Operational Calculus Approach to Nonlocal Cauchy Problems

Let Phi be a linear functional of the space C = C(Delta) of continuous functions on an interval Delta. The nonlocal boundary problem for an arbitrary linear differential equation P (d/dt) y = F(t) with constant coefficients and boundary value conditions of the form Phi{y((k))} = alpha(k), k = 0, 1, 2,..., degP - 1 is said to be a nonlocal Cauchy boundary value problem. For solution of such problems an operational calculus of Mikusinski's type, based on the convolution (f * g)(t) = Phi(tau) {integral(t)(tau) f (t + tau - sigma) g(sigma)} is developed. In the frames of this operational calculus the classical Heaviside algorithm is extended to nonlocal Cauchy problems. The obtaining of periodic, antiperiodic and mean-periodic solutions of linear ordinary differential equations with constant coefficients both in the non-resonance and in the resonance cases reduces to such problems. Here only the non-resonance case is considered. Extensions of the Duhamel principle are proposed.