Two-points problem for an evolutional first order equation in Banach space

Abstract

Two-point nonlocal problem for the first order differential evolution equation with an operator co-

efficient in a Banach space X is considered. An exponentially convergent algorithm is proposed and

justified in assumption that the operator coefficient is strongly positive and some existence and unique-

ness conditions are fulfilled. This algorithm leads to a system of linear equations that can be solved

by fixed-point iteration. The algorithm provides exponentially convergence in time that in combination

with fast algorithms on spatial variables can be efficient treating such problems. The efficiency of the

proposed algorithms is demonstrated by numerical examples.