On Solutions of Variational Inequality Problems via Iterative Methods

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 424875, 10 pages http://dx.doi.org/10.1155/2014/424875 Research Article On Solutions of Variational Inequality Problems via Iterative Methods Mohammed Ali Alghamdi,1 Naseer Shahzad,1 and Habtu Zegeye2 1 2 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana Correspondence should be addressed to Naseer Shahzad; Received 12 May 2014; Revised 24 June 2014; Accepted 30 June 2014; Published 4 August 2014 Academic Editor: Adrian Petrusel Copyright © 2014 Mohammed Ali Alghamdi 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 investigate an algorithm for a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of 𝛾-inverse strongly accretive mappings. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings. 1. Introduction Let 𝐶 be a subset of a real Hilbert space 𝐻. Let 𝐴 : 𝐶 → 𝐻 be a nonlinear mapping. The variational inequality problem for 𝐴 and 𝐶 is to find 𝑥∗ ∈ 𝐶 such that ⟨𝐴𝑥∗ , V − 𝑥∗ ⟩ ≥ 0, ∀V ∈ 𝐶. (1) The set of solutions of variational inequality problem is denoted by VI(𝐶, 𝐴); that is, VI (𝐶, 𝐴) = {𝑥∗ ∈ 𝐶 : ⟨𝐴𝑥∗ , 𝑥 − 𝑥∗ ⟩ ≥ 0, ∀𝑥 ∈ 𝐶} . is, ‖𝐴𝑥 − 𝐴𝑦‖ ≤ (1/𝛾)‖𝑥 − 𝑦‖, for all 𝑥, 𝑦 ∈ 𝐶. If in (3) we have that 𝛾 = 0, then 𝐴 is called accretive (or monotone). Let 𝐶 be a closed and convex subset of a real Hilbert space 𝐻. A mapping 𝑇 : 𝐶 → 𝐻 is called a contraction mapping if there exists 𝐿 ∈ [0, 1) such that ‖𝑇𝑥 − 𝑇𝑦‖ ≤ 𝐿‖𝑥 − 𝑦‖ for all 𝑥, 𝑦 ∈ 𝐶. If 𝐿 = 1, then 𝑇 is called nonexpansive. A mapping 𝑇 : 𝐶 → 𝐸 is called 𝜆-strictly pseudocontractive of BrowderPetryshyn type [6] if and only if there exists 𝜆 ∈ (0, 1) such that (2) 󵄩2 󵄩 󵄩2 󵄩2 󵄩 󵄩󵄩 󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + 𝜆󵄩󵄩󵄩(𝐼 − 𝑇) 𝑥 − (𝐼 − 𝑇) 𝑦󵄩󵄩󵄩 It is well known that variational inequality theory has emerged as an important tool in studying a wide class of numerous problems in variational inequalities, minimax problems, optimization, physics, and the Nash equilibrium problems in noncooperative games. Several numerical methods have been developed for solving variational inequalities and related optimization problems; see, for instance, [1–5] and the references therein. A mapping 𝐴 : 𝐶 ⊆ 𝐻 → 𝐻 is said to be 𝛾-inverse strongly accretive (or 𝛾-inverse strongly monotone) if there exists a positive real number 𝛾 such that ∀𝑥, 𝑦 ∈ 𝐶. 󵄩2 󵄩 ⟨𝑥 − 𝑦, 𝐴𝑥 − 𝐴𝑦⟩ ≥ 𝛾󵄩󵄩󵄩𝐴𝑥 − 𝐴𝑦󵄩󵄩󵄩 , ∀𝑥, 𝑦 ∈ 𝐶. (3) If 𝐴 is 𝛾-inverse strongly accretive, then inequality (3) implies that 𝐴 is Lipschitzian with constant 𝐿 := 1/𝛾; that (4) 𝑇 is called pseudocontractive if 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩 󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 + 󵄩󵄩󵄩(𝐼 − 𝑇) 𝑥 − (𝐼 − 𝑇) 𝑦󵄩󵄩󵄩 , ∀𝑥, 𝑦 ∈ 𝐶. (5) We note that inequalities (4) and (5) can be equivalently written as 󵄩2 󵄩2 󵄩 󵄩 ⟨𝑇𝑥 − 𝑇𝑦, 𝑥 − 𝑦⟩ ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 − 𝑘󵄩󵄩󵄩(𝑥 − 𝑇𝑥) − (𝑦 − 𝑇𝑦)󵄩󵄩󵄩 ∀𝑥, 𝑦 ∈ 𝐶, (6) 2 Abstract and Applied Analysis for some 𝑘 > 0 and 󵄩2 󵄩 ⟨𝑇𝑥 − 𝑇𝑦, 𝑥 − 𝑦⟩ ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∀𝑥, 𝑦 ∈ 𝐶, (7) respectively. We remark that 𝑇 is pseudocontractive if and only if 𝐴 := (𝐼 − 𝑇) is accretive. A point 𝑥 ∈ 𝐶 is a fixed point of 𝑇 if 𝑇𝑥 = 𝑥 and we denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇) = {𝑥 ∈ 𝐶 : 𝑇𝑥 = 𝑥}. We observe that in a real Hilbert space 𝐻 a class of pseudocontractive