Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces
Furmose Mendy, John T. Mendy, Jatta Bah, Gabriel J Mendy · International Journal of Theoretical and Applied Mathematics · 2023
Viscosity’s implicit algorithm for finding a common element of the set of fixed points for nonlinear operators and the set of solutions of variational inequality problems have been investigated by many authors in different settings in Hilbert and Banach space. In most cases, they consider the following study of viscosity implicit double midpoint, generalized viscosity in the class of nonexpansive and asymptotically nonexpansive mappings. The implicit midpoint rule can effectively solve ordinary differential equations. Meanwhile, many authors have used viscosity iterative algorithms for finding common fixed points for nonlinear operators and solutions of variational inequality problems. Recently, the convergence rate and comparison viscosity implicit iterative algorithm has been studied widely. Under suitable conditions imposed on the control parameters, it is shown in this paper that certain two implicit iterative sequences {ωn} and {ξn} converge to the same fixed point of an asymptotically nonexpansive mapping in Hilbert spaces without comparison. It is also proven that {ωn} and {ξn} converge strongly to the same solution, which also solves the variational inequality problem. The results presented in this paper improve and extend some recent corresponding results in the literature.