Operational Calculus Approach to Nonlocal Cauchy Problems

Abstract

Let Φ be a linear functional of the space ${\mathcal{C} =\mathcal{C}(\Delta)}$ of continuous functions on an interval Δ. The nonlocal boundary problem for an arbitrary linear differential equation

$$ P\left(\frac{d}{d t}\right)y = F(t) $$

with constant coefficients and boundary value conditions of the form

$$ \Phi\{\,y^{(k)}\} =\alpha_k,\,\,\,k = 0,\,1,\,2,\, \ldots,\,{\rm deg} P-1 $$

is said to be a nonlocal Cauchy boundary value problem. For solution of such problems an operational calculus of Mikusiński’s type, based on the convolution

$$ (f*g)(t) = \Phi_\tau\, \left\{{\int\limits_\tau^t} f(t+\tau - \sigma)\,g(\sigma)\, d \sigma\, \right\}, $$

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.