Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 129069, pages http://dx.doi.org/10.1155/2014/129069 Research Article A Modified Mixed Ishikawa Iteration for Common Fixed Points of Two Asymptotically Quasi Pseudocontractive Type Non-Self-Mappings Yuanheng Wang and Huimin Shi Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China Correspondence should be addressed to Yuanheng Wang; wangyuanhengmath@163.com Received January 2014; Accepted 21 February 2014; Published 26 March 2014 Academic Editor: Rudong Chen Copyright © 2014 Y Wang and H Shi 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 A new modified mixed Ishikawa iterative sequence with error for common fixed points of two asymptotically quasi pseudocontractive type non-self-mappings is introduced By the flexible use of the iterative scheme and a new lemma, some strong convergence theorems are proved under suitable conditions The results in this paper improve and generalize some existing results Introduction Let 𝐸 be a real Banach space with its dual 𝐸∗ and let 𝐶 be a nonempty, closed, and convex subset of 𝐸 The mapping 𝐽 : ∗ 𝐸 → 2𝐸 is the normalized duality mapping defined by 𝐽 (𝑥) = {𝑥 ∈ 𝐸 : ⟨𝑥, 𝑥 ⟩ = ‖𝑥‖ ⋅ 𝑥∗ , ‖𝑥‖ = 𝑥∗ } , ∗ ∗ ∗ 𝑥 ∈ 𝐸 (1) Let 𝑇 : 𝐶 → 𝐸 be a mapping We denote the fixed point set of 𝑇 by 𝐹(𝑇); that is, 𝐹(𝑇) = {𝑥 ∈ 𝐶 : 𝑥 = 𝑇𝑥} Recall that a mapping 𝑇 : 𝐶 → 𝐸 is said to be nonexpansive if, for each 𝑥, 𝑦 ∈ 𝐶, 𝑇𝑥 − 𝑇𝑦 ≤ 𝑥 − 𝑦 (2) 𝑇 is said to be asymptotically nonexpansive if there exists a sequence 𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → as 𝑛 → ∞ such that 𝑛 𝑛 𝑇 𝑥 − 𝑇 𝑦 ≤ 𝑘𝑛 𝑥 − 𝑦 , ∀𝑥, 𝑦 ∈ 𝐶 (3) A sequence of self-mappings {𝑇𝑖 }∞ 𝑖=1 on 𝐶 is said to be uniform Lipschitzian with the coefficient 𝐿 if, for any 𝑖 = 1, 2, , the following holds: 𝑛 𝑛 𝑇𝑖 𝑥 − 𝑇𝑖 𝑦 ≤ 𝐿 𝑥 − 𝑦 , ∀𝑥, 𝑦 ∈ 𝐶 (4) 𝑇 is said to be asymptotically pseudocontractive if there exist 𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → as 𝑛 → ∞ and 𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) such that 2 ⟨𝑇𝑛 𝑥 − 𝑇𝑛 𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≤ 𝑘𝑛 𝑥 − 𝑦 , ∀𝑥, 𝑦 ∈ 𝐶 (5) It is obvious to see that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is asymptotically pseudocontractive Goebel and Kirk [1] introduced the class of asymptotically nonexpansive mappings in 1972 The class of asymptotically pseudocontractive mappings was introduced by Schu [2] and has been studied by various authors for its generalized mappings in Hilbert spaces, Banach spaces, or generalized