Modified Iterative Solution of Nonlinear Uniformly Continuous Mappings Equation in Arbitrary Real Banach Space Eman M. Nemah Dept. of Mathematics/College of Eduction for Pure Science (Ibn Al-Haitham)/ University of Baghdad. Received in : 12 June 2013 , Accepted in : 24 September 2013 Abstract In this paper, we study the convergence theorems of the Modified Ishikawa iterative sequence with mixed errors for the uniformly continuous mappings and solving nonlinear uniformly continuous mappings equation in arbitrary real Banach space. 379 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Introduction and preliminaries Throughout this paper, we assume that X be a Banach space with norm ‖ ‖, the dual space 𝑋∗ and J denote the normalized duality from X into 2𝑋 ∗ give by: 𝐽(𝑥) = {𝑓 ∈ 𝑋∗: 〈𝑥, 𝑓〉 = ‖𝑥‖‖𝑓‖, ‖𝑓‖ = ‖𝑥‖, ∀𝑥 ∈ 𝑋} … (1.1)[1] Where 〈. , . 〉 denotes the generalized duality pairing. A mapping T with domain D(T) and range R(T) in X is called accretive if the inequality holds: ‖𝑥 − 𝑦‖ ≤ ‖𝑥 − 𝑦 + 𝑠(𝑇𝑥 − 𝑇𝑦)‖ … (1.2)[2] For every 𝑥, 𝑦 ∈ 𝐷(𝑇) and for all 𝑠 > 0. A mapping T is called a strongly pseudocontraction if there exists 𝑡 > 0 such that 𝑥, 𝑦 ∈ 𝐷(𝑇) and 𝑟 > 0, the following inequality holds: ‖𝑥 − 𝑦‖ ≤ ‖(1 + 𝑟)(𝑥 − 𝑦) + 𝑟𝑡(𝑇𝑥 − 𝑇𝑦‖ … (1.3) If t=1 in inequality (1.3), then T is called pseudocontractive. Also, as a consequence of Kato [3], if follows from the inequality (1.3) that T is strongly pseudocontractive if and only if the following inequality holds: 〈(𝐼 − 𝑇)𝑥 − (𝐼 − 𝑇)𝑦, 𝑗(𝑥 − 𝑦)〉 ≥ 𝑘‖𝑥 − 𝑦‖2 … (1.4) For all 𝑥, 𝑦 ∈ 𝐷(𝑇) and for some 𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦), 𝑤ℎ𝑒𝑟𝑒 𝑘 = (𝑡−1) 𝑡 ∈ (0, 1). Consequently, it follows easily, again from Kato [3] and (1.3), that T is strongly pseudocontractive if and only if the inequality holds: ‖𝑥 − 𝑦‖ ≤ ‖𝑥 − 𝑦 + 𝑠[(𝐼 − 𝑇 − 𝑘𝐼)𝑥 − (𝐼 − 𝑇 − 𝑘𝐼)𝑦]‖ … (1.5) For every 𝑥, 𝑦 ∈ 𝐷(𝑇) and for all 𝑠 > 0. Closely related to the class of pseudocontractive mapping is the class of accretive mapping. It is clear that T is strongly accretive mapping if and only if I-T is strongly pseudocontractive mapping. Let us recall the following iterative scheme du to Mann [4], Ishikawa [5], Xu[6] and Cho [7], respectively. Definition 1.1 Let C be a convex