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

Abstract

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: $$\mbox{Find}~x\in C,~y\in Q\hspace{0.4cm}\mbox{such that}~Ax=By, \leqno(1.1)$$ where $H_1,~H_2,~H_3$ are real Hilbert spaces, $C\subset H_1$, $Q\subset H_2$ are two nonempty, closed and convex level sets, and $A:H_1\to H_3$, $B:H_2\to H_3$ are 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.