New two types of semi-implicit viscosity iterations for approximating the fixed points of nonexpansive operators associated with contraction operators and applications

Abstract

Abstract Motivated and inspired by the growing contribution with respect to iterative approximations from some researchers in the literature, we design and investigate two types of brand-new semi-implicit viscosity iterative approximation methods for finding the fixed points of nonexpansive operators associated with contraction operators in complete CAT ( 0 ) ${\operatorname{CAT}(0)}$ spaces and for solving related variational inequality problems. Under some suitable assumptions, strong convergence theorems of the sequences generated by the approximation iterative methods are devised, and a numerical example and some applications to related variational inequality problems are included to verify the effectiveness and practical utility of the convergence theorems. Our main results presented in this paper do not only improve, extend and refine some corresponding consequences in the literature, but also show that the additional variational inequalities, general variational inequality systems and equilibrium problems can be solved via approximation of the iterative sequences. Finally, we provide an open question for future research.

Keywords

• Semi-implicit viscosity approximation method
• Nonexpansive operator associated with contraction operator
• Application to related variational inequality problem
• Complete CAT ( 0 ) $\operatorname{CAT}(0)$ space
• Strong convergence