The implicit midpoint rule for nonexpansive mappings

Abstract

The implicit midpoint rule (IMR) for nonexpansive mappings is established. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a Hilbert space. Applications to the periodic solution of a nonlinear time-dependent evolution equation and to a Fredholm integral equation are included. © 2014 Alghamdi et al.; licensee Springer.