Viscosity Projection Algorithms for Pseudocontractive Mappings in Hilbert Spaces

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 791209, 8 pages http://dx.doi.org/10.1155/2014/791209 Research Article Viscosity Projection Algorithms for Pseudocontractive Mappings in Hilbert Spaces Xiujuan Pan,1,2 Shin Min Kang,3 and Young Chel Kwun4 1 School of Science, Tianjin Polytechnic University, Tianjin 300387, China College of Management and Economics, Tianjin University, Tianjin 300072, China 3 Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea 4 Department of Mathematics, Dong-A University, Busan 614-714, Republic of Korea 2 Correspondence should be addressed to Young Chel Kwun; Received 6 December 2013; Accepted 28 December 2013; Published 9 February 2014 Academic Editor: Chong Li Copyright © 2014 Xiujuan Pan 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. An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping. 1. Introduction Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, respectively. Let C be a nonempty closed convex subset of H. Recall that a mapping T : C → C is said to be (i) 𝐿-Lipschitz ⇔ there exists a constant 𝐿 > 0 such that ‖T𝑢 − TV‖ ≤ 𝐿‖𝑢 − V‖ for all 𝑢, V ∈ C; if 𝐿 ∈ (0, 1), then T is said to be contractive; if 𝐿 = 1, then T is said to be nonexpansive; (ii) pseudocontractive 2 ⇔ ⟨T𝑢 − TV, 𝑢 − V⟩ ≤ ‖𝑢 − V‖ ; ⇔ ‖T𝑢 − TV‖2 ≤ ‖𝑢 − V‖2 +‖(𝐼 − T)𝑢 − (𝐼 − T)V)‖2 ; ⇔ ⟨𝑢 − V, (𝐼 − T)𝑢 − (𝐼 − T)V⟩ ≥ 0, for all 𝑢, V ∈ C. Interest in pseudocontractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling [1], that if T is an accretive operator, then the solutions of the equations T𝑥 = 0 correspond to the equilibrium points of some evolution systems. This explains the importance, from this point of view, of the improvement brought by the Ishikawa iteration which was introduced by Ishikawa [2] in 1974. The original result of Ishikawa is stated in the following. Theorem 1. Let C be a convex compact subset of a Hilbert space H and let T : C → C be an 𝐿-Lipschitzian pseudocontractive mapping with Fix(T) ≠ 0. For any 𝑥0 ∈ C, define the sequence {𝑥𝑛 } iteratively by 𝑦𝑛 = (1 − 𝜂𝑛 ) 𝑥𝑛 + 𝜂𝑛 T𝑥𝑛 , 𝑥𝑛+1 = (1 − 𝜉𝑛 ) 𝑥𝑛 + 𝜉𝑛 T𝑦𝑛 , (1) for all 𝑛 ∈ N, where {𝜉𝑛 } ⊂ [0, 1] and {𝜂𝑛 } ⊂ [0, 1] satisfy the conditions: lim𝑛 → ∞ 𝜂𝑛 = 0 and ∑∞ 𝑛=1 𝜉𝑛 𝜂𝑛 = ∞. Then the sequence {𝑥𝑛 } generated by (1) converges strongly to a fixed point of T. The iteration (1) is now referred to as the Ishikawa iterative sequence. However, we note that 𝐶 is compact subset. Now, we know that strong convergence has not been achieved without compactness assumption on the involved operation or the underlying spaces. A counter example can be found in Chidume and Mutangadura [3]. In order to obtain a strong convergence result for pseudocontractive mappings without the compactness assumption, in [4], Zhou coupled the Ishikawa algorithm with the 2 Abstract and Applied Analysis hybrid technique and presented the following algorithm for Lipschitz pseudocontractive mappings: 𝑦𝑛 = (1 − 𝜉𝑛 ) 𝑥𝑛 + 𝜉𝑛 T𝑥𝑛 , 𝑧𝑛 = (1 − 𝜂𝑛 ) 𝑥𝑛 + 𝜂𝑛 T𝑦𝑛 , 󵄩 󵄩2 󵄩 󵄩2 C𝑛 = {𝑧 ∈ C : 󵄩󵄩󵄩𝑧𝑛 − 𝑧󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑧󵄩󵄩󵄩 