General semi-implicit approximations with errors for common fixed points of nonexpansive-type operators and applications to Stampacchia variational inequality

Abstract

It is meaningful and valuable to find common fixed points of different nonexpansive-type operators, which are associated with variational inequalities, integral equations, image process and other optimization problems in real life. The purpose of this paper is to suggest and consider a class of general semi-implicit iterative methods involving semi-implicit rule and inaccurate computing errors, which extend the iterative algorithm introduced by Ali et al. in 2020. Using Liu’s lemma, we analyze convergence and stability of the new iterative approximations for common fixed points of three different nonexpansive-type operators. Furthermore, we provide convergence rates of the new iterations and some numerical examples to illustrate the efficiency and stability of the new iterative schemes. As an application of our main results presented in this paper, we use the proposed iterative schemes to solve the known Stampacchia variational inequality.

Novel Multistep Implicit Iterative Methods for Solving Common Solution Problems with Asymptotically Demicontractive Operators and Applications

“…On the other hand, it is well known that the implicit rule is one powerful tool in the field of ordinary differential equations and is widely used to construct the iteration scheme for (asymptotically) nonexpansive-type operators; see, for example, [15][16][17][18][19] and the references therein. Of particular note is that Aibinu and Kim [20] compared the convergence rates of the following two viscosity implicit iterations:…”

“…where {a n }, {b n }, {c n }, and {d n } are four sequences satisfied by special conditions, and S and T are two self operators for H. They also proved that iteration (10) converges faster than (9) under some prerequisites. Due to the complexity and effectiveness of the implicit rules (see [19]), there are few pieces of research on implicit iterations for the more general asymptotically demicontractive operators. Thus, the following question comes naturally: Question 1.…”

Iterative Approach for Solving Variational Inequality Problems using Fixed Point Concept

“…Akram et al (5) worked on FPP and split variational inclusion problem. Many researchers introduced different iterative schemes to find out the solutions of VIPs (6)(7)(8)(9)(10)(11) . Lamba and Panwar (12) and Panwar et al (13) introduced iterative schemes for nonexpansive mappings.…”