Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 531859, 19 pages http://dx.doi.org/10.1155/2013/531859 Research Article Relaxed Viscosity Approximation Methods with Regularization for Constrained Minimization Problems Lu-Chuan Ceng,1 Hong-Kun Xu,2,3 and Ching-Feng Wen4 1 Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China 2 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan 3 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 4 Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan Correspondence should be addressed to Ching-Feng Wen; cfwen@kmu.edu.tw Received 13 January 2013; Accepted 26 March 2013 Academic Editor: Luigi Muglia Copyright © 2013 Lu-Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a new relaxed viscosity approximation method with regularization and prove the strong convergence of the method to a common fixed point of finitely many nonexpansive mappings and a strict pseudocontraction that also solves a convex minimization problem and a suitable equilibrium problem. 1. Introduction Let đģ be a real Hilbert space with inner product â¨⋅, ⋅⊠and norm â ⋅ â, đļ a nonempty closed convex subset of đģ, and đđļ the metric projection of đģ, onto đļ. Let đ : đļ → đļ be selfmapping on đļ. We denote by Fix(đ) the set of fixed points of đ and by R the set of all real numbers. A mapping đ : đļ → đļ is called đ-strictly pseudocontractive if there exists a constant đ ∈ [0, 1) such that ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđđĨ − đđĻķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĨ − đĻķĩŠķĩŠķĩŠ + đķĩŠķĩŠķĩŠ(đŧ − đ) đĨ − (đŧ − đ) đĻķĩŠķĩŠķĩŠ , ∀đĨ, đĻ ∈ đļ. (1) In particular, if đ = 0, then đ is called a nonexpansive mapping. A mapping đ´ : đļ → đģ is called đŧ-inverse strongly monotone, if there exists a constant đŧ > 0 such that ķĩŠ2 ķĩŠ â¨đ´đĨ − đ´đĻ, đĨ − đĻ⊠≥ đŧķĩŠķĩŠķĩŠđĨ − đĻķĩŠķĩŠķĩŠ , ∀đĨ, đĻ ∈ đļ. (2) Let đ : đļ → R be a convex and a continuous FreĖchet differentiable functional. Consider the minimization problem (MP) of minimizing đ over the constraint set đļ minđ (đĨ) , đĨ∈đļ (3) where we assume the existence of minimizers. We denote by Γ the set of minimizers of (3). The gradient-projection algorithm (GPA) generates a sequence {đĨđ } determined by the gradient ∇đ and the metric projection đđļ as follows: đĨđ+1 := đđļ (đĨđ − đ∇đ (đĨđ )) , ∀đ ≥ 0, (4) ∀đ ≥ 0, (5) or more generally, đĨđ+1 := đđļ (đĨđ − đ đ ∇đ (đĨđ )) , where, in both (4) and (5), the initial guess đĨ0 is taken from đļ arbitrarily, the parameters đ or đ đ are positive real numbers. The convergence of algorithms (4) and (5) depends on the behavior of the gradient ∇đ. As a matter of fact, it is known that if ∇đ is strongly monotone and Lipschitz continuous, then, for 0 < đ < 2đŧ/đŋ2 , the operator đ := đđļ (đŧ − đ∇đ) (6) is a contraction. Hence, the sequence {đĨđ } defined by the GPA (4) converges in norm to the unique solution of (3). More generally, if the sequence {đ đ } is chosen to satisfy the property 0 < lim inf đ đ ≤ lim supđ đ < đ→∞ đ→∞ 2đŧ , đŋ2 (7) 2 Journal of Applied Mathematics then the sequence {đĨđ } defined by the GPA (5) converges in norm to the unique minimizer of (3). If the gradient ∇đ is only assumed to be a Lipschitz continuous, then {đĨđ } can only be weakly convergent if đģ is infinite dimensional. A counterexample is given by Xu in [1]. Since the Lipschitz continuity of the gradient ∇đ implies that it is inverse strongly monotone (ism), it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping. Consequently, the GPA can be rewritten as the composite of a projectionand an averaged mapping which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Very recently, Xu [1] used averaged mappings to study the convergence analysis of the GPA which is an operator-oriented approach. We observe that the regularization, in particular, the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the following regularized minimization problem: đŧ minđđŧ (đĨ) := đ (đĨ) + âđĨâ2 , (8) đĨ∈đļ 2 where đŧ > 0 is the regularization parameter and again đ is convex with an đŋ-Lipschitz continuous gradient ∇đ. The advantage of a regularization method is that it is possible to get strong convergence to the minimum-norm solution of the optimization problem under investigation. The disadvantage is however its implicity, and hence explicit iterative methods seem more attractive. See, for example, [1]. Given a mapping đ´ : đļ → đģ, the classical variational inequality problem (VIP) is to find đĨ∗ ∈ đļ such that â¨đ´đĨ∗ , đĨ − đĨ∗ ⊠≥ 0, ∀đĨ ∈ đļ. (9) The solution set of VIP (9) is denoted by VI(đļ, đ´). It is well known that đĨ∗ ∈ VI(đļ, đ´) if and only if đĨ∗ = đđļ(đĨ∗ − đđ´đĨ∗ ) for some đ > 0. The variational inequality was first discussed by Lions [2] and now is well known. The variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, and equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. See, for example, [3–10] and the references therein. In this paper, we study the following equilibrium problem (EP) which is to find đĨ∗ ∈ đļ such that đš (đĨ∗ , đĻ) + â (đĨ∗ , đĻ) ≥ 0, ∀đĻ ∈ đļ. (10) The solution set of EP (10) is denoted by EP(đš, â). We will introduce and consider a relaxed viscosity iterative scheme with regularization for finding a common element of the solution set Γ of the minimization problem (3), the solution set EP(đš, â) of the equilibrium problem (10), and the common fixed point set Fix(đ) ∩ (âđ Fix(đđ )) of finitely many nonexpansive mappings đđ : đļ → đļ, đ = 1, . . . , đ, and a strictly pseudocontractive mapping đ in the setting of the infinitedimensional Hilbert space. We will prove that this iterative scheme converges strongly to a common fixed point of the mappings đ, đđ : đļ → đļ, đ = 1, . . . , đ, which is both a minimizer of MP (3) and an equilibrium point of EP (10). 2. Preliminaries Let đģ be a real Hilbert space whose inner product and norm are denoted by â¨⋅, ⋅⊠and â⋅â, respectively. Let đž be a nonempty closed convex subset of đģ. We write đĨđ â đĨ to indicate that the sequence {đĨđ } converges weakly to đĨ and đĨđ → đĨ to indicate that the sequence {đĨđ } converges strongly to đĨ. Moreover, we use đđ¤ (đĨđ ) to denote the weak đ-limit set of the sequence {đĨđ } and đđ (đĨđ ) to denote the strong đ-limit set of the sequence {đĨđ }; that is, đđ¤ (đĨđ ) := {đĨ ∈ đģ : đĨđđ â đĨ for some subsequence {đĨđđ } of {đĨđ }} , đđ (đĨđ ) :={đĨ ∈ đģ : đĨđđ ķŗ¨→ đĨ for some subsequence {đĨđđ } of {đĨđ }} . (11) The metric (or nearest point) projection from đģ onto đž is the mapping đđž : đģ → đž which assigns to each point đĨ ∈ đģ the unique point đđž đĨ ∈ đž satisfying the property ķĩŠ ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđĨ − đđž đĨķĩŠķĩŠķĩŠ = inf ķĩŠķĩŠķĩŠđĨ − đĻķĩŠķĩŠķĩŠ =: đ (đĨ, đž) . đĻ∈đž (12) Some important properties of projections are gathered in the following. Proposition 1. For given đĨ ∈ đģ and đ§ ∈ đž (i) đ§ = đđž đĨ ⇔ â¨đĨ − đ§, đĻ − đ§âŠ ≤ 0, for all đĻ ∈ đž; (ii) đ§ = đđž đĨ ⇔ âđĨ−đ§â2 ≤ âđĨ−đĻâ2 −âđĻ−đ§â2 , for all đĻ ∈ đž; (iii) â¨đđž đĨ − đđž đĻ, đĨ − đĻ⊠≥ âđđž đĨ − đđž đĻâ2 , for all đĻ ∈ đģ, which hence implies that đđž is nonexpansive and monotone. Definition 2. A mapping đ : đģ → đģ is said to be (a) nonexpansive if ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđđĨ − đđĻķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĨ − đĻķĩŠķĩŠķĩŠ , ∀đĨ, đĻ ∈ đģ; (13) (b) firmly nonexpansive if 2đ − đŧ is nonexpansive, or equivalently, ķĩŠ2 ķĩŠ â¨đĨ − đĻ, đđĨ − đđĻ⊠≥ ķĩŠķĩŠķĩŠđđĨ − đđĻķĩŠķĩŠķĩŠ , ∀đĨ, đĻ ∈ đģ; (14) alternatively, đ is firmly nonexpansive if and only if đ can be expressed as đ= 1 (đŧ + đ) , 2 (15) where đ : đģ → đģ is nonexpansive; projections are firmly nonexpansive. Definition 3. Let đ be a nonlinear operator with domain đˇ(đ) ⊆ đģ and range đ (đ) ⊆ đģ. Journal of Applied Mathematics 3 (a) đ is said to be monotone if â¨đĨ − đĻ, đđĨ − đđĻ⊠≥ 0, ∀đĨ, đĻ ∈ đˇ (đ) . (16) (b) Given a number đŊ > 0, đ is said to be đŊ strongly monotone if ķĩŠ2 ķĩŠ (17) â¨đĨ − đĻ, đđĨ − đđĻ⊠≥ đŊķĩŠķĩŠķĩŠđĨ − đĻķĩŠķĩŠķĩŠ , ∀đĨ, đĻ ∈ đˇ (đ) . (c) Given a number ] > 0, đ is said to be ]-inverse strongly monotone (]-ism) if ķĩŠ2 ķĩŠ â¨đĨ − đĻ, đđĨ − đđĻ⊠≥ ]ķĩŠķĩŠķĩŠđđĨ − đđĻķĩŠķĩŠķĩŠ , ∀đĨ, đĻ ∈ đˇ (đ) . (18) It can be easily seen that if đ is nonexpansive, then đŧ − đ is monotone. It is also easy to see that a projection đđž is 1-ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields. Definition 4. A mapping đ : đģ → đģ is said to be an averaged mapping if it can be written as the average of the identity đŧ and a nonexpansive mapping; that is, đ ≡ (1 − đŧ) đŧ + đŧđ, (19) where đŧ ∈ (0, 1) and đ : đģ → đģ is nonexpansive. More precisely, when the last equality holds, we say that đ is đŧaveraged. Thus, firmly nonexpansive mappings (in particular, projections) are (1/2)-averaged maps. Proposition 5 (see [11]). Let đ : đģ → đģ be a given mapping. The notation Fix(đ) denotes the set of all fixed points of the mapping đ, that is, Fix(đ) = {đĨ ∈ đģ : đđĨ = đĨ}. It is clear that, in a real Hilbert space đģ, đ : đļ → đļ is đ-strictly pseudocontractive if and only if there holds the following inequality: ķĩŠ2 1 − đ ķĩŠ â¨đđĨ − đđĻ, đĨ − đĻ⊠≤ ķĩŠķĩŠķĩŠđĨ − đĻķĩŠķĩŠķĩŠ − 2 ķĩŠ2 ķĩŠķĩŠ × ķĩŠķĩŠ(đŧ − đ)đĨ − (đŧ − đ)đĻķĩŠķĩŠķĩŠ , ∀đĨ, đĻ ∈ đļ. (21) This immediately implies that if đ is a đ-strictly pseudocontractive mapping, then đŧ − đ is ((1 − đ)/2)-inverse strongly monotone; for further detail, we refer to [12] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings. Lemma 7 (see [12, Proposition 2.1]). Let đļ be a nonempty closed convex subset of a real Hilbert space đģ and đ : đļ → đļ be a mapping. (i) If đ is a đ-strictly pseudocontractive mapping, then đ satisfies the Lipschitz condition where ķĩŠ ķĩŠ 1 + đ ķĩŠķĩŠ ķĩŠķĩŠ ķĩŠđĨ − đĻķĩŠķĩŠķĩŠ , ķĩŠķĩŠđđĨ − đđĻķĩŠķĩŠķĩŠ ≤ 1−đķĩŠ ∀đĨ, đĻ ∈ đļ. (22) (ii) If đ is a