󵄩 󵄩2 − 𝜉𝑛 𝜂𝑛 (1 − 2𝜉𝑛 − 𝜉𝑛2 𝐿2 ) 󵄩󵄩󵄩𝑥𝑛 − T𝑥𝑛 󵄩󵄩󵄩 } , (2) 2. Preliminaries Q𝑛 = {𝑧 ∈ C : ⟨𝑥𝑛 − 𝑧, 𝑥0 − 𝑥𝑛 ⟩ ≥ 0} , 𝑥𝑛+1 = projC𝑛 ∩Q𝑛 𝑥0 , 𝑛 ∈ N. Zhou proved that the sequence {𝑥𝑛 } generated by (2) converges strongly to the fixed point of T. Further, in [5], Yao et al. introduced the hybrid Mann algorithm as follows. Let C be a nonempty closed convex subset of a real Hilbert space H. Let {𝜉𝑛 } be a sequence in (0, 1). Let 𝑥0 ∈ H. For C1 = C and 𝑥1 = projC1 𝑥0 , define a sequence {𝑥𝑛 } of C as follows: 𝑦𝑛 = (1 − 𝜉𝑛 ) 𝑥𝑛 + 𝜉𝑛 T𝑥𝑛 , 󵄩2 󵄩 C𝑛+1 = {𝑧 ∈ C𝑛 : 󵄩󵄩󵄩𝜉𝑛 (𝐼 − T) 𝑦𝑛 󵄩󵄩󵄩 ≤ 2𝜉𝑛 ⟨𝑥𝑛 − 𝑧, (𝐼 − T) 𝑦𝑛 ⟩} , 𝑥𝑛+1 = projC𝑛+1 𝑥0 , (3) Note that, in iterations (2) and (3), we need to compute the half-spaces C𝑛 (and/or Q𝑛 ). Very recently, Zegeye et al. [6] further studied the convergence analysis of the Ishikawa iteration (1). They proved ingeniously the strong convergence of the Ishikawa iteration (1). However, we have to assume that the interior of Fix(T) is nonempty. This appears very restrictive since even in R with the usual norm, Lipschitz pseudocontractive maps with finite number of fixed points do not enjoy this condition that intFix(T) ≠ 0. For some related works, please refer to [7–19]. On the other hand, we notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset C of a Hilbert space H1 and a bounded linear operator B : H1 → H2 , where H2 is another Hilbert space. The C-constrained pseudoinverse of B, B†C , is then defined as the minimum-norm solution of the constrained minimization problem 𝑥∈C (4) which is equivalent to the fixed point problem 𝑢 = projC (𝑢 − 𝜇B∗ (B𝑢 − 𝑏)) , Recall that the metric projection projC : H → C satisfies ‖𝑢− projC 𝑢‖ = inf{‖𝑢 − V‖ : V ∈ C}. The metric projection projC is a typical firmly nonexpansive mapping. The characteristic inequality of the projection is ⟨𝑢 − projC 𝑢, V − projC 𝑢⟩ ≤ 0 for all 𝑢 ∈ H, V ∈ C. In the sequel we will use the following expressions: (i) Fix(T) denotes the set of fixed points of T; (ii) 𝑥𝑛 ⇀ 𝑥† denotes the weak convergence of 𝑥𝑛 to 𝑥† ; (iii) 𝑥𝑛 → 𝑥† denotes the strong convergence of 𝑥𝑛 to 𝑥† . The following lemmas will be useful for the next section. 𝑛 ∈ N. B†C (𝑏) := arg min ‖B𝑥 − 𝑏‖ point problem. The purpose of this paper is to solve such a problem for pseudocontractions. More precisely, we will introduce an explicit projection algorithm with viscosity technique for finding the fixed points of a Lipschitzian pseudocontractive mapping. Strong convergence theorem is demonstrated. Consequently, as an application, we can find the minimum-norm fixed point of the pseudocontractive mapping. (5) where B∗ is the adjoint of B, 𝜇 > 0 is a constant, and 𝑏 ∈ H2 is such that projB(C) (𝑏) ∈ B(C). It is, therefore, an interesting problem to invent iterative algorithms that can generate sequences which converge strongly to the minimum-norm solution of a given fixed Lemma 2 (see [20]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Then, C ⊃ 𝑢𝑛 ⇀ 𝑢‡ } 󳨐⇒ (I − T) 𝑢‡ = ]. { (I − T) 𝑢𝑛 󳨀→ ] (6) Lemma 3 (see [21]). Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that a mapping A : C → H is monotone and weakly conti (...truncated)