topological vector spaces by using the modified Mann or Ishikawa iteration methods (see, e.g.,[3–21]) In 2003, Chidume et al [22] studied fixed points of an asymptotically nonexpansive non-self-mapping 𝑇 : 𝐶 → 𝐸 and the strong convergence of an iterative sequence {𝑥𝑛 } generated by 𝑥𝑛+1 = 𝑃 ((1 − 𝛼𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇(𝑃𝑇)𝑛−1 𝑥𝑛 ) , 𝑛 ≥ 1, 𝑥1 ∈ 𝐶, (6) where 𝑃 : 𝐸 → 𝐶 is a nonexpansive retraction 2 Abstract and Applied Analysis In 2011, Zegeye et al [23] proved a strong convergence of Ishikawa scheme to a uniformly L-Lipschitzian and asymptotically pseudocontractive mappings in the intermediate sense which satisfies the following inequality (see [24]): 2 lim sup sup (⟨𝑇𝑛 𝑥 − 𝑇𝑛 𝑦, 𝑥 − 𝑦⟩ − 𝑘𝑛 𝑥 − 𝑦 ) ≤ 0, 𝑛 → ∞ 𝑥,𝑦∈𝐶 (7) ∀𝑥, 𝑦 ∈ 𝐶, where 𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → as 𝑛 → ∞ Motivated and inspired by the above results, in this paper, we introduce a new modified mixed Ishikawa iterative sequence with error for common fixed points of two more generalized asymptotically quasi pseudocontractive type non-self-mappings By the flexible use of the iterative scheme and a new lemma (i.e., Lemma in this paper), under suitable conditions, we prove some strong convergence theorems Our results extend and improve many results of other authors to a certain extent, such as [6, 8, 14–23] 𝑇 is said to be asymptotically pseudocontractive (with 𝑃) if there exist 𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → as 𝑛 → ∞ and ∀𝑥, 𝑦 ∈ 𝐶, ∃𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) such that 2 ⟨𝑇(𝑃𝑇)𝑛−1 𝑥 − 𝑇(𝑃𝑇)𝑛−1 𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≤ 𝑘𝑛 𝑥 − 𝑦 (9) 𝑇 is said to be an asymptotically pseudocontractive type (with 𝑃) if there exist 𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → as 𝑛 → ∞ and ∀𝑥, 𝑦 ∈ 𝐶, 𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) such that 𝑛 → ∞ 𝑥,𝑦∈𝐶 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) (⟨𝑇(𝑃𝑇) 𝑛−1 𝑛−1 𝑥 − 𝑇(𝑃𝑇) 𝑦, 2 𝑗 (𝑥 − 𝑦)⟩ − 𝑘𝑛 𝑥 − 𝑦 ) ≤ (10) 𝑇 is said to be an asymptotically quasi pseudocontractive type (with 𝑃) if 𝐹(𝑇) ≠ 0, for 𝑝 ∈ 𝐹(𝑇), there exist 𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → as 𝑛 → ∞, and, ∀𝑥 ∈ 𝐶, 𝑗(𝑥 − 𝑝) ∈ 𝐽(𝑥 − 𝑝) such that lim sup sup lim inf 𝑛 → ∞ 𝑥∈𝐶 𝑗(𝑥−𝑝)∈𝐽(𝑥−𝑝) 𝑛−1 𝑥𝑛+1 = 𝑃 ((1 − 𝛼𝑛 − 𝛾𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇1 (𝑃𝑇1 ) 𝑛−1 × ((1 − 𝛽𝑛 ) 𝑦𝑛 + 𝛽𝑛 𝑇1 (𝑃𝑇1 ) 𝑦𝑛 ) + 𝛾𝑛 𝑢𝑛 ) , 𝑛−1 Definition Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸 𝐶 is said to