New Iterative Algorithm for Solving Constrained Convex Minimization Problem and Split Feasibility Problem

Abstract

The purpose of this paper is to introduce a new iterative algorithm to approximate the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Also, we show that our proposed iterative algorithm converges weakly and strongly to the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. Furthermore, it is proved analytically that our new iterative algorithm converges faster than one of the leading iterative algorithms in the literature for almost contraction mappings. Some numerical examples are also provided and used to show that our new iterative algorithm has better rate of convergence than all of S, Picard-S, Thakur and M iterative algorithms for almost contraction mappings and generalized α-nonexpansive mappings. Again, we show that the proposed iterative algorithm is stable with respect to T and data dependent for almost contraction mappings. Some applications of our main results and new iterative algorithm are considered. The results in this article are improvements, generalizations and extensions of several relevant results existing in the literature.

A Robust Iterative Approach for Solving Nonlinear Volterra Delay Integro–differential Equations

“…Remark 1. Since the notion of w 2 -stability is more general the concept of simple stability considered in [15,24,28], hence, our result improves and generalizes the corresponding in [15,24,28] and several others.…”

“…In this paper, we have studied the stability result of our newly introduced iterative algorithm (1.9) for a wider concept of stability known as w 2 -stability instead of the simple notion of stability considered in [15,24,28]. A numerical example is also used to support the analytical proof of our stability theorem.…”

“…In this section, we will approximate the solution of nonlinear Volterra delay integro-differential equations by utilizing the following iterative algorithm recently introduced by Ofem et al [24]:…”

“…We remark that the iterative algorithm (1.9) is the multivalued conversion of the iterative algorithm (6.1). It is shown in [24] that the iterative algorithm (6.1) has a better speed of convergence than S [3], Picard-S [12], Thakur [34] and M [36] iteration processes for single-valued generalized α-nonexpansive mappings.…”

“…The notion of stability in Definition 6 has been studied by several authors for both single and multivalued mappings (see [15,24,28] and the references in them).…”