subset of X and 𝑇: 𝐶 → 𝐶 be a mapping, then: i . For any 𝑥1 ∈ 𝐶, the sequence {𝑥𝑛} 𝑖𝑠 defined by: 𝑦𝑛 = (1 − 𝛽𝑛)𝑥𝑛 + 𝛽𝑛𝑇𝑥𝑛, 𝑥𝑛+1 = (1 − 𝛼𝑛)𝑥𝑛 + 𝛼𝑛𝑇𝑦𝑛, 𝑛 ≥ 1 Is called Ishikawa iteration sequence, where {𝛼𝑛} 𝑎𝑛𝑑 {𝛽𝑛} are two real sequences in [0, 1] satisfying some conditions. ii. . If 𝛽𝑛 = 0 for all 𝑛 ≥ 1 in (1.6), then the sequence {𝑥𝑛} is defined by: 𝑥1 ∈ 𝐶, 𝑥𝑛+1 = (1 − 𝛼𝑛)𝑥𝑛 + 𝛼𝑛𝑇𝑦𝑛, 𝑛 ≥ 1 … (1.7) Is called Mann iteration sequence. iii. For any 𝑥1 ∈ 𝐶, the sequence {𝑥𝑛} is defined by: 𝑦𝑛 = 𝑎�̀�𝑥𝑛 + 𝑏�̀�𝑇𝑥𝑛 + 𝑐�̀�𝑣𝑛, 𝑥𝑛+1 = 𝑎𝑛𝑥𝑛 + 𝑏𝑛𝑇𝑦𝑛 + 𝑐𝑛𝑢𝑛, 𝑛 ≥ 1 … (1.8) Is called Ishikawa iteration sequence with random errors. Here {𝑢𝑛}, {𝑣𝑛} are two bounded sequences in C; {𝑎�̀�} , �𝑏�̀�� , {𝑐�̀�}, {𝑎𝑛}, {𝑏𝑛}, {𝑐𝑛}are real sequences in [0, 1] satisfying 𝑎�̀� + 𝑏�̀� + 𝑐�̀� = 𝑎𝑛 + 𝑏𝑛 + 𝑐𝑛 =1 for all 𝑛 ≥ 1 v. If 𝑏�̀� = 𝑐�̀� = 0 for all 𝑛 ≥ 1 in (1.8), then the sequence {𝑥𝑛} defined by: 𝑥0 ∈ 𝐶, 𝑥𝑛+1 = 𝑎𝑛𝑥𝑛 + 𝑏𝑛𝑇𝑥𝑛 + 𝑐𝑛𝑢𝑛, 𝑛 ≥ 1 … (1.9) Is called Mann iteration sequence with random errors. iv. . For any 𝑥1 ∈ 𝐶, the sequence {𝑥𝑛} is defined by: 𝑦𝑛 = 𝑎�̀�𝑥𝑛 + 𝑏�̀�𝑇𝑛𝑥𝑛 + 𝑐�̀�𝑣𝑛, 𝑥𝑛+1 = 𝑎𝑛𝑥𝑛 + 𝑏𝑛𝑇𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛, 𝑛 ≥ 1 … (1.10) Is called modified Ishikawa iteration sequence with random errors. Here {𝑢𝑛}, {𝑣𝑛} are two bounded sequences in C; {𝑎�̀�} , �𝑏�̀�� , {𝑐�̀�}, {𝑎𝑛}, {𝑏𝑛}, {𝑐𝑛}are real sequences in [0, 1] satisfying 380 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 𝑎�̀� + 𝑏�̀� + 𝑐�̀� = 𝑎𝑛 + 𝑏𝑛 + 𝑐𝑛 =1 for all 𝑛 ≥ 1. iiv. . If 𝑏�̀� = 𝑐�̀� = 0 for all 𝑛 ≥ 0 in (1.10), then the sequence {𝑥𝑛} is defined by: 𝑥1 ∈ 𝐶, 𝑥𝑛+1 = 𝑎𝑛𝑥𝑛 + 𝑏𝑛𝑇𝑛𝑥𝑛 + 𝑐𝑛𝑢𝑛, 𝑛 ≥ 1 … (1.11) Is called modified Mann iteration sequence with random errors. It is clear that the Mann, Ishikawa iterative sequences and Mann, Ishikawa iterative sequence with random errors are special cases of Modified iterative sequences with errors. Definition 1.2 [8, 9] A mapping T: C→C called: i. Lipschitz if there exists a constant L>0 such that ( ) ( )T x T y L x y− ≤ − …(1.12) for all ,x y C∈ . ii. Uniformly L-Lipschitzian if there exists a constant L>0 such that ( ) ( )n nT x T y L x y− ≤ − …(1.13) for all ,x y C∈ , 𝑛 ≥ 1 Definition 1.3 [10] Let C be a convex subset of vector space and f real valued function defined on