  The $m$-point nonlocal problem for the first order differential equation with
an operator coefficient in a Banach space $X$ is considered. An exponentially
convergent algorithm is proposed and justified provided that the operator
coefficient $A$ is strongly positive and some existence and uniqueness
conditions are fulfilled. This algorithm is based on representations of
operator functions by a Dunford-Cauchy integral along a hyperbola enveloping
the spectrum of $A$ and on the proper quadratures involving short sums of
resolvents. The efficiency of the proposed algorithms is demonstrated by
numerical examples.
