On stability and oscillation of fractional differential equations with a distributed delay

Abstract

In this paper, we study the stability and oscillation of fractional differential equations $^cD^α x(t)$ + $int_0^{1}$x(t) + ∫ 1 0 x(s + [t − 1])dR(s) = 0$. We discretize the fractional differential equation by variation of constant formula and semigroup property of Mittag– Leffler function, and get the difference equation corresponding to the integer points. From the equivalence analogy of qualitative properties between the difference equations and the original fractional differential equations, the necessary and sufficient conditions of oscillation, stability and exponential stability of the equations are obtained.