Fractional Differential Equations in Terms of Comparison Results and Lyapunov Stability with Initial Time Difference

and Applied Analysis 3 In particular, if x τ0 0, we obtain Dx t Dx t . 2.4 Hence, we can see that Caputo’s derivative is defined for functions for which RiemannLiouville fractional-order derivative exists. Let us write that Grünwald-Letnikov’s notion of fractional-order derivative in a convenient form D q 0x t lim lim h→ 0 nh t−τ0 1 hq [ x t − Sx, h, r, q, 2.5 where S x, h, r, q ∑n r 1 −1 r 1 ( q r ) x t − rh . If we know that x t is continuous and dx t /dt exist and integrable, then Riemann-Liouville and Grünwald-Letnikov fractionalorder derivatives are connected by the relation Dx t D 0x t x τ0 t − τ0 −q Γ ( 1 − q ∫ t τ0 t − s −q Γ ( 1 − q d ds x s ds. 2.6 Using 2.3 implies that we have the following relations among the Caputo, RiemannLiouville and Grünwald-Letnikov fractional derivatives Dx t D x t − x τ0 D 0 x t − x τ0 1 Γ ( 1 − q ∫ t τ0 t − s −q dx s ds ds. 2.7 The foregoing equivalent expressions are very useful in the study of qualitative properties of solutions of fractional differential equations. 2.2. Existence of Euler Solution We consider the initial value problem of the fractional-order differential equation with Reimann-Liouville’s derivative Dx f t, x , x t t − t0 |t t0 x0 for t ≥ t0, t0 ∈ R , 2.8 where f is any function from t0, T × R → R. Let π t0, t1, . . . , tN 2.9 be a partition of t0, T . 4 Abstract and Applied Analysis Consider the interval t0, t1 and observe that the right hand side of the initial value problem of fractional-order differential equation with Reimann-Liouville’s derivative Dx f ( t0, x 0 ) , x t t − t0 |t t0 x0 for t ≥ t0 2.10 on t0, t1 is constant. Therefore, the initial value problem has a unique solution of 2.10 of the fractionalorder differential equation with Reimann-Liouville’s derivative given by x t x0 t − t0 q−1 Γ ( q ) f ( t0, x 0 ) t − t0 q Γ ( 1 q ) , t ∈ t0, t1 . 2.11 Define the node x1 x t1 and iterate next by considering on t1, t2 the initial value problem Dx f t1, x1 , x1 t t − t1 |t t1 x0 1 for t ≥ t1. 2.12 The next node is x2 x t2 and we proceed this way till an arc xπ xπ t has been defined on all t0, T . Let us employ the notation xπ to emphasize the role played by the particular partition π in determining xπ which is the Euler curved arc corresponding to the partition π. The diameter of the partition π is given by μπ max ti − ti−1 : 1 ≤ i ≤ N . 2.13 Definition 2.1. An Euler solution is any curved arc x x t which is the uniform limit of Euler curved arcs xπ, corresponding to some sequence πj such that πj → 0, which means the convergence of the diameter μπj → 0 as j → ∞. Now, we can give the following result on existence of an Euler solution of the initial value problem of fractional-order differential equation with Reimann-Liouville’s derivative for 2.8 . Theorem 2.2. Assume that i ‖f t, x ‖ ≤ g t, ‖x‖ , t, x ∈ t0, T × R, where g ∈ C t0, T × R ,R , g t, u is nondecreasing in t, u ; ii The maximal solution r t r t, t0, u0 of the fractional-order scalar differential equation with Reimann-Liouville’s derivative Du g t, u , u t t − t0 |t t0 u0 ≥ 0 for t ≥ t0, t0 ∈ R 2.14 exists on t0, T . Then a there exists at least one Euler solution x t x t, t0, x0 to the initial value