Existence and Ulam–hyers Stability of Odes Involving Two Caputo Fractional Derivatives

In this paper, we study existence of solutions to a Cauchy problem for non-linear ordinary differential equations involving two Caputo fractional derivatives. The existence and uniqueness of solutions are obtained by using monotonicity, continuity and explicit estimation of Mittag-Leffler functions via fixed point theorems. Further, we present Ulam–Hyers stability results by using direct analysis methods. Finally, examples are given to illustrate our theoretical results.

In [17,Chapter 5], Kilbas et al. studied the solvability of a Cauchy problem for nonlinear ordinary differential equations involving two Caputo fractional derivatives of the type: , where λ ∈ R, c D α t and c D β t denote the Caputo fractional derivatives of order α, β with the lower limit zero, respectively (see Definition 2.1).Further, Wang and Li [30] discussed E α -Ulam-Hyers stability of fractional differential equations of the type: c D α t x(t) = λx(t) + f (t, x(t)) on finite time interval via the properties of Mittag-Leffler functions E α (z) := ∑ ∞ Γ(αk+α) for z ≤ 0 (see [29,Lemma 2]) and a singular Gronwall type integral inequality (see [31,Theorem 1]).Very recently, Cong et al. [6] explored some asymptotic behavior on E α (z) and E α,α (z) for z > 0 by using [11,Theorem 2.3], which inspired the reader to study further estimation and asymptotic behavior on E α,β (z).
However, the development of existence and Ulam's type stability theory for nonlinear ordinary differential equations involving two Caputo fractional derivatives is still in its infancy.One of the reasons for this fact might be that asymptotic property of E α,β (z) have not been explored completely.
Motivated by [6,17,30], we consider the following Cauchy problem for nonlinear differential equations involving two Caputo fractional derivatives: where with the two parameter Mittag-Leffler function . Before we deal with existence of solutions and Ulam-Hyers stability, the key step is to discuss the elementary properties of Mittag-Leffler functions.By virtue of integrable expansion of Mittag-Leffler functions in [6], we give monotonicity, continuity and explicit estimation of Mittag-Leffler functions E α (z) and E α,β (z) for z > 0 and z < 0, which extend the previous results in [29, Lemma 2] and [6, Lemma 3].
The first purpose of this paper is to discuss existence of solutions to the equation (1.1) by using fixed point theorems.The second purpose of this paper is to present that the equation (1.1) is Ulam-Hyers stable on the time interval J.When we discuss existence theorems and Ulam-Hyers stability theorems, the new derived properties of Mittag-Leffler functions E α (z) and E α,β (z) for z > 0 and z < 0 are widely used in this paper.Meanwhile, these properties will help the researcher to study other fractional ODEs with constant coefficients.
The rest of this paper is organized as follows.In Section 2, we recall some notations and give some useful properties of the two-parameter Mittag-Leffler function.In Section 3, we apply fixed point theorems to derive the existence of solutions.In Section 4, Ulam-Hyers stability theorems are presented.Examples are given in Section 5 to demonstrate the application of our main results.

Preliminaries
Let C(J, R) be the Banach space of all continuous functions from J into R with the norm z ∞ = sup {|z(t)| : t ∈ J}.

Definition 2.1 ([17]
).The Caputo derivative of order γ for a function f : [0, ∞) → R can be written as where L D γ t f denotes the Riemann-Liouville derivative of order γ with the lower limit zero for a function f , which given by We recall the famous integrable expansion of two differential parameters Mittag-Leffler function.
(i) For all t > 0, we have In particular, (ii) For all t > 0, we have In particular, Proof.(i) By virtue of Lemma 2.2 (i), we have It follows the fact we obtain In particular, (ii) By virtue of Lemma 2.2 (ii) for the case z < 0, we obtain that In particular, The proof is completed.
To end this section, we recall the famous Krasnoselskii-Zabreiko fixed point theorem.

Lemma 2.6 ([18]
).Let (X, • ) be a Banach space, and K : X → X be a completely continuous operator.Assume that L : X → X is a bounded linear operator such that 1 is not an eigenvalue of L and Then K has a fixed point in X.
3 Existence results

