Exponentially convergent method for m-point nonlocal problem for the first order differential equation in Banach space

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 algorithm is demonstrated by numerical examples.