Existence and uniqueness of solutions for a class of higher-order fractional boundary value problems with the nonlinear term satisfying some inequalities

This paper focuses on a class of hider-order nonlinear fractional boundary value problems. The boundary conditions contain Riemann–Stieltjes integral and nonlocal multipoint boundary conditions. It is worth mentioning that the nonlinear term and the boundary conditions contain fractional derivatives of different orders. Based on the Schauder fixed point theorem, we obtain the existence of solutions under the hypothesis that the nonlinear term satisfies the Carathéodory conditions. We apply the Banach contraction mapping principle to obtain the uniqueness of solutions. Moreover, by using the theory of spectral radius we prove the uniqueness and nonexistence of positive solutions. Finally, we illustrate our main results by some examples.

Fractional calculus and fractional boundary value problems have been researched extensively to apply them in various areas, including image processing, rheology, electrical networks, virus infection models, and so on. Some interesting results can be found in [1][2][3][4][5][6][7][8][9][10][11] and the references therein. For example, in [1] the authors discovered that the motion frequency of a class of neurons should be characterized by noninteger derivatives. Therefore fractional derivatives are introduced to characterize this behavior, which is not possible by integer-derivative models. In [4] the authors introduced the Riemann-Liouville fractional derivative of order α (0.5 < α ≤ 1) into a model of HIV infection of CD4 + T-cells. By using stability analysis the authors obtained a sufficient condition on the parameters for the stability of the infected steady state. It should be noted that this fractional model possessed positive solutions, which is desired in any population dynamics. Indeed, there are many definitions of fractional derivatives. Because the Riemann-Liouville fractional derivative avoids seeking limits, it is widely used in mathematical studies. The definition of Riemann-Liouville fractional derivative shows that it has some important properties such as globality. In fact, the Riemann-Liouville fractional derivative is very suitable for describing viscoelastic material models and processes with memory properties. It has the advantages of simple modeling and accurate description. Recently, the research on the properties of solutions of fractional boundary value problems has received substantial attention. Some interesting results can be found in  and the references therein. For example, in [15] the authors have obtained the existence of one and two solutions by using the fixed point index theory. In [16], based on the Schaefer fixed point theorem and Banach contraction principle, the existence and uniqueness of solutions for a class of fractional boundary value problem are obtained. Moreover, the higher-order fractional boundary value problems have attracted more attention. We refer to [13-15, 19, 31, 32, 34-37, 44, 45, 58, 59]. For example, in [36] the existence and uniqueness of solutions are obtained by applying the Krasnoselskii theorem and Banach fixed point theorem. Based on the Leggett-Williams and Krasnoselskii fixed point theorems, Zhang and Zhong [31] showed the existence of positive solutions for the following nonlinear fractional boundary value problem: The nonlinearity f may be singular at t = 0, 1 and u = 0, and h ∈ L 1 ([0, 1], [0, +∞)) may be singular at t = 0, 1.
Motivated by the papers mentioned, in this paper, we are lead to study problem (1.1). Evidently, our discussion is novel and meaningful. Firstly, problem (1.1) is more general; especially, the boundary conditions include two types of Riemann-Stieltjes integral boundary conditions and nonlocal multipoint boundary conditions. Secondly, the nonlinear term f contains the fractional derivatives of different orders of the unknown function. Thirdly, the existence of solutions is obtained under the hypothesis that f satisfies the Carathéodory condition, which is weaker than the continuity conditions. Fourthly, we show the uniqueness and nonexistence of positive solutions by using appropriate methods. Moreover, in this paper, our approach in obtaining the corresponding integral operator is the reduction method of fractional order on account of semigroup properties of the Riemann-Liouville derivative. We also illustrate the relationship between higher-and lower-order fractional derivatives.
An outline of this paper is as follows. In Sect. 2, we give some preliminaries and lemmas. We transform problem (1.1) into a relatively low-order problem by using the reduction method and obtain the relevant Green's function. In Sect. 3, we construct two results, one handing the existence of solutions and the other one managing the uniqueness of solutions under two different assumptions. In Sect. 4, we obtain the uniqueness of positive solutions by using spectral radius theory. In Sect. 5, we prove the nonexistence of positive solutions. In Sect. 6, we illustrate the main results by some examples.
Remark 2.4 In view of Lemma 2.4, we infer that researching solutions of problem (1.1) is equivalent to the work on considering solutions of problem (2.1) under the premise that 1 < αα n-2 ≤ 2. Note that the corresponding integral operator of problem (2.1) can be considered in the space C[0, 1], which avoids doing the work in a complex space. Therefore our work focusses on problem (2.1) in the following: Proof By Lemma 2.2 problem (2.11) can be rewritten as where d i ∈ (-∞, +∞) (i = 1, 2) are arbitrary constants. The condition D (2.15) By Lemma 2.3 we have we have (2.20) Thus problem (2.11) has a unique solution where H(t, s) is defined by (2.13).

22)
(3) In a similar manner, we have We omit the details.
Then the Green's function H(t, s) (defined in (2.13)) has the following properties: Proof The conclusion can be directly deduced from Lemma 2.6. So, we omit the details. Lemma 2.9 ([9, 10]) Let P be a generating cone in the Banach space E, that is, E = P -P. Let A is a completely continuous u 0 -bounded linear operator. If the spectral radius r(A) = 0, then A has an eigenfunction ψ ∈ P \{θ } that belongs to the first eigenvalue λ 1 = (r(A)) -1 such that λ 1 Aψ = ψ, and A has no other positive eigenvalue that has positive eigenfunctions.
In this paper, we define the Banach space It is easy to check that P is generating in E, that is, E = P -P. Now, we define the nonlinear operator T : Observe that v is a solution of problem (2.1) if and only if v is a fixed point of the operator T in E.
In the following, we prove that v * is the unique fixed point of T in P. If not, there exists an element v * * ∈ P such that v * * = Tv * * . Similarly, there exists β(|v * *v * |) > 0 such that Thus we get that, for any t
Proof The proof is similar to that of Lemma 4.1. So, we omit the details.
Proof The proof is similar to that of Lemma 4.2. So, we omit the details.
Proof The proof is similar to that of Theorem 5.1. So, we omit the details.

Examples
Now, we give five explicit examples illustrating the main results.  . Then problem (6.1) can be transformed into problem (1.1).