C, then f is called generalized convex if for any three points x, y, z ∈ C and 𝑎, 𝑏, 𝑐 ∈ [0, 1] such that 𝑓(𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧) = 𝑎𝑓(𝑥) + 𝑏 𝑓(𝑦) + 𝑐 𝑓(𝑧). Example 1.4 Let f: [0, ∞) → [0, ∞) such that 𝑓(𝑡) = 𝑡2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡 ∈ [0, ∞). Show that f is generalized convex. Solution: for any three points x, y, z ∈ [0, ∞) and 𝑎, 𝑏, 𝑐 ∈ [0, 1] such that 𝑓(𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧) = (𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧)2 ≤ 𝑎𝑥2 + 𝑏 𝑦2 + 𝑐 𝑧2 Thus f is a generalized convex. Several researchers proved that the Mann iterative scheme, Ishikawa iterative scheme , the Mann iterative scheme with random errors and Ishikawa iterative scheme with random errors can be used to approximate solutions of the equations Tx=f where T is strongly pseudocontractive mapping or ø- strongly pseudocontractive mapping. Very recently, Arifiq [10], proved a related result that deals with the Mann iterative approximation of the fixed point for the class of strongly pseudocontractive mapping in arbitrary real Banach space. At the same time, he puts forth an open problem: It is not Known whether or not the modified Ishikawa iteration method converges uniformly continuous mapping. This open problem has been studied extensively in case of Mann iterative scheme with random errors and Ishikawa iterative scheme with random errors by many of the researchers (see, [1-3], [6], [10-16]). The objective of this paper is to introduce the modified Ishikawa iterative method a class of sequence which much more general than the important class of Mann iterative scheme with random errors and Ishikawa iterative scheme with random errors, and to study problem of approximation fixed point by modified Ishikawa iterative processes with random errors for uniformly continuous mappings and this mapping satisfies some conditions. We will prove that the answer of Airfiq, s open problem is affirmative if X is an arbitrary real Banach space and 𝑇 ∶ 𝑋 → 𝑋 is an uniformly continuous mapping and satisfying some conditions. The results presented in this paper improve, generalize and unify results of [1-2], [11-16]. The following two lemmas