Abstract
We introduce a new iterative scheme by modifying Mann’s iteration method to find a common element for the set of common fixed points of an infinite family of asymptotically strictly pseudocontractive mappings in the intermediate sense, the set of solutions of the cocoercive quasivariational inclusions problems, and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem of the iterative scheme to a common element of the three aforementioned sets is obtained based on the shrinking projection method which extends and improves that of Ezeora and Shehu (Thai J. Math. 9(2):399-409, 2011) and many others. MSC:46C05, 47H09, 47H10, 49J30, 49J40.
Highlights
1 Introduction 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
A mapping T : C → C is called nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ C, and uniformly L-Lipschitzian if there exists a constant L > such that for each n ∈ N, Tnx – Tny ≤ L x – y for all x, y ∈ C, and a mapping f : C → C is called a contraction if there exists a constant α ∈ (, ) such that f (x) – f (y) ≤ α x – y for all x, y ∈ C
We denote by F(T) the set of fixed points of T; that is, F(T) = {x ∈ C : Tx = x}
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. Theorem KX Let C be a nonempty closed convex subset of a real Hilbert space H and let T : C → C be an asymptotically κ-SPC mapping for some ≤ κ < with a bounded sequence {γn} ⊂ [ , ∞) such that limn→∞ γn = .
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