Existence and multiplicity of positive solutions for a new class of singular higher-order fractional differential equations with Riemann–Stieltjes integral boundary value conditions

In this work, the aim is to discuss a new class of singular nonlinear higher-order fractional boundary value problems involving multiple Riemann–Liouville fractional derivatives. The boundary conditions are constituted by Riemann–Stieltjes integral boundary conditions. The existence and multiplicity of positive solutions are derived via employing the Guo–Krasnosel’skii fixed point theorem. In addition, the main results are demonstrated by some examples to show their validity.

Fractional differential equations appear naturally in various fields of science and engineering. This is due to the fact that the differential equations of arbitrary order provide an excellent instrument for the description of memory and hereditary properties of various materials and processes and they have numerous applications in multifarious fields of science and engineering including physics, blood flow phenomena, rheology, diffusive transport akin to diffusion, electrical networks, probability, etc. The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials. For some recent work on this branch of differential equations, see [1][2][3][4] and the references therein. Fractional-order differential equations have been addressed by several researchers with the sphere of study ranging from the theoretical aspects to the analytic and numerical methods for finding solutions (see  and the references therein). The existence of positive solutions for fractional differential equation boundary value problems has attracted much attention, and a great deal of results have been developed for differential and integral boundary value problems. For example, in [5], Ma and Yang studied the following higher-order boundary value problems with a sign-changing nonlinear term: (t, u, u , u , u (3) , . . . , u (n-2) ) = 0, 0 < t < 1, αu (n-2) (0)βu (n-1) (0) = 0, γ u (n-2) (1) + δu (n-1) (1) = 0, where n ≥ 2, 0 < λ, α, β, γ and δ are constants satisfying α, γ > 0 and β, δ ≥ 0, f : [0, 1] × R n-1 → R 1 = (-∞, +∞) is continuous. Intervals of λ are determined to ensure the existence of a positive solution of the boundary value problem according to the signs of a and f . By using the Schauder fixed point theorem, the authors obtained the existence of a positive solution.
In [6], by means of the fixed point index theory, Zhang et al. investigated the existence of positive solutions for the fractional differential equation with integral boundary conditions: x are the standard Riemann-Liouville derivatives, 1 0 x(s) dA(s) denotes a Riemann-Stieltjes integral, A is a function of bounded variation, and dA can be a signed measure, f : (0, 1) × (0, +∞) × (0, +∞) → R 1 + is continuous, f (t, x, y) may be singular at both t = 0, 1 and x = y = 0.
Inspired by the works illustrated above, we are committed to establishing the existence and multiplicity of positive solutions for the fractional differential equation boundary value problem (BVP for short) (1.1). The novelty of this article is as follows: Firstly, fractional derivatives are involved in the nonlinear terms and boundary conditions; what is new is that the orders of the fractional derivatives in the nonlinear terms and boundary conditions are different. Moreover, the orders of the fractional derivatives in the boundary conditions can be different, but up to now, there have been few papers dealing with this case where the Riemann-Stieltjes integral boundary conditions contain fractional derivative of different orders. Since fractional derivatives of different orders and high-order fractional derivatives are taken into account in BVP (1.1), it makes the research more complicated. In order to reduce the complexity, we need to use the reduced-order method for fractional differential equation and overcome the difficulties in finding the properties of Green's function. Secondly, the boundary value conditions involving high-order fractional derivatives of unknown function are more general as they contain multi-point boundary conditions and integral boundary conditions [6,7,20,[28][29][30]32] as special cases. Thirdly, the given conditions f 0 , f ∞ and (H 3 ), (H 5 ) are quite different from those in other papers such as  and are weaker and wider.
The remaining part of article is structured as follows. In Sect. 2, we present some preliminaries and lemmas which are required in later considerations. We also develop and prove some properties of Green's function. In Sect. 3, we discuss the existence and multiplicity of positive solutions for BVP (1.1). In Sect. 4, two examples are presented to illustrate our fundamental results.

Preliminaries and lemmas
In this section, some notations and lemmas, which will be used in the proof of our main results, are stated. They can be found in the literature, see [2,3,21].
provided that the right-hand side is pointwise defined on (0, ∞).

Lemma 2.4 (Auxiliary lemma) Let l
Given y ∈ C(0, 1) ∩ L 1 (0, 1) and the following condition is satisfied: The unique solution of Proof We may apply Lemma 2.2 to reduce (2.1) to an equivalent integral equation for some d i ∈ R 1 (i = 1, 2). Consequently, the general solution of (2.1) is (2.4)
Using similar arguments as those used in the proof of Lemma 3 from [19], we obtain the following properties of the functions L i (i = 1, 2, 3).
(2) The function L 1 is nondecreasing in the first variable. In fact, from For the proof of properties (4)-(9) is similar to the proof of properties (1)-(3), we omit it here.
(10) This property is evident, it follows from the definitions of L i (i = 1, 2, 3) and from properties (3), (6), and (9). The proof is complete. Now, from the definitions of the Green's functions G and the properties of functions L i (i = 1, 2, 3), we obtain the following lemma. ( which implies that From u(t) = I α n-2 0 + x(t) and the above expression, we have Hence, we demonstrate that u(t) = I The proof is complete.
Remark 2.1 Under the assumptions of Lemma 2.8, for c ∈ (0, 1/2), the solutions of problem (2.1) satisfy the inequality min t∈ [c,1] It is easy to see that P is a normal cone of E. We define an operator A : E → E by Obviously, if x is a fixed point of operator A, then x is a solution of problem (2.9). We present the basic assumptions that we shall use in the sequel.

Lemma 2.9
If conditions (H 1 ) and (H 2 ) hold, then A : P → P is completely continuous.
Proof We denote by Q i = 1 0 J i (s)p(s) ds (i = 1, 2, 3), where J i (i = 1, 2, 3) are defined in Lemma 2.6. Using (H 2 ) and Lemma 2.5, we deduce that Q i > 0 (i = 1, 2, 3) and By Lemma 2.6, we also conclude that A : P → P. We prove that A maps bounded sets into relatively compact sets. Suppose that D ⊂ P is an arbitrary bounded set of E, then there exists M 1 > 0 such that x ≤ M 1 for all x ∈ D. By the continuity of q, we obtain

Thus, A(D) is bounded in E.
In what follows, we prove that A(D) is equicontinuous. By using Lemma 2.4, for any x ∈ D and t ∈ [0, 1], we have Thus, for any t ∈ (0, 1), we conclude (2.12) We denote For the integral of the function g, by exchanging the order of integration, we obtain For the integral of the function μ, we have (2.13) We deduce that μ ∈ L 1 (0, 1). Thus, for any t 1 , t 2 ∈ [0, 1] with t 1 ≤ t 2 and x ∈ D, by (2.12) and (2.13), we conclude (2.14) From (2.13), (2.14) and the absolute continuity of the integral function, we obtain that A(D) is equicontinuous. By the Ascoli-Arzela theorem, we conclude that A(D) is a relatively compact set of E. Therefore A is a compact operator. Besides, we can show that A is continuous on P (see the proof of Lemma 1.4.1 in [21]). Hence A : P → P is completely continuous.
Under assumptions (H 1 ), (H 2 ) and Remark 2.1, we have A(P) ⊂ P 0 , and so A| P 0 : P 0 → P 0 (denoted again by A) is also a completely continuous operator.