Stability of Ulam–Hyers and Ulam–Hyers–Rassias for a class of fractional differential equations

*Correspondence: gaorm1983@163.com 1School of Science, Changchun University of Science and Technology, Changchun, China Abstract In this paper, we investigate a class of nonlinear fractional differential equations with integral boundary condition. By means of Krasnosel’skĭi fixed point theorem and contraction mapping principle we prove the existence and uniqueness of solutions for a nonlinear system. By means of Bielecki-type metric and the Banach fixed point theorem we investigate the Ulam–Hyers and Ulam–Hyers–Rassias stability of nonlinear fractional differential equations. Besides, we discuss an example for illustration of the main work.


Introduction
Fractional derivatives provide an effective instrument in the modeling of many physical phenomena. Fractional differential equations and fractional integral equations appeared in various fields such as polymer rheology, blood flow phenomena, electrodynamics of complex medium, modeling and control theory, signal processing, and so on; see [1,2]. In recent years, many researchers proved the existence and uniqueness of solutions to fractional differential equations [3][4][5][6][7][8]. Moreover, integral boundary problems had a variety of applications in real-life problems such as blood flow, underground water flow, population dynamics, thermoplasticity, chemical engineering, and so on; see [9][10][11].
On the other hand, S.M. Ulam presented the stability problem of the solutions of functional equations (of group homomorphisms) in 1940 in a talk given at Wisconsin University [12]. In 1941, Hyers [13] gave the first answer to the question in Banach spaces. Since then, many researchers were interested in Ulam-type stability. With a wide expansion of the fractional calculus, the study of stability for fractional differential equations also attracted the attention of researchers [14,15].
Abbas, Benchohra et al. [17] researched the existence and Ulam stability for the fractional differential equation is the Hilfer-Hadamard fractional derivative, and H I α, 1-γ 1+ (·) is the Hadamard integral. Chalishajar et al. [18] discussed the existence, uniqueness, and Ulam-Hyers stability of solutions for the following coupled system of fractional differential equations with integral boundary conditions: where c D α 0+ (·) and c D β 0+ (·) are the Caputo fractional derivatives, p, p > 0, q, q ≥ 0 are real numbers, and a 1 , a 2 , a 1 , a 2 are continuous functions.
Vanterler da C. Sousa et al. [19][20][21] studied the ψ-Hilfer fractional derivative and the stability of Hyers-Ulam-Rassias and Hyers-Ulam of the following Volterra integrodifferential equation [14]: where f (t, u) is a continuous function with respect to the variables t and u on I × R, K(t, s, u) is continuous with respect to t, s, and u on I × I × R, σ is a given constant, H D α,β;ψ 0+ (·) is the right-sided ψ-Hilfer fractional derivative with α ∈ (0, 1) and β ∈ [0, 1], and I 1-γ 0+ (·) is the ψ-Riemann-Liouville fractional integral with γ ∈ [0, 1). In this paper, we consider the following class of fractional differential equations with integral boundary condition: where c D α 0+ (·) is the Caputo derivative with 0 < α < 1, I β 0+ (·) is the Riemann-Liouville fractional integral with β > 0, η ∈ (0, 1] is a fixed real number, u ∈ C 1 [0, 1], and f : [0, 1] × R → R is a continuous function.  [22] can also be used to describe macroscopic models II for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding, or trapping. This paper is organized as follows. In the second section, we recall some basic definitions of fractional calculus, the concepts of Ulam-Hyers and Ulam-Hyers-Rassias stability for Eq. (1.1) and fixed point theorems. In the third section, we investigate the existence and uniqueness of solutions for problem (1.1)-(1.2). Moreover, we discuss the Ulam-Hyers and Ulam-Hyers-Rassias stability for Eq. (1.1). In the last section, we provide an illustrative example.

Preliminaries
In this section, we recall some useful definitions, notations, and the fundamental results about fractional derivatives (refer to [23,24] and [25]). Also, we present the concepts of where Γ is the gamma function.

Definition 2.3 ([23-25]) The two-parametric Mittag-Leffler function is defined as
The Laplace transform of the Caputo derivative c D α The Laplace transform of the two-parametric Mittag-Leffler function is ,

Definition 2.4 ([26]
) A function f is of exponential order λ if there exist constants M > 0 and λ such that for some t 0 > 0, Next, we present the concepts of the Ulam-and Ulam-Hyers-Rassias stability for Eq. (1.1). The following Definitions 2.5 and 2.6 are adapted from [14].

Definition 2.5 If x(t) is a continuously differentiable function satisfying
where θ > 0, and there are a solution u(t) of Eq. (1.1) and a constant C > 0 independent of x(t) and u(t) such that then we say that the Eq. (1.1) has the Ulam-Hyers stability.

Definition 2.6 If x(t) is a continuously differentiable function satisfying
where σ : [0, 1] → [0, +∞) is a continuous function, and there exist a solution u(t) of Eq. (1.1) and a constant C > 0 independent of x(t) and u(t) such that then we say that the Eq. (1.1) has the Ulam-Hyers-Rassias stability.  (Λ k+1 x, Λ k x) < ∞ for some x ∈ X, then the following three propositions hold: (1) The sequence {Λ n x} converges to a fixed point x * of Λ;

Main results
In this section, we derive the existence and uniqueness of solutions for the integral boundary problem (1.1)-(1.2). Moreover, we study the Ulam-Hyers and Ulam-Hyers-Rassias stability for Eq. (1.1).

Existence and uniqueness results
In this subsection, by means of the Krasnosel'skiȋ fixed point theorem and contraction mapping principle, we investigate the existence and uniqueness of solutions for problem For any g ∈ C[0, 1] and η ∈ (0, 1], the solution of the boundary value problem is given by where G(t, s) is called the Green's function of problem (3.1)-(3.2)and is given by Therefore the unique solution of problem (3.1)-(3.2) is where G(t, s) is given by (3.3). This completes the proof.
Remark 3.1 By the definition of the two-parameter Mittag-Leffler function we get which is a convergent series of real numbers. Therefore there exists a constant E 1-α,2 > 0 such that Moreover, by Eq.
Consider the operator T defined on C 1 [0, 1] by
We define the set Hence TB ⊆ B.
Next, we show that T is a contraction operator. For u 1 , u 2 ∈ C 1 [0, 1] and t ∈ [0, 1], from Eq. (3.6), using the Lipschitz condition on f , we have As η β Γ (β+1) + LM < 1, T is a contraction mapping. By the contraction mapping principle it has a unique fixed point, which is the unique solution of problem (1.1)-(1.2). Proof We consider the operators A and B on C 1 [0, 1] defined by For any u, v ∈ W r , having in mind Remark 3.1, Remark 3.2, and the definitions of the operators A and B, we conclude that .
It follows from the proof of the operator T. By the continuity of the two-parameter Mittag-Leffler function and f (t.u(t)), for any continuous function u ∈ W r , the operator A is continuous.
For any u ∈ W r , from Remarks 3.1 and 3.2 we have Hence A is uniformly bounded on W r . For any u ∈ W r and t 1 , t 2 ∈ [0, 1] such that t 1 < t 2 , The constant NE 1-α,1 (-(1θ ) 1-α ) is independent of u, so A is relatively compact on W r . Therefore by the Arzelà-Ascoli theorem the operator A is compact on W r . By Theorem 2.1 problem (1.1)-(1.2) has at least one solution on [0, 1].
This means that under the above conditions, the fractional differential Eq. (1.1) has the Ulam-Hyers-Rassias stability. We consider the operator Λ : By the continuity of the two-parameter Mittag-Leffler function and f the operator Λ is continuous.
On the other hand, let x ∈ C 1 [0, 1] satisfy Eq. (3.8). By the Laplace transform and the inverse Laplace transform we obtain that x satisfies (3.11) which follows from the proof of Lemma 3.1. By Eq. (3.7), Eq. (3.11), and the definition of the operator Λ we get Therefore we conclude that with L > 0. Moreover, let σ : [0, 1] → (0, ∞) be a nondecreasing continuous function, and suppose that there exists a constant ξ ∈ [0, 1) such that and Lξ < 1.
Proof: The first part of the proof follows the same steps as in the proof of Theorem 3.3. Consider the operator Λ : Since Lξ < 1, we conclude that the operator Λ is strictly contractive in (C 1 [0, 1], d), which follows from the proof of Theorem 3.3.
The Ulam-Hyers and Ulam-Hyers-Rassias stability for Eq. (4.1) is independent of the initial value condition. Using MATLAB, the solution u(t) of Eq. (4.1) with initial value condition u(0) = 0 is computed and depicted in Fig. 1.
We conclude that x satisfies Eq. (3.8). Therefore we have