mappings includes the class of 𝜆-strictly pseudocontractive mappings and hence the classes of nonexpansive and contraction mappings. Closely related to the variational inequality problems is the problem of finding fixed points of nonexpansive mappings, 𝜆-strict pseudocontraction mappings or pseudocontractive mappings which is the current interest in functional analysis. Several researchers considered a unified approach that approximates a common point of fixed point of nonlinear problems and solutions of variational inequality problems and solutions of variational inequality problems; see, for example, [7–18] and the references therein. In [19], Takahashi and Toyoda studied the problem of finding a common point of fixed points of a nonexpansive mapping and solutions of a variational inequality problem (1) by considering the following iterative algorithm: 𝑥0 ∈ 𝐶, 𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝐴𝑥𝑛 ) , 𝑛 = 0, 1, . . . , (8) where {𝛼𝑛 } is a sequence in (0, 1), {𝜆 𝑛 } is a positive sequence, 𝑇 : 𝐶 → 𝐶 is a nonexpansive mapping, and 𝐴 : 𝐶 → 𝐻 is an 𝛾-inverse strongly accretive mapping. They showed that the sequence {𝑥𝑛 } generated by (8) converges weakly to some 𝑧 ∈ VI(𝐶, 𝐴) ∩ 𝐹(𝑆) provided that the control sequences satisfy some restrictions. Iiduka and Takahashi [20] reconsidered the common element problem via the following iterative algorithm: 𝑥1 = 𝑥 ∈ 𝐶, 𝑥𝑛+1 = 𝛼𝑛 𝑥 + (1 − 𝛼𝑛 ) 𝑇𝑃𝐶 (𝑥𝑛 − 𝜆 𝑛 𝐴𝑥𝑛 ) , 𝑛 = 0, 1, . . . , (9) where 𝑇 : 𝐶 → 𝐶 is a nonexpansive mapping, 𝐴 : 𝐶 → 𝐻 is a 𝛾-inverse-strongly accretive mapping, {𝛼𝑛 } is a sequence in (0, 1), and {𝜆 𝑛 } is a sequence in (0, 2𝛼). They proved that the sequence {𝑥𝑛 } strongly converges to some point 𝑧 ∈ 𝐹(𝑇) ∩ VI(𝐶, 𝐴). Recently, Zegeye and Shahzad [21] investigated the problem of finding a common point of fixed points of a Lipschitz pseudocontractive mapping 𝑇 and solutions of a variational inequality problem for 𝛾-inverse strongly accretive mapping 𝐴 by considering the following iterative algorithm: 𝑦𝑛 = (1 − 𝛽𝑛 ) 𝑥𝑛 + 𝛽𝑛 𝑇𝑥𝑛 , 𝑥𝑛+1 = 𝑃𝐶 [(1 − 𝛼𝑛 ) (𝛿𝑛 𝑇𝑦𝑛 + 𝜃𝑛 𝑥𝑛 + 𝛾𝑛 𝑃𝐶 [𝐼 − 𝛾𝐴] 𝑥𝑛 )] , (10) where 𝑃𝐶 is a metric projection from 𝐻 onto 𝐶 and {𝛿𝑛 }, {𝜃𝑛 }, {𝛾𝑛 }, {𝛼𝑛 }, {𝛽𝑛 } are in (0, 1) satisfying certain conditions. Then, they proved that the sequence {𝑥𝑛 } converges strongly to the minimum-norm point of 𝐹(𝑇) ∩ VI(𝐶, 𝐴). A natural question arises whether we can obtain an iterative scheme which converges strongly to a common point of fixed points of a finite family of pseudocontractive mappings and solutions of a finite family of variational inequality problems for 𝛾-inverse strongly accretive mappings or not. It is our purpose in this paper to introduce an algorithm and prove that the algorithm converges strongly to a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of variational inequality problems for 𝛾-inverse strongly accretive mappings. The results obtained in this paper improve and extend the results of Takahashi and Toyoda [19], Iiduka and Takahashi [20], and Zegeye and Shahzad [21], Theorem 3.2 of Yao et al. [22], and some other results in this direction. 2. Preliminaries In what follows we will make use of the following lemmas. Lemma 1. Letting 𝐻 be a real Hilbert space, the following identity holds: 󵄩2 󵄩󵄩 2 󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩 ≤ ‖𝑥‖ + 2 ⟨𝑦, 𝑥 + 𝑦⟩ , ∀𝑥, 𝑦 (...truncated)