Asymptotic stability for mixed fractional delay differential equations

This paper is concerned with the stability analysis of nonlinear mixed fractional delay differential equations using Krasnoselskii’s fixed point theorem in a weighted Banach space.


Introduction
Fractional differential equations have become an important field of applied mathematics and modeling of many physical phenomena associated to very rapid and very short changes, for more details we refer to the books ( [7,11,15,16,20,21]) and the references therein. In particular, initial value problems and boundary value problems related to the qualitative theory of the existence, uniqueness and stability of solutions for fractional differential equations have been mainly discussed by a lot of authors especially in last three decades or so. Beside, several methods have been employed to prove the existence, uniqueness and stability of solutions for fractional boundary value problems among the spectra theory, the critical point theory, method of upper and lower solutions and the fixed point theorems [1,2,3,4,5,6,9,12,13,14], among others.
In [5], Bashir et al. studied the qualitative theory of the following boundary value problem with delay where D α and D β are the Caputo-Hadamard fractional derivatives, 0 < α, β < 1.
Ge and Kou [9], by utilizing the Krasnoselskii's fixed point theorem, discussed the stability and asymptotic stability of the zero solution to the following Caputo type fractional differential equation Furthermore, in [6], Boulares et al. discussed the stability and asymptotic stability of the zero solution of the following boundary value problem with delay where m 0 = inf t≥0 {t − r (t)}. Motivated by the works mentioned above and the papers [1,2,3,8,10,12,13,14,17,18,19,23,24] and the references therein, We aim to enrich the field of differential equations by talking about the analysis of qualitative theory of the subjects of the stability and asymptotic stability of the zero solution to the following initial value problem of mixed Riemann-Liouville-Caputo fractional differential equations with delay on unbounded interval where RL D α and C D β are the left Riemann Liouville and left Caputo fractional derivatives respectively, 0 < α ≤ 1, 1 < β ≤ 2, r > 0, f, g : R + × R → R are continuous functions with f (t, 0) = g(t, 0) = 0 and we denote the solution of (1) by u (t, Φ, u 0 ).
To this aim, we start by transformation (1) into fixed point problem using some mathematical skills of fractional integral and derivative, then we use Krasnoselskii's fixed point theorem in appropriate weighted Banach space.

preliminaries
In this section, we introduce some notations, definitions, and preliminary concepts that which we need in later and can be found in [16,17,21,22] as well as we present the equivalent integral equation of (1).
Let C λ be the class of all continuous functions defined on [−r, +∞) with the norm for all positive real number λ > 1. Also C r = C ([−r, 0]) is endowed with norm We present in the following some basic concepts of fractional calculus.
Definition 2.1 ( [16,21]). The Riemann-Liouville fractional integral of the function u of order α > 0 is defined by where Γ is the Euler gamma function defined by Γ (α) = ∞ 0 e −t t α−1 dt. 16,21]). The Riemann-Liouville fractional derivative of the function u of order α ∈ (n − 1, n] is defined by 16,21]). The Caputo fractional derivative of the function u of order α ∈ (n − 1, n] is defined by Let α > 0 be a real number, we have two following lemmas. Lemma 2.4. The unique solution of linear fractional differential equation Lemma 2.6. Problem (1) is equivalent to the following Caputo type fractional differential equation with delay Proof. Using Lemma 2.4, equation one of (1) can be written as using condition lim t→0 t 1−α C D β u(t) = 0, we get c 0 = 0. Then we obtain the desired result.
Lemma 2.7. Let f and g are continuous functions. Then u ∈ C ([−r, +∞)) is a solution of the problem (2) if and only if u is a solution of the delay Cauchy type problem Proof. Let u ∈ C ([−r, +∞)) be a solution of the problem (2), for any t ≥ 0, we have According to Lemma 2.5 and the condition u (0) = u 0 , one gets which means that u is a solution of the problem (3). Conversely, let u be a solution of the problem (3). Then, for any t ≥ 0, it is easy to see that Besides, we have u (0) = u 0 . (3) is equivalent to the following Volterra type integral equation Proof. Let k defined above. It is clear that (3) can be written as follow By the variation of constants formula, we obtain (4).
Conversely, it is clear that using this fact, we get e ks u(s)ds This means that On the other hand, if (4) holds we have u(0) = Φ (0). From the argument above, we get that the system (3) can be equivalently written as (4).
(ii) asymptotically stable, if it is stable in C λ and there exists a number σ > 0 such that Our main results based on the Krasnoselskii fixed point theorem.
Lemma 2.10 (Krasnoselskii fixed point theorem [22]). If M is a nonempty bounded, closed and convex subset of a Banach space E, A and B two operators defined on M with values in E such as (i) Au + Bv ∈ M , for all u, v ∈ M, (ii) A is continuous and compact, (iii) B is a contraction. Then there exists w ∈ M such as: w = Aw + Bw.
In order to prove (ii), we need to the following modified compactness criterion. iii) e −λt u(t) : u ∈ M is equiconvergent at infinity, i.e. for any given > 0, there exists a T 0 > 0 such that for all u ∈ M and t 1 , t 2 > T 0 , it holds e −λt 2 u(t 2 ) − e −λt 1 u(t 1 ) < .