Case of λ > 0
We introduce the following assumptions: for each t ∈ J, and all x, y ∈ R. Proof.Define an operator Note that Q is well defined on C(J, R) due to (H 1 ).
Step 1.We prove that Q(B r ) ⊂ B r .Now, take t ∈ J and x ∈ B r .By using (H 2 ) via Lemma 2.3 and Lemma 2.5 (i), we obtain Step 2. We check that Q is a contraction mapping.For x, y ∈ B r and for each t ∈ J, by using Lemma 2.3 and Lemma 2.5 (i), we obtain From (H 3 ), one can obtain the conclusion of theorem by the contraction mapping principle.The proof is completed.
Next, we apply Krasnoselskii's fixed point theorem to derive the existence result.
(H 4 ) There exists a nondecreasing function ∈ Theorem 3.2.Assume that (H 1 ) and (H 4 ) are satisfied.Then the equation (1.1) has at least one solution.
Proof.For some r > 0, define two operators G and H on B r given by We show that (G + H)(B r ) ⊂ B r .If it is not true, then for each r > 0, there would exist x r ∈ B r and t r ∈ J such that |(Gx r )(t r ) + (Hx r )(t r )| > r .By repeating the same process of Step 1 of Theorem 3.1, we have Dividing both sides by r and taking the lower limit as r → +∞, we obtain 1 ≤ ρ lim r →∞ inf (r ) r , which contradicts with (H 4 ).Thus, for some positive number r , (G + H)(B r ) ⊂ B r .
We observe that H is a contraction with the constant zero and the continuity of f implies that the operator G is continuous.Moreover, G is uniformly bounded on B r .Now we need to prove the compactness of the operator G. Define f max = sup{| f (t, x)| : t ∈ J, x ∈ B r }.For any t 2 < t 1 , by using Lemma 2.3 (ii), we have which tends to zero as This yields that G is equicontinuous.So G is relatively compact.Hence, G is compact.At last, we can conclude that G + H is a condensing map on B r .By using the Krasnoselskii fixed point theorem, the problem has at least one solution.The proof is completed.
Next, we apply the Krasnoselskii-Zabreiko fixed point theorem to derive the existence result.
(H 5 ) The function f (t, 0) = 0 for some t ∈ J and lim If not, one can derive the fact which contradicts with (H 6 ).Therefore, (3.3) implies that 1 is not an eigenvalue of the operator L.

Finally we will show that
This means that Then, we can get lim Consequently, the proof is completed by virtue of Lemma 2.6.

Symmetrical results for λ < 0
In this section, we give symmetrical existence results for Section 3.
Recall the above definition of M and B r , where Now we are ready to give the following result.
Proof.Like in Theorem 3.1, consider Q : B r → C(J, R) again, where r is chosen in (3.4).We prove that Q(B r ) ⊂ B r .Now, take t ∈ J and x ∈ B r .By using (H 2 ) via Lemma 2.5 (ii), we obtain We check that Q is a contraction mapping.For x, y ∈ B r and for each t ∈ J.By using Lemma 2.5 (ii), we obtain By (H 7 ) and the contraction mapping principle, one can complete the proof.
(H 8 ) There exists a nondecreasing function The rest of the proof is the same as that of Theorem 3.2.So we omit it here.

Case of λ > 0
In this part, we will discuss Ulam-Hyers stability of the equation (1.1) for the case λ > 0 on the time interval J.
Let > 0. Consider the equation (1.1) and below inequality Then we have the following estimation.
Remark 4.3.Let y ∈ C(J, R) be a solution of the inequality (4.1).Then y is a solution of the following integral inequality where we use Remark 4.2, Lemma 2.5 (i) and the fact ρ is defined in (H 3 ).Now we are ready to state our Ulam-Hyers stability result.The proof is completed.

Symmetrical results for λ < 0
Next, we apply the same method to investigate Ulam-Hyers stability of the equation (1.1) for the case λ < 0 on the time interval J.By Remark 4.2 and Lemma 2.5 (ii), one can give a similar result according to Remark 4.3.where is defined in (H 7 ).

Lemma 2.3 (
[24, Lemma 2.3]).Let α ∈ (0, 1], β ∈ R and β < 1 + α be arbitrary.The functions E α (•) and E α,β (•) are nonnegative and have the following properties.(i)For all λ > 0 and t 1 , t 2 ∈ J and t 1 ≤ t 2 , and 0 < lim Assume that (H 1 ) and (H 8 ) are satisfied.Then the equation (1.1) has at least one solution.Proof.We consider the operators G and H in Theorem 3.2 again.We use proof by contradiction to show that (G + H)(B r ) ⊂ B r for some positive number r .By repeating the same process of Step 1 of Theorem 3.4, we have r < |(Gx r )(t r ) + (Hx r )(t r )| ≤ |x 0 | − , contradicts (H 8 ).To prove the compactness of the operator G, we only need to check equicontinuity, for any t 2 < t 1 , by using Remark 2.4 ([29, Lemma 2 (ii)]), we have r Remark 4.2.A function y ∈ C(J, R) is a solution of the inequality (4.1) if and only if there exists a function g ∈ C(J, R) (which depend on y) such that (i) |g(t)| ≤ , t ∈ J, (ii) c D α t y(t) − λ c D β t y(t) = f (t, y(t)) + g(t), t ∈ J.
each > 0 and for each solution y ∈ C(J, R) of the inequality (4.1) there exists a solution x ∈ C(J, R) of the equation (1.1) with |y(t) − x(t)| ≤ c , t ∈ J.