be a nonexpansive retract (with 𝑃) of 𝐸 if there exists a nonexpansive mapping 𝑃 : 𝐸 → 𝐶 such that, for all 𝑥 ∈ 𝐶, 𝑃𝑥 = 𝑥 And 𝑃 is called a nonexpansive retraction Let 𝑇 : 𝐶 → 𝐸 be a non-self-mapping (maybe selfmapping) 𝑇 is called uniformly L-Lipschitzian (with 𝑃) if there exists a constant 𝐿 > such that 𝑇(𝑃𝑇)𝑛−1 𝑥 − 𝑇(𝑃𝑇)𝑛−1 𝑦 ≤ 𝐿 𝑥 − 𝑦 , ∀𝑥, 𝑦 ∈ 𝐶, 𝑛 ≥ (8) lim inf Definition Let 𝐶 be a nonexpansive retract (with 𝑃) of 𝐸, let 𝑇1 , 𝑇2 : 𝐶 → 𝐸 be two uniformly L-Lipschitzian non-self-mappings and let 𝑇1 be an asymptotically quasi pseudocontractive type (with 𝑃) The sequence {𝑥𝑛 } is called the new modified mixed Ishikawa iterative sequence with error (with 𝑃), if {𝑥𝑛 } is generated by 𝑦𝑛 = 𝑃 ((1 − 𝛼𝑛 − 𝛾𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇2 (𝑃𝑇2 ) Preliminaries lim sup sup Remark It is clear that every asymptotically pseudocontractive mapping (with 𝑃) is asymptotically pseudocontractive type (with 𝑃) and every asymptotically pseudocontractive type (with 𝑃) is asymptotically quasi pseudocontractive type (with 𝑃) If 𝑇 : 𝐶 → 𝐶 is a self-mapping, then we can choose 𝑃 = 𝐼 as the identical mapping and we can get the usual definition of asymptotically pseudocontractive mapping, and so forth (⟨𝑇(𝑃𝑇)𝑛−1 𝑥 − 𝑝, 𝑗 (𝑥 − 𝑦)⟩ 2 − 𝑘𝑛 𝑥 − 𝑝 ) ≤ (11) 𝑛−1 × ((1 − 𝛽𝑛 ) 𝑥𝑛 + 𝛽𝑛 𝑇2 (𝑃𝑇2 ) (12) 𝑥𝑛 ) + 𝛾𝑛 V𝑛 ) , where 𝑥1 ∈ 𝐶 is arbitrary, {𝑢𝑛 } and {V𝑛 } ⊂ 𝐶 are bounded, and 𝛼𝑛 , 𝛽𝑛 , 𝛾𝑛 , 𝛼𝑛 , 𝛽𝑛 , 𝛾𝑛 ∈ [0, 1], 𝑛 = 1, 2, If 𝛼𝑛 = 𝛽𝑛 = 𝛾𝑛 = 0, (12) turns to 𝑛−1 𝑥𝑛+1 = 𝑃 ((1 − 𝛼𝑛 − 𝛾𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇1 (𝑃𝑇1 ) 𝑛−1 × ((1 − 𝛽𝑛 ) 𝑥𝑛 + 𝛽𝑛 𝑇1 (𝑃𝑇1 ) 𝑥𝑛 ) + 𝛾𝑛 𝑢𝑛 ) , (13) and it is called the new modified mixed Mann iterative sequence with error (with 𝑃) If 𝛾𝑛 = 𝛾𝑛 = 0, (12) becomes 𝑛−1 𝑥𝑛+1 = 𝑃 ( (1 − 𝛼𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇1 (𝑃𝑇1 ) 𝑛−1 × ((1 − 𝛽𝑛 ) 𝑦𝑛 + 𝛽𝑛 𝑇1 (𝑃𝑇1 ) 𝑦𝑛 )) , 𝑛−1 𝑦𝑛 = 𝑃 ( (1 − 𝛼𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇2 (𝑃𝑇2 ) 𝑛−1 × ((1 − 𝛽𝑛 ) 𝑥𝑛 + 𝛽𝑛 𝑇2 (𝑃𝑇2 ) (14) 𝑥𝑛 )) , and it is called the new modified mixed Ishikawa iterative sequence (with 𝑃) If 𝛽𝑛 = 𝛽𝑛 = 0, (14) turns to 𝑛−1 𝑥𝑛+1 = 𝑃 ((1 − 𝛼𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇1 (𝑃𝑇1 ) 𝑦𝑛 ) , 𝑛−1 𝑦𝑛 = 𝑃 ((1 − 𝛼𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇2 (𝑃𝑇2 ) 𝑥𝑛 ) , (15) and it is called the new mixed Ishikawa iterative sequence (with 𝑃) If 𝑇1 = 𝑇2 = 𝑇 : 𝐶 → 𝐶 is a self-mapping and 𝑃 = 𝐼 is the identical mapping, then (15) is just the modified Ishikawa iterative sequence 𝑥𝑛+1 = (1 − 