FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Boundary Value Problems for Differential Equations of Fractional Order 67(1) The operator RLa D γ t defined byRLa D γ t y(t) = dn 1dt n Γ(n − γ)∫ ta(t − τ) n−γ−1 y(τ) dτ (3.11)for t ∈ [a, b] and y(t) ∈ C n [a, b], is called the Riemann–Liouville differentialoperator of order γ.(2) The operator C a D γ t defined byCa D γ 1t y(t) =Γ(n − γ)∫ ta(t − τ) n−γ−1( d) ny(τ)dτ (3.12)dτfor t ∈ [a, b] and y(t) ∈ C n [a, b], is called the Caputo differential operator oforder γ.Definition 3.2 (see [29]). Provided γ > 0, the operator Ja γ , definedon L 1 [a, b] byJ γ a y(t) = 1Γ(γ)∫ ta(t − τ) γ−1 y(τ) dτ (3.13)for t ∈ [a, b], is called the Riemann–Liouville fractional integral operator oforder γ, where L 1 [a, b] := { y(t) : [a, b] → R; y(t) is measurable on [a, b] and∫ ba|y(t)| dt < ∞ } .Lemma 3.1 (see [31]). (1) Let γ > 0. Then, for every f ∈ L 1 [a, b],RLa D γ t J γ a f = f (3.14)almost everywhere.(2) Let γ > 0 and n − 1 < γ ≤ n. Assume that f is such that Jan−γ f ∈A n [a, b]. Then,n−1∑Ja γ RLa D γ (t − a) γ−k−1t f(t) = f(t) −Γ(γ − k)k=0limz→a +(3) Let γ > θ > 0 and f be continuous. ThenCa D γ t J γ a f = f,d n−k−1n−γJdzn−k−1 a f(z). (3.15)Ca Dt θ Ja γ f = Ja γ−θ f. (3.16)(4) Let γ ≥ 0, n − 1 < γ ≤ n and f ∈ A n [a, b]. Thenn−1∑Ja γ C a D γ t f(t) = f(t) −k=0D k f(a)k!(t − a) k , (3.17)where A n [a, b] is the set of functions with absolutely continuous derivativeof order n − 1.