Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability

Abstract In this article, we discuss the nonlinear boundary value problems involving both left Riemann-Liouville and right Caputo-type fractional derivatives. By using some new techniques and properties of the Mittag-Leffler functions, we introduce a formula of the solutions for the aforementioned problems, which can be regarded as a novelty item. Moreover, we obtain the existence result of solutions for the aforementioned problems and present the Ulam-Hyers stability of the fractional differential equation involving two different fractional derivatives. An example is given to illustrate our theoretical result.

On the other hand, Ulam's stability problem [15] 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: where + ∈ ( ) α β α β , , 0, 1 , > λ ρ q , , 0, + > α ρ 1. 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.

Preliminaries
In this section, we introduce some notations and definitions of fractional calculus.
the space of all the absolutely continuous functions defined on [ ] a b , .
Definition 2.1. [3,4] The left-sided and the right-sided fractional integrals of order γ for a function ( ) ∈ x t L 1 are defined as , and can be written as respectively.
(1) By using Lemma 3.3, we find Similarly, we have , we arrive at The proof of (3) is similar to that of (2).

Solutions for BVP
In this section, we present the formulas of solutions to problem (1.1) and (1.2).
[4] For > θ 0, a general solution of the fractional differential equation denotes the integer part of the real number θ.
To prove our results, we make the following assumptions.

K t τ K t τ φ τ τ k t s k t s φ s s k t s φ s s k t s k t s φ s s k t s φ s s
, d . By (4.5), (4.6), (4.1) and (4.2), we observe that Now we can see that for ∈ t J. Clearly, the boundary value condition (1.2) holds and hence the necessity is proved. □

Existence and uniqueness result
In this section, we deal with the existence and uniqueness of solutions to problem (1.1) and (1.2).