Asymptotic stability in Caputo-Hadamard fractional dynamic equations

In this work, we investigate the asymptotic stability of the zero solution for Caputo-Hadamard fractional dynamic equations on a time scale. We will make use of the Krasnoselskii (cid:28)xed point theorem in a weighted Banach space to show new stability results.


Introduction
Fractional dynamic equations without and with delay arise from a variety of applications including in various elds of science and engineering. In particular, problems concerning qualitative analysis of fractional dynamic equations have received the attention of many authors, see [1]- [27], [29]- [31] and the references therein.
Fractional dynamic equations involving Riemann-Liouville and Caputo ∆-fractional derivatives have been studied extensively by several researchers, see [2], [11], [30], [31]. However, the literature on Hadamard dynamic equations is not yet as enriched. The ∆-fractional derivative due to Hadamard diers from the aforementioned derivatives in the sense that the kernel of the integral in the denition of Hadamard ∆fractional derivative contains a logarithmic function of arbitrary exponent, see [21].
In [11], Belaid et al. investigated the following Caputo fractional dynamic equation where C T D α 0 + is the Caputo fractional derivative on T of order 1 < α < 2. By employing the Krasnoselskii xed point theorem, the asymptotic stability of the zero solution has been established.
In this paper, we extend the results in [11] by proving the stability and asymptotic stability of the zero solution for the following Caputo-Hadamard fractional dynamic equation where T is an unbounded above time scale with 1 ∈ T, CH To prove the stability and asymptotic stability of the trivial solution, we transform (1.1) into an equivalent integral equation and then use the Krasnoselskii xed point theorem. The obtained integral equation is the sum of two mappings, one is a compact and the other is a contraction.

Preliminaries
In this section, We use C rd ([1, ∞) T ) for a space of rd-continuous functions where [1, ∞) T is an interval. Denition 2.1 ([13]). A time scale T is an arbitrary nonempty closed subset of the real numbers. Denition 2.2 ([13]). For t ∈ T, the forward jump operator σ : T → T is dened by Denition 2.3 ([13]). A function f : T → R is rd-continuous provided that it is continuous at all right-dense points of T and its left-sided limits exist at left-dense points of T. The set of all rd-continuous functions on T is denoted by C rd (T). Denition 2.4 ([13]). Let t ∈ T and f : T → R be a function. Then ∆-derivative of f at the point t is dened to be the number f ∆ (t) with the property that for each ε > 0 there exists a neighborhood U of t in T such that Remark 2.6. All rd-continuous bounded functions on [a, b) T are delta integrable from a to b. Denition 2.7 ([2, 21]). Assume T is a time scale, [1, b] T ⊆ T and the function x is an integrable function on [1, b] T , then Hadamard ∆-fractional integral of x is dened by where Γ (α) is the Gamma function.
In our discussion, the following Banach space plays a fundamental role.
. For more properties of this Banach space, see [26]. Also, let is a solution of the following Cauchy type problem if and only if x is a solution of the following Cauchy type problem By using Lemma 2.9, we have which means that x is a solution of (2.2).
2) Let x be a solution of (2.2). For any t ∈ [1, ∞) T , by Remark 2.10, it is easy to show that In addition, note that r ∈ C rd ([1, ∞) T ), we have be a solution of (1.1). By Lemma 2.11, we get Then So, we can write (2.4) as We obtain (2.3) by using the variation of constants formula. The converse follows easily because each step is reversible. This completes the proof.
(ii) asymptotically stable, if it is stable in Banach space E and there is a number σ > 0 such that |x 0 | + |x 1 | ≤ σ implies lim t→∞ x (t) = 0.
Lastly in this section, we state the Krasnoselskii xed point theorem which enables us to prove the stability and asymptotic stability of the trivial solution to (1.1). Theorem 2.14 (Krasnoselskii [28]). Let Ω be a closed convex nonempty subset of a Banach space (S, . ). Assume that A and B map Ω into S such that (i) Ax + By ∈ Ω for all x, y ∈ Ω, (ii) A is continuous and AΩ is contained in a compact set of S, (iii) B is a contraction with constant l < 1. Then, there is a x ∈ Ω with Ax + Bx = x.
The following modied compactness criterion is needed in order to show (ii). Theorem 2.15 ([26]). Let M be a subset of the Banach space E. Then, M is relatively compact in E if the following conditions are satised x ∈ M} is equiconvergent at innity i.e. for any given ε > 0, there is a T 0 > 1 such that for all x ∈ M and t 1 , t 2 > T 0 , if holds
Step 2. It is easy to prove that A is continuous. Now, we only show that A (ε) is a relatively compact in E. By (3.8), we obtain that {x(t)/h(log t) : x ∈ (ε)} is uniformly bounded in E. Also, a classical theorem states the fact that the convolution of an L 1 -function with a function tending to zero, does also tend to zero. Then we conclude that for log t u ≥ 0, we get because lim t→∞ Together with the continuity of K and h, we obtain that there is a constant and for any T 0 ∈ [1, ∞) T , the function K(log t u )h(log u)/h(log t) is uniformly continuous on as t 2 → t 1 , which implies that {x(t)/h(log t) : x ∈ (ε)} is equicontinuous on any compact interval of [1, ∞) T . By using Theorem 2.15, in order to prove that A (ε) is a relatively compact set of E, we only need to show that {x(t)/h(log t) : x ∈ (ε)} is equiconvergent at innity. In fact, for any ε 1 > 0, there is a L > 1 such that By (3.10), we obtain that So, there is T > L such that t 1 , t 2 ≥ T , we get Thus, for t 1 , t 2 ≥ T , Hence, the required conclusion is true.
Step 3. We prove that B : (ε) → E is a contraction mapping.
For any x, y ∈ (ε), by (3.1), we get that By using Theorem 2.14, we know that there is a xed point of the operator A + B in (ε). Finally, for any Then, the zero solution of (1.1) is stable in Banach space E.
Proof. It follows by Theorem 3.1 that the zero solution of (1.1) is stable in the Banach space E. Next, we shall prove that the zero solution x = 0 of (1.1) is attractive. For any r > 0, we dene * (r) = x ∈ (r) , lim t→∞ x(t)/h(log t) = 0 .
We only need to show that Ax + By ∈ * (r) for any x, y ∈ * (r), i.e.