  Two-points nonlocal problem for the first order differential evolution
equation with an operator coefficient 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 uniqueness
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.