play crucial roles in the proofs of our main results: Lemma 1.5 [2] Let : 2 XJ X ∗ → be the normalized duality mapping. Then for the ,x y X∈ , we have 2 2 2 , ( ) , ( ) ( ).x y x y j x y j x y J x y+ ≤ + + ∀ + ∈ + 381 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Lemma 1.6 [2] If there exists a positive integer N such that for all , ,n N n N≥ ∈ 𝛼𝑛+1 ≤ (1 − 𝑏𝑛)𝛼𝑛 + 𝑐𝑛 𝑛 ≥ 1 Then lim𝑛 →∞ 𝛼𝑛 = 0, 𝑤ℎ𝑒𝑟𝑒 𝑏𝑛 ∈ [0,1] 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑛 ∈ 𝑁, ∑ 𝑏𝑛 = ∞∞𝑛=1 𝑎𝑛𝑑 𝑐𝑛 = 0(𝑏𝑛), Main result Theorem 2.1 Let X be an arbitrary real Banach space , C be a convex subset of X, T:C → 𝐶 be an uniformly continuous mapping with bounded range and T satisfies the condition ‖𝑥 − 𝑦‖ ≤ ‖𝑥 − 𝑦 + 𝑟[(𝐼 − 𝑇𝑛 − 𝑘𝐼)𝑥 − (𝐼 − 𝑇𝑛 − 𝑘𝐼)𝑦]‖ … (2.1) where I is the identity mapping on C, for all 𝑥, 𝑦 ∈ 𝐶, 𝑘 ∈ (0, 1), 𝑟 > 0. If q is a fixed point of T and for arbitrary x∈ 𝐶, then modified Ishikawa iterative scheme with random errors defined by ( 1.10) which satisfies the conditions: i. ∑ 𝑏𝑛 = ∞∞𝑛=1 ii. 𝑐𝑛 = 0(𝑏𝑛), iii. lim𝑛→∞ 𝑏𝑛 = lim𝑛→∞ 𝑏�́� = lim𝑛→∞ 𝑐�́� = 0. is converges strongly to unique fixed point q of T. Proof: Since T bounded range, we set 𝑀1 = ‖𝑥1 − 𝑞‖ + 𝑠𝑢𝑝𝑛≥1‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑠𝑢𝑝𝑛≥1‖𝑢𝑛 − 𝑞‖ Obviously 𝑀1 < ∞. It is clear that ‖𝑥1 − 𝑞‖ ≤ 𝑀1. Let ‖𝑥𝑛 − 𝑞‖ ≤ 𝑀1. 𝑁𝑒𝑥𝑡 𝑤𝑒 𝑤𝑖𝑙𝑙 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 ‖𝑥𝑛+1 − 𝑞‖ ≤ 𝑀1. Consider ‖𝑥𝑛+1 − 𝑞‖ = ‖𝑎𝑛𝑥𝑛 + 𝑏𝑛𝑇𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛 − 𝑞‖ = ‖𝑎𝑛𝑥𝑛 + 𝑏𝑛𝑇𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛 − (𝑎𝑛 + 𝑏𝑛 + 𝑐𝑛)𝑞‖ = ‖𝑎𝑛(𝑥𝑛 − 𝑞) + 𝑏𝑛(𝑇𝑛𝑦𝑛 − 𝑞) + 𝑐𝑛(𝑢𝑛 − 𝑞)‖ ≤ 𝑎𝑛‖𝑥𝑛 − 𝑞‖ + 𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑐𝑛‖𝑢𝑛 − 𝑞‖ ≤ (1 − 𝑏𝑛)‖𝑥𝑛 − 𝑞‖ + 𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑐𝑛‖𝑢𝑛 − 𝑞‖ ≤ (1 − 𝑏𝑛)𝑀1 + 𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑐𝑛‖𝑢𝑛 − 𝑞‖ ≤ (1 − 𝑏𝑛)(‖𝑥1 − 𝑞‖ + 𝑠𝑢𝑝𝑛≥1‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑠𝑢𝑝𝑛≥1‖𝑢𝑛 − 𝑞‖) + 𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑐𝑛‖𝑢𝑛 − 𝑞‖ ≤ ‖𝑥1 − 𝑞‖ + 𝑠𝑢𝑝𝑛≥1‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑠𝑢𝑝𝑛≥1‖𝑢𝑛 − 𝑞‖ − 𝑏𝑛(‖𝑥1 − 𝑞‖ + 𝑠𝑢𝑝𝑛≥1‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑠𝑢𝑝𝑛≥1‖𝑢𝑛 − 