problem 2.8 , which satisfies a Hölder condition; Abstract and Applied Analysis 5 b any Euler solution x t of 2.8 satisfies the relation ∥ ∥ ∥x t − x0 t ∥ ∥ ∥ ≤ r ( t, t0, u 0 ) − u0, t ∈ t0, T , 2.15and Applied Analysis 5 b any Euler solution x t of 2.8 satisfies the relation ∥ ∥ ∥x t − x0 t ∥ ∥ ∥ ≤ r ( t, t0, u 0 ) − u0, t ∈ t0, T , 2.15 where u0 ‖x0‖ and x0 t x0 t − t0 q−1/Γ q . For proof of Theorem 2.2, please see in 6 . If f t, x in 2.8 is assumed to be continuous, then x t x t, t0, x0 , an Euler solution, is actually a solution of the initial value problem 2.8 . Theorem 2.3. Under the assumptions of Theorem 2.2 and if we suppose that f ∈ C t0, t0 T × R ,R , then x t is a solution of initial value problem 2.8 . For proof Theorem 2.3, please see in 6 . 2.3. Fractional-Order Differential Equations with Caputo’s Derivative Consider the initial value problems of the fractional-order differential equations with Caputo’s derivative Dx f t, x , x t0 x0 for t ≥ t0, t0 ∈ R , 2.16 Dx f t, x , x τ0 y0 for t ≥ τ0, τ0 ∈ R , 2.17 where x0 limt→ t0D q−1x t and y0 limt→ τ0D q−1x t exist, and the perturbed system of initial value problem of the fractional-order differential equation with Caputo’s derivative of 2.17 Dy F ( t, y ) , y τ0 y0 for t ≥ τ0 ≥ t0, 2.18 where y0 limt→ τ0D q−1y t , exists, and f, F ∈ C t0, τ0 T ×Rn,Rn ; satisfy a local Lipschitz condition on the setR ×Sρ, Sρ x ∈ R : ‖x‖ < ρ < ∞ and f t, 0 0 for t ≥ 0. In particular, F t, y f t, y R t, y ,we have a special case of 2.18 and R t, y is said to be perturbation term. Corollary 2.4. Let 0 < q < 1, and f : t0, t0 T ×Sρ → R a function such that f t, x ∈ L t0, t0 T for any x ∈ Sρ. If x t ∈ L t0, t0 T , then x t satisfies a.e. the initial value problems of the fractionalorder differential equations with Reimann-Liouville’s derivative 2.19 if, and only if, x t satisfies a.e. the Volterra fractional-order integral equation 2.20 . For proof of Corollary 2.4, please see in 2 . We assume that we have sufficient conditions to the existence and uniqueness of solutions through t0, x0 and τ0, y0 . If f ∈ C t0, t0 T × R,R and x t is the solution of Dx f t, x , x t t − t0 |t t0 x0 for t ≥ t0, t0 ∈ R , 2.19 6 Abstract and Applied Analysis where Dx is the Reimann-Liouville fractional-order derivative of x as in 2.2 , then it also satisfies the Volterra fractional-order integral equation x t x0 t − t0 q−1 Γ ( q ) 1 Γ ( q ) ∫ t t0 t − s q−1f s, x s ds, t0 ≤ t ≤ t0 T 2.20 that is, every solution of 2.20 is also a solution of 2.19 ; for detail please see 2 . We will only give the basic existence and uniqueness result with the Lipschitz condition by using contraction mapping theorem and a weighted norm with Mittag-Leffler function in 6 . Theorem 2.5. Assume that i f ∈ C R,R and bounded byM on R, where R t, x : t0 ≤ t ≤ t0 T, ‖x − x0‖ ≤ b ; ii ‖f t, x −f t, y ‖ ≤ L‖x−y‖, L > 0, t, x ∈ R, where the inequalities are componentwise. Then there exists a unique solution x t x t, t0, x0 on t0, t0 α for the initial value problem of the fractional-order differential equation with Caputo’s derivative of 2.16 , where α min T, bΓ q 1 /M 1/q . For proof of Theorem 2.5, please see in 6 . 