Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications

Abstract

In this article, we study a fractional lower-order differential equation, $-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)+\mathrm{a}\left(\xi \right){\rm Y}\left(\xi \right)=\mathrm{y}\left(\xi \right),\xi \in (0,1),\alpha \in (1,2),$ with a Dirichlet-type boundary condition, where $\mathrm{a}\left(\xi \right)\in L^1[0,1]$ permits singularity. When the coefficient of perturbation term $\mathrm{a}\left(\xi \right)$ is continuous on $[0,1]$, Graef et al. derived the associated Green’s function under certain conditions on $\mathrm{a}$, but failed to obtain its positivity. We obtain some positive properties of the Green’s function of this problem under certain conditions. The results will fill the gap in the above literature. As for application of the theoretical results, a singular fractional differential equation with a perturbation term is considered. The unique positive solution is obtained by using the fixed point theorem, and an iterative sequence is established to approximate it. In addition, a semipositone fractional differential equation, $-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)=\mu \mathcal{F}(\xi ,{\rm Y}\left(\xi \right)),\xi \in (0,1),\alpha \in (1,2),$ is also considered. The existence of positive solutions is determined under a more general condition, $\mathcal{F}(\xi ,\mathrm{x})\ge -\mathrm{b}\left(\xi \right)\mathrm{x}-\mathrm{e}\left(\xi \right)$, where $\mathrm{b}\left(\xi \right),\mathrm{e}\left(\xi \right)\in L^1[0,1]$ are non-negative functions. Relevant examples are listed to manifest the theoretical results.

1. Introduction

In this paper, we consider the positivity of the Green’s function (GF) of lower-order fractional differential equations (FDEs) containing the variable coefficient

$$-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)+\mathrm{a}\left(\xi \right){\rm Y}\left(\xi \right)=\mathrm{y}\left(\xi \right),\xi \in (0,1),\alpha \in (1,2),$$

(1)

with the Dirichlet-type boundary condition (BC)

$${\rm Y}\left(1\right)=0,{\rm Y}\left(0\right)=0,$$

(2)

where $D_{0+}^{\alpha }$ is the Riemann–Liouville (R-L) derivative, and $\mathrm{a}\left(\xi \right)\in L^1[0,1]$.

In view of their excellent performance in describing the dynamics of complex systems, more and more models of fractional order have emerged, and some models of integer order have been modified to fractional-order models for research (see [1,2,3,4,5]). For instance, Area et al. [3] incorporated a fractional derivative into compartmental models of Ebola epidemics. Ndaïrou et al. [4] indicated that the compartmental model of fractional order fitted well with the data published by the WHO. Due to their extensive applications in the field of science, fractional boundary value problems (FBVPs) have been a popular topic of research for decades. A primary problem of boundary value problems (BVPs) is the existence result of solutions. Many researchers have been devoted to the study of BVPs consisting of certain BCs and FDEs with only one term in the linear operator; see [6,7,8,9] and references therein.

In [6], Cui studied the BVP

$$\left\{\begin{array}{c}-D_{0+}^{\beta }{\rm Y}\left(\xi \right)=g(\xi ,{\rm Y}\left(\xi \right)),\xi \in (0,1),2<\beta \le 3,\hfill \\ {\rm Y}\left(1\right)=0,{\rm Y}\left(0\right)={{\rm Y}}^{\prime }\left(0\right)=0.\hfill \end{array}\right.$$

in which $D_{0+}^{\beta }$ is an R-L derivative. Zhang and Zhong [7] studied the FDE

$$-D_{0+}^{\beta }{\rm Y}\left(\xi \right)=g(\xi ,{\rm Y}\left(\xi \right)),\xi \in (0,1),\beta \in (n-1,n),n\ge 3,$$

subject to integral BCs. In [8], the authors considered the BVP

$$\left\{\begin{array}{c}{D}_{0+}^{\beta }{\rm Y}\left(\xi \right)+\lambda g(\xi ,{\rm Y}\left(\xi \right))=0,0<\xi <1,n-1<\beta \le n,n\ge 3\hfill \\ {{\rm Y}}^{\left(j\right)}\left(0\right)=0,j=0,1,\cdots ,n-2,D_{0+}^{{\beta }_1}{\rm Y}\left(1\right)=\sum _{i=1}^m{\eta }_i{D}_{0+}^{{\beta }_2}{\rm Y}\left({\xi }_i\right),\hfill \end{array}\right.$$

where the parameter $\lambda >0$, and $g(\xi ,x)$ is a sign-changing continuous function. In [9], Xu et al. studied

$$\left\{\begin{array}{c}{-}^H{D}_{{\xi }_1+}^{\gamma }{\rm Y}\left(\xi \right)=g(t,{\rm Y}\left(\xi \right)),\xi \in ({\xi }_1,{\xi }_2),0<{\xi }_1<{\xi }_2<\infty ,\gamma \in (2,3),\hfill \\ {\rm Y}\left({\xi }_1\right)={{\rm Y}}^{\prime }\left({\xi }_1\right)=0,{\rm Y}\left({\xi }_2\right)={\int }_{{\xi }_1}^{{\xi }_2}\omega \left(\xi \right){\rm Y}\left(\xi \right){\displaystyle \frac{d\xi }{\xi }},\hfill \end{array}\right.$$

where ${}^{H}D_{{\xi }_1+}^{\gamma }$ is the Hadamard-type derivative.

