Some Results on Strictly Pseudocontractive Nonself-Mappings and Equilibrium Problems in Hilbert Spaces

and Applied Analysis 3 It is clear that 1.7 is equivalent to 〈 Sx − Sy, x − y ≤ ∥x − y∥2, ∀x, y ∈ C. 1.8 The class of κ-strict pseudocontractions which was introduced by Browder and Petryshyn 17 in 1967 has been considered by many authors. It is easy to see that the class of strict pseudocontractions falls into the one between the class of nonexpansive mappings and the class of pseudocontractions. For studying the class of strict pseudocontractions, Zhou 18 proposed the following convex combination method: define a mapping St : C → H by Stx tx 1 − t Sx, ∀x ∈ C. 1.9 He showed that St is nonexpansive if t ∈ κ, 1 ; see 18 for more details. Recently, many authors considered the problem of finding a common element in the fixed point set of a nonexpansive mapping and in the solution set of the equilibrium problem 1.1 based on iterative methods; see, for instance, 19–27 . In 2007, Tada and Takahashi 23 considered an iterative method for the equilibrium problem 1.1 and a nonexpansive nonself-mapping. To be more precise, they obtained the following results. Theorem TT. Let C be a closed convex subset of a real Hilbert space H, let f : C × C → R be a bifunction satisfying (A1)–(A4), and let S be a nonexpansive mapping of C into H such that F S ∩ EP f / ∅. Let {xn} and {un} be sequences generated by x1 x ∈ H, and let un ∈ C such that f ( un, y ) 1 rn 〈 y − un, un − xn 〉 ≥ 0, ∀y ∈ C, wn 1 − αn xn αnSun, Cn {z ∈ H : ‖wn − z‖ ≤ ‖xn − z‖}, Qn {z ∈ H : 〈xn − z, x − xn〉 ≥ 0}, xn 1 PCn∩Qnx, 1.10 for every n ∈ N, where {αn} ⊂ a, 1 , for some a ∈ 0, 1 and {rn} ⊂ 0,∞ satisfies lim infn→∞rn > 0. Then the sequence {xn} converges strongly to PF S ∩EP f x . We remark that the iterative process 1.10 is called the hybrid projection iterative process. Recently, the hybrid projection iterative process which was first considered by Haugazeau 28 in 1968 has been studied for fixed point problem of nonlinear mappings and equilibrium problems bymany authors. Since the sequence generated in the hybrid projection iterative process depends on the sets Cn andQn, the hybrid projection iterative process is also known as “CQ” iterative process; see 29 and the reference therein. Recently, Takahashi et al. 30 considered the shrinking projection process for the fixed point problem of nonexpansive self-mapping. More precisely, they obtain the iterative sequence monontonely without the help of the set Qn; see 30 for more details. In this paper, we reconsider the same shrinking projection process for the equilibrium problem 1.1 and a strictly pseudocontractive nonself-mapping. We show that the sequence 4 Abstract and Applied Analysis generated in the proposed iterative process converges strongly to some common element in the fixed point set of a strictly pseudocontractive nonself-mapping and in the solution set of the equilibrium problem 1.1 . The main results presented in this paper mainly improved the corresponding results in Tada and Takahashi 23 . 2. Preliminaries Let C be a nonempty closed and convex subset of a real Hilbert spaceH. Let PC be the metric projection fromH onto C. That is, for x ∈ H, PCx is the only point in C such that ‖x −PCx‖ inf{‖x − z‖ : z ∈ C}. We know that the mapping PC is firmly nonexpansive, that is, ∥ ∥PCx − PCy ∥ ∥2 ≤ PCx − PCy, x − y 〉 , ∀x, y ∈ H. 2.1 The following lemma can be found in 1, 2 . Lemma 2.1. Let C be a nonempty closed convex subset ofH, and let F : C ×C → R be a bifunction satisfying (A1)–(A4). Then, for any r > 0 and x ∈ H, there exists z ∈ C such that F ( z, y ) 1 r 〈 y − z, z − x ≥ 0, ∀y ∈ C. 2.2