Existence and Ulam stability results for nonlinear hybrid implicit Caputo fractional differential equations

In this paper, we study the existence, uniqueness and estimate of solutions for nonlinear hybrid implicit Caputo fractional differential equations by using the contraction mapping principle and the generalization of Gronwall’s inequality. After that, we also establish the Ulam stability for the problem at hand. Finally, an example is given to illustrate this work.


Introduction
The concept of fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non integer order. Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1]- [28], [30]- [36], [38]- [40] and the references therein. The study of Ulam stability for fractional differential equations was initiated by Wang et al. [39]. An overview on the development of theory of the Ulam-Hyers and the Ulam-Hyers-Rassias stability for fractional differential equations can be found in [39,40] and the references given therein. Subsequently, many authors discussed various Ulam-Hyers stability problem for different kinds of fractional integral and fractional differential equations by using different techniques, see [10,14,32,39,40] and the references therein.
Hybrid differential equations arise from a variety of different areas of applied mathematics and physics, e.g., in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [4,20,22,21,38].
Dhage and Lakshmikantham [21] discussed the existence of solutions for the following first-order hybrid differential equation R → R are continuous functions. By using the fixed point theorem in Banach algebra, the authors obtained the existence results. The hybrid fractional differential equation has been investigated in [38], where D α is the Riemann-Liouville fractional derivative of order 0 < α < 1, g : are continuous functions. By employing the fixed point theorem in Banach algebra, the authors obtained the existence of a solution.
In [14], Benchohra and Lazreg established four types of Ulam stability for the following implicit fractional differential equation where C D α is the Caputo fractional derivative, f : [0, T ] × R × R → R is a given continuous function.
In [10], Ardjouni and Djoudi studied the existence, interval of existence and uniqueness of solutions for the hybrid implicit Caputo fractional differential equation where f : [0, T ] × R × R → R and g : [0, T ] × R → R\{0} are nonlinear continuous functions and C D α denotes the Caputo fractional derivative of order 0 < α < 1.
Motivated by the above works, we study the existence and uniqueness of solution and the Ulam stability for the following nonlinear hybrid implicit Caputo fractional differential equation are nonlinear continuous functions and C D α denotes the Caputo fractional derivative of order 0 < α < 1. To show the existence, uniqueness and estimate of solutions of (1), we transform (1) into an integral equation and then use the contraction mapping principle and the generalization of Gronwall's inequality. Further, we obtain Ulam-Hyers and Ulam-Hyers-Rassias stability results of (1). Finally, we provide an example to illustrate our obtained results.
The rest of this paper is organized as follows. Some definitions from fractional calculus theory are recalled in Section 2. In Section 3, we will prove the existence, uniqueness and estimate of solutions for (1). In Section 4, we discuss the Ulam stability results. Finally, an example is given in Section 5 to illustrate the usefulness of our main results.

Preliminaries
In this section we present some basic definitions, notations and results of fractional calculus which are used throughout this paper.

Definition 1 ([30]
). The fractional integral of order α > 0 of a function x : R + → R is given by provided the right side is pointwise defined on R + .

Definition 2 ([30]
). The Caputo fractional derivative of order α > 0 of a function x : R + → R is given by where n = [α] + 1, provided the right side is pointwise defined on R + .
) and x (n) exists almost every-where on any bounded interval of R + . Then In particular, when 0 < α < 1, 112

Fractional differential equations
The following generalization of Gronwall's lemma for singular kernels plays an important role in obtaining our main results.

Lemma 3 ([29])
. Let x : [0, T ] → [0, ∞) be a real function and w is a nonnegative locally integrable function on [0, T ]. Assume that there is a constant a > 0 such that for 0 < α < 1 then, there exists a constant k = k(α) such that To define Ulam's stability, we consider the following fractional differential equation

Definition 3 ([36]
). The equation (3) is said to be Ulam-Hyers stable if there exists a real number k > 0 such that for each > 0 and for each there exists a solution u of the equation (3) with

Definition 4 ([36]
). The equation (3) is said to be Ulam-Hyers-Rassias stable with respect to φ ∈ C ([0, T ] , R + ) if there exists a real number k φ,f > 0 such that for each > 0 and for each y ∈ C ([0, T ] , R) solution of the inequality there exists a solution u ∈ C ([0, T ] , R) of the equation (3) with Then a function y ∈ C ([0, T ] , R) is a solution of the inequality (4).
Theorem 1 (Banach's fixed point theorem [37]). Let Ω be a non-empty closed convex subset of a Banach space (S, . ), then any contraction mapping P of Ω into itself has a unique fixed point.

Existence and Uniqueness
In this section, we give the equivalence of the initial value problem (1) and prove the existence, uniqueness and estimate of solution of (1).
Proof. Let then by Lemma 4, we have That is x (t) = f (t, x(t)) + θg(t, x (t)) + g (t, x (t)) I α z x (t). Define the map- It is clear that the fixed points of P are solutions of (1). Let x, y ∈ C ([0, T ] , R), then we have and By replacing (8) in the inequality (7), we get x − y .
Then P x − P y ≤ β x − y .
By (6), the mapping P is a contraction in C ([0, T ] , R). Hence P has a unique fixed point x ∈ C ([0, T ] , R). Therefore (1) has a unique solution.
Proof. Theorem 2 shows that the problem (1) has a unique solution. Let Then by (H1), (H2) and (H3), for any t ∈ [0, T ] we have On the other hand, for any t ∈ [0, T ] we get Thus By Lemma 3, there is a constant K = K (α) such that . Hence .
This completes the proof.

Ulam Stability
In this section, we study two types of Ulam stability of the hybrid implicit Caputo fractional differential equation (1) which are Ulam-Hyers and Ulam-Hyers-Rassias stabilities.
Lemma 5. Assume that g satisfies (H1). If y ∈ C ([0, T ] , R) is a solution of the fractional differential inequality for each > 0 (9) then, y is a solution of the following inequality Proof. Let y ∈ C ([0, T ] , R) be a solution of the inequality (9) for each > 0. Then, from Remark 1 and Lemma 4 for some continuous function Ψ (t) such that |Ψ (t)| < , t ∈ [0, T ], we have Then, by Remark 1 and (H1), we obtain which is satisfied inequality (10). This completes the proof. Then .
This completes the proof.
In the next, we introduce the following function.