𝛼𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇𝑛 𝑦𝑛 , 𝑦𝑛 = (1 − 𝛼𝑛 ) 𝑥𝑛 + 𝛼𝑛 𝑇𝑛 𝑥𝑛 (16) Abstract and Applied Analysis If 𝛼𝑛 = 0, (15) becomes (6), obviously So, iterative method (12) is greatly generalized The following lemmas will be needed in what follows to prove our main results Lemma (see [19]) Let 𝐸 be a real Banach space Then, for all 𝑥, 𝑦 ∈ 𝐸, 𝑗(𝑥 + 𝑦) ∈ 𝐽(𝑥 + 𝑦), the following inequality holds: 2 𝑥 + 𝑦 ≤ ‖𝑥‖ + ⟨𝑥, 𝑗 (𝑥 + 𝑦)⟩ (17) Lemma (see [6, 7]) Let {𝑎𝑛 }, {𝑏𝑛 }, {𝑐𝑛 } be three sequences of nonnegative numbers satisfying the recursive inequality: 𝑎𝑛+1 ≤ (1 + 𝑏𝑛 ) 𝑎𝑛 + 𝑐𝑛 , ∀𝑛 ≥ 𝑛0 , (18) ∞ where 𝑛0 is some nonnegative integer If Σ∞ 𝑛=1 𝑏𝑛 < ∞, Σ𝑛=1 𝑐𝑛 < ∞, then lim𝑛 → ∞ 𝑎𝑛 exists Lemma Suppose that 𝜙 : [0, +∞) → [0, +∞) is a strictly increasing function with 𝜙(0) = Let {𝑎𝑛 }, {𝑏𝑛 }, {𝑐𝑛 }, {𝜆 𝑛 } (0 ≤ 𝜆 𝑛 ≤ 1) be four sequences of nonnegative numbers satisfying the recursive inequality: 𝑎𝑛+1 ≤ (1 + 𝑏𝑛 ) 𝑎𝑛 − 𝜆 𝑛 𝜙 (𝑎𝑛+1 ) + 𝑐𝑛 , ∀𝑛 ≥ 𝑛0 , (19) ∞ where 𝑛0 is some nonnegative integer If Σ∞ 𝑛=1 𝑏𝑛 < ∞, Σ𝑛=1 𝑐𝑛 < ∞ ∞, Σ𝑛=1 𝜆 𝑛 = ∞, then lim𝑛 → ∞ 𝑎𝑛 = Proof From (19), we get Lemma Suppose that 𝜙 : [0, +∞) → [0, +∞) is a strictly increasing function with 𝜙(0) = Let {𝑎𝑛 }, {𝑏𝑛 }, {𝑐𝑛 }, {𝜆 𝑛 } (0 ≤ 𝜆 𝑛 ≤ 1), {𝜀𝑛 } be five sequences of nonnegative numbers satisfying the recursive inequality: 𝑎𝑛+1 ≤ (1 + 𝑏𝑛 ) 𝑎𝑛 − 𝜆 𝑛 𝜙 (𝑎𝑛+1 ) + 𝑐𝑛 + 𝜆 𝑛 𝜀𝑛 , ∞ where 𝑛0 is some nonnegative integer If Σ∞ 𝑛=1 𝑏𝑛 < ∞, Σ𝑛=1 𝑐𝑛 < ∞ ∞, Σ𝑛=1 𝜆 𝑛 = ∞, lim𝑛 → ∞ 𝜀𝑛 = 0, then lim𝑛 → ∞ 𝑎𝑛 = Proof Firstly, we show lim inf 𝑛 → ∞ 𝑎𝑛 = 𝑎 = If 𝑎 > 0, then, for arbitrary 𝑟 ∈ (0, 𝑎), ∃𝑛1 ≥ 𝑛0 , such that 𝑎𝑛+1 ≥ 𝑟 > when 𝑛 ≥ 𝑛1 Because 𝜙 is a strictly increasing function and lim𝑛 → ∞ 𝜀𝑛 = 0, so 𝜙(𝑎𝑛+1 ) ≥ 𝜙(𝑟) > and 𝜀𝑛 ≤ (1/2)𝜙(𝑟) when 𝑛 ≥ 𝑛1 From (22), we have 𝑎𝑛+1 ≤ (1 + 𝑏𝑛 ) 𝑎𝑛 − 𝜆 𝑛 𝜙 (𝑎𝑛+1 ) + 𝑐𝑛 + 𝜆 𝑛 𝜙 (𝑎𝑛+1 ) = (1 + 𝑏𝑛 ) 𝑎𝑛 − 𝜆 𝑛 𝜙 (𝑎𝑛+1 ) + 𝑐𝑛 , ∀𝑛 ≥ 𝑛0 ∞ ∀𝑛 ≥ 𝑛1 ∞ ∑ 𝑏𝑛 ≤ ln 2, ∑ 𝑐𝑛 ≤ 𝜀 𝑛=𝑛2 (20) By Lemma 5, we know that lim𝑛 → ∞ 𝑎𝑛 = 𝑎 ≥ exists Let 𝑀 = sup1≤𝑛≤∞ {𝑎𝑛 } < ∞ Now we show 𝑎 = Otherwise, if 𝑎 > 0, then ∃𝑛1 ≥ 𝑛0 , such that 𝑎𝑛+1 ≥ (1/2)𝑎 > when 𝑛 ≥ 𝑛1 Because 𝜙 is a strictly increasing function, so 𝜙(𝑎𝑛+1 ) ≥ 𝜙((1/2)𝑎) > From (19) again, we have (23) By Lemma 6, we get = lim𝑛 → ∞ 𝑎𝑛 = lim inf 𝑛 → ∞ 𝑎𝑛 = 𝑎 > This is contradictory So, lim inf 𝑛 → ∞ 𝑎𝑛 = Secondly, ∀𝜀 > 0, from the given conditions in Lemma 7, ∃𝑛2 ≥ 𝑛0 , when ∀𝑛 ≥ 𝑛2 , we have 𝜀𝑛 ≤ 𝜙 (𝜀) , 𝑎𝑛+1 ≤ (1 + 𝑏𝑛 ) 𝑎𝑛 + 𝑐𝑛 , ∀𝑛 ≥ 𝑛0 , (22) 𝑛=𝑛2 (24) On the other hand, since lim inf 𝑛 → ∞ 𝑎𝑛 = 0, ∃𝑁 ≥ 𝑛2 such that 𝑎𝑁 ≤ 𝜀 