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