2.4. Stability Criteria with ITD and Lyapunov-Like Function Before we can establish our comparison theorem and Lyapunov stability criteria for initial time difference, we need to introduce the following definitions of ITD stability and Lyapunovlike functions. Definition 2.6. The solution y t, τ0, y0 of the initial value problems of fractional-order differential equation with Caputo’s derivative of 2.18 through τ0, y0 is said to be initial time difference stable with respect to the solution x̃ t, τ0, x0 x t−η, t0, x0 ,where x t, t0, x0 is any solution of the initial value problems of fractional-order differential equation with Caputo’s derivative of 2.16 for t ≥ τ0, τ0 ∈ R and η τ0 − t0 if and only if given any > 0 there exist δ1 δ1 , τ0 > 0 and δ2 δ2 , τ0 > 0, such that ∥y ( t, τ0, y0 ) − xt − η, t0, x0 )∥ < , whenever ∥y0 − x0 ∥ < δ1, |τ0 − t0| < δ2 for t ≥ τ0. 2.21 If δ1, δ2 are independent of τ0, then the solution y t, τ0, y0 of the initial value problems of fractional-order differential equation with Caputo’s derivative of 2.18 is initial time difference uniformly stable with respect to the fractional solution x t−η, t0, x0 . If the solution of initial value problems of fractional-order differential equation with Caputo’s derivative of y t, τ0, y0 of the fractional system 2.18 through τ0, y0 is initial time difference stable and there exist γ1 τ0 > 0 and γ2 τ0 > 0 such that lim t→∞ ∥y ( t, τ0, y0 ) − xt − η, t0, x0 )∥ 0 2.22 Abstract and Applied Analysis 7 for all y t, τ0, y0 and x t − η, t0, x0 with ‖y0 − x0‖ < γ1 and |τ0 − t0| < γ2 for t ≥ τ0, then it is said to be initial time difference asymptotically stable with respect to the fractional solution x t − η, t0, x0 . It is initial time difference uniformly asymptotically stable with respect to the fractional solution x t − η, t0, x0 if γ1 and γ2 are independent of τ0. Definition 2.7. A function φ r is said to belong to the class K if φ ∈ C 0, ρ ,R , φ 0 0, and φ r is strictly monotone increasing in r. Definition 2.8. For any Lyapunov-like function V t, x ∈ C R × R,R , we define the fractional-order Dini derivatives in Caputo’s sense cD V t, x and DV t, x as followsand Applied Analysis 7 for all y t, τ0, y0 and x t − η, t0, x0 with ‖y0 − x0‖ < γ1 and |τ0 − t0| < γ2 for t ≥ τ0, then it is said to be initial time difference asymptotically stable with respect to the fractional solution x t − η, t0, x0 . It is initial time difference uniformly asymptotically stable with respect to the fractional solution x t − η, t0, x0 if γ1 and γ2 are independent of τ0. Definition 2.7. A function φ r is said to belong to the class K if φ ∈ C 0, ρ ,R , φ 0 0, and φ r is strictly monotone increasing in r. Definition 2.8. For any Lyapunov-like function V t, x ∈ C R × R,R , we define the fractional-order Dini derivatives in Caputo’s sense cD V t, x and DV t, x as follows D q V t, x lim h→ 0 sup 1 hq [ V t, x − V t − h, x − hf t, x , 2.23 DV t, x lim h→ 0− inf 1 hq [ V t, x − V t − h, x − hf t, x )] 2.24 for t, x ∈ R × R. Definition 2.9. For a real-valued function V t, x ∈ C R × R,R , we define the generalized fractional-order derivatives Dini-like derivatives in Caputo’s sense ∗D q V t, y − x̃ and c ∗D V t, y − x̃ as follows c ∗D q V ( t, y − x̃ lim h→ 0 sup 1 hq [ V ( t, y − x̃−V ( t − h, y −x̃ − h ( F ( t, y − f̃ t, x̃ ))] , 2.25 c ∗D V ( t, y − x̃ lim h→ 0− inf 1 hq [ V ( t, y − x̃ − V ( t − h, y − x̃ − h ( F ( t, y ) − f̃ t, x̃ ))] 2.26