Stability of nonlinear Volterra integro-differential equations with Caputo fractional derivative and bounded delays

Abstract

We use Lyapunov functions to study stability of the first-order Volterra integro-differential equation with Caputo fractional derivative Ct0Dqtx(t) = -α(t)ƒ(x(t)) + ∫tt-r B(t, s)g(s, x(s))ds + h(t, x(t), x(t - τ(t))). For the Lyapunov functions, we consider three types of fractional derivatives. By means of these derivatives, we obtain new sufficient conditions for stability and uniformly stability of solutions. We consider both constant and time variable bounded delays, and illustrated our results with an example.