The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces

Abstract

The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces are established. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. The results presented in this paper extend and improve the main results of Refs. (Moudafi in J. Math. Anal. Appl. 241:46-55, 2000; Xu et al. in Fixed Point Theory Appl. 2015:41, 2015. Moreover, applications to a more general system of variational inequalities, the constrained convex minimization problem and K-mapping are included. MSC: 47H09

Strong convergence theorems for the generalized viscosity implicit rules of nonexpansive mappings in uniformly smooth Banach spaces

“…Very recently, iterative sequence for the implicit midpoint rule has been studied by many authors, because it is a powerful method for solving ordinary differential equations; see [1,2,5,8,13,18,19] and the references therein. Recently, Xu et al [18] considered the following viscosity implicit midpoint rule:…”

“…Very recently, Ke et al [8] applied the viscosity technique to the implicit rules of nonexpansive mappings in Hilbert spaces. More precisely, they proposed the following two viscosity implicit rules: 5) and…”

Viscosity implicit iterative algorithms based on generalized contractions for strictly pseudo-contractive mappings in Banach spaces

“…They obtained that the sequence {x n } generated by (1.4) and (1.5) converges strongly to a fixed point of nonexpansive mapping T , which also solves variational inequality (1.2). In 2016, Yan et al [19] extended the main results of Ke et al [6] from Hilbert spaces to uniformly smooth Banach spaces and replaced strict contractions by generalized contractions. They proved that the sequence generated by (1.5) converges strongly to a fixed point x * of nonexpansive mapping T , which is also the solution of the following variational inequality in Banach spaces:…”

“…In the same year, Ke et al [6] generalized the viscosity implicit midpoint rule of Xu et al [18] to the following two viscosity implicit rules: 5) where f and T are still contractions and nonexpansive mappings from H onto itself, respectively. They obtained that the sequence {x n } generated by (1.4) and (1.5) converges strongly to a fixed point of nonexpansive mapping T , which also solves variational inequality (1.2).…”

“…The implicit midpoint rule is a powerful method for solving ordinary differential equations; see [1,2,4,6,12,18,20] and the references therein. Recently, Xu et al [18] considered the following viscosity implicit midpoint rule:…”