Resonance mean-periodic solutions of Euler differential equations

Abstract

The Euler operator δ = tddt is considered in the space C = C(R+) of the continuous functions on R+ = (0,∞). Nonlocal operational calculi for it are developed and used for solving nonlocal Cauchy boundary value problems for Euler differential equations of the form P(δ)y = f with a polynomial P. A function f∈C(R+) is said to be mean-periodic for the Euler operator with respect to the linear functional Φ (or simply Φ-mean-periodic) if Φτ{f(tτ)} = 0 identically on R+. The solution of Euler differential equations in mean-periodic functions for δ with respect to an arbitrary linear functional Φ reduces to non-local homogeneous Cauchy problems. Denoting the algebraic equivalent of the Euler differential operator δ by S, the solution of an Euler differential equation P(δ)y = f in Φ-mean-periodic functions reduces to the interpretation of y = 1P(S)f as a function. This is done both in the non-resonance and in the resonance cases.