Relative regularity conditions and linear regularity properties for split feasibility problems in normed linear spaces

Abstract

The bounded linear regularity property plays a key role in the study of the strong convergence and/or convergence rate of the CQ algorithm for solving split feasibility problems. To establish some sufficient conditions ensuring the bounded linear regularity property for split feasibility problems in normed linear spaces, we introduce the notion of a relative regularity condition and its associated relative regularity constant in spirit of the regularity condition used in Burke and Ferris (1995) [12]. Based on convex analysis techniques, we explore equivalent characterizations of the relative regularity condition, which in particular extend the classical results in Burke and Ferris (1995) [12] from the Euclidean space to general normed linear spaces, and then establish some important and useful properties in terms of the related relative regularity constant. Consequently, we develop a new technique to establish some sufficient conditions ensuring the bounded linear regularity property for split feasibility problems in normed linear spaces. The sufficient conditions presented in this paper are in terms of the relative regularity constant, which seem completely new. Applied to the case of Hilbert spaces, our results extend and improve the corresponding ones in Wang et al. (2017) [35] by relaxing the relevant assumptions.