Abstract
This paper is concerned with a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.
Highlights
Throughout this paper, we always assume that C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by ·, · and ·, respectively, 2H denoting the family of all the nonempty subsets of H.Let B : H → H be a single-valued nonlinear mapping and M : H → 2H a set-valued mapping
Recall that PC is the metric projection of H onto C; that is, for each x ∈ H, there exists the unique point in PCx ∈ C such that x − PCx miny∈C x − y
A mapping T : C → C is called nonexpansive if T x − T y ≤ x − y for all x, y ∈ C
Summary
Throughout this paper, we always assume that C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by ·, · and · , respectively, 2H denoting the family of all the nonempty subsets of H. We introduce an iterative scheme 1.15 for finding a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems by the shrinking projection method in Hilbert spaces as follows: yn αnWnxn. It is well known that the class of strictly pseudocontractive mappings contains the class of nonexpansive mappings, and it follows that, if k 0, the iterative scheme 1.15 is reduced to find a common element of the set of common fixed points for an infinite family of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature
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