Now we claim 𝑘−1 𝑘−1 𝑛=𝑁 𝑛=𝑁 𝑎𝑘 ≤ (𝜀 + ∑ 𝑐𝑛 ) exp ( ∑ 𝑏𝑛 ) , ∀𝑘 ≥ 𝑁 (25) In fact, when 𝑘 = 𝑁, (25) holds Suppose that (25) holds for 𝑘 dose not for 𝑘 + Then ∞ < 𝜙 ( 𝑎) ∑ 𝜆 𝑛 𝑛=1 𝑛 ∞ 1 = 𝜙 ( 𝑎) ∑ 𝜆 𝑛 + 𝜙 ( 𝑎) ∑ 𝜆 𝑛 𝑛=1 𝑛=𝑛1 +1 𝑛 ∞ 1 ≤ 𝜙 ( 𝑎) ∑ 𝜆 𝑛 + ∑ (𝑎𝑛 − 𝑎𝑛+1 ) 𝑛=1 𝑛=𝑛1 +1 ∞ ∞ 𝑛=𝑛1 +1 𝑛=𝑛1 +1 𝑘 𝑛=𝑁 𝑛=𝑁 (26) Furthermore, 𝑎𝑘+1 > 𝜀, 𝜙(𝑎𝑘+1 ) > 𝜙(𝜀) But by (22), (24), and the inductive hypothesis, we have 𝑛 ∞ 1 ≤ 𝜙 ( 𝑎) ∑ 𝜆 𝑛 + ∑ 𝜆 𝑛 𝜙 (𝑎𝑛+1 ) 𝑛=1 𝑛=𝑛1 +1 𝑘 𝑎𝑘+1 > (𝜀 + ∑ 𝑐𝑛 ) exp ( ∑ 𝑏𝑛 ) (21) + ∑ 𝑏𝑛 𝑎𝑛 + ∑ 𝑐𝑛 𝑛 ∞ ∞ 1 ≤ 𝜙 ( 𝑎) ∑ 𝜆 𝑛 + 𝑎𝑛1 +1 + 𝑀 ∑ 𝑏𝑛 + ∑ 𝑐𝑛 < ∞ 𝑛=1 𝑛=1 𝑛=1 This is a contradiction with the given condition Σ∞ 𝑛=1 𝜆 𝑛 = ∞ Therefore lim𝑛 → ∞ 𝑎𝑛 = 𝑎𝑛+1 ≤ (1 + 𝑏𝑛 ) 𝑎𝑛 − 𝜆 𝑛 𝜙 (𝑎𝑛+1 ) + 𝑐𝑛 + 𝜆 𝑛 𝜀𝑛 ≤ (1 + 𝑏𝑛 ) 𝑎𝑛 − 𝜆 𝑛 𝜙 (𝜀) + 𝑐𝑛 + 𝜆 𝑛 𝜙 (𝜀) 𝑘−1 𝑘−1 𝑛=𝑁 𝑛=𝑁 ≤ (1 + 𝑏𝑛 ) (𝜀 + ∑ 𝑐𝑛 ) exp ( ∑ 𝑏𝑛 ) + 𝑐𝑛 𝑘−1 𝑘 𝑛=𝑁 𝑛=𝑁 𝑘 𝑘 𝑛=𝑁 𝑛=𝑁 ≤ (𝜀 + ∑ 𝑐𝑛 ) exp ( ∑ 𝑏𝑛 ) + 𝑐𝑛 ≤ (𝜀 + ∑ 𝑐𝑛 ) exp ( ∑ 𝑏𝑛 ) (27) Abstract and Applied Analysis This is a contradiction with (26) So, (25) holds Whereupon, ∞ ∞ 𝑛=𝑁 𝑛=𝑁 lim sup 𝑎𝑘 ≤ (𝜀 + ∑ 𝑐𝑛 ) exp ( ∑ 𝑏𝑛 ) 𝑘→∞ (28) ≤ (𝜀 + 𝜀) = 4𝜀 Now, we are in a position to state and prove the main results of this paper Theorem Let 𝐶 be nonexpansive retract (with 𝑃) of a real Banach space 𝐸 Assume that 𝑇1 , 𝑇2 : 𝐶 → 𝐸 are two uniformly L-Lipschitzian non-self-mappings (with 𝑃) and 𝑇1 is an asymptotically quasi pseudocontractive type with coefficient numbers {𝑘𝑛 } ⊂ [1, +∞) : 𝑘𝑛 → satisfying 𝐹 = 𝐹(𝑇1 ) ∩ 𝐹(𝑇2 ) ≠ Suppose that {𝑢𝑛 }, {V𝑛 } ⊂ 𝐶 are two bounded sequences; {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 }, {𝛼𝑛 }, {𝛽𝑛 }, {𝛾𝑛 } ⊂ [0, 1] are six number sequences satisfying the following: ∞ ∞ (C1) Σ∞ 𝑛=1 𝛼𝑛 = +∞, Σ𝑛=1 𝛼𝑛 < +∞, Σ𝑛=1 𝛼𝑛 (𝑘𝑛 − 1) < +∞; (C2) 𝛼𝑛 + 𝛾𝑛 ≤ 1, 𝛼𝑛 + 𝛾𝑛 ≤ 1, Σ∞ 𝑛=1 𝛾𝑛 < +∞; ∞ ∞ (C3) Σ∞ 𝑛=1 𝛼𝑛 𝛽𝑛 < +∞, Σ𝑛=1 𝛼𝑛 𝛼𝑛 < +∞, Σ𝑛=1 𝛼𝑛 𝛾𝑛 < +∞ If 𝑥1 ∈ 𝐶 is arbitrary, then the iterative sequence {𝑥𝑛 } generated by (12) converges strongly to the fixed point 𝑥∗ ∈ 𝐹 if and only if there exists a strictly increasing function 𝜙 : [0, +∞) → [0, +∞) with 𝜙(0) = such that 𝑛−1 𝑛 → ∞ 𝑗(𝑥𝑛+1 [⟨𝑇1 (𝑃𝑇1 ) inf )∈𝐽(𝑥𝑛+1 −𝑥∗ ) 𝑥𝑛+1 − 𝑥∗ , 2 𝑗 (𝑥𝑛+1 − 𝑥∗ ) ⟩− 𝑘𝑛 𝑥𝑛+1 − 𝑥∗ + 𝜙 (𝑥𝑛+1 − 𝑥∗ ) ] ≤ (29) Proof (Adequacy) Let 𝜀𝑛 = 𝑛−1 inf 𝑗(𝑥𝑛+1 −𝑥∗ )∈𝐽(𝑥𝑛+1 −𝑥∗ ) [⟨𝑇1 (𝑃𝑇1 ) 𝑛−1 𝑥𝑛 , (32) + 𝜙 (𝑥𝑛+1 − 𝑥∗ ) ] , 𝜀𝑛 = max {𝜀𝑛 , 0} + 𝑛 ∗ (30) ∗ ∗ ∗ 𝑥𝑛+1 − 𝑥 , 𝑗 (𝑥𝑛+1 − 𝑥 )⟩ 2 − 𝑘𝑛 𝑥𝑛+1 − 𝑥∗ + 𝜙 (𝑥𝑛+1 − 𝑥∗ ) ≤ 𝜀𝑛 + 𝛼𝑛 𝐿 [(1 + 𝛽𝑛 𝐿) 𝑥𝑛 − 𝑥∗ ] + (𝛼𝑛 + 𝛼𝑛 + 𝛾𝑛 + 𝛾𝑛 ) 𝑥𝑛 − 𝑥∗ + (𝛾𝑛 + 𝛾𝑛 ) 𝑀 ≤ [𝛼𝑛 𝐿 (1 + 𝐿) (1 + 𝐿 + 𝐿2 ) + 𝛼𝑛 𝐿 (1 + 𝛽𝑛 𝐿) Then there exists 𝑗(𝑥𝑛+1 − 𝑥 ) ∈ 𝐽(𝑥𝑛+1 − 𝑥 ) such that 𝑛−1 ∗ ∗ ∗ 𝛿𝑛 − 𝑥 ≤ 𝛽𝑛 𝑇2 (𝑃𝑇2 ) 𝑥𝑛 − 𝑥 + (1 − 𝛽𝑛 ) 𝑥𝑛 − 𝑥 ≤ 𝛽𝑛 𝐿 𝑥𝑛 − 𝑥∗ + 𝑥𝑛 − 𝑥∗ ; 𝑦𝑛 − 𝑥∗ ≤ (1 − 𝛼 − 𝛾 ) 𝑥𝑛 − 𝑥∗ 𝑛 𝑛 ∗ + 𝛼𝑛 𝐿 𝛿𝑛 − 𝑥 + 𝛾𝑛 ]𝑛 − 𝑥∗ ≤ 𝑥𝑛 − 𝑥∗ + 𝛼𝑛 𝛽𝑛 𝐿2 𝑥𝑛 − 𝑥∗ + 𝛼𝑛 𝐿 𝑥𝑛 − 𝑥∗ + 𝛾𝑛 𝑀 = (1 + 𝛼𝑛 𝛽𝑛 𝐿2 + 𝛼𝑛 𝐿) 𝑥𝑛 − 𝑥∗ + 𝛾𝑛 𝑀 ≤ (1 + 𝐿 + 𝐿2 ) 𝑥𝑛 − 𝑥∗ + 𝑀; 𝑛−1 ∗ ∗ 𝜎𝑛 − 𝑥 ≤ 𝛽𝑛 𝑇1 (𝑃𝑇1 ) 𝑦𝑛 − 𝑥 + (1 − 𝛽𝑛 ) 𝑦𝑛 − 𝑥∗ ≤ 𝛽𝑛 𝐿 𝑦𝑛 − 𝑥∗ + 𝑦𝑛 − 𝑥∗ ≤ (1 + 𝐿) (1 + 𝐿 + 𝐿2 ) 𝑥𝑛 − 𝑥∗ + (1 + 𝐿) 𝑀; ∗ ∗ 𝑦𝑛 − 𝑥𝑛+1 ≤ 𝛼𝑛 𝐿 𝜎𝑛 − 𝑥 + 𝛼𝑛 𝑥𝑛 − 𝑥 + 𝛼𝑛 𝐿 𝛿𝑛 − 𝑥∗ + 𝛼𝑛 𝑥𝑛 − 𝑥∗ + (𝛾𝑛 + 𝛾𝑛 ) 𝑥𝑛 − 𝑥∗ + (𝛾𝑛 + 𝛾𝑛 ) 𝑀 ≤ 𝛼𝑛 𝐿 [(1 + 𝐿) (1 + 𝐿 + 𝐿2 ) 𝑥𝑛 − 𝑥∗ + (1 + 𝐿) 𝑀] 𝑥𝑛+1 − 𝑥∗ , 2 𝑗 (𝑥𝑛+1 − 𝑥∗ ) ⟩ − 𝑘𝑛 𝑥𝑛+1 − 𝑥∗ ⟨𝑇1 (𝑃𝑇1 ) 𝑦𝑛 , and 𝑀 = sup𝑛≥1 {‖𝜇𝑛 − 𝑥∗ ‖, ‖]𝑛 − 𝑥∗ ‖} < ∞ Then Main Results −𝑥∗ 𝑛−1 𝜎𝑛 = (1 − 𝛽𝑛 ) 𝑦𝑛 + 𝛽𝑛 𝑇1 (𝑃𝑇1 ) 𝛿𝑛 = (1 − 𝛽𝑛 ) 𝑥𝑛 + 𝛽𝑛 𝑇2 (𝑃𝑇2 ) Therefore, lim sup𝑘 → ∞ 𝑎𝑘 = = lim𝑛 → ∞ 𝑎𝑛 lim sup From (29), we know that lim sup𝑛 → ∞ 𝜀𝑛 ≤ So, lim𝑛 → ∞ 𝜀𝑛 = Now, from the given conditions and (12), we can let (31) + 𝛼𝑛 + 𝛼𝑛 + 𝛾𝑛 + 𝛾𝑛 ] 𝑥𝑛 − 𝑥∗ + (𝛼𝑛 𝐿 (1 + 𝐿) + 𝛾𝑛 + 𝛾𝑛 ) 𝑀; 𝑛−1 𝜎𝑛 − 𝑥𝑛+1 ≤ 𝑦𝑛 − 𝑥𝑛+1 + 𝛽𝑛 𝑇1 (𝑃𝑇1 ) 𝑦𝑛 − 𝑦𝑛 ≤ 𝑠𝑛 𝑥𝑛 − 𝑥∗ + 𝑡𝑛 , (33) Abstract and Applied Analysis Let 𝑎𝑛 = ‖𝑥𝑛 − 𝑥∗ ‖ , 𝜑(𝑡) = 2𝜙(√𝑡), and where 𝑠𝑛 = 𝛼𝑛 𝐿 (1 + 𝐿) (1 + 𝐿 + 𝐿2 ) + 𝛼𝑛 𝐿 (1 + 𝛽𝑛 𝐿) + 𝛼𝑛 + 𝛼𝑛 + 𝛾𝑛 + 𝛾𝑛 + 𝛽𝑛 (1 + 𝐿) (1 + 𝐿 + 𝐿2 ) ; 𝜉𝑛 = 𝐿𝛼𝑛 𝑠𝑛 = 𝐿2 𝛼𝑛2 (1 + 𝐿) (1 + 𝐿 + 𝐿2 ) (34) + 𝛼𝑛 𝛼𝑛 𝐿2 (1 + 𝛽𝑛 𝐿) + 𝛼𝑛2 𝐿 + 𝛼𝑛 𝛼𝑛 𝐿 + 𝐿𝛼𝑛 𝛾𝑛 𝑡𝑛 = [𝛼𝑛 𝐿 (1 + 𝐿) + 𝛾𝑛 + 𝛾𝑛 + 𝛽𝑛 (1 + 𝐿)] 𝑀 + 𝐿𝛼𝑛 𝛾𝑛 + 𝐿𝛼𝑛 𝛽𝑛 (1 + 𝐿) (1 + 𝐿 + 𝐿2 ) , So, by Lemma 4, 𝑛−1 2𝛼𝑛 ⟨𝑇1 (𝑃𝑇1 ) 𝑛−1 𝜎𝑛 − 𝑇1 (𝑃𝑇1 ) 𝜌𝑛 = 𝐿𝛼𝑛 𝑡𝑛 + 𝑀𝛾𝑛 𝑥𝑛+1 , 𝑗 (𝑥𝑛+1 − 𝑥∗ )⟩ ≤ 2𝛼𝑛 𝐿 𝑥𝑛+1 − 𝑥∗ 𝜎𝑛 − 𝑥𝑛+1 = [𝛼𝑛2 𝐿2 (1 + 𝐿) + 𝐿𝛼𝑛 𝛾𝑛 + 𝐿𝛼𝑛 𝛾𝑛 + 𝛼𝑛 𝛽𝑛 (𝐿 + 𝐿2 )] 𝑀 (35) + 𝛾𝑛 𝑀 ≤ 2𝛼𝑛 𝐿 𝑥𝑛+1 − 𝑥∗ [𝑠𝑛 𝑥𝑛 − 𝑥∗ + 𝑡𝑛 ] ; (41) ∗ 2 𝑥𝑛+1 − 𝑥 2 Then (39) becomes ∗ 2 ≤ (1 − 𝛼𝑛 − 𝛾𝑛 ) 𝑥𝑛 − 𝑥 𝑛−1 + 2𝛼𝑛 ⟨𝑇1 (𝑃𝑇1 ) 𝑎𝑛+1 ≤ (1 − 𝛼𝑛 ) 𝑎𝑛 + 2𝛼𝑛 𝜀𝑛 + 2𝛼𝑛 𝑘𝑛 𝑎𝑛+1 − 𝛼𝑛 𝜑 (𝑎𝑛+1 ) 𝜎𝑛 − 𝑥∗ , 𝑗 (𝑥𝑛+1 − 𝑥∗ )⟩ + (𝜉𝑛 𝑥𝑛 − 𝑥∗ + 𝜌𝑛 ) 𝑥𝑛+1 − 𝑥∗ + 2𝛾𝑛 ⟨𝜇𝑛 − 𝑥∗ , 𝑗 (𝑥𝑛+1 − 𝑥∗ )⟩ 2 𝑛−1 + 2𝛼𝑛 ⟨𝑇1 (𝑃𝑇1 ) (36) 𝑛−1 𝜎𝑛 − 𝑇1 (𝑃𝑇1 ) 𝑥𝑛+1 , 𝑎𝑛+1 ≤ (1 − 𝛼𝑛 ) 𝑎𝑛 + 2𝛼𝑛 𝜀𝑛 + 2𝛼𝑛 𝑘𝑛 𝑎𝑛+1 − 𝛼𝑛 𝜑 (𝑎𝑛+1 ) + 𝜉𝑛 (𝑎𝑛 + 𝑎𝑛+1 ) + 𝜌𝑛 (1 + 𝑎𝑛+1 ) 𝑗 (𝑥𝑛+1 − 𝑥∗ ) ⟩ 𝑛−1 + 2𝛼𝑛 ⟨𝑇1 (𝑃𝑇1 ) = (1 − 2𝛼𝑛 + 𝛼𝑛2 + 𝜉𝑛 ) 𝑎𝑛 + (2𝛼𝑛 𝑘𝑛 + 𝜉𝑛 + 𝜌𝑛 ) 𝑎𝑛+1 𝑥𝑛+1 − 𝑥∗ , 𝑗 (𝑥𝑛+1 − 𝑥∗ )⟩ From (40), (41), and the given conditions, we know For the third in (36), we have ∞ ∑ 𝛼𝑛2 < +∞, 𝑥𝑛+1 − 𝑥∗ , 𝑗 (𝑥𝑛+1 − 𝑥∗ )⟩ 𝑛=1 2 = 2𝛼𝑛 𝑑𝑛 + 2𝛼𝑛 [𝑘𝑛 𝑥𝑛+1 − 𝑥∗ − 𝜙 (𝑥𝑛+1 − 𝑥∗ )] (37) 2 ≤ 2𝛼𝑛 𝜀𝑛 + 2𝛼𝑛 [𝑘𝑛 𝑥𝑛+1 − 𝑥∗ − 𝜙 (𝑥𝑛+1 − 𝑥∗ )] , where 𝑛−1 𝑑𝑛 = ⟨𝑇1 (𝑃𝑇1 ) 𝑥𝑛+1 − 𝑥∗ , 𝑗 (𝑥𝑛+1 − 𝑥∗ )⟩ 2 − 𝑘𝑛 𝑥𝑛+1 − 𝑥∗ + 𝜙 (𝑥𝑛+1 − 𝑥∗ ) ≤ 𝜀𝑛 (38) ∞ ∑ 𝜉𝑛 < +∞, 𝑛=1 ∞ ∑ 𝜌𝑛 < +∞ (44) 𝑛=1 Then, lim𝑛 → ∞ (2𝛼𝑛 𝑘𝑛 + 𝜉𝑛 + 𝜌𝑛 ) = Therefore ∃𝑛0 , when 𝑛 ≥ 𝑛0 , 2𝛼𝑛 𝑘𝑛 + 𝜉𝑛 + 𝜌𝑛 ≤ 1/2 Let 𝑏𝑛 = − 2𝛼𝑛 + 𝛼𝑛2 + 𝜉𝑛 2𝛼 (𝑘 − 1) + 𝛼𝑛2 + 2𝜉𝑛 + 𝜌𝑛 −1= 𝑛 𝑛 ; − 2𝛼𝑛 𝑘𝑛 − 𝜉𝑛 − 𝜌𝑛 − 2𝛼𝑛 𝑘𝑛 − 𝜉𝑛 − 𝜌𝑛 𝑐𝑛 = 𝜌𝑛 − 2𝛼𝑛 𝑘𝑛 − 𝜉𝑛 − 𝜌𝑛 (45) So, when 𝑛 ≥ 𝑛0 , we get Substituting (35) into (36), we get 2 ∗ 2 ∗ 2 𝑥𝑛+1 − 𝑥 ≤ (1 − 𝛼𝑛 ) 𝑥𝑛 − 𝑥 + 2𝛼𝑛 𝜀𝑛 2 + 2𝛼𝑛 𝑘𝑛 𝑥𝑛+1 − 𝑥∗ − 2𝛼𝑛 𝜙 (𝑥𝑛+1 − 𝑥∗ ) + 2𝛼𝑛 𝐿 (𝑠𝑛 𝑥𝑛 − 𝑥∗ + 𝑡𝑛 ) 𝑥𝑛+1 − 𝑥∗ + 2𝛾𝑛 𝑀 𝑥𝑛+1 − 𝑥∗ (43) − 𝛼𝑛 𝜑 (𝑎𝑛+1 ) + 2𝛼𝑛 𝜀𝑛 + 𝜌𝑛 + 2𝛾𝑛 𝑀 𝑥𝑛+1 − 𝑥∗ 𝑛−1 (42) By using 2𝑎𝑏 ≤ 𝑎2 + 𝑏2 , we have ∗ 2 ≤ (1 − 𝛼𝑛 − 𝛾𝑛 ) 𝑥𝑛 − 𝑥 2𝛼𝑛 ⟨𝑇1 (𝑃𝑇1 ) (40) (39) ≤ 𝑏𝑛 ≤ [2𝛼𝑛 (𝑘𝑛 − 1) + 𝛼𝑛2 + 2𝜉𝑛 + 𝜌𝑛 ] , ≤ 𝑐𝑛 ≤ 2𝜌𝑛 (46) From (44) and the given conditions, we have ∑∞ 𝑛=𝑛0 𝑏𝑛 < +∞, 𝑐 < +∞ On the other hand, from (43), we have ∑∞ 𝑛=𝑛0 𝑛 𝑎𝑛+1 ≤ (1 + 𝑏𝑛 ) 𝑎𝑛 − 𝛼𝑛 𝜑 (𝑎𝑛+1 ) + 4𝛼𝑛 𝜀𝑛 + 𝑐𝑛 , ∀𝑛 ≥ 𝑛0 (47) Abstract and Applied Analysis By Lemma 7, we at last get lim 𝑎 𝑛→∞ 𝑛 2 = lim 𝑥𝑛 − 𝑥∗ = 0; 𝑛→∞ (48) for example, lim𝑛 → ∞ 𝑥𝑛 = 𝑥∗ ∈ 𝐹 = 𝐹(𝑇1 ) ∩ 𝐹(𝑇2 ) (Necessity) Suppose that lim𝑛 → ∞ 𝑥𝑛 = 𝑥∗ ∈ 𝐹 Then we can choose an arbitrary continuous strictly increasing function 𝜙 : [0, +∞) → [0, +∞) with 𝜙(0) = 0, such as 𝜙(𝑡) = 𝑡 We can get lim𝑛 → ∞ 𝜙(‖𝑥𝑛+1 − 𝑥∗ ‖) = Because 𝑇1 is an asymptotically quasi pseudocontractive type (with 𝑃), by (11) in Definition 1, for any 𝑝 ∈ 𝐹(𝑇1 ) ⊇ 𝐹, we have lim sup sup lim inf 𝑛→∞ 𝑥∈𝐶 𝑗(𝑥−𝑝)∈𝐽(𝑥−𝑝) (⟨𝑇(𝑃𝑇)𝑛−1 𝑥 − 𝑝, 𝑗 (𝑥 − 𝑦)⟩ 2 − 𝑘𝑛 𝑥 − 𝑝 ) ≤ (49) So, lim sup 𝑛−1 inf 𝑛 → ∞ 𝑗(𝑥𝑛+1 −𝑥∗ )∈𝐽(𝑥𝑛+1 −𝑥∗ ) [⟨𝑇1 (𝑃𝑇1 ) 2 𝑗 (𝑥𝑛+1 − 𝑥 ) ⟩− 𝑘𝑛 𝑥𝑛+1 − 𝑥∗ + 𝜙 (𝑥𝑛+1 − 𝑥∗ ) ] inf 𝑛 → ∞ 𝑗(𝑥𝑛+1 −𝑥∗ )∈𝐽(𝑥𝑛+1 −𝑥∗ ) 𝑛−1 [⟨𝑇1 (𝑃𝑇1 ) ∞ ∞ (C1) Σ∞ 𝑛=1 𝛼𝑛 = +∞, Σ𝑛=1 𝛼𝑛 < +∞, Σ𝑛=1 𝛼𝑛 (𝑘𝑛 − 1) < +∞; (C2) Σ∞ 𝑛=1 𝛼𝑛 𝛼𝑛 < +∞ If 𝑥1 ∈ 𝐶 is arbitrary, then the iterative sequence {𝑥𝑛 } generated by (15) converges strongly to the fixed point 𝑥∗ ∈ 𝐹 if and only if there exists a strictly increasing function 𝜙 : [0, +∞) → [0, +∞) with 𝜙(0) = such that (29) holds Theorem 11 Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸 Assume that 𝑇 : 𝐶 → 𝐶 is uniformly L-Lipschitzian self-mappings and asymptotically quasi pseudocontractive type with coefficient numbers {𝑘𝑛 } ⊂ [1, +∞) : 𝑘𝑛 → satisfying 𝐹 = 𝐹(𝑇) ≠ Suppose that {𝛼𝑛 }, {𝛼𝑛 } ⊂ [0, 1] are two number sequences satisfying the following: ∞ ∞ (C1) Σ∞ 𝑛=1 𝛼𝑛 = +∞, Σ𝑛=1 𝛼𝑛 < +∞, Σ𝑛=1 𝛼𝑛 (𝑘𝑛 − 1) < +∞; 𝑥𝑛+1 − 𝑥∗ , ∗ = lim sup uniformly L-Lipschitzian non-self-mappings (with 𝑃) and 𝑇1 is an asymptotically quasi pseudocontractive type with coefficient numbers {𝑘𝑛 } ⊂ [1, +∞) : 𝑘𝑛 → satisfying 