On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 284937, 5 pages http://dx.doi.org/10.1155/2013/284937 Research Article On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces Shin Min Kang,1 Arif Rafiq,2 and Sun Young Cho3 1 Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea School of CS and Mathematics, Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore 54660, Pakistan 3 Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea 2 Correspondence should be addressed to Sun Young Cho; Received 14 December 2012; Accepted 26 March 2013 Academic Editor: Luigi Muglia Copyright © 2013 Shin Min Kang 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 study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of Ćirić et al. (2009). 1. Introduction and Preliminaries Let 𝐸 be a real Banach space with dual 𝐸∗ . The symbol 𝐷(𝑇) stands for the domain of 𝑇. Let 𝑇 : 𝐷(𝑇) → 𝐸 be a mapping. Definition 1. The mapping 𝑇 is said to be Lipschitzian if there exists a constant 𝐿 > 0 such that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 (1) for all 𝑥, 𝑦 ∈ 𝐷(𝑇). Definition 2. The mapping 𝑇 is called strongly pseudocontractive if there exists 𝑡 > 1 such that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(1 + 𝑟) (𝑥 − 𝑦) − 𝑟𝑡 (𝑇𝑥 − 𝑇𝑦)󵄩󵄩󵄩 (2) for all 𝑥, 𝑦 ∈ 𝐷(𝑇) and 𝑟 > 0. If 𝑡 = 1 in inequality (2), then 𝑇 is called pseudocontractive. We will denote by 𝐽 the normalized duality mapping from ∗ 𝐸 to 2𝐸 defined by 󵄩 󵄩2 𝐽 (𝑥) = {𝑓∗ ∈ 𝐸∗ : ⟨𝑥, 𝑓∗ ⟩ = ‖𝑥‖2 = 󵄩󵄩󵄩𝑓∗ 󵄩󵄩󵄩 } , (3) where ⟨⋅, ⋅⟩ denotes the generalized duality pairing. It follows from inequality (2) that 𝑇 is strongly pseudocontractive if and only if there exists 𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) such that 󵄩2 󵄩 (4) ⟨(𝐼 − 𝑇) 𝑥 − (𝐼 − 𝑇) 𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≥ 𝑘󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 for all 𝑥, 𝑦 ∈ 𝐷(𝑇), where 𝑘 = (𝑡 − 1)/𝑡 ∈ (0, 1). Consequently, from inequality (4) it follows easily that 𝑇 is strongly pseudocontractive if and only if 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦 + 𝑠 [(𝐼 − 𝑇 − 𝑘𝐼) 𝑥 − (𝐼 − 𝑇 − 𝑘𝐼) 𝑦]󵄩󵄩󵄩 (5) for all 𝑥, 𝑦 ∈ 𝐷(𝑇) and 𝑠 > 0. Closely related to the class of pseudocontractive maps is the class of accretive operators. Let 𝐴 : 𝐷(𝐴) → 𝐸 be an operator. Definition 3. The operator 𝐴 is called accretive if 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑥 − 𝑦󵄩󵄩 ≤ 󵄩󵄩𝑥 − 𝑦 + 𝑠 (𝐴𝑥 − 𝐴𝑦)󵄩󵄩 for all 𝑥, 𝑦 ∈ 𝐷(𝐴) and 𝑠 > 0. (6) Also, as a consequence of Kato [1], this accretive condition can be expressed in terms of the duality mapping as follows. For each 𝑥, 𝑦 ∈ 𝐷(𝐴), there exists 𝑗(𝑥−𝑦) ∈ 𝐽(𝑥−𝑦) such that ⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≥ 0. (7) 2 Journal of Applied Mathematics Consequently, inequality (2) with 𝑡 = 1 yields that 𝐴 is accretive if and only if 𝑇 := (𝐼 − 𝐴) is pseudocontractive. Furthermore, from setting 𝐴 := (𝐼 − 𝑇), it follows from inequality (5) that 𝑇 is strongly pseudocontractive if and only if (𝐴 − 𝑘𝐼) is accretive, and, using (7), this implies that 𝑇 = (𝐼 − 𝐴) is strongly pseudocontractive if and only if there exists 𝑘 ∈ (0, 1) such that 󵄩2 󵄩 ⟨𝐴𝑥 − 𝐴𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≥ 𝑘󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 (8) for all 𝑥, 𝑦 ∈ 𝐷(𝐴). The operator 𝐴 satisfying inequality (8) is called strongly accretive. It is then clear that 𝐴 is strongly accretive if and only if 𝑇 = (𝐼 − 𝐴) is strongly pseudocontractive. Thus, the mapping theory for strongly accretive operators