Strong convergence of a viscosity iterative algorithm in Banach spaces with applications

Abstract

We present the strong convergence theorems for the viscosity iterative scheme for finding a common element of the solution set of the system of general variational inequalities for two arbitrary nonlinear mappings and the fixed point set of a nonexpansive mapping in real 2uniformly smooth and uniformly convex Banach spaces. Furthermore, we apply our main result with the problem of approximating a zero point of accretive operators and a fixed point of strictly pseudocontractive mappings in Banach spaces. The main results presented in this paper improve and extend some results in the literature.

The implicit midpoint rule of nonexpansive mappings and applications in uniformly smooth Banach spaces

“…Let L be the 2-uniformly smooth constant of a 2-uniformly smooth Banach space and A 1 , A 2 : K → E be µ 1 -inverse strongly accretive and µ 2 -inverse strongly accretive, respectively. If 0 < α 1 < µ 1 L 2 and 0 < α 2 < µ 2 L 2 , then I − α 1 A 1 and I − α 2 A 2 are nonexpansive [12].…”