đ-strictly pseudocontractive mapping, then the mapping đŧ − đ is semiclosed at 0; that is, if {đĨđ } is a sequence in đļ such that đĨđ → đĨĖ weakly and (đŧ − đ)đĨđ → 0 strongly, then (đŧ − đ)đĨĖ = 0. (i) đ is nonexpansive if and only if the complement đŧ−đ is (1/2)-ism. (ii) If đ is ]-ism, then for đž > 0, đžđ is (]/đž)-ism. (iii) đ is averaged if and only if the complement đŧ − đ is ]ism for some ] > 1/2. Indeed, for đŧ ∈ (0, 1), đ is đŧaveraged if and only if đŧ − đ is (1/2đŧ)-ism. (iii) If đ is a đ-(quasi-)strict pseudocontraction, then the fixed point set Fix(đ) of đ is closed and convex so that the projection đFix(đ) is well defined. Proposition 6 (see [11]). Let đ, đ, đ : đģ → đģ be given operators. The following lemma is an immediate consequence of an inner product. (i) If đ = (1 − đŧ)đ + đŧđ for some đŧ ∈ (0, 1) and if đ is averaged and đ is nonexpansive, then đ is averaged. (ii) đ is firmly nonexpansive if and only if the complement đŧ − đ is firmly nonexpansive. (iii) If đ = (1 − đŧ)đ + đŧđ for some đŧ ∈ (0, 1) and if đ is firmly nonexpansive and đ is nonexpansive, then đ is averaged. (iv) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {đđ }đ đ=1 is averaged, then so is the composite đ1 ⋅ ⋅ ⋅ đđ. In particular, if đ1 is đŧ1 -averaged and đ2 is đŧ2 -averaged, where đŧ1 , đŧ2 ∈ (0, 1), then the composite đ1 đ2 is đŧaveraged, where đŧ = đŧ1 + đŧ2 − đŧ1 đŧ2 . Lemma 8. In a real Hilbert space đģ, there holds the following inequality: (v) If the mappings {đđ }đ đ=1 are averaged and have a common fixed point, then đ â Fix (đđ ) = Fix (đ1 ⋅ ⋅ ⋅ đđ) . đ=1 (20) ķĩŠ2 ķĩŠķĩŠ 2 ķĩŠķĩŠđĨ + đĻķĩŠķĩŠķĩŠ ≤ âđĨâ + 2 â¨đĻ, đĨ + đĻ⊠, ∀đĨ, đĻ ∈ đģ. (23) The following elementary result on real sequences is quite well known. Lemma 9 (see [13]). Let {đđ } be a sequence of nonnegative real numbers satisfying the property đđ+1 ≤ (1 − đ đ ) đđ + đ đ đĄđ + đđ , ∀đ ≥ 0, (24) where {đ đ } ⊂ (0, 1] and {đĄđ } are the real sequences such that (i) ∑∞ đ=0 đ đ = ∞; (ii) either lim supđ → ∞ đĄđ ≤ 0 or ∑∞ đ=0 đ đ |đĄđ | < ∞; (iii) ∑∞ đ=0 đđ < ∞ where đđ ≥ 0, for all đ ≥ 0. Then, limđ → ∞ đđ = 0. 4 Journal of Applied Mathematics Lemma 10 (see [14]). Let đļ be a nonempty closed convex subset of a real Hilbert space đģ. Let đ : đļ → đļ be a đ-strictly pseudocontractive mapping. Let đž and đŋ be two nonnegative real numbers such that (đž + đŋ)đ ≤ đž. Then, ķĩŠķĩŠķĩŠđž (đĨ − đĻ) + đŋ (đđĨ − đđĻ)ķĩŠķĩŠķĩŠ ≤ (đž + đŋ) ķĩŠķĩŠķĩŠđĨ − đĻķĩŠķĩŠķĩŠ , ∀đĨ, đĻ ∈ đļ. ķĩŠ ķĩŠ ķĩŠ ķĩŠ (25) The following lemma appears implicitly in the paper of Reinermann [15]. Lemma 11 (see [15]). Let đģ be a real Hilbert space. Then, for all đĨ, đĻ ∈ đģ and đ ∈ [0, 1], ķĩŠ2 ķĩŠķĩŠ ķĩŠķĩŠđđĨ + (1 − đ) đĻķĩŠķĩŠķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ = đâđĨâ2 + (1 − đ) ķĩŠķĩŠķĩŠđĻķĩŠķĩŠķĩŠ − đ (1 − đ) ķĩŠķĩŠķĩŠđĨ − đĻķĩŠķĩŠķĩŠ . (26) (a) if âđĨđ − đđđ đĨđ â → 0 as đ → ∞, each weak cluster point of {đĨđ } satisfies the problem: đš (đĨ, đĻ) + â (đĨ, đĻ) ≥ 0, ∀đĻ ∈ đļ, (29) that is, đđ¤ (đĨđ ) ⊆ EP(đš, â). (b) The demiclosedness principle holds in the sense that, if đĨđ â đĨ∗ and âđĨđ − đđđ đĨđ â → 0 as đ → ∞, then (đŧ − đđđ )đĨ∗ = 0 for all đ ≥ 1. 3. Main Results Lemma 12 (see [16]). Let đļ be a nonempty closed convex subset of a real Hilbert space đģ. Let đš : đļ × đļ → R be a bifunction such that (f1) đš(đĨ, đĨ) = 0 for all đĨ ∈ đļ; (f2) đš is monotone and upper hemicontinuous in the first variable; (f3) đš is lower semicontinuous and convex in the second variable. We now propose the following relaxed viscosity iterative scheme with regularization: đš (đĸđ , đĻ) + â (đĸđ , đĻ) + 1 â¨đĻ − đĸđ , đĸđ − đĨđ ⊠≥ 0, đđ ∀đĻ ∈ đļ, đĻđ,1 = đŊđ,1 đ1 đĸđ + (1 − đŊđ,1 ) đĸđ , đĻđ,đ = đŊđ,đ đđ đĸđ + (1 − đŊđ,đ ) đĻđ,đ−1 , Let â : đļ × đļ → R be a bifunction such that (h1) â(đĨ, đĨ) = 0 for all đĨ ∈ đļ; (h2) â is monotone and weakly upper semicontinuous in the first variable; (h3) â is convex in the second variable. Moreover, let one suppose that (H) for fixed đ > 0 and đĨ ∈ đļ, there exists a bounded đž ⊂ đļ Ė đ§) + and đĨĖ ∈ đž such that for all đ§ ∈ đļ \ đž, −đš(đĨ, Ė + (1/đ)â¨đĨĖ − đ§, đ§ − đĨ⊠< 0. â(đ§, đĨ) For đ > 0 and đĨ ∈ đģ, let đđ : đģ → 2đļ be a mapping defined by đđ đĨ = {đ§ ∈ đļ : đš (đ§, đĻ) + â (đ§, đĻ) Lemma 14 (see [17]). Suppose that the hypotheses of Lemma 12 are satisfied. Let {đđ } be a sequence in (0, ∞) with lim inf đ → ∞ đđ > 0. Suppose that {đĨđ } is a bounded sequence. Then, the following statements are equivalent and true. (27) 1 + â¨đĻ − đ§, đ§ − đĨ⊠≥ 0, ∀đĻ ∈ đļ} đ called the resolvent of đš and â. Then, (1) đđ đĨ =Ė¸ 0; (2) đđ đĨ is a singleton; (3) đđ is firmly nonexpansive; (4) đ¸đ(đš, â) = Fix(đđ ) and it is closed and convex. Lemma 13 (see [16]). Let one suppose that (f1)–(f3), (h1)–(h3) and (H) hold. Let đĨ, đĻ ∈ đģ, đ1 , đ2 > 0. Then, ķĩ¨ ķĩ¨ ķĩŠķĩŠķĩŠđ đĻ − đ đĨķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĻ − đĨķĩŠķĩŠķĩŠ + ķĩ¨ķĩ¨ķĩ¨ķĩ¨ đ2 − đ1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđ đĻ − đĻķĩŠķĩŠķĩŠ . (28) ķĩŠ ķĩŠ đ1 ķĩŠ ķĩŠ ķĩ¨ķĩ¨ đ ķĩ¨ķĩ¨ ķĩŠķĩŠ đ2 ķĩŠķĩŠ đ2 ķĩŠķĩŠ ķĩ¨ 2 ķĩ¨ đ = 2, . . . , đ, (30) đĻđ = đŊđ đđĻđ,đ + (1 − đŊđ ) đđļ (đĻđ,đ − đ∇đđŧđ (đĻđ,đ)) , đĨđ+1 = đđ đĻđ + đžđ đđļ (đĻđ − đ∇đđŧđ (đĻđ )) + đŋđ đđđļ (đĻđ − đ∇đđŧđ (đĻđ )) , for all đ ≥ 0, where the mapping đ : đļ → đļ is a đcontraction; the mapping đ : đļ → đļ is a đ-strict pseudocontraction; đđ : đļ → đļ is a nonexpansive mapping for each đ = 1, . . . , đ; ∇đ : đļ → đģ satisfies the Lipschitz condition (10) with 0 < đ < (2/đŋ); đš, â : đļ × đļ → R are two bifunctions satisfying the hypotheses of Lemma 12; {đŧđ } is a sequence in (0, ∞) with ∑∞ đ=0 đŧđ < ∞; {đŊđ }, {đđ } are sequences in (0, 1) with 0 < lim inf đ → ∞ đđ ≤ lim supđ → ∞ đđ < 1; {đžđ }, {đŋđ } are sequences in [0, 1] with đđ + đžđ + đŋđ = 1, for all đ ≥ 0; {đŊđ,đ }đ đ=1 are sequences in (0, 1) and (đžđ + đŋđ )đ ≤ đžđ , for all đ ≥ 0; {đđ } is a sequence in (0, ∞) with lim inf đ → ∞ đđ > 0 and lim inf đ → ∞ đŋđ > 0. Before stating and proving the main convergence results, we first establish the following lemmas. Lemma 15. Let one suppose that Ω = Fix(đ) ∩ (âđ Fix(đđ )) ∩ EP(đš, â) ∩ Γ =Ė¸ 0. Then, the sequences {đĨđ }, {đĻđ }, {đĻđ,đ } for all đ, and {đĸđ } are bounded. Proof. First of all, we can show as in [18] that đđļ(đŧ − đ∇đđŧ ) is nonexpansive for đ ∈ (0, 2/(đŧ + đŋ)), and đđļ(đŧ − đ∇đđŧđ ) is nonexpansive for all đ ≥ 0 and đ ∈ (0, 2/đŋ). We observe that if đ ∈ Ω, then ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠ (31) ķĩŠķĩŠđĻđ,1 − đķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĸđ − đķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ . Journal of Applied Mathematics 5 For all, from đ = 2 to đ = đ, by induction, one proves that ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ≤ đŊđ,đ ķĩŠķĩŠķĩŠđĸđ − đķĩŠķĩŠķĩŠ + (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđĻđ,đ−1 − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĸđ − đķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ . (32) Thus, we obtain that for every đ = 1, . . . , đ, ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĸđ − đķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ . (33) For simplicity, put đĻĖđ,đ = đđļ(đĻđ,đ − đ∇đđŧđ (đĻđ,đ)) and đĻĖđ = đđļ(đĻđ − đ∇đđŧđ (đĻđ )) for every đ ≥ 0. Then, đĻđ = đŧđ đđĻđ,đ + (1 − đŧđ )đĻĖđ,đ and đĨđ+1 = đđ đĻđ + đžđ đĻĖđ + đŋđ đđĻĖđ for every đ ≥ 0. Taking into consideration that đđ = đ and đđļ(đŧ − đ∇đ)đ = đ for đ ∈ (0, 2/đŋ), we have ķĩŠ ķĩŠķĩŠ Ė ķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đĻđ,đ − đđļ (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đĻđ,đ − đđļ (đŧ−đ∇đđŧđ ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đ − đđļ (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đ − đđļ (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ . (34) Similarly, we get âđĻĖđ − đâ ≤ âđĻđ − đâ + đđŧđ âđâ. Thus, from (34) we have ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ = ķĩŠķĩŠķĩŠđŊđ (đđĻđ,đ − đ) + (1 − đŊđ ) (đĻĖđ,đ − đ)ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đđķĩŠķĩŠķĩŠ + đŊđ ķĩŠķĩŠķĩŠđđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + (1 − đŊđ ) (ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đŊđ đ ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đŊđ ķĩŠķĩŠķĩŠđđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + (1 − đŊđ ) (ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ = (1 − đŊđ (1 − đ)) ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đŊđ ķĩŠķĩŠķĩŠđđ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ = (1 − đŊđ (1 − đ)) ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠķĩŠ ķĩŠđđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + đŊđ (1 − đ) ķĩŠ + (1 − đŊđ ) đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ 1−đ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđđ − đķĩŠķĩŠķĩŠ ≤ max {ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ , ķĩŠ } + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ 1−đ Since (đžđ + đŋđ )đ ≤ đžđ for all đ ≥ 0, utilizing Lemma 10, we derive from (35) ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđĨđ+1 − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠđ (đĻđ − đ) + đžđ (đĻĖđ − đ) + đŋđ (đđĻĖđ − đ)ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđžđ (đĻĖđ − đ) + đŋđ (đđĻĖđ − đ)ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (đžđ + đŋđ ) ķĩŠķĩŠķĩŠđĻĖđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (đžđ + đŋđ ) (ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđđ − đķĩŠķĩŠķĩŠ ≤ max {ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ , ķĩŠ } + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ 1−đ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđđ − đķĩŠķĩŠķĩŠ = max {ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ , ķĩŠ } + 2đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ . 1−đ (36) By induction, we get ķĩŠ ķĩŠ ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđđ − đķĩŠķĩŠķĩŠ } ķĩŠķĩŠđĨđ+1 − đķĩŠķĩŠķĩŠ ≤ max {ķĩŠķĩŠķĩŠđĨ0 − đķĩŠķĩŠķĩŠ , ķĩŠ 1−đ đ ķĩŠ ķĩŠ + 2đ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ ⋅ ∑đŧđ , (37) ∀đ ≥ 0. đ=0 This implies that {đĨđ } is bounded and so are {đĻđ }, {đĸđ }, and {đĻđ,đ } for each đ = 1, . . . , đ. It is clear that both {đĻĖđ,đ} and {đĻĖđ } are also bounded. Since âđđĻĖđ − đâ ≤ ((1 + đ)/(1 − đ))âđĻĖđ − đâ, {đđĻĖđ } is also bounded. Lemma 16. Let one suppose that Ω =Ė¸ 0. Moreover, let one suppose that the following hold: (H1) limđ → ∞ đŊđ = 0 and ∑∞ đ=0 đŊđ = ∞; (H2) ∑∞ đ=1 |đŊđ − đŊđ−1 | < ∞ or limđ → ∞ (|đŊđ − đŊđ−1 |/đŊđ ) = 0; (H3) ∑∞ đ=1 |đŊđ,đ −đŊđ−1,đ | < ∞ or limđ → ∞ (|đŊđ,đ −đŊđ−1,đ |/đŊđ ) = 0 for each đ = 1, . . . , đ; (H4) ∑∞ đ=1 |đđ − đđ−1 | < ∞ or limđ → ∞ (|đđ − đđ−1 |/đŊđ ) = 0; (H5) ∑∞ đ=1 |đđ − đđ−1 | < ∞ or limđ → ∞ (|đđ − đđ−1 |/đŊđ ) = 0; (H6) ∑∞ < ∞ or đ=1 |đžđ /(1 − đđ ) − đžđ−1 /(1 − đđ−1 )| limđ → ∞ (1/đŊđ )|đžđ /(1 − đđ ) − đžđ−1 /(1 − đđ−1 )| = 0. ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđđ − đķĩŠķĩŠķĩŠ ≤ max {ķĩŠķĩŠķĩŠđĸđ − đķĩŠķĩŠķĩŠ , ķĩŠ } + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ 1−đ Then, limđ → ∞ âđĨđ+1 − đĨđ â = 0, that is, {đĨđ } is asymptotically regular. ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđđ − đķĩŠķĩŠķĩŠ ≤ max {ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ , ķĩŠ } + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ . 1−đ (35) ≤ Proof. Taking into account 0 < lim inf đ → ∞ đđ lim supđ → ∞ đđ < 1, we may assume, without loss of generality, that {đđ } ⊂ [đ, đ] for some đ, đ ∈ (0, 1). First, we write 6 Journal of Applied Mathematics đĨđ = đđ−1 đĻđ−1 + (1 − đđ−1 )Vđ−1 , for all đ ≥ 1, where Vđ−1 = (đĨđ − đđ−1 đĻđ−1 )/(1 − đđ−1 ). It follows that for all đ ≥ 1 Vđ − Vđ−1 = đĨđ+1 − đđ đĻđ đĨđ − đđ−1 đĻđ−1 − 1 − đđ 1 − đđ−1 ķĩ¨ ķĩ¨ķĩŠ ķĩŠ ķĩŠ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨đŊđ −đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđđĻđ−1,đ − đĻĖđ−1,đķĩŠķĩŠķĩŠ +đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đđĻđ−1,đķĩŠķĩŠķĩŠ ķĩŠ ķĩ¨ ķĩ¨ķĩŠ ķĩŠ ķĩŠ ≤ (1 − đŊđ ) (ķĩŠķĩŠķĩŠđĻđ,đ − đĻđ−1,đķĩŠķĩŠķĩŠ +đ ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđĻđ−1,đķĩŠķĩŠķĩŠ) đžđ (đĻĖđ − đĻĖđ−1 ) + đŋđ (đđĻĖđ − đđĻĖđ−1 ) 1 − đđ ķĩ¨ ķĩ¨ķĩŠ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđđĻđ−1,đ − đĻĖđ−1,đķĩŠķĩŠķĩŠ đž đžđ−1 +( đ − ) đĻĖ 1 − đđ 1 − đđ−1 đ−1 +( đŋđ−1 đŋđ − ) đđĻĖđ−1 . 1 − đđ 1 − đđ−1 ķĩŠ ķĩŠ + đŊđ đ ķĩŠķĩŠķĩŠđĻđ,đ − đĻđ−1,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĻđ,đ − đĻđ−1,đķĩŠķĩŠķĩŠ (38) Since (đžđ + đŋđ )đ ≤ đžđ for all đ ≥ 0, utilizing Lemma 10, we have ķĩŠķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđžđ (đĻĖđ − đĻĖđ−1 ) + đŋđ (đđĻĖđ − đđĻĖđ−1 )ķĩŠķĩŠ (39) ķĩŠ ķĩŠ ≤ (đžđ + đŋđ ) ķĩŠķĩŠķĩŠđĻĖđ − đĻĖđ−1 ķĩŠķĩŠķĩŠ . Next, we estimate âđĻđ −đĻđ−1 â. Observe that for every đ ≥ 1 ķĩŠķĩŠ Ė ķĩŠ ķĩŠķĩŠđĻđ,đ − đĻĖđ−1,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đĻđ,đ − đđļ (đŧ − đ∇đđŧđ ) đĻđ−1,đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đĻđ−1,đ − đđļ (đŧ − đ∇đđŧđ−1 ) đĻđ−1,đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ,đ − đĻđ−1,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đĻđ−1,đ − đđļ (đŧ − đ∇đđŧđ−1 ) đĻđ−1,đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ,đ − đĻđ−1,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ−1,đ − (đŧ − đ∇đđŧđ−1 ) đĻđ−1,đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠđĻđ,đ − đĻđ−1,đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠķĩŠđ∇đđŧđ (đĻđ−1,đ) − đ∇đđŧđ−1 (đĻđ−1,đ)ķĩŠķĩŠķĩŠķĩŠ ķĩ¨ ķĩŠ ķĩŠ ķĩ¨ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠđĻđ,đ − đĻđ−1,đķĩŠķĩŠķĩŠ + đ ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđĻđ−1,đķĩŠķĩŠķĩŠ , (40) and similarly, ķĩŠķĩŠ Ė ķĩ¨ ķĩŠ ķĩŠ ķĩŠ ķĩ¨ķĩŠ ķĩŠ ķĩŠķĩŠđĻđ − đĻĖđ−1 ķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ − đĻđ−1 ķĩŠķĩŠķĩŠ + đ ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđĻđ−1 ķĩŠķĩŠķĩŠ . Also, from (30), we have ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđĻđ − đĻđ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ (1 − đŊđ ) ķĩŠķĩŠķĩŠđĻĖđ,đ − đĻĖđ−1,đķĩŠķĩŠķĩŠ đž đĻĖ + đŋđ đđĻĖđ đžđ−1 đĻĖđ−1 + đŋđ−1 đđĻĖđ−1 = đ đ − 1 − đđ 1 − đđ−1 = Then, passing to the norm we get from (40) that (41) ķĩ¨ ķĩ¨ķĩŠ ķĩŠ + đ ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđĻđ−1,đķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ķĩŠ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđđĻđ−1,đ − đĻĖđ−1,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĻđ,đ − đĻđ−1,đķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + đ1 (ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨) , where đâđĻđ,đâ + âđđĻđ,đ − đĻĖđ,đâ ≤ đ1 , for all đ ≥ 0 for some đ1 ≥ 0. Furthermore, by the definition of đĻđ,đ one obtains that, for all đ = đ, . . . , 2 ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđĻđ,đ − đĻđ−1,đ ķĩŠķĩŠķĩŠ ≤ đŊđ,đ ķĩŠķĩŠķĩŠđĸđ − đĸđ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ķĩŠķĩŠķĩŠđđ đĸđ−1 − đĻđ−1,đ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ ķĩŠ ķĩŠ + (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđĻđ,đ−1 − đĻđ−1,đ−1 ķĩŠķĩŠķĩŠ . (42) ∀đ ≥ 1. ķĩŠķĩŠ ķĩŠ ķĩŠķĩŠđĻđ,1 − đĻđ−1,1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ đŊđ,1 ķĩŠķĩŠķĩŠđĸđ − đĸđ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ķĩŠķĩŠķĩŠđ1 đĸđ−1 − đĸđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,1 − đŊđ−1,1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ ķĩŠ + (1 − đŊđ,1 ) ķĩŠķĩŠķĩŠđĸđ − đĸđ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩ¨ ķĩ¨ = ķĩŠķĩŠķĩŠđĸđ − đĸđ−1 ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđ1 đĸđ−1 − đĸđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,1 − đŊđ−1,1 ķĩ¨ķĩ¨ķĩ¨ . Substituting (46) in all (45) type one obtains for đ = 2, . . . , đ đ ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ∑ ķĩŠķĩŠķĩŠđđ đĸđ−1 − đĻđ−1,đ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đĻđ − đĻđ−1 = (1 − đŊđ ) (đĻĖđ,đ − đĻĖđ−1,đ) + đŊđ (đđĻđ,đ − đđĻđ−1,đ) . (46) ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđĻđ,đ − đĻđ−1,đ ķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĸđ − đĸđ−1 ķĩŠķĩŠķĩŠ Simple calculations show that + (đŊđ − đŊđ−1 ) (đđĻđ−1,đ − đĻĖđ−1,đ) (45) In the case of đ = 1, we have đĻđ = đŊđ đđĻđ,đ + (1 − đŊđ ) đĻĖđ,đ, đĻđ−1 = đŊđ−1 đđĻđ−1,đ+ (1 − đŊđ−1 ) đĻĖđ−1,đ, (44) đ=2 (43) ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ķĩŠķĩŠķĩŠđ1 đĸđ−1 − đĸđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,1 − đŊđ−1,1 ķĩ¨ķĩ¨ķĩ¨ . (47) Journal of Applied Mathematics 7 ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ đ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + đ2 [ ķĩ¨ đ đ−1 ķĩ¨ + ∑ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ,1 − đŊđ−1,1 ķĩ¨ķĩ¨ķĩ¨ đđ đ=2 This together with (44) implies that ķĩŠķĩŠ ķĩŠ ķĩŠķĩŠđĻđ − đĻđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ] ≤ (1 − (1 − đ) đŊđ ) ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ đ ķĩ¨ ķĩ¨ + đ2 [ ķĩ¨ đ đ−1 ķĩ¨ + ∑ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ đ=1 ķĩŠ ķĩŠ × [ ķĩŠķĩŠķĩŠđĸđ − đĸđ−1 ķĩŠķĩŠķĩŠ đ ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ∑ ķĩŠķĩŠķĩŠđđ đĸđ−1 − đĻđ−1,đ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ] , đ=2 ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ ķĩŠķĩŠ ķĩ¨ķĩ¨ ķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨ + ķĩŠķĩŠđ1 đĸđ−1 − đĸđ−1 ķĩŠķĩŠ ķĩ¨ķĩ¨đŊđ,1 − đŊđ−1,1 ķĩ¨ķĩ¨ķĩ¨] ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + đ1 (ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨) ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĸđ − đĸđ−1 ķĩŠķĩŠķĩŠ đ ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ∑ ķĩŠķĩŠķĩŠđđ đĸđ−1 − đĻđ−1,đ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ=2 ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ķĩŠķĩŠķĩŠđ1 đĸđ−1 − đĸđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,1 − đŊđ−1,1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + đ1 (ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨) . By Lemma 13, we know that ķĩ¨ķĩ¨ đ ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđĸđ − đĸđ−1 ķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ + đ ķĩ¨ķĩ¨ķĩ¨1 − đ−1 ķĩ¨ķĩ¨ķĩ¨ , đđ ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ (49) where đ = supđ≥0 âđĸđ − đĨđ â. So, substituting (49) in (48) we obtain ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđĻđ − đĻđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ đ ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) (ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ + đ ķĩ¨ķĩ¨ķĩ¨1 − đ−1 ķĩ¨ķĩ¨ķĩ¨) ķĩ¨ķĩ¨ đđ ķĩ¨ķĩ¨ đ ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ∑ ķĩŠķĩŠķĩŠđđ đĸđ−1 − đĻđ−1,đ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ=2 ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ķĩŠķĩŠķĩŠđ1 đĸđ−1 − đĸđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,1 − đŊđ−1,1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + đ1 (ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨) ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ + đ ķĩ¨ đ đ−1 ķĩ¨ đ đ đ ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ∑ ķĩŠķĩŠķĩŠđđ đĸđ−1 − đĻđ−1,đ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ=2 ķĩŠ ķĩŠķĩ¨ ķĩ¨ + ķĩŠķĩŠķĩŠđ1 đĸđ−1 − đĸđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đŊđ,1 − đŊđ−1,1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + đ1 (ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨) ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ (50) (48) where đ > 0 is a minorant for {đđ } and đ + đ1 + ∑đ đ=2 âđđ đĸđ − đĻđ,đ−1 â + âđ1 đĸđ − đĸđ â ≤ đ2 , for all đ ≥ 0 for some đ2 ≥ 0. This together with (38)-(39), implies that ķĩŠ ķĩŠ ķĩŠķĩŠ ķĩŠ ķĩŠķĩŠđžđ (đĻĖđ − đĻĖđ−1 ) + đŋđ (đđĻĖđ − đđĻĖđ−1 )ķĩŠķĩŠķĩŠ ķĩŠķĩŠVđ − Vđ−1 ķĩŠķĩŠķĩŠ ≤ ķĩŠ 1 − đđ ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠ ķĩ¨ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ ķĩŠđĻĖ ķĩŠķĩŠ ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ đ−1 ķĩŠ ķĩ¨ķĩ¨ đŋ đŋđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠ ķĩ¨ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ķĩ¨ ķĩŠķĩŠđđĻĖđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ ķĩŠ ķĩŠ (đžđ + đŋđ ) ķĩŠķĩŠķĩŠđĻĖđ − đĻĖđ−1 ķĩŠķĩŠķĩŠ ≤ 1 − đđ ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠ ķĩ¨ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ ķĩŠđĻĖ ķĩŠķĩŠ ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ đ−1 ķĩŠ ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠ ķĩ¨ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ ķĩŠđđĻĖ ķĩŠķĩŠ ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ đ−1 ķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠđĻĖđ − đĻĖđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠ ķĩ¨ ķĩŠ ķĩŠ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ (ķĩŠđĻĖ ķĩŠķĩŠ + ķĩŠķĩŠđđĻĖ ķĩŠķĩŠ) ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ đ−1 ķĩŠ ķĩŠ đ−1 ķĩŠ ķĩ¨ ķĩŠ ķĩŠ ķĩ¨ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ − đĻđ−1 ķĩŠķĩŠķĩŠ + đ ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđĻđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠ ķĩ¨ ķĩŠ ķĩŠ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ (ķĩŠđĻĖ ķĩŠķĩŠ + ķĩŠķĩŠđđĻĖ ķĩŠķĩŠ) ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ đ−1 ķĩŠ ķĩŠ đ−1 ķĩŠ ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨đ − đ ķĩ¨ķĩ¨ķĩ¨ đ ķĩ¨ ķĩ¨ + đ2 [ ķĩ¨ đ đ−1 ķĩ¨ + ∑ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ đ=1 ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ] ķĩ¨ ķĩ¨ķĩŠ ķĩŠ + đ ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđĻđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠ ķĩ¨ ķĩŠ ķĩŠ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ (ķĩŠđĻĖ ķĩŠķĩŠ + ķĩŠķĩŠđđĻĖ ķĩŠķĩŠ) ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ đ−1 ķĩŠ ķĩŠ đ−1 ķĩŠ ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ 8 Journal of Applied Mathematics ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ đ ķĩ¨ ķĩ¨ + đ3 [ ķĩ¨ đ đ−1 ķĩ¨ + ∑ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ đ=1 ķĩ¨ķĩŠ ķĩŠ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđĻđ−1 − Vđ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + đđ { (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ] ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + đ3 ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + đ3 ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ ķĩŠ = (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ đ ķĩ¨ķĩ¨ ķĩ¨ + đ3 [ + ∑ ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ đ=1 ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + 2 ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨], ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ where đ2 + đâđĻđ â + âđĻĖđ â + âđđĻĖđ â ≤ đ3 , for all đ ≥ 0 for some đ3 ≥ 0. Further, we observe that đĨđ+1 = đđ đĻđ + (1 − đđ ) Vđ , (52) đĨđ+1 − đĨđ = (1 − đđ ) (Vđ − Vđ−1 ) Then, passing to the norm, we get from (51) ķĩŠķĩŠ ķĩŠ ķĩŠķĩŠđĨđ+1 − đĨđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ (1 − đđ ) ķĩŠķĩŠķĩŠVđ − Vđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ķĩŠ ķĩŠ ķĩŠ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđĻđ−1 − Vđ−1 ķĩŠķĩŠķĩŠ + đđ ķĩŠķĩŠķĩŠđĻđ − đĻđ−1 ķĩŠķĩŠķĩŠ ≤ (1 − đđ ) ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ đ ķĩ¨ ķĩ¨ + đ3 [ ķĩ¨ đ đ−1 ķĩ¨ + ∑ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ đ=1 ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨] ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ķĩŠ ķĩŠ + ķĩ¨ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠķĩŠķĩŠđĻđ−1 − Vđ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ đ ķĩ¨ ķĩ¨ + đ [ ķĩ¨ đ đ−1 ķĩ¨ + ∑ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ đ=1 ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ ] + 2đ ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ , ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ Simple calculations show that + đđ (đĻđ − đĻđ−1 ) . ķĩŠ ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ∀đ ≥ 1. + (đđ − đđ−1 ) (đĻđ−1 − Vđ−1 ) ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ]} ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + 2 ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ (51) đĨđ = đđ−1 đĻđ−1 + (1 − đŊđ−1 ) Vđ−1 , ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ đ ķĩ¨ķĩ¨ ķĩ¨ + đ2 [ + ∑ ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ đ=1 (54) (53) where đ3 + âđĻđ − Vđ â ≤ đ, for all đ ≥ 0 for some đ ≥ 0. By hypotheses (H1)–(H6) and Lemma 9, from ∑∞ đ=0 đŧđ < ∞, we obtain the claim. Lemma 17. Let one suppose that Ω =Ė¸ 0. Let one suppose that {đĨđ } is asymptotically regular. Then, âđĨđ − đĻđ â → 0 and âđĨđ − đĸđ â = âđĨđ − đđđ đĨđ â → 0 as đ → ∞. Proof. We recall that, by the firm nonexpansivity of đđđ , a standard calculation (see [17]) shows that if V ∈ EP(đš, â), then ķĩŠķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠķĩŠđĸđ − VķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĨđ − VķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĨđ − đĸđ ķĩŠķĩŠķĩŠ . ķĩŠ ķĩŠ × { (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ đ ķĩ¨ ķĩ¨ + đ3 [ ķĩ¨ đ đ−1 ķĩ¨ + ∑ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ đ đ=1 ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ + 2 ķĩ¨ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ + ķĩ¨ķĩ¨ķĩ¨ đ − ķĩ¨ ]} ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ (55) Let đ ∈ Ω. Then by Lemma 11, we have from (33)–(34) the following ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠķĩŠ Ė ķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ = ķĩŠķĩŠķĩŠđŊđ (đđĻđ,đ − đ) + (1 − đŊđ ) (đĻĖđ,đ − đ)ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + [ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ] Journal of Applied Mathematics 9 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 = đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) , ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ = đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (1 − đđ ) ķĩŠķĩŠķĩŠđĻĖđ − đķĩŠķĩŠķĩŠ − ķĩŠ ķĩŠ2 ≤ đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (1 − đđ ) (56) ķĩŠ2 ķĩŠķĩŠ Ė ķĩŠ2 ķĩŠ ķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ = ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đĻđ − đđļ (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ ≤ (ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ = ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) . (57) Since (đžđ + đŋđ )đ ≤ đžđ for all đ ≥ 0, utilizing Lemma 10, we have ķĩŠķĩŠ ķĩŠ2 ķĩŠķĩŠđĨđ+1 − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 = ķĩŠķĩŠķĩŠđđ (đĻđ − đ) + đžđ (đĻĖđ − đ) + đŋđ (đđĻĖđ − đ)ķĩŠķĩŠķĩŠ ķĩŠķĩŠ 1 ķĩŠ = ķĩŠķĩŠķĩŠđđ (đĻđ − đ) + (đžđ + đŋđ ) ķĩŠķĩŠ đžđ + đŋđ ķĩŠķĩŠ2 ķĩŠ × [đžđ (đĻĖđ − đ) + đŋđ (đđĻĖđ − đ)] ķĩŠķĩŠķĩŠ ķĩŠķĩŠ ķĩŠ ķĩŠ2 = đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (đžđ + đŋđ ) ķĩŠķĩŠ2 ķĩŠķĩŠķĩŠ 1 ķĩŠ [đžđ (đĻĖđ − đ) + đŋđ (đđĻĖđ − đ)]ķĩŠķĩŠķĩŠ × ķĩŠķĩŠķĩŠ ķĩŠķĩŠ ķĩŠķĩŠ đžđ + đŋđ ķĩŠķĩŠ 1 ķĩŠ − đđ (đžđ + đŋđ ) ķĩŠķĩŠķĩŠ(đĻđ − đ) − ķĩŠķĩŠ đžđ + đŋđ ķĩŠķĩŠ2 ķĩŠ × [đžđ (đĻĖđ − đ) + đŋđ (đđĻĖđ − đ)] ķĩŠķĩŠķĩŠ ķĩŠķĩŠ ķĩŠ ķĩŠ2 = đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (đžđ + đŋđ ) ķĩŠķĩŠ2 ķĩŠķĩŠ 1 ķĩŠ ķĩŠ [đžđ (đĻĖđ − đ) + đŋđ (đđĻĖđ − đ)]ķĩŠķĩŠķĩŠ × ķĩŠķĩŠķĩŠ ķĩŠķĩŠ ķĩŠķĩŠ đžđ + đŋđ ķĩŠķĩŠ 1 ķĩŠķĩŠ2 ķĩŠ ķĩŠ − đđ (đžđ + đŋđ ) ķĩŠķĩŠķĩŠ [đžđ (đĻĖđ − đĻđ ) + đŋđ (đđĻĖđ − đĻđ )]ķĩŠķĩŠķĩŠ ķĩŠķĩŠ đžđ + đŋđ ķĩŠķĩŠ ķĩŠ ķĩŠ2 = đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (đžđ + đŋđ ) ķĩŠķĩŠ 1 ķĩŠķĩŠ2 ķĩŠ ķĩŠ × ķĩŠķĩŠķĩŠ [đžđ (đĻĖđ − đ) + đŋđ (đđĻĖđ − đ)]ķĩŠķĩŠķĩŠ ķĩŠķĩŠ đžđ + đŋđ ķĩŠķĩŠ đđ ķĩŠķĩŠ ķĩŠ2 − ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ đžđ + đŋđ ķĩŠ đ+1 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (đžđ + đŋđ ) ķĩŠķĩŠķĩŠđĻĖđ − đķĩŠķĩŠķĩŠ − đđ ķĩŠķĩŠ ķĩŠ2 ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ đžđ + đŋđ ķĩŠ đ+1 đđ ķĩŠķĩŠ ķĩŠ2 ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ 1 − đđ ķĩŠ đ+1 ķĩŠ2 ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ × [ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ)] − đđ ķĩŠķĩŠ ķĩŠ2 ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ 1 − đđ ķĩŠ đ+1 ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) − đđ ķĩŠķĩŠ ķĩŠ2 ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ 1 − đđ ķĩŠ đ+1 ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) − đđ ķĩŠķĩŠ ķĩŠ2 ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ 1 − đđ ķĩŠ đ+1 ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 = ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + 2đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) đđ ķĩŠķĩŠ ķĩŠ2 − ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ . 1 − đđ ķĩŠ đ+1 (58) Taking into account 0 < lim inf đ → ∞ đđ ≤ lim supđ → ∞ đđ < 1, we may assume that {đđ } ⊂ [đ, đ] for some đ, đ ∈ (0, 1). So, we deduce that đ ķĩŠķĩŠ ķĩŠ2 ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ 1 − đ ķĩŠ đ+1 đđ ķĩŠķĩŠ ķĩŠ2 ≤ ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ 1 − đđ ķĩŠ đ+1 ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ2 ≤ ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĨđ+1 − đķĩŠķĩŠķĩŠ + đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + 2đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ≤ (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ+1 − đķĩŠķĩŠķĩŠ) ķĩŠķĩŠķĩŠđĨđ − đĨđ+1 ķĩŠķĩŠķĩŠ + đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + 2đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) . (59) Since đŧđ → 0, đŊđ → 0 and âđĨđ − đĨđ+1 â → 0 as đ → ∞, we conclude from the boundedness of {đĨđ }, {đĻđ }, and {đĻđ,đ} that âđĨđ+1 −đĻđ â → 0 as đ → ∞. This together with âđĨđ −đĨđ+1 â → 0, implies that ķĩŠķĩŠ lim ķĩŠđĨ đ→∞ ķĩŠ đ ķĩŠ − đĻđ ķĩŠķĩŠķĩŠ = 0. (60) 10 Journal of Applied Mathematics Furthermore, from (33), (55), and (56), we have ķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĸđ − đķĩŠķĩŠķĩŠ (61) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĨđ − đĸđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) , which hence implies that ķĩŠķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠķĩŠđĨđ − đĸđ ķĩŠķĩŠķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 (62) ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ) ķĩŠķĩŠķĩŠđĨđ − đĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) . (H8) there exists a constant đ > 0 such that (1/đŊđ )|1/đŊđ,đ0 − 1/đŊđ−1,đ0 | < đ for all đ ≥ 1. Then, ķĩŠķĩŠ ķĩŠ ķĩŠđĨ − đĨđ ķĩŠķĩŠķĩŠ lim ķĩŠ đ+1 = 0. đ→∞ đŊđ,đ0 Proof. We start by (54). Dividing both the terms by đŊđ,đ0 we have ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđĨđ+1 − đĨđ ķĩŠķĩŠķĩŠ đŊđ,đ0 ķĩŠ ķĩŠķĩŠ ķĩŠđĨ − đĨđ−1 ķĩŠķĩŠķĩŠ ≤ [1 − (1 − đ) đŊđ ] ķĩŠ đ đŊđ,đ0 ķĩ¨ đ ķĩ¨ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ∑đ=1 ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ +đ [ + đđŊđ,đ0 đŊđ,đ0 ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ 2 ķĩ¨ķĩ¨đŧ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ đ + + đŊđ,đ0 đŊđ,đ0 đŊđ,đ0 Since đŧđ → 0, đŊđ → 0 and âđĨđ − đĻđ â → 0 as đ → ∞, we deduce from the boundedness of {đĨđ }, {đĻđ }, and {đĻđ,đ} that ķĩŠķĩŠ lim ķĩŠđĨ đ→∞ ķĩŠ đ ķĩŠ − đĸđ ķĩŠķĩŠķĩŠ = 0. (66) ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đž / (1 − đđ ) − đžđ−1 / (1 − đđ−1 )ķĩ¨ķĩ¨ķĩ¨ +ķĩ¨ đ ]. đŊđ,đ0 (63) (67) So, by (H8) we have Remark 18. By the last lemma we have đđ¤ (đĨđ ) = đđ¤ (đĸđ ) and đđ (đĨđ ) = đđ (đĸđ ); that is, the sets of strong/weak cluster points of {đĨđ } and {đĸđ } coincide. Of course, if đŊđ,đ → đŊđ =Ė¸ 0, as đ → ∞, for all index đ, the assumptions of Lemma 16 are enough to assure that ķĩŠķĩŠ ķĩŠ ķĩŠđĨ − đĨđ ķĩŠķĩŠķĩŠ = 0, lim ķĩŠ đ+1 đ→∞ đŊđ,đ ∀đ ∈ {1, . . . , đ} . (64) In the next lemma, we examine the case in which at least one sequence {đŊđ,đ0 } is a null sequence. Lemma 19. Let one suppose that Ω =Ė¸ 0. Let one suppose that (H1) holds. Moreover, for an index đ0 ∈ {1, . . . , đ}, limđ → ∞ đŊđ,đ0 = 0, and the following hold: (H7) for all đ, ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨đŧđ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ = lim = lim lim đ→∞ đ→∞ đŊ đŊ đ→∞ đŊ đŊ đŊđ đŊđ,đ0 đ đ,đ0 đ đ,đ0 ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨đ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ = lim ķĩ¨ đ = lim ķĩ¨ đ đ−1 ķĩ¨ đ→∞ đŊ đŊ đ→∞ đŊ đŊ đ đ,đ0 đ đ,đ0 1 đ→∞đŊ đŊ đ đ,đ0 = lim ķĩ¨ķĩ¨ đž đžđ−1 ķĩ¨ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ đ − ķĩ¨ķĩ¨ ķĩ¨ = 0; ķĩ¨ķĩ¨ 1 − đđ 1 − đđ−1 ķĩ¨ķĩ¨ķĩ¨ (65) ķĩŠķĩŠ ķĩŠ ķĩŠķĩŠđĨđ+1 − đĨđ ķĩŠķĩŠķĩŠ đŊđ,đ0 ķĩŠ ķĩŠķĩŠ ķĩŠđĨ − đĨđ−1 ķĩŠķĩŠķĩŠ ≤ [1 − (1 − đ) đŊđ ] ķĩŠ đ đŊđ−1,đ0 ķĩ¨ ķĩ¨ 1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ ķĩ¨ķĩ¨ 1 ķĩŠ ķĩ¨ķĩ¨ + [1 − (1 − đ) đŊđ ] ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨ķĩ¨ − ķĩ¨ķĩ¨ đŊđ,đ0 đŊđ−1,đ0 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ đ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ ∑ ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ + đ [ ķĩ¨ đ đ−1 ķĩ¨ + đ=1 ķĩ¨ đđŊđ,đ0 đŊđ,đ0 ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đž / (1 − đđ ) − đžđ−1 / (1 − đđ−1 )ķĩ¨ķĩ¨ķĩ¨ +ķĩ¨ đ đŊđ,đ0 ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ 2 ķĩ¨ķĩ¨đŧ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ đ + + ] đŊđ,đ0 đŊđ,đ0 đŊđ,đ0 ķĩŠ ķĩŠķĩŠ ķĩŠđĨ − đĨđ−1 ķĩŠķĩŠķĩŠ ≤ [1 − (1 − đ) đŊđ ] ķĩŠ đ đŊđ−1,đ0 ķĩ¨ ķĩ¨ 1 ķĩ¨ķĩ¨ķĩ¨ ķĩŠ ķĩ¨ķĩ¨ 1 ķĩŠ ķĩ¨ķĩ¨ + ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩ¨ķĩ¨ķĩ¨ķĩ¨ − ķĩ¨ķĩ¨ đŊđ,đ0 đŊđ−1,đ0 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ķĩ¨đ − đ ķĩ¨ķĩ¨ķĩ¨ ∑đ ķĩ¨ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ + đ [ ķĩ¨ đ đ−1 ķĩ¨ + đ=1 ķĩ¨ đđŊđ,đ0 đŊđ,đ0 Journal of Applied Mathematics 11 ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đž / (1 − đđ ) − đžđ−1 / (1 − đđ−1 )ķĩ¨ķĩ¨ķĩ¨ +ķĩ¨ đ đŊđ,đ0 Proof. Let đ ∈ Ω. Then, by Lemma 11 we have ķĩ¨ ķĩ¨ 2 ķĩ¨ķĩ¨đŧ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ đ đŊđ,đ0 ķĩŠ2 ķĩŠķĩŠ ķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 = ķĩŠķĩŠķĩŠđŊđ (đđĻđ,đ − đ) + (1 − đŊđ ) (đĻĖđ,đ − đ)ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 = đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩŠ ķĩŠ2 × ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đĻđ,đ − đđļ(đŧ − đ∇đ)đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đŊ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ +ķĩ¨ đ + ] đŊđ,đ0 đŊđ,đ0 ķĩŠ ķĩŠķĩŠ ķĩŠđĨ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + đŊđ đ ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ≤ [1 − (1 − đ) đŊđ ] ķĩŠ đ đŊđ−1,đ0 ķĩ¨ đ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ ∑ ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ + đ [ ķĩ¨ đ đ−1 ķĩ¨ + đ=1 ķĩ¨ đđŊđ,đ0 đŊđ,đ0 ķĩŠ ķĩŠ2 × ķĩŠķĩŠķĩŠ(đŧ − đ∇đ) đĻđ,đ − (đŧ − đ∇đ)đ − đđŧđ đĻđ,đķĩŠķĩŠķĩŠ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đž / (1 − đđ ) − đžđ−1 / (1 − đđ−1 )ķĩ¨ķĩ¨ķĩ¨ +ķĩ¨ đ đŊđ,đ0 ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ 2 ķĩ¨ķĩ¨đŧ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ķĩ¨đđ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ đ + + ] đŊđ,đ0 đŊđ,đ0 đŊđ,đ0 ķĩŠ ķĩŠķĩŠ ķĩŠđĨ − đĨđ−1 ķĩŠķĩŠķĩŠ 1 + (1 − đ) đŊđ ⋅ = [1 − (1 − đ) đŊđ ] ķĩŠ đ đŊđ−1,đ0 1−đ ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩŠ2 ķĩŠ × [ķĩŠķĩŠķĩŠ(đŧ − đ∇đ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠ −2đđŧđ â¨đĻđ,đ, (đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đ⊠] ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) 2 ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 × [ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + 2đ (đ − ) ķĩŠķĩŠķĩŠ∇đ(đĻđ,đ) − ∇đ(đ)ķĩŠķĩŠķĩŠ đŋ ķĩŠ ķĩŠ × {đ ķĩŠķĩŠķĩŠđĨđ − đĨđ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠ +2đđŧđ ķĩŠķĩŠķĩŠđĻđ,đķĩŠķĩŠķĩŠ ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ] ķĩ¨ đ ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨ ķĩ¨đ − đ ķĩ¨ķĩ¨ ∑ ķĩ¨ķĩ¨đŊđ,đ − đŊđ−1,đ ķĩ¨ķĩ¨ķĩ¨ + đ [ ķĩ¨ đ đ−1 ķĩ¨ + đ=1 ķĩ¨ đđŊđ đŊđ,đ0 đŊđ đŊđ,đ0 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đž / (1 − đđ ) − đžđ−1 / (1 − đđ−1 )ķĩ¨ķĩ¨ķĩ¨ +ķĩ¨ đ đŊđ đŊđ,đ0 2 ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 × [ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + 2đ (đ − ) ķĩŠķĩŠķĩŠ∇đ(đĻđ,đ) − ∇đ(đ)ķĩŠķĩŠķĩŠ đŋ ķĩ¨ ķĩ¨ ķĩ¨ ķĩ¨ 2 ķĩ¨ķĩ¨đŧ − đŧđ−1 ķĩ¨ķĩ¨ķĩ¨ ķĩ¨ķĩ¨ķĩ¨đŊđ − đŊđ−1 ķĩ¨ķĩ¨ķĩ¨ + ķĩ¨ đ + đŊđ đŊđ,đ0 đŊđ đŊđ,đ0 ķĩŠ ķĩŠ ķĩŠķĩŠ +2đđŧđ ķĩŠķĩŠķĩŠđĻđ,đķĩŠķĩŠķĩŠ ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ] . (71) ķĩ¨ ķĩ¨ķĩ¨ ķĩ¨đ − đđ−1 ķĩ¨ķĩ¨ķĩ¨ +ķĩ¨ đ ]} . đŊđ đŊđ,đ0 So, we obtain (68) Therefore, utilizing Lemma 9, from (H1), (H7), and the asymptotical regularity of {đĨđ } (due to Lemma 16), we deduce that ķĩŠķĩŠ ķĩŠ ķĩŠķĩŠđĨđ+1 − đĨđ ķĩŠķĩŠķĩŠ = 0. lim đ→∞ đŊđ,đ0 (69) Lemma 20. Let one suppose that Ω =Ė¸ 0. Let one suppose that (H1)–(H6) hold. Then, ķĩŠ ķĩŠ ķĩŠ ķĩŠ lim ķĩŠķĩŠđĻĖ − đĻđ,đķĩŠķĩŠķĩŠ = lim ķĩŠķĩŠķĩŠđĻĖđ − đĻđ ķĩŠķĩŠķĩŠ = 0. đ → ∞ ķĩŠ đ,đ đ→∞ (70) 2 ķĩŠ2 ķĩŠ − đ) ķĩŠķĩŠķĩŠ∇đ(đĻđ,đ) − ∇đ(đ)ķĩŠķĩŠķĩŠ đŋ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + (1 − đŊđ ) 2đđŧđ ķĩŠķĩŠķĩŠđĻđ,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ × ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ × (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠķĩŠ + 2đđŧđ ķĩŠķĩŠķĩŠđĻđ,đķĩŠķĩŠķĩŠ ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ) (ķĩŠķĩŠķĩŠđĨđ − đĻđ ķĩŠķĩŠķĩŠ) (1 − đŊđ ) 2đ ( ķĩŠ ķĩŠ ķĩŠķĩŠ + 2đđŧđ ķĩŠķĩŠķĩŠđĻđ,đķĩŠķĩŠķĩŠ ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ . (72) 12 Journal of Applied Mathematics Since đŧđ → 0, đŊđ → 0, âđĨđ − đĻđ â → 0, and 0 < đ < 2/đŋ, from the boundedness of {đĨđ }, {đĻđ }, and {đĻđ,đ} it follows that limđ → ∞ â∇đ(đĻđ,đ) − ∇đ(đ)â = 0, and hence ķĩŠ ķĩŠ lim ķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ = 0. (73) đ→∞ ķĩŠ Moreover, from the firm nonexpansiveness of đđļ we obtain ķĩŠķĩŠ Ė ķĩŠ2 ķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 = ķĩŠķĩŠķĩŠķĩŠđđļ(đŧ − đ∇đđŧđ )đĻđ,đ − đđļ(đŧ − đ∇đ)đķĩŠķĩŠķĩŠķĩŠ 1 ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ≤ {ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + 2đ ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ 2 ķĩŠ ķĩŠ × ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĻđ,đ − đĻĖđ,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 − (1 − đŊđ ) đ2 ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 − ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + 2đ ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ × ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 − (1 − đŊđ ) đ2 ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ) ķĩŠķĩŠķĩŠđĨđ − đĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + 2đ ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ 2ķĩŠ ķĩŠ (74) ķĩŠ ķĩŠ × ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠ + 2đ ķĩŠķĩŠķĩŠđĻđ,đ − đĻĖđ,đķĩŠķĩŠķĩŠ ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ . and so ķĩŠķĩŠ Ė ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĻđ,đ − đĻĖđ,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + 2đ ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ × ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ (77) Since đŊđ → 0, âđĨđ − đĻđ â → 0, and â∇đđŧđ (đĻđ,đ) − ∇đ(đ)â → 0, from the boundedness of {đĨđ }, {đĻđ }, {đĻđ,đ}, and {đĻĖđ,đ}, it follows that limđ → ∞ âđĻđ,đ − đĻĖđ,đâ = 0. Observe that + 2đ â¨đĻđ,đ − đĻĖđ,đ, ∇đđŧđ (đĻđ,đ) − ∇đ (đ)⊠(75) Thus, we have ķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ2 ķĩŠ − (1 − đŊđ ) ķĩŠķĩŠķĩŠđĻđ,đ − đĻĖđ,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + 2đ ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ × ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđĻđ − đĻĖđ,đķĩŠķĩŠķĩŠ = đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đĻĖđ,đķĩŠķĩŠķĩŠ ķŗ¨→ 0 as đ ķŗ¨→ ∞, (78) and hence ķĩŠķĩŠ Ė ķĩŠ ķĩŠķĩŠđĻđ − đĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻĖđ − đĻĖđ,đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠķĩŠđđļ (đŧ − đ∇đđŧđ ) đĻđ −đđļ (đŧ − đ∇đđŧđ ) đĻđ,đķĩŠķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ − đĻđ,đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĻđ − đĻĖđ,đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đĻđ,đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ = 2 ķĩŠķĩŠķĩŠđĻđ − đĻĖđ,đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đĻđ,đķĩŠķĩŠķĩŠ ķŗ¨→ 0 as đ ķŗ¨→ ∞. (79) Thus, limđ → ∞ âđĻĖđ − đĻđ â = 0. + 2 (1−đŊđ ) đ â¨đĻđ,đ − đĻĖđ,đ, ∇đđŧđ (đĻđ,đ)−∇đ (đ)⊠ķĩŠ ķĩŠ2 − (1 − đŊđ ) đ2 ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ , ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 − ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + 2đ ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ + 2 (1 − đŊđ ) đ â¨đĻđ,đ − đĻĖđ,đ, ∇đđŧđ (đĻđ,đ) − ∇đ (đ)⊠+ 2đ â¨đĻđ,đ − đĻĖđ,đ, ∇đđŧđ (đĻđ,đ) − ∇đ (đ)⊠ķĩŠ ķĩŠ2 − đ2 ķĩŠķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ . ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + 2 (1 − đŊđ ) đâ¨đĻđ,đ − đĻĖđ,đ, ∇đđŧđ (đĻđ,đ) − ∇đ (đ)⊠1 ķĩŠķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 {ķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ 2 ķĩŠ ķĩŠ ķĩŠ2 − ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đ − (đĻĖđ,đ − đ)ķĩŠķĩŠķĩŠķĩŠ } ķĩŠ2 − đ ķĩŠķĩŠķĩŠ∇đđŧđ (đĻđ,đ) − ∇đ (đ)ķĩŠķĩŠķĩŠķĩŠ } , ķĩŠ2 ķĩŠ (1 − đŊđ ) ķĩŠķĩŠķĩŠđĻđ,đ − đĻĖđ,đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ × ķĩŠķĩŠķĩŠķĩŠ(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đķĩŠķĩŠķĩŠķĩŠ ≤ â¨(đŧ − đ∇đđŧđ ) đĻđ,đ − (đŧ − đ∇đ) đ, đĻĖđ,đ − đ⊠= which implies that (76) Lemma 21. Let one suppose that Ω =Ė¸ 0. Let one suppose that 0 < lim inf đ → ∞ đŊđ,đ ≤ lim supđ → ∞ đŊđ,đ < 1 for each đ = 1, . . . , đ. Moreover, suppose that (H1)–(H6) are satisfied. Then, limđ → ∞ âđđ đĸđ − đĸđ â = 0 for each đ = 1, . . . , đ. Journal of Applied Mathematics 13 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ ķĩŠ2 ķĩŠ − đŊđ,đ (1 − đŊđ,đ) ķĩŠķĩŠķĩŠđđđĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) . Proof. First of all, observe that đĨđ+1 − đĻđ = đžđ (đĻĖđ − đĻđ ) + đŋđ (đđĻĖđ − đĻđ ) = đžđ (đĻĖđ − đĻđ ) + đŋđ (đđĻĖđ − đđĻđ ) + đŋđ (đđĻđ − đĻđ ) . (80) By Lemmas 16 and 20, we know that âđĨđ+1 − đĨđ â → 0 and âđĻĖđ − đĻđ â → 0 as đ → ∞. Hence, utilizing Lemma 7(i), we have ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ) ķĩŠķĩŠķĩŠđĨđ − đĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ 1 + đ ķĩŠķĩŠ Ė ķĩŠ ≤ ķĩŠķĩŠķĩŠđĨđ+1 − đĨđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đĻđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ − đĻđ ķĩŠķĩŠķĩŠ + ķĩŠđĻ − đĻđ ķĩŠķĩŠķĩŠ 1−đķĩŠ đ 2 ķĩŠķĩŠ ķĩŠ ķĩŠđĻĖ − đĻđ ķĩŠķĩŠķĩŠ , 1−đķĩŠ đ (81) which together with âđĨđ − đĻđ â → 0 implies that limđ → ∞ âđŋđ (đđĻđ − đĻđ )â = 0. Taking into account lim inf đ → ∞ đŋđ > 0, we have ķĩŠ ķĩŠ lim ķĩŠķĩŠđđĻđ − đĻđ ķĩŠķĩŠķĩŠ = 0. đ→∞ ķĩŠ So, we have ķĩŠ2 ķĩŠ đŊđ,đ (1 − đŊđ,đ) ķĩŠķĩŠķĩŠđđđĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ ķĩŠķĩŠ ķĩŠ ķĩŠķĩŠđŋđ (đđĻđ − đĻđ )ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠđĨđ+1 − đĻđ − đžđ (đĻĖđ − đĻđ ) − đŋđ (đđĻĖđ − đđĻđ )ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĨđ+1 − đĻđ ķĩŠķĩŠķĩŠ + đžđ ķĩŠķĩŠķĩŠđĻĖđ − đĻđ ķĩŠķĩŠķĩŠ + đŋđ ķĩŠķĩŠķĩŠđđĻĖđ − đđĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđĨđ+1 − đĻđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ − đĻđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđđĻĖđ − đđĻđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠđĨđ+1 − đĨđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đĻđ ķĩŠķĩŠķĩŠ + (83) (82) Let us show that for each đ ∈ {1, . . . , đ}, one has âđđ đĸđ − đĻđ,đ−1 â → 0 as đ → ∞. Let đ ∈ Ω. When đ = đ, by Lemma 11, we have from (33)-(34) the following: ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠķĩŠ ķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 = đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 = đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đŊđ,đķĩŠķĩŠķĩŠđđđĸđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 + (1 − đŊđ,đ) ķĩŠķĩŠķĩŠđĻđ,đ−1 − đķĩŠķĩŠķĩŠ ķĩŠ2 ķĩŠ − đŊđ,đ (1 − đŊđ,đ) ķĩŠķĩŠķĩŠđđđĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĸđ − đķĩŠķĩŠķĩŠ ķĩŠ2 ķĩŠ − đŊđ,đ (1 − đŊđ,đ) ķĩŠķĩŠķĩŠđđđĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) . (84) Since đŧđ → 0, đŊđ → 0, 0 < lim inf đ → ∞ đŊđ,đ ≤ lim supđ → ∞ đŊđ,đ < 1, and limđ → ∞ âđĨđ − đĻđ â = 0, it is known that {âđđđĸđ − đĻđ,đ−1 â} is a null sequence. Let đ ∈ {1, . . . , đ − 1}. Then, one has ķĩŠ2 ķĩŠķĩŠ ķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻĖđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + (ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 = đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đŊđ,đķĩŠķĩŠķĩŠđđđĸđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 + (1 − đŊđ,đ) ķĩŠķĩŠķĩŠđĻđ,đ−1 − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đŊđ,đķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 + (1 − đŊđ,đ) ķĩŠķĩŠķĩŠđĻđ,đ−1 − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 + đŊđ,đķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 + (1 − đŊđ,đ) [đŊđ,đ−1 ķĩŠķĩŠķĩŠđđ−1 đĸđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 + (1 − đŊđ,đ−1 ) ķĩŠķĩŠķĩŠđĻđ,đ−2 − đķĩŠķĩŠķĩŠ ] ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) 14 Journal of Applied Mathematics ķĩŠ2 ķĩŠ + (đŊđ,đ + (1 − đŊđ,đ) đŊđ,đ−1 ) ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ To conclude, we have that đ ķĩŠ ķĩŠ2 + ∏ (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđĻđ,đ−2 − đķĩŠķĩŠķĩŠ , đ=đ−1 (85) and so, after (đ − đ + 1) iterations, ķĩŠ2 ķĩŠķĩŠ ķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ķĩŠ2 ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) đ đ đ=đ+2 đ=đ + (đŊđ,đ + ∑ (∏ (1 − đŊđ,đ )) đŊđ,đ−1 ) đ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 × ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ∏ (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ đ=đ+1 ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) đ đ đ=đ+2 đ=đ + (đŊđ,đ + ∑ (∏ (1 − đŊđ,đ )) đŊđ,đ−1 ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđ2 đĸđ − đĸđ ķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđ2 đĸđ − đĻđ,1 ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ,1 − đĸđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠđ2 đĸđ − đĻđ,1 ķĩŠķĩŠķĩŠ + đŊđ,1 ķĩŠķĩŠķĩŠđ1 đĸđ − đĸđ ķĩŠķĩŠķĩŠ , (89) from which âđ2 đĸđ − đĸđ â → 0. Thus, by induction âđđ đĸđ − đĸđ â → 0 for all đ = 2, . . . , đ since it is enough to observe that ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđđ đĸđ − đĸđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđđ đĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ,đ−1 − đđ−1 đĸđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđđ−1 đĸđ − đĸđ ķĩŠķĩŠķĩŠ (90) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđđ đĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ + (1 − đŊđ,đ−1 ) ķĩŠķĩŠķĩŠđđ−1 đĸđ − đĻđ,đ−2 ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ + ķĩŠķĩŠķĩŠđđ−1 đĸđ − đĸđ ķĩŠķĩŠķĩŠ . Remark 22. As an example, we consider đ = 2 and the following sequences: (a) đđ = 1/2 + 2/đ, đžđ = đŋđ = 1/4 − 1/đ for all đ > 4; (b) đŊđ = 1/√đ, đđ = 2 − 1/đ, for all đ > 1; (c) đŊđ,1 = 1/2 − 1/đ, đŊđ,2 = 1/2 − 1/đ2 , for all đ > 2. đ ķĩŠ ķĩŠ2 × ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ∏ (1 − đŊđ,đ ) Then, they satisfy the hypotheses on the parameter sequences in Lemma 21. đ=đ+1 ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 × [đŊđ,đ ķĩŠķĩŠķĩŠđđ đĸđ − đķĩŠķĩŠķĩŠ + (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđĻđ,đ−1 − đķĩŠķĩŠķĩŠ ķĩŠ2 ķĩŠ −đŊđ,đ (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđđ đĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ ] ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) đ Lemma 23. Let one suppose that Ω =Ė¸ 0 and đŊđ,đ → đŊđ for all đ as đ → ∞. Suppose there exists đ ∈ {1, . . . , đ} such that đŊđ,đ → 0 as đ → ∞. Let đ0 ∈ {1, . . . , đ} be the largest index such that đŊđ,đ0 → 0 as đ → ∞. Suppose that (i) (đŧđ + đŊđ )/đŊđ,đ0 → 0 as đ → ∞; ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − đŊđ,đ ∏ (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđđ đĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ . đ=đ (86) Again, we obtain that (ii) if đ ≤ đ0 and đŊđ,đ → 0, then đŊđ,đ0 /đŊđ,đ → 0 as đ → ∞; (iii) if đŊđ,đ → đŊđ =Ė¸ 0, then đŊđ lies in (0, 1). đ ķĩŠ2 ķĩŠ đŊđ,đ ∏ (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđđ đĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ Moreover, suppose that (H1), (H7), and (H8) hold. Then, limđ → ∞ âđđ đĸđ − đĸđ â = 0 for each đ = 1, . . . , đ. đ=đ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ) ķĩŠķĩŠķĩŠđĨđ − đĻđ ķĩŠķĩŠķĩŠ . (87) Since đŧđ → 0, đŊđ → 0, 0 < lim inf đ → ∞ đŊđ,đ ≤ lim supđ → ∞ đŊđ,đ < 1 for each đ = 1, . . . , đ − 1, and limđ → ∞ âđĨđ − đĻđ â = 0, it is known that ķĩŠķĩŠ lim ķĩŠđ đĸ đ→∞ ķĩŠ đ đ ķĩŠ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ = 0. Obviously, for đ = 1, we have âđ1 đĸđ − đĸđ â → 0. (88) Proof. First of all, we note that if (H7) holds, then also (H2)– (H6) are satisfied. So {đĨđ } is asymptotically regular. Let đ0 be as in the hypotheses. As in Lemma 21, for every index đ ∈ {1, . . . , đ} such that đŊđ,đ → đŊđ =Ė¸ 0 (which leads to 0 < lim inf đ → ∞ đŊđ,đ ≤ lim supđ → ∞ đŊđ,đ < 1), one has âđđ đĸđ − đĻđ,đ−1 â → 0 as đ → ∞. For all the other indexes đ ≤ đ0 , we can prove that âđđ đĸđ − đĻđ,đ−1 â → 0 as đ → ∞ in a similar manner. By the following relation (due to (86)): ķĩŠ2 ķĩŠķĩŠ ķĩŠķĩŠđĨđ+1 − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ≤ đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (đžđ + đŋđ ) ķĩŠķĩŠķĩŠđĻĖđ − đķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 = đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (1 − đđ ) ķĩŠķĩŠķĩŠđĻĖđ − đķĩŠķĩŠķĩŠ Journal of Applied Mathematics 15 ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ2 ķĩŠ ≤ đđ ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + (1 − đđ ) (ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) Proof. By Remark 18, we have đđ¤ (đĨđ ) = đđ¤ (đĸđ ) and đđ (đĨđ ) = đđ (đĸđ ). Observe that ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ≤ (ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđĨđ − đĻđ,1 ķĩŠķĩŠķĩŠ ≤ ķĩŠķĩŠķĩŠđĨđ − đĸđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ,1 − đĸđ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ = ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠđĨđ − đĸđ ķĩŠķĩŠķĩŠ + đŊđ,1 ķĩŠķĩŠķĩŠđ1 đĸđ − đĸđ ķĩŠķĩŠķĩŠ . ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ,đ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) đ ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − đŊđ,đ ∏ (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđđ đĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ By Lemmas 17 and 21, âđĨđ − đĸđ â → 0 and âđ1 đĸđ − đĸđ â → 0 as đ → ∞, and hence đ=đ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠķĩŠ lim ķĩŠđĨ đ→∞ ķĩŠ đ ķĩŠ ķĩŠ ķĩŠ ķĩŠ2 ≤ đŊđ ķĩŠķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + 2đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ × (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) đ ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ − đŊđ,đ ∏ (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđđ đĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ , đ=đ (91) we immediately obtain that ķĩŠ2 ķĩŠ ∏ (1 − đŊđ,đ ) ķĩŠķĩŠķĩŠđđ đĸđ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ (96) So, we get đđ¤ (đĨđ ) = đđ¤ (đĻđ,1 ) and đđ (đĨđ ) = đđ (đĻđ,1 ). Let đ ∈ đđ¤ (đĨđ ). Since đ ∈ đđ¤ (đĸđ ), by Lemma 21 and Lemma 7(ii) (demiclosedness principle), we have đ ∈ Fix(đđ ) for all index đ, that is, đ ∈ âđ Fix(đđ ). Taking into consideration that đ is đ-strictly pseudocontractive, by Lemma 7(i), we get ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ≤ ķĩŠķĩŠķĩŠđđĨđ − đđĻđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđđĻđ − đĻđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đĨđ ķĩŠķĩŠķĩŠ đ=đ đŊđ ķĩŠķĩŠ ķĩŠ2 đŧ ķĩŠ ķĩŠ ķĩŠķĩŠđđĻđ,đ − đķĩŠķĩŠķĩŠ + đ 2đ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ đŊđ,đ đŊđ,đ ķĩŠ − đĻđ,1 ķĩŠķĩŠķĩŠ = 0. ķĩŠ ķĩŠķĩŠ ķĩŠķĩŠđđĨđ − đĨđ ķĩŠķĩŠķĩŠ đ ≤ (95) (92) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ × (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđĨ − đĨđ+1 ķĩŠķĩŠķĩŠ + (ķĩŠķĩŠķĩŠđĨđ − đķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ+1 − đķĩŠķĩŠķĩŠ) ķĩŠ đ . đŊđ,đ By Lemma 19 or by hypothesis (ii) of the sequences, we have ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠķĩŠđĨđ − đĨđ+1 ķĩŠķĩŠķĩŠ ķĩŠķĩŠķĩŠđĨđ − đĨđ+1 ķĩŠķĩŠķĩŠ đŊđ,đ0 = ⋅ ķŗ¨→ 0. (93) đŊđ,đ đŊđ,đ0 đŊđ,đ So, the thesis follows. Remark 24. Let us consider đ = 3 and the following sequences: (a) đŧđ = 1/đ5/4 , đŊđ = 1/đ1/2 , đđ = 2 − 1/đ2 , for all đ > 1; (b) đđ = 1/2 + 2/đ2 , đžđ = đŋđ = 1/4 − 1/đ2 , for all đ > 2; (c) đŊđ,1 = 1/đ1/4 , đŊđ,2 = 1/2 − 1/đ2 , đŊđ,3 = 1/đ1/3 , for all đ > 1. It is easy to see that all hypotheses (i)–(iii), (H1), (H7), and (H8) of Lemma 23 are satisfied. Remark 25. Under the hypotheses of Lemma 23, analogously to Lemma 21, one can see that ķĩŠ ķĩŠ lim ķĩŠķĩŠđ đĸ − đĻđ,đ−1 ķĩŠķĩŠķĩŠ = 0, ∀đ ∈ {2, . . . , đ} . (94) đ→∞ ķĩŠ đ đ Corollary 26. Let one suppose that the hypotheses of either Lemma 21 or Lemma 23 are satisfied. Then, đđ¤ (đĨđ ) = đđ¤ (đĸđ ) = đđ¤ (đĻđ,1 ), đđ (đĨđ ) = đđ (đĸđ ) = đđ (đĻđ,1 ), and đđ¤ (đĨđ ) ⊂ Ω. ≤ 1 + đ ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đđĻđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĨđ − đĻđ ķĩŠķĩŠķĩŠ 1−đķĩŠ đ = 2 ķĩŠķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠđĨ − đĻđ ķĩŠķĩŠķĩŠ + ķĩŠķĩŠķĩŠđĻđ − đđĻđ ķĩŠķĩŠķĩŠ , 1−đķĩŠ đ (97) which together with âđĨđ − đĻđ â → 0 (by Lemma 17) and âđĻđ − đđĻđ â → 0 (by (82)) implies that ķĩŠķĩŠ lim ķĩŠđĨ đ→∞ ķĩŠ đ ķĩŠ − đđĨđ ķĩŠķĩŠķĩŠ = 0. (98) Utilizing Lemma 7(ii) (demiclosedness principle), we have đ ∈ Fix(đ). Furthermore, by Lemmas 14 and 17, we know that đ ∈ EP(đš, â). Finally, by similar argument as in [18], we can show that đ ∈ Γ, and as a result đ ∈ Ω. Theorem 27. Let one suppose that Ω =Ė¸ 0. Let {đŊđ }, {đŊđ,đ }, đ = 1, . . . , đ, be sequences in (0, 1) such that 0 < lim inf đ → ∞ đŊđ,đ ≤ lim supđ → ∞ đŊđ,đ < 1 for all index đ. Moreover, Let one suppose that (H1)–(H6) hold. Then, the sequences {đĨđ }, {đĻđ }, and {đĸđ }, explicitly defined by scheme (30), all converge strongly to the unique solution đĨ∗ ∈ Ω of the following variational inequality: â¨đđĨ∗ − đĨ∗ , đ§ − đĨ∗ ⊠≤ 0, ∀đ§ ∈ Ω. (99) Proof. Since the mapping đΩ đ is a đ-contraction, it has a unique fixed point đĨ∗ ; it is the unique solution of (99). Since (H1)–(H6) hold, the sequence {đĨđ } is asymptotically regular (by Lemma 16). In terms of Lemma 17, âđĨđ − đĻđ â → 0 and 16 Journal of Applied Mathematics âđĨđ − đĸđ â → 0 as đ → ∞. Moreover, utilizing Lemmas 8 and 10, we have from (33)-(34) the following: we know that đđ¤ (đĨđ ) ⊂ Ω, and hence đ§ ∈ Ω. Taking into consideration that đĨ∗ = đΩ đđĨ∗ we obtain from (101) that lim sup â¨đđĨ∗ − đĨ∗ , đĻđ − đĨ∗ ⊠ķĩŠķĩŠ ∗ ķĩŠ2 ķĩŠķĩŠđĨđ+1 − đĨ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ2 ķĩŠ ≤ đđ ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + (1 − đđ ) ķĩŠķĩŠķĩŠđĻĖđ − đĨ∗ ķĩŠķĩŠķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ đđ ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + (1 − đđ ) (ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) đ→∞ = lim sup [â¨đđĨ∗ − đĨ∗ , đĨđ − đĨ∗ ⊠+ â¨đđĨ∗ − đĨ∗ , đĻđ − đĨđ âŠ] đ→∞ = lim sup â¨đđĨ∗ − đĨ∗ , đĨđ − đĨ∗ ⊠đ→∞ ķĩŠ ķĩŠ ķĩŠ2 ķĩŠ ≤ (ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) = lim â¨đđĨ∗ − đĨ∗ , đĨđđ − đĨ∗ ⊠ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ = ķĩŠķĩŠķĩŠđĻđ − đĨ ķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ ķĩŠ2 ≤ ķĩŠķĩŠķĩŠđŊđ (đđĻđ,đ − đđĨ∗ ) + (1 − đŊđ ) (đĻĖđ,đ − đĨ∗ )ķĩŠķĩŠķĩŠ = â¨đđĨ∗ − đĨ∗ , đ§ − đĨ∗ ⊠≤ 0. đ→∞ ∗ķĩŠ ķĩŠ2 ∗ ∗ (102) ∞ Since ∑∞ đ=0 đŧđ < ∞ and ∑đ=0 đŊđ = ∞, we deduce that ∞ ∗ ∑đ=0 đđŧđ âđâ(2âđĻđ − đĨ â + đđŧđ âđâ) < ∞ and ∑∞ đ=0 (1 − đ)đŊđ = ∞. In terms of Lemma 9 we derive đĨđ → đĨ∗ as đ → ∞. ∗ + 2đŊđ â¨đđĨ − đĨ , đĻđ − đĨ ⊠ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) In a similar way, we can derive the following result. ķĩŠ ķĩŠ2 ķĩŠ ķĩŠ ≤ (đŊđ đ ķĩŠķĩŠķĩŠđĻđ,đ − đĨ∗ ķĩŠķĩŠķĩŠ + (1 − đŊđ ) ķĩŠķĩŠķĩŠđĻđ,đ − đĨ∗ ķĩŠķĩŠķĩŠ) + 2đŊđ â¨đđĨ∗ − đĨ∗ , đĻđ − đĨ∗ ⊠ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) 2ķĩŠ ķĩŠ2 = (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĻđ,đ − đĨ∗ ķĩŠķĩŠķĩŠ (100) + 2đŊđ â¨đđĨ∗ − đĨ∗ , đĻđ − đĨ∗ ⊠(i) (đŧđ + đŊđ )/đŊđ,đ0 → 0 as đ → ∞; ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ2 ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĸđ − đĨ∗ ķĩŠķĩŠķĩŠ (ii) if đ ≤ đ0 and đŊđ,đ → 0, then đŊđ,đ0 /đŊđ,đ → 0 as đ → ∞; (iii) if đŊđ,đ → đŊđ =Ė¸ 0, then đŊđ lies in (0, 1). + 2đŊđ â¨đđĨ∗ − đĨ∗ , đĻđ − đĨ∗ ⊠Then, the sequences {đĨđ }, {đĻđ }, and {đĸđ } explicitly defined by scheme (30) all converge strongly to the unique solution đĨ∗ ∈ Ω of the following variational inequality: ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ2 ķĩŠ ≤ (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨ∗ ķĩŠķĩŠķĩŠ â¨đđĨ∗ − đĨ∗ , đ§ − đĨ∗ ⊠≤ 0, + 2đŊđ â¨đđĨ∗ − đĨ∗ , đĻđ − đĨ∗ ⊠∀đ§ ∈ Ω. (103) Remark 29. According to the above argument processes for Theorems 27 and 28, we can readily see that if in scheme (30), the iterative step đĻđ = đŊđ đđĻđ,đ +(1−đŊđ )đđļ(đĻđ,đ −đ∇đđŧđ (đĻđ,đ)) is replaced by the iterative one đĻđ = đŊđ đđĨđ +(1−đŊđ )đđļ(đĻđ,đ − đ∇đđŧđ (đĻđ,đ)), then Theorems 27 and 28 remain valid. ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) ķĩŠ2 ķĩŠ = (1 − (1 − đ) đŊđ ) ķĩŠķĩŠķĩŠđĨđ − đĨ∗ ķĩŠķĩŠķĩŠ 2 â¨đđĨ∗ − đĨ∗ , đĻđ − đĨ∗ ⊠1−đ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ ķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ (2 ķĩŠķĩŠķĩŠđĻđ − đĨ∗ ķĩŠķĩŠķĩŠ + đđŧđ ķĩŠķĩŠķĩŠđķĩŠķĩŠķĩŠ) . + (1 − đ) đŊđ ⋅ Remark 30. Theorems 27 and 28 improve, extend, supplement, and develop [17, Theorems 3.12 and 3.13] and [1, Theorems 5.2 and 6.1] in the following aspects: Now, let {đĨđđ } be a subsequence of {đĨđ } such that lim sup â¨đđĨ∗ − đĨ∗ , đĨđ − đĨ∗ ⊠= lim â¨đđĨ∗ − đĨ∗ , đĨđđ − đĨ∗ ⊠. đ→∞ Theorem 28. Let one suppose that Ω =Ė¸ 0. Let {đŊđ }, {đŊđ,đ }, đ = 1, . . . , đ, be sequences in (0, 1) such that đŊđ,đ → đŊđ for all đ as đ → ∞. Suppose that there exists đ ∈ {1, . . . , đ} for which đŊđ,đ → 0 as đ → ∞. Let đ0 ∈ {1, . . . , đ} the largest index for which đŊđ,đ0 → 0. Moreover, let one suppose that (H1), (H7), and (H8) hold, and đ→∞ (101) By the boundedness of {đĨđ }, we may assume, without loss of generality, that đĨđđ â đ§ ∈ đđ¤ (đĨđ ). According to Corollary 26, (a) the multistep iterative scheme (30) of [17] is extended to develop our relaxed viscosity iterative scheme (30) with regularization for MP (3), EP (10), and strict pseudocontraction đ by virtue of Xu iterative schemes in [1]; (b) the argument techniques in Theorems 27 and 28 are very different from the ones in [17, Theorems 3.12 and 3.13] and the ones in [1, Theorems 5.2 and 6.1] because we use the properties of strict pseudocontractive mappings and maximal monotone mappings (see, e.g., Lemmas 7 and 10); Journal of Applied Mathematics 17 (c) compared with the proof of Theorems 5.2 and 6.1 in [1], the proof of Theorems 27 and 28 shows that limđ → ∞ âđĻđ,đ − đđļ(đŧ − đ∇đđŧđ )đĻđ,đâ = limđ → ∞ âđĻđ − đđļ(đŧ − đ∇đđŧđ )đĻđ â = 0 via the argument of limđ → ∞ â∇đđŧđ (đĻđ,đ) − ∇đ(đ)â = 0, for all đ ∈ Ω (see Lemma 20 and its proof); (d) the problem of finding an element of Fix(đ) ∩ (âđ Fix(đđ )) ∩ EP(đš, â) ∩ Γ in Theorems 27 and 28 is more general than the one of finding an element of Fix(đ) ∩ (âđ Fix(đđ )) ∩ EP(đš, â) in [17, Theorems 3.12 and 3.13] and the one of finding an element of Γ in [1, Theorems 5.2 and 6.1]. us consider the following mixed problem of finding đĨ∗ ∈ Fix(đ) ∩ EP(đš, â) ∩ Γ such that â¨(đŧ − đĖ1 ) đĨ∗ , đĻ − đĨ∗ ⊠≥ 0, ∀đĻ ∈ Fix (đ) ∩ EP (đš, â) ∩ Γ, â¨(đŧ − đĖ2 ) đĨ∗ , đĻ − đĨ∗ ⊠≥ 0, ∀đĻ ∈ Fix (đ) ∩ EP (đš, â) ∩ Γ, .. . â¨(đŧ − đĖđ) đĨ∗ , đĻ − đĨ∗ ⊠≥ 0, â¨đ´ 1 đĨ∗ , đĻ − đĨ∗ ⊠≥ 0, ∀đĻ ∈ đļ, â¨đ´ 2 đĨ∗ , đĻ − đĨ∗ ⊠≥ 0, ∀đĻ ∈ đļ, .. . 