Existence and Ulam stability for fractional di ﬀ erential equations of mixed Caputo-Riemann derivatives

: In this paper, we study the existence, uniqueness, and stability theorems of solutions for a di ﬀ erential equation of mixed Caputo-Riemann fractional derivatives with integral initial conditions in a Banach space. Our analysis is based on an application of the Shauder ﬁxed point theorem with Ulam-Hyers and Ulam-Hyers-Rassias theorems. A couple of examples are presented to illustrate the obtained results.


Introduction
The fractional differential equations have drawn much attention due to their applications in a number of fields such as physics, mechanics, chemistry, biology, economics, biophysics, etc, see [16,32]. Some physical phenomena such as the fractional oscillator equations and fractional Euler-Lagrange equations with mixed fractional derivatives can be found in [10,12,31]. Once a model of fractional differential equation for the real problem have been constructed, people faced the issue of how to solve this model. In many circumstances, finding the exact solution to the fractional differential equation is quite challenging. As a result, researchers must identify as many aspects of the problem's solution as possible. Is there a solution to the problem, for example? Is the solution unique if there is a one? Hence the study of existence and uniqueness solutions for fractional differential equations with initial and boundary conditions appealed many scientists and mathematicians [2-4, 19-21, 25, 27, 30]. Some existence results for fractional differential equations with integral boundary conditions can be found in [17,28,29]. Recently, the existence theorem for fractional differential equations involving mixed fractional derivatives have been studied by many authors [5,6,8]. More specifically, Abbas [1] proved the existence and uniqueness of solution for a boundary value problem of fractional differential equation of the form C D α y(t) = f (t, y(t), C D β y(t)), β > 0, y(0) = λ 1 y(η), y (0) = 0, y (0) = 0, ..., y (m−2) (0) = 0, y(1) = λ 2 y(η), where α ∈ (m − 1, m], m ≥ 2, and C D α , C D β are the Caputo fractional derivatives. Alghamdi et al. [7] studied new existence and uniqueness results for three-point boundary value problem of sequential fractional differential equations given by where C D β is Caputo fractional derivative. Song et al. [34] used the coincidence degree theory while proving the existence of solutions of the following nonlinear mixed fractional differential equation with the integral boundary value problem: where C D α 1− and D β 0+ are respectively the left Caputo fractional derivative and the right Riemann-Liouville fractional derivative. Sousa et al. [35] investigated the existence and uniqueness of mild and strong solutions of fractional semilinear evolution equations, by means of the Banach fixed point theorem and the Gronwall inequality. The notion of Ulam stability has been studied and expanded in many ways. There have been a number of articles published on this subject that have yielded a number of conclusions [11,23,24]. Ibrahim [18] examined Ulam stability for the Cauchy differential equation of fractional order in the unit disk. Chen et al. [13] studied the Ulam-Hyers stability of solutions for linear and nonlinear nabla fractional Caputo difference equations when 0 < v ≤ 1 on finite intervals. The linear case has the form x(a) = y(a), and the non-linear case has the form x(a) = y(a). [26] discussed Hyers-Ulam and Hyers-Ulam-Rassias stability for the following fractional differential equation with boundary condition D α y(t) = f (t, y(t)), 0 < α ≤ 1,

Muniyappan and Rajan
where D α is Caputo fractional derivative of order α. Dai et al. [14] researched the Ulam-Hyers and Ulam-Hyers-Rassias stability of nonlinear fractional differential equations with integral boundary condition which has the form where D α is Caputo derivative and I β 0+ (.) is the Riemann-Liouville fractional integral. In this paper, we consider the nonlinear fractional differential equations wich has the form with initial conditions

