Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces

Abstract

Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E , and let T : K ⟶ E be a continuous pseudocontraction which satisfies the weakly inward condition. For f : K ⟶ K any contraction map on K , and every nonempty closed convex and bounded subset of K having the fixed point property for nonexpansive self-mappings, it is shown that the path x → x t , t ∈ [ 0 , 1 ) , in K , defined by x t = t T x t + ( 1 − t ) f ( x t ) is continuous and strongly converges to the fixed point of T , which is the unique solution of some co-variational inequality. If, in particular, T is a Lipschitz pseudocontractive self-mapping of K , it is also shown, under appropriate conditions on the sequences of real numbers { α n } , { μ n } , that the iteration process: z 1 ∈ K , z n + 1 = μ n ( α n T z n + ( 1 − α n ) z n ) + ( 1 − μ n ) f ( z n ) , n ∈ N , strongly converges to the fixed point of T , which is the unique solution of the same co-variational inequality. Our results propose viscosity approximation methods for Lipschitz pseudocontractions.