is closely related to the fixed point theory of strongly pseudocontractive mappings. We will exploit this connection in the sequel. The notion of accretive operators was introduced independently in 1967 by Kato [1] and Browder [2]. An early fundamental result in the theory of accretive operators, due to Browder, states that the initial value problem 𝑑𝑢 + 𝐴𝑢 = 0, 𝑑𝑡 𝑢 (0) = 𝑢0 (9) is solvable if 𝐴 is locally Lipschitzian and accretive on 𝐸. If 𝑢 is independent of 𝑡, then 𝐴𝑢 = 0 and the solution of this equation corresponds to the equilibrium points of the system (9). Consequently, considerable research efforts have been devoted, especially within the past 15 years or so, to developing constructive techniques for the determination of the kernels of accretive operators in Banach spaces (see, e.g., [3–19]. Two well-known iterative schemes, the Mann iterative method (see, e.g., [20]) and the Ishikawa iterative scheme (see, e.g., [21]), have successfully been employed. The Mann and Ishikawa iterative schemes are global and their rate of convergence is generally of the order 𝑂(𝑛−1/2 ). It is clear that if, for an operator 𝑈, the classical iterative sequence of the form, 𝑥𝑛+1 = 𝑈𝑥𝑛 , 𝑥0 ∈ 𝐷(𝑈) (the socalled Picard iterative sequence) converges, then it is certainly superior and preferred to either the Mann or the Ishikawa sequence since it requires less computations and, moreover, its rate of convergence is always at least as fast as that of a geometric progression. In [22, 23], Chidume proved the following results. Theorem 4. Let 𝐸 be an arbitrary real Banach space and 𝐴 : 𝐸 → 𝐸 Lipschitz (with constant 𝐿 > 0) and strongly accretive with a strong accretive constant 𝑘 ∈ (0, 1). Let 𝑥∗ denote a solution of the equation 𝐴𝑥 = 0. Set 𝜖 := (1/2)(𝑘/(1 + 𝐿(3 + 𝐿 − 𝑘))) and define 𝐴 𝜖 : 𝐸 → 𝐸 by 𝐴 𝜖 𝑥 := 𝑥 − 𝜖𝐴𝑥 for each 𝑥 ∈ 𝐸. For arbitrary 𝑥0 ∈ 𝐸, define the sequence {𝑥𝑛 }∞ 𝑛=0 in 𝐸 by 𝑥𝑛+1 = 𝐴 𝜖 𝑥𝑛 , 𝑛 ≥ 0. ∗ Then {𝑥𝑛 }∞ 𝑛=0 converges strongly to 𝑥 with 󵄩󵄩 ∗󵄩 𝑛󵄩 ∗󵄩 󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩 ≤ 𝛿 󵄩󵄩󵄩𝑥0 − 𝑥 󵄩󵄩󵄩 , where 𝛿 = (1 − (1/2)𝑘𝜖) ∈ (0, 1). Moreover, 𝑥∗ is unique. Corollary 5. Let 𝐸 be an arbitrary real Banach space and 𝐾 a nonempty convex subset of 𝐸. Let 𝑇 : 𝐾 → 𝐾 be Lipschitz (with constant 𝐿 > 0) and strongly pseudocontractive (i.e., 𝑇 satisfies inequality (5) for all 𝑥, 𝑦 ∈ 𝐾). Assume that 𝑇 has a fixed point 𝑥∗ ∈ 𝐾. Set 𝜖0 := (1/2)(𝑘/(1 + 𝐿(3 + 𝐿 − 𝑘))) and define 𝑇𝜖0 : 𝐾 → 𝐾 by 𝑇𝜖0 𝑥 = (1 − 𝜖0 )𝑥 + 𝜖0 𝑇𝑥 for each 𝑥 ∈ 𝐾. For arbitrary 𝑥0 ∈ 𝐾, define the sequence {𝑥𝑛 }∞ 𝑛=0 in 𝐾 by 𝑥𝑛+1 = 𝑇𝜖0 𝑥𝑛 , 𝑛 ≥ 0. (12) ∗ Then {𝑥𝑛 }∞ 𝑛=0 converges strongly to 𝑥 with 󵄩󵄩 ∗󵄩 𝑛󵄩 ∗󵄩 󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩 ≤ 𝛿 󵄩󵄩󵄩𝑥0 − 𝑥 󵄩󵄩󵄩 , (13) where 𝛿 = (1 − (1/2)𝑘𝜖0 ) ∈ (0, 1). Moreover, 𝑥∗ is unique. Recently, Ćirić et al. [24] improved the results of Chidume [22, 23], Liu [14], and Sastry and Babu [18] as in the following results. Theorem 6. Let 𝐸 be an arbitrary real Banach space and 𝐴 : 𝐸 → 𝐸 a Lipschitz (with constant 𝐿 > 0) and strongly accretive with a strong accretive constant 𝑘 ∈ (0, 1). Let 𝑥∗ denote a solution of the equation 𝐴𝑥 = 0. Set 𝜀 := (𝑘 − 𝜂)/𝐿(2 + 𝐿), 𝜂 ∈ (0, 𝑘) and define 𝐴 𝜀 : 𝐸 → 𝐸 by 𝐴 𝜀 𝑥 := 𝑥 − 𝜀𝐴𝑥 for each 𝑥 ∈ 𝐸. For arbitrary 𝑥0 ∈ 𝐸, define the sequence {𝑥𝑛 }∞ 𝑛=0 in 𝐸 by 𝑥𝑛+1 = 𝐴 𝜀 𝑥𝑛 , 𝑛 ≥ 0. (14) ∗ Then {𝑥𝑛 }∞ 𝑛=0 converges strongly to 𝑥 with 󵄩󵄩 ∗󵄩 𝑛󵄩 ∗󵄩 󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩 ≤ 𝜃 󵄩󵄩󵄩𝑥0 − 𝑥 󵄩󵄩󵄩 , (15) where 𝜃 = (1 − ((𝑘 − 𝜂)/ (...truncated)