The Green’s function of a class of two-term fractional differential equation boundary value problem and its applications

In this paper, we consider a Riemann–Liouville type two-term fractional differential equation boundary value problem. Some positive properties of the Green’s function are deduced by using techniques of analysis. As application, we obtain the existence and multiplicity of positive solutions for a fractional boundary value problem under conditions that the nonlinearity f (t, x) may change sign and may be singular at t = 0, 1 and x = 0, and we also obtain the uniqueness results of positive solution for a singular problem by means of the monotone iterative technique.


Introduction
In this paper, we study properties of the Green's function of the following two-term fractional differential equation boundary value problem (FBVP): -D α 0+ u(t) + au(t) = y(t), 0 < t < 1, where 2 < α < 3, a > 0, D α 0+ is the standard Riemann-Liouville derivative. During the past decades, much attention has been paid to the study of fractional differential equations (FDEs) due to the more accurate effect in describing important phenomena in biology, engineering, and so on. It has been proved that a multi-term FDE can be used to describe various types of visco-elastic damping [1,2]. Most of the model equations proposed can be expressed by the linear form where a i ∈ R, i = 0, 1, . . . , N -1, equipped with initial conditions (see [3][4][5][6][7] and the references therein). For example, Elshehawey et al. [5] considered the endolymph equation which can be used to describe the response of the semicircular canals to the angular acceleration.
Recently, many authors have focused on the existence of solutions to nonlinear FBVPs by using the techniques of nonlinear analysis such as fixed point theorems, Leray-Schauder theory, etc. (see ). Since only positive solutions are meaningful in most practical problems, the existence of positive solutions for FBVPs has particularly attracted a great deal of attention, e.g., the nonlocal FBVPs [10,22,25], singular FBVPs [16,21,28], semipositone FBVPs [15,18,27].
It is known that the cone which usually depends on the positive properties of the Green's function plays a very important role in discussing positive solutions. When 1 < α < 2, Jiang and Yuan [14] obtained some properties of the Green's function for the FDE: with Dirichlet type boundary value condition. Xu and Fei [30] investigated (2) with threepoint boundary value condition. In [19], we established some new positive properties of the corresponding Green's function for (2) with multi-point boundary value condition. When α > 2, Zhang et al. [27,28] obtained triple positive solutions for (2) with conjugate type integral conditions by employing height functions on special bounded sets which were derived from properties of the Green's function. While there are a lot of works dealing with multi-term FDEs with initial conditions, the results dealing with boundary value problems of multi-term FDEs are relatively scarce. For some recent literature on Caputo type multi-term FBVPs, we mention the papers [8,9] and the references therein. In [20], we established some new positive properties of the Green's function for the Riemann-Liouville type FBVP, in which the linear operator contains two terms: where 1 < α < 2, b > 0. As application, the existence and uniqueness of positive solution are obtained under singular conditions. Inspired by the above work, in this paper, we aim to deduce some positive properties of the Green's function for FBVP (1). As application, we investigate the existence and multiplicity of positive solutions for a singular FBVP with changing sign nonlinearity, and we also consider the uniqueness results of positive solution for a singular FBVP. Compared with the existing works, this paper has the following features. Firstly, the fractional derivative discussed in this paper is the standard Riemann-Liouville derivative, which is different from [8,9], and the linear operator of the FBVP we are considered with contains two terms, which is different from [14,19,27,28,30]; in other words, we discuss different problem which has been seldom studied before. Secondly, some meaningful properties of the Green's function for the case that 2 < α < 3 are established; this is different from [20] since Ref. [20] considered the case that 1 < α < 2. Thirdly, we consider a multiplicity of positive solutions under conditions that the nonlinearity f (t, x) may change sign and possess singularity at x = 0; this is different from [15,18]. It should be noted that there are relatively few results on multiple solutions for FBVPs under this circumstance, not to mention two-term FBVPs. Finally, we obtain the uniqueness results of positive solution for a singular two-term FBVP by means of the monotone iterative technique, and the rate of convergence for the iterative sequence is considered.

Basic definitions and preliminaries
Definition 2.1 ([33]) The fractional integral of a function u : (0, +∞) → R is given by provided that the right-hand side is point-wise defined on (0, +∞).
For convenience, we introduce the following notations: Therefore, h(x) has a unique positive root a * , that is, h(a * ) = 0. Throughout this paper, we always assume that the following assumption holds: (H 1 ) a ∈ (0, a * ] is a constant. Then the unique solution of the two-term FBVP (1) is Proof It follows from [33] that the general solution of the equation can be expressed by By direct calculation, we have By It follows from u(1) = 0 that Therefore, the solution of (1) is Proof In fact, we have It follows from [32, Lemma 2.1] that I α 0+ y(t) ∈ AC 1 [0, 1] and

Theorem 3.1 The Green's function G(t, s) satisfies the following properties:
Proof Since (p 2 ) is trivially true and (p 1 ) can be derived from (p 3 ), it remains to verify (p 3 ) and (p 4 ).

Semipositone problem
In this section, we consider the existence and multiplicity of positive solutions to the semipositone FBVP: For convenience, we list here the hypotheses to be used in this section: (H 2 ) f ∈ C((0, 1) × (0, +∞), (-∞, +∞)) and satisfies  Let E = C[0, 1] be endowed with the maximum norm u = max 0≤t≤1 |u(t)|. Define a cone Denote B r = {u(t) ∈ E : u(t) < r} and P r = P ∩ B r .

Lemma 4.1 The unique solution of the FBVP
Proof The lemma can be deduced from Lemma 2.1 and Corollary 3.1, so we omit it.
Next we consider the auxiliary FBVP: where Proof For any u ∈ P with u ≥ r, one has The rest of the proof is similar to Lemma 2.6 in [21], we omit it here.
By the extension theorem of completely continuous operator (see [34]), there exists an extension operator A : P → P, which is still completely continuous. Without loss of generality, we still write it as A.

Uniqueness results
In this section, we consider the uniqueness results of positive solution to the singular FBVP: For convenience, we assume that the following assumptions hold in the rest of this paper: x, y) is nondecreasing on x, nonincreasing on y, and there exists μ ∈ (0, 1) such that f t, rx, y r ≥ r μ f (t, x, y), ∀x, y > 0, r ∈ (0, 1).
This and (25) yield T : Proof Let w ∈ Q 1 , it follows from Lemma 4.4 that T(w, w) ∈ Q 1 . Then we can select r 0 ∈ (0, 1) such that Set where It is easy to see that u i , v i ∈ Q 1 , i = 0, 1, . . . , and It follows from (23) and (24) that Then we have By induction, we can get u n ≥ r μ n 0 v n , n = 1, 2, . . . .
Therefore, (28) and (29) yield Then {u n } is a Cauchy sequence. Similarly, we can get {v n } is a Cauchy sequence. It follows from (28) that there exist u * , v * ∈ Q 1 such that {u n } and {v n } converge to u * and v * respectively. Moreover, This and (29) imply that v *u * ≤ v nu n ≤ 1r μ n 0 v 0 , n = 1, 2, . . . . Hence By (30), we have Let n → +∞, we get Then we have u * = T(u * , u * ), that is, u * is a positive fixed point of T.
Next, we will show that the positive fixed point of T is unique. In fact, if u = u * is a positive fixed point of T, by Lemma 4.4, we have u ∈ Q 1 . Denote r 1 = sup r ∈ (0, 1) : ru * ≤ u ≤ r -1 u * .