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Two-points nonlocal problem for the first order differential evolutionequation with an operator coefficient in a Banach space $X$ is considered. Anexponentially convergent algorithm is proposed and justified in assumption thatthe operator coefficient is strongly positive and some existence and uniquenessconditions are fulfilled. This algorithm leads to a system of linear equationsthat can be solved by fixed-point iteration. The algorithm providesexponentially convergence in time that in combination with fast algorithms onspatial variables can be efficient treating such problems. The efficiency ofthe proposed algorithms is demonstrated by numerical examples.

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