𝐹 = 𝐹(𝑇1 ) ∩ 𝐹(𝑇2 ) ≠ Suppose that {𝛼𝑛 }, {𝛼𝑛 } ⊂ [0, 1] are two number sequences satisfying the following: 𝑥𝑛+1 − 𝑥∗ , 𝑗 (𝑥𝑛+1 − 𝑥∗ ) ⟩ 2 − 𝑘𝑛 𝑥𝑛+1 − 𝑥∗ ] + lim 𝜙 (𝑥𝑛+1 − 𝑥∗ ) ≤ + = 0; 𝑛→∞ (50) that is, (29) holds This completes the proof of Theorem Combining with Theorem and Definition 3, we have some results as follows Theorem Let 𝐶 be nonexpansive retract (with 𝑃) of a real Banach space 𝐸 Assume that 𝑇1 , 𝑇2 : 𝐶 → 𝐸 are two uniformly L-Lipschitzian non-self-mappings (with 𝑃) and 𝑇1 is an asymptotically quasi pseudocontractive type with coefficient numbers {𝑘𝑛 } ⊂ [1, +∞) : 𝑘𝑛 → satisfying 𝐹 = 𝐹(𝑇1 ) ∩ 𝐹(𝑇2 ) ≠ Suppose that {𝛼𝑛 }, {𝛽𝑛 }, {𝛼𝑛 }, {𝛽𝑛 } ⊂ [0, 1] are four number sequences satisfying the following: ∞ ∞ (C1) Σ∞ 𝑛=1 𝛼𝑛 = +∞, Σ𝑛=1 𝛼𝑛 < +∞, Σ𝑛=1 𝛼𝑛 (𝑘𝑛 − 1) < +∞; ∞ (C2) Σ∞ 𝑛=1 𝛼𝑛 𝛽𝑛 < +∞, Σ𝑛=1 𝛼𝑛 𝛼𝑛 < +∞ If 𝑥1 ∈ 𝐶 is arbitrary, then the iterative sequence {𝑥𝑛 } generated by (14) converges strongly to the fixed point 𝑥∗ ∈ 𝐹 if and only if there exists a strictly increasing function 𝜙 : [0, +∞) → [0, +∞) with 𝜙(0) = such that (29) holds Theorem 10 Let 𝐶 be nonexpansive retract (with 𝑃) of a real Banach space 𝐸 Assume that 𝑇1 , 𝑇2 : 𝐶 → 𝐸 are two (C2) Σ∞ 𝑛=1 𝛼𝑛 𝛼𝑛 < +∞ If 𝑥1 ∈ 𝐶 is arbitrary, then the iterative sequence {𝑥𝑛 } generated by (16) converges strongly to the fixed point 𝑥∗ ∈ 𝐹 if and only if there exists a strictly increasing function 𝜙 : [0, +∞) → [0, +∞) with 𝜙(0) = such that (29) holds Remark 12 Our research and results in this paper have the following several advantaged characteristics (a) The iterative scheme is the new modified mixed Ishikawa iterative scheme with error on two mappings 𝑇1 , 𝑇2 (b) The common fixed point 𝑥∗ ∈ 𝐹 = 𝐹(𝑇1 ) ∩ 𝐹(𝑇2 ) is studied (c) The research object is the very generalized asymptotically quasi pseudocontractive type (with 𝑃) non-self-mapping (d) The tool used by us is the very powerful tool: Lemma So, our results here extend and improve many results of other authors to a certain extent, such as [6, 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Journal of Mathematical Analysis and Applications, vol 321, no 2, pp 661– 674, 2006 L.-C Zeng, T Tanaka, and J.-C Yao, “Iterative construction of fixed points of nonself -mappings in Banach spaces,”... Cholamjiak and S Suantai, ? ?A new hybrid algorithm for variational inclusions, generalized equilibrium problems, and a finite family of quasi- nonexpansive mappings, ” Fixed Point Theory and Applications,... that every asymptotically pseudocontractive mapping (with