Exponentially Convergent Method for the First-Order Differential Equation in a Banach Space with Integral Nonlocal Condition

Abstract

For the first-order differential equation with unbounded operator coefficient in a Banach space, we study the nonlocal problem with integral condition. An exponentially convergent algorithm for the numerical solution of this problem is proposed and justified under the assumption that the operator coefficient A is strongly positive and certain existence and uniqueness conditions are satisfied. The algorithm is based on the representations of operator functions via the Dunford–Cauchy integral along a hyperbola covering the spectrum of A and the quadrature formula containing a small number of resolvents. The efficiency of the proposed algorithm is illustrated by several examples.