4. Applications For a given nonlinear mapping đ´ : đļ → đģ, we consider again the variational inequality problem (VIP) of finding đĨ ∈ đļ such that â¨đ´đĨ, đĻ − đĨ⊠≥ 0, ∀đĻ ∈ Fix (đ) ∩ EP (đš, â) ∩ Γ, ∀đĻ ∈ đļ. (104) Recall that if đĸ is a point in đļ, then the following relation holds: đĸ ∈ VI (đļ, đ´) ⇐⇒ đĸ = đđļ (đŧ − đđ´) đĸ, for some đ > 0, (105) from which we have the following relation: â¨đ´ đđĨ∗ , đĻ − đĨ∗ ⊠≥ 0, (108) We denote by (SVI) the set of solutions of the above (đ + đ) system. This problem is equivalent to finding a common đ fixed point of đ, {đFix(đ)∩EP(đš,â)∩Γ đĖđ }đ đ=1 , {đđļ (đŧ − đđ´ đ )}đ=1 . The following results are then consequences of Theorems 27 and 28. Theorem 31. Let one suppose that Ω = Fix(đ) ∩ (SVI) ∩ EP(F, h) ∩ Γ =Ė¸ 0. Fix đ ∈ (0, 2đŧ], and đ ∈ (0, 2/đŋ). Let {đŧđ }, {đŊđ,đ }, đ = 1, . . . , (đ + đ), be sequences in (0, 1) such that 0 < lim inf đ → ∞ đŊđ,đ ≤ lim supđ → ∞ đŊđ,đ < 1 for all index đ. Moreover, Let one suppose that (H1)–(H6) hold. Then the sequences {đĨđ }, {đĻđ }, and {đĸđ } explicitly defined by the following scheme: đš (đĸđ , đĻ) + â (đĸđ , đĻ) + đĸ ∈ Γ ⇐⇒ đĸ ∈ VI (đļ, ∇đ) ⇐⇒ đĸ = đđļ (đŧ − đ∇đ) đĸ, for some đ > 0. (106) ∀đĨ, đĻ ∈ đļ. 1 â¨đĻ − đĸđ , đĸđ − đĨđ ⊠≥ 0, đđ ∀đĻ ∈ đļ, đĻđ,1 = đŊđ,1 đFix(đ)∩EP(đš,â)∩Γ đĖ1 đĸđ + (1 − đŊđ,1 ) đĸđ , đĻđ,đ = đŊđ,đ đFix(đ)∩EP(đš,â)∩Γ đĖđ đĸđ An operator đ´ : đļ → đģ is said to be an đŧ-inverse strongly monotone operator if there exists a constant đŧ > 0 such that ķĩŠ2 ķĩŠ â¨đ´đĨ − đ´đĻ, đĨ − đĻ⊠≥ đŧķĩŠķĩŠķĩŠđ´đĨ − đ´đĻķĩŠķĩŠķĩŠ , ∀đĻ ∈ đļ. (107) As an example, we recall that the đŧ-inverse strongly monotone operators are firmly nonexpansive mappings if đŧ ≥ 1 and that every đŧ-inverse strongly monotone operator is also a (1/đŧ) Lipschitz continuous (see [19]). We observe that, if đ´ is đŧ-inverse strongly monotone, the mapping đđļ(đŧ − đđ´) is nonexpansive for all đ ∈ (0, 2đŧ] since they are compositions of nonexpansive mappings (see [19, page 419]). Let us consider đĖ1 , . . . , đĖđ a finite number of nonexpansive self-mappings on đļ and đ´ 1 , . . . , đ´ đ be a finite number of đŧ-inverse strongly monotone operators. Let đ : đļ → đļ be a đ-strict pseudocontraction on đļ with fixed points. Let + (1 − đŊđ,đ ) đĻđ,đ−1 , đ = 2, . . . , đ, đĻđ,đ+đ = đŊđ,đ+đ đđļ (đŧ − đđ´ đ ) đĸđ + (1 − đŊđ,đ+đ ) đĻđ,đ+đ−1 , đ = 1, . . . , đ, đĻđ = đŊđ đđĻđ,đ+đ + (1 − đŊđ ) × đđļ (đĻđ,đ+đ − đ∇đđŧđ (đĻđ,đ+đ)) , đĨđ+1 = đđ đĻđ + đžđ đđļ (đĻđ − đ∇đđŧđ (đĻđ )) + đŋđ đđđļ (đĻđ − đ∇đđŧđ (đĻđ )) , ∀đ ≥ 0, (109) all converge strongly to the unique solution đĨ∗ ∈ Ω of the following variational inequality: â¨đđĨ∗ − đĨ∗ , đ§ − đĨ∗ ⊠≤ 0, ∀đ§ ∈ Ω. (110) 18 Journal of Applied Mathematics Theorem 32. Let one suppose that Ω =Ė¸ 0. Fix đ ∈ (0, 2đŧ] and đ ∈ (0, 2/đŋ). Let {đŊđ }, {đŊđ,đ }, đ = 1, . . . , (đ + đ), be sequences in (0, 1) and đŊđ,đ → đŊđ for all đ as đ → ∞. Suppose that there exists đ ∈ {1, . . . , đ + đ} such that đŊđ,đ → 0 as đ → ∞. Let đ0 ∈ {1, . . . , đ + đ} be the largest index for which đŊđ,đ0 → 0. Moreover, let one suppose that (H1), (H7), and (H8) hold, and (âđ Fix(đđ )) ∩ EP(đš, â) ∩ Γ = (âđ Fix(đđ )) ∩ EP(đš, â) ∩ Γ =Ė¸ 0, (đžđ + đŋđ )đ ≤ đžđ , for all đ ≥ 0, and đĨđ+1 = đđ đĻđ + đžđ đđļ (đĻđ − đ∇đđŧđ (đĻđ )) + đŋđ đđđļ (đĻđ − đ∇đđŧđ (đĻđ )) (i) (đŧđ + đŊđ )/đŊđ,đ0 → 0 as đ → ∞; = đđ đĻđ + đŋđ đđļ (đĻđ − đ∇đđŧđ (đĻđ )) (ii) if đ ≤ đ0 and đŊđ,đ → 0, then đŊđ,đ0 /đŊđ,đ → 0 as đ → ∞; = đđ đĻđ + (1 − đđ ) đđļ (đĻđ − đ∇đđŧđ (đĻđ )) . (iii) if đŊđ,đ → đŊđ =Ė¸ 0, then đŊđ lies in (0, 1). Then, the sequences {đĨđ }, {đĻđ }, and {đĸđ } explicitly defined by scheme (109) all converge strongly to the unique solution đĨ∗ ∈ Ω of the following variational inequality: â¨đđĨ∗ − đĨ∗ , đ§ − đĨ∗ ⊠≤ 0, ∀đ§ ∈ Ω. (111) Remark 33. If we choose ∇đ = đ´ 1 = ⋅ ⋅ ⋅ = đ´ đ = 0 in system (108), we obtain a system of hierarchical fixed point problems introduced by Moudafi and MaingeĖ [20, 21]. Further, utilizing Theorems 27 and 28, we again give the following strong convergence theorems for finding a common element of the solution set Γ of MP (3), the solution set EP(đš, â) of EP (10), and the common fixed point set (âđ Fix(đđ )) of a finite family of nonexpansive mappings đđ : đļ → đļ, đ = 1, . . . , đ. Theorem 34. Let one suppose that Ω = (âđ Fix(đđ )) ∩ EP(đš, â) ∩ Γ =Ė¸ 0. Let {đŊđ }, {đŊđ,đ }, đ = 1, . . . , đ, be sequences in (0, 1) such that 0 < lim inf đ → ∞ đŊđ,đ ≤ lim supđ → ∞ đŊđ,đ < 1 for all index đ. Moreover, Let one suppose that there hold (H1)– (H6) with đžđ = 0, for all đ ≥ 0. Then, the sequences {đĨđ }, {đĻđ }, and {đĸđ } generated explicitly by đš (đĸđ , đĻ) + â (đĸđ , đĻ) + 1 â¨đĻ − đĸđ , đĸđ − đĨđ ⊠≥ 0, đđ ∀đĻ ∈ đļ, đĻđ,1 = đŊđ,1 đ1 đĸđ + (1 − đŊđ,1 ) đĸđ , đĻđ,đ = đŊđ,đ đđ đĸđ + (1 − đŊđ,đ ) đĻđ,đ−1 , đĻđ = đŊđ đđĻđ,đ + (1 − đŊđ ) đđļ (đĻđ,đ − đ∇đđŧđ (đĻđ,đ)) , ∀đ ≥ 0, (112) all converge strongly to the unique solution đĨ∗ ∈ Ω of the following variational inequality: â¨đđĨ∗ − đĨ∗ , đ§ − đĨ∗ ⊠≤ 0, ∀đ§ ∈ Ω. Thus, the conditions in Theorem 27 are all satisfied. and from which we obtain the desired result. Theorem 35. Let one suppose that Ω = (âđ Fix(đđ )) ∩ EP(đš, â) ∩ Γ =Ė¸ 0. Let {đŊđ }, {đŊđ,đ }, đ = 1, . . . , đ, be sequences in (0, 1) such that đŊđ,đ → đŊđ for all đ as đ → ∞. Suppose that there exists đ ∈ {1, . . . , đ} for which đŊđ,đ → 0 as đ → ∞. Let đ0 ∈ {1, . . . , đ} be the largest index for which đŊđ,đ0 → 0. Moreover, let one suppose that there hold (H1), (H7), and (H8) with đžđ = 0, for all đ ≥ 0, and (i) (đŧđ + đŊđ )/đŊđ,đ0 → 0 as đ → ∞; (ii) if đ ≤ đ0 and đŊđ,đ → 0, then đŊđ,đ0 /đŊđ,đ → 0 as đ → ∞; (iii) if đŊđ,đ → đŊđ =Ė¸ 0, then đŊđ lies in (0, 1). Then the sequences {đĨđ }, {đĻđ }, and {đĸđ } generated explicitly by (112) all converge strongly to the unique solution đĨ∗ ∈ Ω of the following variational inequality: â¨đđĨ∗ − đĨ∗ , đ§ − đĨ∗ ⊠≤ 0, ∀đ§ ∈ Ω. (115) Acknowledgments This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Leading Academic Discipline Project of Shanghai Normal University (DZL707). This research was partially supported by the Grant from the NSC 101-2115-M-037-001. References đ = 2, . . . , đ, đĨđ+1 = đđ đĻđ + (1 − đđ ) đđļ (đĻđ − đ∇đđŧđ (đĻđ )) , (114) (113) Proof. In Theorems 27, put đ = đŧ the identity mapping and đžđ = 0, for all đ ≥ 0. Then, đ is a đ-strictly pseudocontractive mapping with đ = 0. Hence, we deduce that Ω = Fix(đ) ∩ [1] H.-K. Xu, “Averaged mappings and the gradient-projection algorithm,” Journal of Optimization Theory and Applications, vol. 150, no. 2, pp. 360–378, 2011. [2] J.-L. Lions, Quelques MeĖthodes de ReĖsolution des ProbleĖmes aux Limites non LineĖaires, Dunod, Paris, France, 1969. [3] L. C. Zeng, “Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 187, no. 2, pp. 352–360, 1994. [4] J. C. Yao, “Variational inequalities with generalized monotone operators,” Mathematics of Operations Research, vol. 19, no. 3, pp. 691–705, 1994. [5] L. C. Zeng, S. Schaible, and J. C. Yao, “Iterative algorithm for generalized set-valued strongly nonlinear mixed variationallike inequalities,” Journal of Optimization Theory and Applications, vol. 124, no. 3, pp. 725–738, 2005. Journal of Applied Mathematics [6] L.-C. Ceng and J.-C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 205–215, 2007. [7] L.-C. Ceng and S. Huang, “Modified extragradient methods for strict pseudo-contractions and monotone mappings,” Taiwanese Journal of Mathematics, vol. 13, no. 4, pp. 1197–1211, 2009. [8] L.-C. Ceng, C.-Y. Wang, and J.-C. Yao, “Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,” Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 375–390, 2008. [9] N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 128, no. 1, pp. 191–201, 2006. [10] L.-C. Zeng and J.-C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293–1303, 2006. [11] C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004. [12] G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007. [13] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240–256, 2002. [14] Y. Yao, Y.-C. Liou, and S. M. Kang, “Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3472–3480, 2010. [15] J. Reinermann, “UĖber Fixpunkte kontrahierender Abbildungen und schwach konvergente Toeplitz-Verfahren,” Archiv der Mathematik, vol. 20, pp. 59–64, 1969. [16] F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem,” Fixed Point Theory and Applications, vol. 2010, Article ID 383740, 19 pages, 2010. [17] G. Marino, L. Muglia, and Y. Yao, “Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 4, pp. 1787–1798, 2012. [18] L. C. Ceng, A. PetrusĖ§el, and J. C. Yao, “Relaxed extragradient methods with regularization for general system of variational inequalities with constraints of split feasibility and fixed point problems,” Abstract and Applied Analysis, vol. 2013, Article ID 891232, 25 pages, 2013. [19] W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417– 428, 2003. [20] A. Moudafi and P.-E. MaingeĖ, “Towards viscosity approximations of hierarchical fixed-point problems,” Fixed Point Theory and Applications, vol. 2006, Article ID 95453, 10 pages, 2006. [21] P.-E. MaingeĖ and A. 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