𝑞‖) + 𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑐𝑛‖𝑢𝑛 − 𝑞‖ ≤ 𝑀1. So from the above discussion, we can conclude that the sequence {𝑥𝑛}𝑛=1∞ is bounded. Let 𝑀2 = 𝑠𝑢𝑝𝑛≥1‖𝑦𝑛 − 𝑞‖. Denote 𝑀 = 𝑀1 + 𝑀2. Obviously 𝑀 < ∞. Consider ‖𝑥𝑛+1 − 𝑞‖2 = ‖𝑎𝑛𝑥𝑛 + 𝑏𝑛𝑇𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛 − 𝑞‖2 = ‖𝑎𝑛𝑥𝑛 + 𝑏𝑛𝑇𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛 − (𝑎𝑛 + 𝑏𝑛 + 𝑐𝑛)𝑞‖2 = ‖𝑎𝑛(𝑥𝑛 − 𝑞) + 𝑏𝑛(𝑇𝑛𝑦𝑛 − 𝑞) + 𝑐𝑛(𝑢𝑛 − 𝑞)‖2 ≤ � 𝑎𝑛‖𝑥𝑛 − 𝑞‖ + 𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑐𝑛‖𝑢𝑛 − 𝑞‖ � 2 ≤ �(1 − 𝑏𝑛)‖𝑥𝑛 − 𝑞‖ + 𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑞‖ + 𝑐𝑛‖𝑢𝑛 − 𝑞‖ � 2 From def.(1.3) ≤ (1 − 𝑏𝑛)‖𝑥𝑛 − 𝑞‖2 + 𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑞‖2 + 𝑐𝑛‖𝑢𝑛 − 𝑞‖2 ≤ (1 − 𝑏𝑛)‖𝑥𝑛 − 𝑞‖2 + 𝑏𝑛𝑀2 + 𝑐𝑛𝑀2 … (2.2) Now from lemma (1.5) for all 𝑛 ≥ 1, we obtain ‖𝑥𝑛+1 − 𝑞‖2 = ‖𝑎𝑛𝑥𝑛 + 𝑏𝑛𝑇𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛 − 𝑞‖2 = ‖𝑎𝑛(𝑥𝑛 − 𝑞) + 𝑏𝑛(𝑇𝑛𝑦𝑛 − 𝑞) + 𝑐𝑛(𝑢𝑛 − 𝑞)‖2 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑞‖2 + 𝑐𝑛‖𝑢𝑛 − 𝑞‖2 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2〈𝑏𝑛(𝑇𝑛𝑦𝑛 − 𝑞) + 𝑐𝑛(𝑢𝑛 − 𝑞), 𝑗(𝑥𝑛+1 + 𝑞)〉 382 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2〈𝑏𝑛(𝑇𝑛𝑦𝑛 − 𝑞), 𝑗(𝑥𝑛+1 − 𝑞)〉 +2〈𝑐𝑛(𝑢𝑛 − 𝑞), 𝑗(𝑥𝑛+1 − 𝑞)〉 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛〈𝑇𝑛𝑦𝑛 − 𝑞, 𝑗(𝑥𝑛+1 − 𝑞)〉 + 2𝑐𝑛〈𝑢𝑛 − 𝑞, 𝑗(𝑥𝑛+1 − 𝑞)〉 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛〈𝑇𝑛𝑥𝑛+1 − 𝑞, 𝑗(𝑥𝑛+1 − 𝑞)〉 + 2𝑏𝑛〈𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1, 𝑗(𝑥𝑛+1 − 𝑞)〉 + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛〈𝑇𝑛𝑥𝑛+1 − 𝑞, 𝑗(𝑥𝑛+1 − 𝑞)〉 + 2𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖‖𝑥𝑛+1 − 𝑞‖ + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛(1 − 𝑘)‖𝑥𝑛+1 − 𝑞‖2 + 2𝑏𝑛‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖‖𝑥𝑛+1 − 𝑞‖ + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛(1 − 𝑘)‖𝑥𝑛+1 − 𝑞‖2 + 2𝑏𝑛𝑀‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖ + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛(1 − 𝑘)‖𝑥𝑛+1 − 𝑞‖2 + 2𝑏𝑛𝑑𝑛 + 2𝑀2𝑐𝑛 …(2.3) Where 𝑑𝑛 = 𝑀‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖ … (2.4) From (1.10) we have ‖𝑦𝑛 − 𝑥𝑛+1‖ = �(�̀�𝑛−𝑎𝑛)𝑥𝑛 + �̀�𝑛𝑇𝑛𝑥𝑛 + 𝑐�̀��̀�𝑛 + 𝑏𝑛𝑇𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛)� …(2.5) By