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

Abstract

Let K be a nonempty closed convex subset of a Banach space E and T : K → K be a nonexpansive mapping. Using a viscosity approximation method, we study the implicit midpoint rule of a nonexpansive mapping T. We establish a strong convergence theorem for an iterative algorithm in the framework of uniformly smooth Banach spaces and apply our result to obtain the solutions of an accretive mapping and a variational inequality problem. The numerical example which compares the rates of convergence shows that the iterative algorithm is the most efficient. Our result is unique and the method of proof is of independent interest.

The Implicit Midpoint Procedures for Asymptotically Nonexpansive Mappings

“…Suppose f: K ⟶ K is a contraction with constant α ∈ [0, 1) and assume that a n satisfies conditions (i) − (iii). For an arbitrary x 1 ∈ K, the iterative sequence x n given by (10) is bounded.…”

“…Noting the hypothesis that z t n ⟶ z as n ⟶ ∞, coupled with the continuity of f and uniform continuity of the duality mapping J on bounded subsets of E, it gives Assume that a n satisfies conditions (i) − (iii) and in addition (iv) lim n⟶∞ ‖x n − T n x n ‖ � 0 Then, for an arbitrary x 1 ∈ K, the iterative sequence x n given by (10) converges in norm to a fixed point p of T which is the unique solution of variational inequality:…”

“…In 2015, Yao et al in [7] introduced x n+1 � a n f x n + b n x n + c n T x n + x n+1 2 , n ∈ N, (9) which gives a faster approximation compared with (7), where T is a nonexpansive mapping in a Hilbert space and a n + b n + c n � 1, ∀n ∈ N. Aibinu et al in [10] compared the rate of convergence of the iteration procedures (6), (7), and (9). Aibinu and Kim in [11] recently introduced a new scheme of the viscosity implicit iterative algorithms for nonexpansive mappings in Banach spaces.…”

Solutions of Nonlinear Operator Equations by Viscosity Iterative Methods

“…where J is a normalized duality mapping and h,i is the duality pairing. Aibinu and Kim [7] used an analytical method to compare the rate of convergence of the sequences (4) and (5). Due to the important roles which nonlinear operator equations play in modeling many phenomena in scientific fields, research on the solution and application of nonlinear operator equations are ever green (see e.g., [8][9][10][11][12]).…”

“…where h,i is the inner product. Aibinu et al [4] studied the convergence of the sequence (2) in uniformly smooth Banach spaces. Ke and Ma [5] introduced generalized viscosity implicit rules which extend the results of Xu et al [3].…”