Semilinear ordinary differential equation coupled with distributed order fractional differential equation

Abstract

The system D2 y (t) + z (t) = f (t, y), ∫02 φ{symbol}1 (γ) Dγ y (t) d γ = ∫02 φ{symbol}2 (γ) Dγ z (t) d γ, t > 0, where Dγ, γ ∈ [0, 2] are operators of fractional differentiation, is investigated and the existence of a mild and classical solution is proven. Also, a necessary and sufficient condition for the existence and uniqueness of a solution to a general linear fractional differential equation ∑i = 1k ai Dγi z = g, γi ∈ R, i = 1, ..., k in S+′ is given. © 2010 Elsevier Ltd. All rights reserved.