conditions (ii-iii) and (2.5) then; lim𝑛→∞‖𝑦𝑛 − 𝑥𝑛+1‖ = 0 ⇒ lim𝑛→∞‖𝑇𝑦𝑛 − 𝑇𝑥𝑛+1‖ = 0 ⇒ lim𝑛→∞‖𝑇𝑛𝑦𝑛 − 𝑇𝑛𝑥𝑛+1‖ = 0 lim 𝑛→∞ 𝑑𝑛 = 0. …(2.6) Substi ‖𝑥𝑛+1 − 𝑞‖2 ≤ (1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛(1 − 𝑘)‖𝑥𝑛+1 − 𝑞‖2 + 2𝑏𝑛𝑑𝑛 + 2𝑀2𝑐𝑛 ≤ �1 − 𝑏𝑛)2‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛((1 − 𝑏𝑛) ‖𝑥𝑛 − 𝑞‖2 + 𝑀2𝑏𝑛 + 𝑀2𝑐𝑛� +2𝑏𝑛𝑑𝑛 + 2𝑀2𝑐𝑛 ≤ (1 − 𝑏𝑛)2 ‖𝑥𝑛 − 𝑞‖2 + 2𝑏𝑛(1 − 𝑘)(1 − 𝑏𝑛) ‖𝑥𝑛 − 𝑞‖2 +2𝑏𝑛(1 − 𝑘)𝑀2𝑏𝑛 + 2𝑏𝑛(1 − 𝑘)𝑀2𝑐𝑛 + 2𝑏𝑛𝑑𝑛 + 2𝑀2𝑐𝑛 = ‖𝑥𝑛 − 𝑞‖2[�1 − 𝑏𝑛)2 + 2𝑏𝑛(1 − 𝑘)(1 − 𝑏𝑛) ] + 2𝑏𝑛[𝑀2(1 − 𝑘)(𝑏𝑛 + 𝑐𝑛� +𝑑𝑛] + 2𝑀2𝑐𝑛 …(2.7) By lim𝑛→∞ 𝑏𝑛 = 0, 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑛𝑜 ∈ 𝑁 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≥ 𝑛𝑜 We have 𝑏𝑛 ≤ 1 2 . 𝐹𝑟𝑜𝑚 (2.7), 𝑤𝑒 𝑔𝑒𝑡 ‖𝑥𝑛+1 − 𝑞‖2 ≤ �1 − 𝑘𝑎𝑛) ‖𝑥𝑛 − 𝑞‖2 +2𝑏𝑛[𝑀2(1 − 𝑘)(𝑏𝑛 + 𝑐𝑛� + 𝑑𝑛 + 2𝑀2𝑡𝑛 (2.8) where 𝑡𝑛 = 𝑏𝑛 𝑐𝑛 Now with the help of (i-ii), (2.6) and lemma 1.6 lim 𝑛→∞ ‖𝑥𝑛 − 𝑞‖ = 0. Then {𝑥𝑛} converges strongly to fixed point 𝑞 ∈ 𝐹(𝑇). If p also is a fixed point of T; we will show that q is uniqueness ‖𝑞 − 𝑝‖2 = 〈𝑞 − 𝑝, 𝑗(𝑞 − 𝑝)〉 = 〈𝑇𝑞 − 𝑇𝑝, 𝑗(𝑞 − 𝑝)〉 ≤ (1 − 𝑘)‖𝑞 − 𝑝‖2 Since 𝑘 ∈ (0,1), it follows that ‖𝑞 − 𝑝‖2 ≤ 0, which implies the uniqueness. ■ Theorem 2.2 Let X be an arbitrary Banach space , C be a convex subset of X, T:C → 𝐶 be L- Lipschitzian mapping with bounded range and T satisfies the condition ‖𝑥 − 𝑦‖ ≤ ‖𝑥 − 𝑦 + 𝑟[(𝐼 − 𝑇𝑛 − 𝑘𝐼)𝑥 − (𝐼 − 𝑇𝑛 − 𝑘𝐼)𝑦]‖ where I is the identity mapping on C, for all 𝑥, 𝑦 ∈ 𝐶, 𝑘 ∈ (0, 1), 𝑟 > 0. If q is a fixed point of T, then the modified Ishikawa iterative scheme with random errors defined by ( 1.10) which satisfies the conditions: i ∑ 𝑏𝑛 = ∞∞𝑛=1 383 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 𝑖𝑖. 𝑐𝑛 = 0(𝑏𝑛), 𝑖𝑖𝑖. lim 𝑛→∞𝜕 𝑏𝑛 = lim𝑛→∞ 𝑏�́� = lim𝑛→∞ 𝑐�́� = 0. Then the sequence {𝑥𝑛}is converges strongly to unique fixed point q of T. Theorem 2.3 Let X be an arbitrary Banach space , T: X → 𝑋 be an uniformly continuous mapping and T satisfies the condition 〈𝑇𝑛𝑥 − 𝑇𝑛𝑦, 𝑗(𝑥 − 𝑦)〉 ≥ 𝑘‖𝑥 − 𝑦‖2, 𝑛 ≥ 1 for all 𝑥, 𝑦 ∈ 𝑋, 𝑘 ∈ (0, 1). For given 𝑓 ∈ 𝑋∗; let 𝑥∗denote the unique solution of the equation 𝑇𝑛𝑥 = 𝑓. Define the mapping H: X → 𝑋 such that 𝐻𝑛𝑥 = 𝑓+𝑥 − 𝑇𝑛𝑥, and suppose that the range of H is bounded 𝐿𝑒𝑡 {𝑥𝑛}𝑛=1∞ the modified Ishikawa iterative