Razumikhin Stability Theorem for Fractional Systems with Delay

and Applied Analysis 3 For 0 < q ≤ 1 we have t0D q t x t 1 Γ ( 1 − q d dt (∫ t t0 x s t − s q ds ) , ( 0 < q ≤ 1, c t0D q t x t 1 Γ ( 1 − q ∫ t t0 x′ s t − s q ds, ( 0 < q ≤ 1. 2.2 Some properties of Riemann-Liouville and Caputo derivatives are recalled below 1–3 . Property 1. When 0 < q ≤ 1, we have c t0D q t x t t0D q t x t − x t0 Γ ( 1 − q t − t0 −q. 2.3 In particular, if x t0 0, we have c t0D q t x t t0D q t x t . 2.4 Property 2. For any v > −1, we have t0D q t t − t0 v Γ 1 v Γ ( 1 v − q t − t0 v−q. 2.5 In particular, if 0 < q < 1, v > 0, and x t t − t0 , then from Property 1, we have c t0D q t t − t0 v Γ 1 v Γ ( 1 v − q t − t0 v−q. 2.6