General modified viscosity implicit rules for generalized asymptotically nonexpansive mappings in complete CAT(0) $\operatorname{CAT}(0)$ spaces

Abstract

Abstract It is well known that the concept (generalized) asymptotically nonexpansive is closely related to the theory of fixed points in Banach spaces, and the implicit midpoint rule is one of the powerful numerical methods for solving differential equations. The purpose of this paper is to introduce a class of new general modified viscosity implicit rules of generalized asymptotically nonexpansive mappings in complete CAT(0) $\operatorname{CAT}(0)$ spaces, and to prove some strong convergence theorems of the procedure generated by the new general modified viscosity implicit rules under some suitable conditions. The results presented in this paper improve and extend varieties of results in the recent literature.

Keywords

• General modified viscosity implicit rule
• Fixed point
• Strong convergence
• Generalized asymptotically nonexpansive mapping
• CAT ( 0 ) $\operatorname{CAT}(0)$ space