scheme with random errors defined by : 𝑥1 ∈ 𝑋 , 𝑦𝑛 = 𝑎�̀�𝑥𝑛 + 𝑏�̀�𝐻𝑛𝑥𝑛 + 𝑐�̀�𝑣𝑛, 𝑥𝑛+1 = 𝑎𝑛𝑥𝑛 + 𝑏𝑛𝐻𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛, 𝑛 ≥ 1 Here {𝑢𝑛}, {𝑣𝑛} are two bounded sequences in X; {𝑎�̀�} , �𝑏�̀�� , {𝑐�̀�}, {𝑎𝑛}, {𝑏𝑛}, {𝑐𝑛}are real sequences in [0, 1] satisfying 𝑎�̀� + 𝑏�̀� + 𝑐�̀� = 𝑎𝑛 + 𝑏𝑛 + 𝑐𝑛 =1 for all 𝑛 ≥ 1 .which satisfying the conditions: i ∑ 𝑏𝑛 = ∞∞𝑛=1 𝑖𝑖. 𝑐𝑛 = 0(𝑏𝑛), 𝑖𝑖𝑖. lim 𝑛→∞𝜕 𝑏𝑛 = lim𝑛→∞ 𝑏�́� = lim𝑛→∞ 𝑐�́� = 0. Then the sequence {𝑥𝑛} is converges strongly to unique solution of 𝑇𝑛𝑥 = 𝑓. Theorem 2.4 Let X be arbitrary Banach space, T: X → 𝑋 be L-Lipschitizan mapping and T satisfies the condition 〈𝑇𝑛𝑥 − 𝑇𝑛𝑦, 𝑗(𝑥 − 𝑦)〉 ≥ 𝑘‖𝑥 − 𝑦‖2, 𝑛 ≥ 1 for all 𝑥, 𝑦 ∈ 𝑋, 𝑘 ∈ (0, 1). For given 𝑓 ∈ 𝑋; let 𝑥∗denotes the unique solution of the equation 𝑇𝑛𝑥 = 𝑓. Define the mapping H: X → 𝑋 such that 𝐻𝑛𝑥 = 𝑓+𝑥 − 𝑇𝑛𝑥, and suppose that the range of H is bounded For any 𝑥1 ∈ 𝑋, 𝑙𝑒𝑡 {𝑥𝑛}𝑛=1∞ the modified Ishikawa iterative scheme with random errors defined by 𝑦𝑛 = 𝑎�̀�𝑥𝑛 + 𝑏�̀�𝐻𝑛𝑥𝑛 + 𝑐�̀�𝑣𝑛, 𝑥𝑛+1 = 𝑎𝑛𝑥𝑛 + 𝑏𝑛𝐻𝑛𝑦𝑛 + 𝑐𝑛𝑢𝑛, 𝑛 ≥ 1 Here {𝑢𝑛}, {𝑣𝑛} are two bounded sequences in X; {𝑎�̀�} , �𝑏�̀�� , {𝑐�̀�}, {𝑎𝑛}, {𝑏𝑛}, {𝑐𝑛}are real sequences in [0, 1] satisfying 𝑎�̀� + 𝑏�̀� + 𝑐�̀� = 𝑎𝑛 + 𝑏𝑛 + 𝑐𝑛 =1 for all 𝑛 ≥ 1 .which satisfying the conditions: i ∑ 𝑏𝑛 = ∞∞𝑛=1 𝑖𝑖. 𝑐𝑛 = 0(𝑏𝑛), 𝑖𝑖𝑖. lim 𝑛→∞𝜕 𝑏𝑛 = lim𝑛→∞ 𝑏�́� = lim𝑛→∞ 𝑐�́� = 0. Then the sequence {𝑥𝑛} 𝑖𝑠 converges strongly to unique solution of 𝑇𝑛𝑥 = 𝑓. References 1- Yongjin L., (2005), “Ishikawa Iterative Sequence with Errors for ø-Strongly Accretive Operator ”, Commune Korean Math. Soc. ,20, 1, pp.71-78. 2- Yougjin L., (2004), “Ishikawa Iterative Sequence with Errors for Strongly-Pseudocontative Operators in Arbitrary Banach Space” IJMMS, 33, pp.1771-1775. 3- Kato T., (1967), "Nonlinear Semigroups and Evolution Equation", J. Math. Soc. Jpn., 19, pp.508-520. 4- Mann R.W., (1953), “Mean Value Method in Iteration”, Proc. Math. Soc., 4, pp.506-510. 5- Ishikawa S., (1976), “Fixed Points and Iteration of a Nonexpansive Mapping a Banach Space”, Amer. Math. Soc., 59, pp.65-71. 6- Xu Y., (1998), “Ishikawa and Mann Iteration Process with Errors for Nonlinear Sttrongly Accretive Operator Equations”, J. Math. Anal. Appl., 224, pp. 91-101. 