Main results
This section devoted to presenting and proving our main results. Before this end, we introduce the following hypotheses.
(H1) f, g : I × C r → R are continuous functions.
(H2) There exists a constant l > 0 and a bounded continuous function η > 0 so that if |u| , |v| ≤ l then (H3) There exist a constant γ > 0 and two continuous functions ζ : holds for all t ≥ 0, 0 < |u| ≤ γ, where Ψ is nondecreasing function and ζ ∈ L 1 ([0, ∞)) .  (1) is stable in C λ , provided that there exist constants M 1 , M 2 > 0 such that for all z ∈ (0, γ], and where Proof. In the proof of this theorem, we use the fact that e −λt = e −λ(t−τ ) e −λτ for all t ≥ τ . For any given > 0, we first prove the existence of δ > 0 such that and Clearly, for u ∈ B , both Au and Bu are continuous functions on [−r, +∞). Also, for u ∈ B , for any t ≥ 0, we have and Then AB ⊂ C λ and BB ⊂ C λ . Now we shall to prove that there exists at least one fixed point of the operator A + B. To this end, we divide the proof into three claims. Claim 1. We show that Au + Bv ∈ B for all u, v ∈ B , we combine (9) and (10) to get this means that Au + Bv ∈ B , for all u, v ∈ B . Claim 2. Obviously, A is continuous operator, it remains to prove that AB is relatively compact in C λ . In fact, from (11), we get that e −λt u(t) : u ∈ B is uniformly bounded in C λ . Moreover, 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 t ≥ τ , we have Together with the continuity of functions K and t −→ e −λt , we get that there exists a constant M 3 > 0 such that Also, for any fixed T 0 ≥ 0 and any t 1 , t 2 ∈ [0, T 0 ] , t 1 < t 2 , we have Thus this means that e −λt u(t) : u ∈ B is equicontinuous on any compact interval of R + , it remains to show that the set e −λt u(t) : u ∈ B is equiconvergent at infinity. In fact, for any 1 > 0 such that ≤ λ+k 6|k| 1 , there exists a L > 0 satisfies .
According to (12), we get that Then, there exists T > L such that for t 1 , t 2 ≥ T , we obtain Furthermore, for t ≥ s, we have lim t→∞ e −(λ+k)(t−s) = 0, Therefore, for t 1 , t 2 ≥ T , we have this achieves the proof. Claim 3. We show that B : B → C λ is a contraction mapping. In fact, for any u, v ∈ B , from (H2) , we have (7) A is a contraction mapping. By 2.10, there exists at least one fixed point of the operator A + B. Finally, for any 2 > 0, if 0 < δ 1 ≤ |k| this means that Thus, we know that the trivial solution of (1) is stable in Banach space C λ .
Proof. First, according to the Theorem 3.1, the trivial solution of (1) is stable in the Banach space C λ . Next, we shall prove that the trivial solution u = 0 of (1) is attractive. To this purpose, we define the subset of B R It is a nought to prove that Au In fact, for u, v ∈ B * r , based on the fact that used in the proof of Theorem 3.1 (Claim 2), it follows that as t → ∞. Together with the hypothesis ϕ R , ψ R ∈ L 1 ([0, +∞)) and using same way of (12) on the function H, we obtain as t → ∞. Furthermore, it is easy to see that Thus, the trivial solution of (1) is asymptotically stable.  Then α = 1/2, β = 3/2, r = 2, u 0 = 1, Φ (t) = sin (t), g (t, u (t − 2)) = 1 4 e −4t sin (u (t − 2)), f (t, u (t − 2)) =  (14) is stable in C 3 follows from Theorem 3.1.

Conclusion
In this paper, by utilizing the Krasnoselskii fixed point theorem in a weighted Banach space, we investigate the stability and asymptotic stability of the trivial solution for nonlinear fractional differential equations with the left Riemann Liouville and left Caputo fractional derivatives of orders α ∈ (0, 1] and β ∈ (1, 2] respectively. We establish the equivalence between the fractional differential equation and the integral equation on an infinite interval. Two main theorems are obtained. We also put an example to illustrate our results. However, we still have works to improve our constraint conditions for they are a little complicated in reality. Acknowledgement. The authors wish to thank deeply the anonymous referee for useful remarks and careful reading of the proofs presented in this paper.