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Problem for the first order differential equation with an unbounded operatorcoefficient in Banach space and integral nonlocal condition is considered. Anexponentially convergent algorithm is proposed and justified for the numericalsolution of this problem in assumption that an operator coefficient $A$ isstrongly positive and some existence and uniqueness conditions are fulfilled.This algorithm is based on the representations of operator functions by aDunford-Cauchy integral along a hyperbola, enveloping the spectrum of $A$, andon the proper quadratures involving short sums of resolvents. The efficiency ofthe proposed algorithms is demonstrated by several numerical examples.

Exponentially convergent method for integral nonlocal problem for the first order differential equation with unbounded coefficient in Banach space