Nonlinear fractional differential equation involving two mixed fractional orders with nonlocal boundary conditions and Ulam–Hyers stability

In this paper, we study a nonlinear fractional differential equation involving two mixed fractional orders with nonlocal boundary conditions. By using some new techniques, we introduce a formula of solutions for above problem, which can be regarded as a novelty item. Moreover, under the weak assumptions and using Leray–Schauder degree theory, we obtain the existence result of solutions for above problem. Furthermore, we discuss the Ulam–Hyers stability of the above fractional differential equation. Three examples illustrate our results.

Different from the previous results, the boundary conditions considered in this paper include the nonlocal Katugampola fractional integral, moreover, under the weak assumptions and using Leray-Schauder degree theory, we obtain the existence result of solutions for the above problem (Theorem 5.3). However, to the best of our knowledge, few papers can be found in the literature dealing with the existence result and the Ulam-Hyers stability of differential equation involving the forward and backward fractional derivatives.
The rest of this paper is organized as follows. In Sect. 2, we collect some concepts of fractional calculus. In Sect. 3, we prove some properties of classical and generalized Mittag-Leffler functions. In Sect. 4, we present the definition of solution to (1.1)-(1.2). In Sect. 5, we obtain the existence and uniqueness of solutions to problem (1.1)-(1.2). In Sect. 6, we present Ulam-Hyers stability result for Eq. (1.1). Three examples are given in Sect. 7 to demonstrate the applicability of our result.

Preliminaries
In this section, we introduce some notations and definitions of fractional calculus. Throughout this paper, we denote by C(J, R) the Banach space of all continuous functions from J to R, by AC ([a, b], R) the space of absolutely continuous functions on [a, b]. Γ (·) and B(·, ·) are the gamma and beta functions, respectively. Definition 2.1 ([3, 4]) The left-sided and the right-sided fractional integrals of order δ for a function x(t) ∈ L 1 are defined by respectively.
a + x(t) of order δ exists almost everywhere on [a, b] and can be written as and can be written as

Solutions for problem (1.1)-(1.2)
In this section, we present the formula of the solution to the problem (1.1)-(1.2).
Similar to the arguments in [3], we can obtain the following result.
Formally, by Lemma 4.1, for c 0 ∈ R, we have ( Based on the arguments in Sect. 4.1.1 of [3], we obtain We need the following assumptions.
For convenience of the following presentation, we set Proof By the mean value theorem and Lemma 3.2, we obtain Then, using the inequality t α 2t α 1 ≤ (t 2t 1 ) α and Eqs. (4.2), (4.7) and (4.8), we get

Existence results for problem (1.1)-(1.2)
In this section, we deal with the existence and uniqueness of solutions to the problem (1.1)-(1.2).
We just need to prove the existence of at least one solution u ∈ C 1-α satisfying u = Fu. Hence, we show that F : B R → C 1-α satisfies the condition u = θ Fu, ∀u ∈ ∂B R , ∀θ ∈ [0, 1], (5.6) where By the Arzela-Ascoli theorem, a continuous map h θ defined by h θ (u) = u -H(θ , u) = uθ Fu is completely continuous. If (5.6) is true, then the Leray-Schauder degrees are well defined and from the homotopy invariance of topological degree, it follows that Let v(t) = t 1-α u(t), we obtain the following estimate: Applying (5.2) and (5.3), we obtain Then from Lemma 5.1, we find u 1-α ≤ M, where M satisfies Set R = M + 1, then (5.6) holds. This completes the proof.
Next, we study the uniqueness of solution, for this purpose, we give the following assumptions. ( (H2 ) There exists a constant L f > 0 such that Theorem 5.4 Assume that (H1 ) and (H2 ) hold, then the problem (1.1)-(1.2) has a unique solution u ∈ C 1-α , provided that Proof By (H1 ) and the proof of Theorem 5.3, it is not difficult to see that (1.1)-(1.2) has a solution u(·) ∈ C 1-α . Let u(·) be another solution of the problem (1.1)-(1.2). According to (H2 ), we find In order to obtain another result for uniqueness of the solutions, we make the following assumption: (H3) There exists a constant L f > 0 such that Theorem 5.5 Assume that (H3) holds, then the problem (1.1)-(1.2) has a unique solution u ∈ C 1-α , provided that Proof We consider an operator F : Clearly, F is well defined. According to (H3), we find .
this means that F is a contraction, and by the Banach fixed point theorem there exists a unique solution u ∈ C 1-α .

Ulam-Hyers stability
Let be a positive real number. We consider Eq. (1.1) with inequality Remark 6.2 A function x ∈ C 1-α is a solution of the inequality (6.1) if and only if there exists a function g ∈ C 1-α such that (i) .