Preliminaries
Let us give some definitions and lemmas that are basic and needed at various places in this work.
provided that this integral (Lebesgue) exists.
Moreover, if f (t) ∈ C[a, b], then the above identity is true for all t ∈ [a, b].
Definition 2.13. The Eq (1.1) is Ulam-Hyers stable if there exists a real number c f > 0 such that for each ε > 0 and for each solution z ∈ C 1 (J, R) of the inequality Definition 2.14. The Eq (1.1) is Ulam-Hyers-Rassias stable with respect to ϕ ∈ C(J, R + ) if there exists a real number c f > 0 such that for each ε > 0 and for each solution z ∈ C 1 (J, R) of the inequality , then the initial value problem (1.1) and (1.2) has a solution where G(t, s) is the Green's function described by where dτds, y(r))drdτds, dτds, Proof. By applying the Lemma 2.5 and 2.7, we may reduce Eq (1.1) to an equivalent equation Operate both sides of Eq (2.4) by operator D, we get Then the solution of Eq (2.5) is Using the initial conditions (1.2), we find that Substituting the values of c 1 and k 1 in Eq (2.6), we have  The converse of the lemma follows from a direct computation. Hence, the proof is completed.

Main results
In this section, we prove the existence and uniqueness of solution for the problem (1.1) and (1.2) in the Banach space C by applying Banach contraction principle and Schauder fixed point theorem. To prove the main results, we need the following assumptions: (H1) There exists a positive constants γ 1 , γ 2 such that f (t, y(t)) ≤ γ 1 + γ 2 |y(t)|, for each t ∈ J and all y ∈ R.
(H2) There exists a positive constant k such that f (t, x(t)) − f (t, y(t)) ≤ k|x(t) − y(t)|, for each t ∈ J and all x, y ∈ R. (H3) There exists an increasing function ϕ ∈ C(J, R + ) and there exists ν ϕ > 0 such that for any t ∈ J, we have For convenience, we define the following notations: , The existence result can be obtained by the Schauder fixed point theorem.  By the continuity of the functions G(t, s) and f (t, y(t)), we have T y ∈ C(J) for any y ∈ C(J). We define the set B r = {y(t) ∈ C(J, R) : y ≤ r} and choose r ≥ . First, we have to show that T Br ⊆ Br, for y ∈ B r . Now, consider ||(T y)(t)|| = sup .

Therefore, we have
Hence, T Br ⊆ Br. Next, we need to prove that T is a completely continuous operator. For this purpose we fix, Q = sup t∈J | f (s, y(s))|, where y ∈ B r , and t, τ ∈ J with t < τ. Then Therefore, and so Let t → τ, the right-hand side of the above inequality tends to zero. Thus, T is uniformly bounded and equicontinuous. Therefore by th Arzela-Ascoli implies that T is completely continuous. Hence, by Schauder's fixed point theorem, the problem (1.1) and (1.2) has a solution on C(J, R). Now, we use the contraction principle mapping to investigate uniqueness results for (1.1) and (1.2).
Proof. Let x, y ∈ C(J, R). Then Therefore, Then Using the condition (3.1), we conclude that T is a contraction mapping. Hence Banach contraction principle guarantees that T has a fixed point which is the unique solution of the problem (1.1) and (1.2). The proof is complete.

Stability theorems
In this section, we study Ulam-Hyers and Ulam-Hyers-Rassias stability of our problem (1.1) and (1.2). From inequality (2.1), for each t ∈ J, we get by (H2), for each t ∈ J, we obtain Then from Eq (4.1) we conclude that holds. Thus the problem (1.1) and (1.2) is Ulam-Hyers stable.
by using the hypothesis (H2), for each t ∈ J, we get Then the use of Eq (4.2) implies that

Examples
In this section, we give two examples to illustrate the usefulness of our main results.
By Theorem 4.1, we have which shows the problem (5.1) is Ulam-Hyers stable.

Conclusions
In this research, we examined the solution of nonlinear fractional differential equations with integral initial conditions. By means of the Shauder fixed point theorem and contraction mapping principle, we proved the existence and uniqueness of solutions for a nonlinear problem. In addition, the Hyers-Ulam and Hyers-Ulam-Rassias stability of the problem (1.1) and (1.2) are studied. Lastly, we presented several examples to demonstrate the use of our main theorems.

Conflict of interest
The authors declare no conflict of interest.