The fractional differential equations can model some engineering and scientific disciplines in the fields of physics, chemistry, electrodynamics of complex medium, polymer rheology, etc. In particular, the forward and backward fractional derivatives provide an excellent tool for the description of some physical phenomena such as the fractional oscillator equations and the fractional Euler-Lagrange equations [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Recently, a linear boundary value problem (BVP) involving both the right Caputo and the left Riemann-Liouville fractional derivatives has been studied by some authors [10,11,12,13,14].
On the other hand, Ulam’s stability problem  has been attracted by many famous researchers (see [16,17] and references therein). Recently, studying the stability of Ulam-Hyers for fractional differential equations is gaining much importance and attention [18,19]. However, the Ulam-Hyers stabilities of differential equations involving with the forward and backward fractional derivatives have not yet been investigated.
In this article, we investigate the following BVP of the fractional differential equation with two different fractional derivatives:
The rest of this article is organized as follows. In Section 2, we collect some concepts of fractional calculus. In Section 3, we prove some properties of classical and generalized Mittag-Leffler functions. In Section 4, we present the definition of solution to (1.1) and (1.2). In Section 5, we obtain the existence and uniqueness of solutions to problem (1.1) and (1.2). In Section 6, we present the Ulam-Hyers stability result for Eq. (1.1). An example is given in Section 7 to demonstrate the application of our result.
In this section, we introduce some notations and definitions of fractional calculus. Throughout this article, we denote by
3 Properties of Mittag-Leffler functions
In this section, we prove some properties of the Mittag-Leffler functions.
, , , .
- (1)By using Lemma 3.3, we findSimilarly, we have
- (2)Using (3.1) and
, we arrive at
- (3)The proof of (3) is similar to that of (2).
- (4)Clearly, for
, the integral exists, then we get
4 Solutions for BVP
Formally, by Lemma 4.1, for
To prove our results, we make the following assumptions.
be a function such that is measurable for all and is continuous for a.e. , and there exists a function ) such that
For convenience of the following presentation, we set
Let us define
For the sake of convenience, we adopt the following notation:
Assume that (H1) and (H2) hold. For
It follows from (4.17) that
Assume that (H1) and (H2) hold. A function u is a solution of the following fractional integral equation:
5 Existence and uniqueness result
We consider an operator
6 Ulam-Hyers stability
Indeed, by Remark 6.2, we have
Assume that (H1) and (H2) are satisfied, then Eq. (
1.1) is Ulam-Hyers stable if
As an example, we consider here the following BVP of the fractional differential equation with two different fractional derivatives:
It is not difficult to obtain
By direct computation, we find
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