384 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 7- Cho Y.J., Guo G.T., and Zhou H.Y., (2004), “ Approximation Fixed Points of Asymptotically Quasi- Nonexpansive with Errors”, Turkey-Dynamical Syatems and Applications, Proceeding, 5, 10, pp.262-272,. 8- Moore C., and Nnoli B, V. C., (2005), "Iteration sequence asymptotically demicontractive mapping in Banach space", J. Math Anal. Appl., 129, 3, pp. 557-562. 9- Chan, S.S., (2001), "Some results for asymplotically pseudocontractive mapping and asymptotically nonexpansive mapping", Proc. Amer. Soc., 302, 2, pp. 845-853. 10- Rafiq A., (2007), “Mann Iterative Scheme for Nonlinear Equations” Math. Comm., 12, pp. 25-31. 11- Chang S. S., Cho Y. J., Lee B. S., and Kang S. M., (1998),."Iterative Approximation of Fixed Points and Solutions for Strongly Accretive and Strongly Pseudocontractive Mappings in Banach Spaces", J. Math. Anal. Appl.,224, pp.149-165. 12- Chidume C.E., (1994), “Approximation off Fixed Points of Strongly Pseudocontractive Mappings”, Proc. Amer. Math. Soc.,120, 2, pp. 545-551. 13- Chidume C.E.,(1998), “Convergence Theorems for Strongly Pseudocontractive and Strongly Accretive Mappings ”,J. Math. Anal. Appl.,228, pp. 254-264. 14- Osilike M. O., (1999), "Iterative Solutions of Nonlinear ø-Strongly Accretive Operator Equations in Arbitrary Banach Space", Nonlinear Analysis,36, pp. 1-9. 15- Lin L. S., (1995), "Ishikawa and Mann Iterative Process with Random Errors for Nonlinear Strongly Accretive mapping in Banach Space", J. Math. Anal. Appl.,194, pp. 114- 125. 16- Yao Y., Chen R., and Yao J.C., (2005), "Strongly Convergence and Certain Control Conditions for Modified Mann Iteration", Nonlinear Analy.,68, pp.1687-1693. 385 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 الحل التكراري المطور لمعادلة غیر خطیة على دوال مستمرة بانتظام في فضاء بناخ الحقیقي األختیاري إیمان محمد نعمھ ) / جامعة بغدادابن الھیثم(/ كلیة التربیھ للعلوم الصرفھ الریاضیاتقسم 2013ایلول 24، قبل في : 2013حزیران 12أستلم البحث في : ا لخالصة في ھذا البحث نقوم بدراسة نظریات التقارب لمتتابعة ایشكاوا المطورة الممزوجة بالخطأ الى دوال مستمره بانتظام وحل معادالت غیر خطیة على دوال مستمرة بانتظام في فضاء بناخ الحقیقي األختیاري. 386 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013