Strong convergence result of split feasibility problems in Banach spaces

Abstract

The purpose of this paper is to introduce and study an iterative scheme for solving the split feasibility problems in the setting of p-uniformly convex and uniformly smooth Banach spaces. Under suitable conditions a strong convergence theorem is established. The main result presented in this paper extends some recent results done by Jitsupa Deepho and Poom Kumam [Jitsupa Deepho and Poom Kumam, A Modified Halperns Iterative Scheme for Solving Split Feasibility Problems, Abstract and Applied Analysis, Volume 2012, Article ID 876069, 8 pages] and some others.Recently, Deepho and Kumam [7] introduced and studied a modified Halperns iterative scheme for solving the split feasibility problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, they established the following strong convergence theorem. Theorem 1.1. Suppose that the SFP is consistent and 0 < ξ < 2 ||A|| 2 . Let {x n } be a sequence defined by x n+1 = α n u + β n x n + γ n P C (I − ξA * (I − P Q )A)x n , ∀n ≥ 1,where {α n }, {β n } and {γ n } are three sequences in [0,1] and satisfy α n + β n + γ n = 1. If the following assumptions are satisfied:

An inertial extrapolation method for multiple-set split feasibility problem

“…It is clear that problem (1) has a nonempty solution set since 0 ∈ . In this example, we compare scheme (48) with the strong convergence result of SFP proposed by Shehu [57]. In the iterative scheme (48), for x 0 , x 1 ∈ C, we take ρ n = 3.5, θ n = 0.75, and α n = 1 √ n+1 .…”

“…In the iterative scheme (48), for x 0 , x 1 ∈ C, we take ρ n = 3.5, θ n = 0.75, and α n = 1 √ n+1 . The iterative scheme (27) in [57] for u, x 1 ∈ C, with α n = 1 n+1 , β n = n 2(n+1) = γ n , and t n = 1 A 2 was reduced into the following form:…”

Half-Space Relaxation Projection Method for Solving Multiple-Set Split Feasibility Problem

“…Additionally, we compare HSRPA with et al ([48], Algorithm 2.1). Also, a comparison of Algorithm 2 with the strong convergence result of SFP proposed by Shehu et al [56] is given in Example 4. Finally in Section 4.1, we present a sparse signal recovery experiment to illustrate the efficiency of Algorithm 2 by comparing with algorithms proposed by Lopez [2] and Yang [37].…”

“…We use x n+1 − x n < 10 −3 as stopping criteria for both algorithms and the outcome of the numerical experiment is reported in Figure 4. It can be observed from Figure 4 that, for different choices of u and x 1 , Algorithm 2 is faster in terms of less number of iterations and CPU-run time than the algorithm proposed by Shehu et al [56].…”

“…In this example, we compare Algorithm 2 with the strong convergence result of SFP proposed by Shehu et al [56]. The iterative scheme (27) in [56]…”