ON AN OPEN QUESTION OF MOUDAFI FOR CONVEX FEASIBILITY PROBLEMS IN HILBERT SPACES

• Published in: Taiwanese journal of mathematics Vol. 18; no. 2; pp. 371 - 408
• Main Author: Naraghirad, Eskandar
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.04.2014
• Subjects: Banach space; Hilbert spaces; Inner products; Linear transformations; Perceptron convergence procedure; Real numbers
• ISSN: 1027-5487; 2224-6851; 2224-6851
• DOI: 10.11650/tjm.18.2014.3463

Very recently, Moudafi (Nonlinear Analysis 79 (2013) 117-121) introduced a relaxed alternating CQ-algorithm (RACQA) with weak convergence for the following convex feasibility problem: (1.1) Find   x ∈ C ,   y ∈ Q   such   that   A x = B y , whereH 1,H 2,H 3are real Hilbert spaces,C⊂H 1,Q⊂H 2are two nonempty, closed and convex level sets, andA:H 1→H 3,B:H 2→H 3are two bounded linear operators. In this paper, we will continue to consider the problem (1.1) and obtain a strongly convergent iterative sequence of Halpern-type to a solution of the problem and provide an affirmative answer to an open question posed by Moudafi in his recent work for convex feasibility problems in real Hilbert spaces. Furthermore, we study Halpern-type iterative schemes for finding common solutions of a convex feasibility problem and common fixed points of an infinite family of quasi-nonexpansive mappings in Hilbert spaces. Our results improve and generalize many known results in the current literature. 2010Mathematics Subject Classification: 47H10, 37C25. Key words and phrases: Halpern iterative scheme, Convex feasibility problem, Split common fixed-point problem, Quasi-nonexpansive mapping, Fixed point, Strong convergence.