∗,y −x∗〉≥ 0, ∀y ∈ F(T). Proof. It is sufficient that assume Tn = T in Theorem 3.4. � Acknowledgment Vahid Dadashi and Somayeh Amjadi are supported by Sari Branch, Islamic Azad University. References [1] M. Alimohammady and V. Dadashi, Convergence of a generalized iterations for a countable family of nonexpansive mappings, TJMM. 4(1) (2012), 15–24. [2] F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, On a two-step algorithm for hierarchical fixed point problems and variational inequalities, J. Inequal. Appl., 2009 (2009), Article ID 208692. [3] V. Dadashi, S. Ghafari, Convergence theorems of iterative approximation for finding zeros of accretive operator and fixed points problems, Int. J. Nonlinear Anal. Appl., 4(2) (2013) , 53–61. [4] Y. Dong and X. Zhang, New step lengths in projection method for variational inequality problems, Appl. Math. Comput. 220(1) (2013), 239–245. A GENERALIZED ITERATIVE ALGORITHM FOR HIERARCHICAL FIXED POINTS PROBLEMS 63 [5] K. Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, 1990. [6] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 220(1) (1967), 957–961. [7] P.E. Mainge and A. Moudafi, Strong convergence of an iterative method for hierarchical fixed point problems, Pacific J. Optim. 3 (2007), 529–538. [8] A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241 (2000), 46–55. [9] X.Qin, M. Shang, H. Zhou, Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces, Appl. Math. Comput., 200(1) (2008), 242–253. [10] Y. Yao, Y.J. Cho, and Y.C. Liou, Iterative algorithms for hierarchical fixed points problems and variational in- equalities, Mathematical and Computer Modelling, 52(9) (2010), 1697–1705. [11] H. K. Xu, Iterative algorithm for nonlinear operators, J. Lond. Math. Soc., 2 (2002), 1–17. Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran ∗Corresponding author: Vahid.Dadashi@iausari.ac.ir 1. Introduction 2. Preliminaries 3. Main results Acknowledgment References