Abstract
AbstractWe introduce an iterative scheme for finding a common element of the set of common fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. As applications, at the end of the paper we utilize our results to study the problem of finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of "Equation missing"-strictly pseudocontractive mappings. The results presented in the paper improve some recent results of Qin and Cho (2008).
Highlights
Throughout this paper, we always assume that H is a real Hilbert space with inner product ·, · and norm ·, respectively, C is a nonempty closed convex subset of H, and PC is the metric projection of H onto C
We denote by F T the set of fixed points of T
It is well known that PC is a nonexpansive mapping of H onto C and satisfies x − y, PCx − PCy ≥ PCx − PCy 2
Summary
Throughout this paper, we always assume that H is a real Hilbert space with inner product ·, · and norm · , respectively, C is a nonempty closed convex subset of H, and PC is the metric projection of H onto C. The second iteration process is referred to as Ishikawa’s iteration process 12 which is defined recursively by x1 x ∈ C chosen arbitrary, yn βnxn 1 − βn T xn, 1.11 xn 1 1 − αn xn αnT yn, n ≥ 1, where {αn} and {βn} are sequences in the interval 0, 1 Both 1.16 and 1.11 have only weak convergence in general see , e.g. Very recently, Qin and Cho introduced a composite iterative algorithm {xn} defined as follows: x1 x ∈ C chosen arbitrary, zn γnxn 1 − γn T xn, 1.12 yn βnxn 1 − βn T zn, xn 1 αnγ f xn δnxn 1 − δn I − αnA yn, n ≥ 1, where f is a contraction, T is a nonexpansive mapping, and A is a strongly positive linear bounded self-adjoint operator, proved that, under certain appropriate assumptions on the parameters, {xn} defined by 1.12 converges strongly to a fixed point of T , which solves some variational inequality and is the optimality condition for the minimization problem 1.9. In this paper, motivated and inspired by Su et al 25 , Marino and Xu 9 , Takahashi and Toyoda , and Iiduka and Takahashi , we will introduce a new iterative scheme: x1 x ∈ C chosen arbitrary, zn γnxn 1 − γn Wnxn, 1.16 yn βnxn 1 − βn Wnzn, xn 1 αnγ f xn δnxn 1 − δn I − αnA PC yn − λnByn , where Wn is a mapping defined by 1.15 , f is a contraction, A is strongly positive linear bounded self-adjoint operator, B is a α-inverse strongly monotone, and we prove that under certain appropriate assumptions on the sequences {αn}, {βn}, {γn}, and {δn}, the sequences {xn} defined by 1.16 converge strongly to a common element of the set of common fixed points of a family of {Tn} and the set of solutions of the variational inequality for an inverse-strongly monotone mapping, which solves some variational inequality and is the optimality condition for the minimization problem 1.9
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