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Exponentially Convergent Method for the m-Point Nonlocal Problem for a 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 algorithms is demonstrated by numerical examples.