In cases where the nonlinearity g contains fractional-order derivatives of unknown function, the FDE can be transformed into a lower-order one using the method of substitution, in which the nonlinearity no longer contains fractional-order derivatives. In [10], the authors studied

$$D_{0+}^{\beta }{\rm Y}\left(\xi \right)+g(\xi ,{\rm Y}\left(\xi \right),D_{0+}^{{\beta }_1}{\rm Y}\left(\xi \right),\dots ,D_{0+}^{{\beta }_{n-2}}{\rm Y}\left(\xi \right))=0,0<\xi <1,n-1<\beta \le n,n\ge 3,$$

with certain BCs. Using the method of reduction order, the above FDE is transformed into the following FDE:

$$D_{0+}^{\beta -{\beta }_{n-2}}\mathcal{Y}\left(\xi \right)+g(\xi ,I_{0+}^{{\beta }_{n-2}}\mathcal{Y}\left(\xi \right),\dots ,I_{0+}^{{\beta }_{n-2}-{\beta }_{n-3}}\mathcal{Y}\left(\xi \right),\mathcal{Y}\left(\xi \right))=0,\xi \in (0,1),\beta -{\beta }_{n-2}\in (1,2].$$

By drawing upon the Banach fixed point theorem and Schauder fixed point theorem, the authors discussed the existence of solutions to the above lower-order BVP. It should be pointed out that some nonlinear analysis tools cannot be used due to the properties of the GF obtained by the authors. In [11], the authors studied

$${}^{H}D_{{\xi }_1+}^{\gamma }{\rm Y}\left(\xi \right)+g(\xi ,{\rm Y}\left(\xi \right),{}^{H}D_{{\xi }_1+}^{{\beta }_1}{\rm Y}\left(\xi \right))=0,{\xi }_1<\xi <{\xi }_2,2.5<\beta \le 3,0<{\beta }_1\le 0.5,$$

with infinite-point BCs. With the method of reduction order and Laplace transform, the above FBVP was converted to an integral operator. Then, they obtained the existence of positive solutions by drawing upon the fixed point index theory.

The usual methodology to prove the existence of solutions is to transform the BVP into a Hammerstein integral operator by drawing upon Laplace transform, and seek fixed points of the operator. In cases where the kernel is non-negative, various methodologies of nonlinear analysis are utilized in a cone to investigate positive fixed points of the operator. It is clear that a smaller and more accurate cone is crucial to the study of positive solutions to BVPs, particularly for singular problems and semipositone problems. On the other hand, some properties of the solutions can also be represented by the cone. In general, the selection of the cone depends on the properties of the GF. It should be noted that compared with high-order FBVPs (i.e., the order $\beta >2$), there are more difficulties in the study of GF for FBVPs of the order $\beta \in (1,2)$ (see [12]).

When a more general FDE with the so-called perturbation term is considered, the Laplace transform cannot be used due to the variable coefficient. Based on the spectral theory, Graef et al. developed an approach for converting these problems into integral equations (see [13,14]). Graef et al. [14] and Zou [15] studied the high-order FBVP

$$\left\{\begin{array}{c}-D_{0+}^{\gamma }{\rm Y}\left(\xi \right)+a\left(\xi \right){\rm Y}\left(\xi \right)=g(\xi ,{\rm Y}\left(\xi \right)),0<\xi <1,\gamma \in (2,3),\hfill \\ {{\rm Y}}^{\prime }\left(1\right)=0,{\rm Y}\left(0\right)={{\rm Y}}^{\prime }\left(0\right)=0,\hfill \end{array}\right.$$

where $a\left(\xi \right)$ is continuous on $C[0,1]$ with $\overline{a}:=\max \left\{\right|a\left(\xi \right)|:\xi \in [0,1]\}<{\displaystyle \frac{(\gamma -1)\Gamma (\gamma +1)}{\gamma +1}}$. Graef et al. [14] and Zou [15] deduced some positive properties of GF of the above FBVP. Using the perturbation approach proposed by [13,14], Gao and Wang [16] derived the GF for the following high-order FBVP:

$$\left\{\begin{array}{c}-D_{0+}^{\gamma }{\rm Y}\left(\xi \right)+a\left(\xi \right){\rm Y}\left(\xi \right)=g(\xi ,{\rm Y}\left(\xi \right)),0<\xi <1,\gamma >2,\gamma \notin \mathbb{N},\hfill \\ {{\rm Y}}^{\left(i\right)}\left(0\right)=0,i=0,1,\dots ,\left[\gamma \right]-1,{\rm Y}\left(1\right)={\int }_0^1{\rm Y}\left(\xi \right)d\mathcal{A}\left(\xi \right).\hfill \end{array}\right.$$

The authors obtained the existence of solutions to the FBVP by drawing upon Schauder’s fixed point theorem. In [17], the authors studied the following high-order FBVP:

$$\left\{\begin{array}{c}-D_{0+}^{\gamma }{\rm Y}\left(\xi \right)+a\left(\xi \right){\rm Y}\left(\xi \right)=g(\xi ,{\rm Y}\left(\xi \right)),0<\xi <1,\gamma >2,\gamma \notin \mathbb{N},\hfill \\ {{\rm Y}}^{\left(i\right)}\left(0\right)=0,i=0,1,\dots ,\left[\gamma \right]-1,D_{0+}^{\beta }{\rm Y}\left(1\right)=0,1\le \beta \le \left[\gamma \right]-1.\hfill \end{array}\right.$$

They derived the associated GF and some estimates of it, and then obtained the existence of solutions by utilizing Schauder’s fixed point theorem. In [18], we studied the following high-order FBVP:

$$\left\{\begin{array}{c}-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)+\lambda a\left(t\right){\rm Y}\left(\xi \right)=\mathrm{y}\left(\xi \right),t\in (0,1),\alpha \in (2,3]\hfill \\ {\rm Y}\left(0\right)={\rm Y}\left(1\right)={{\rm Y}}^{\prime }\left(0\right)=0.\hfill \end{array}\right.$$

where the parameter $\lambda >0$, $a\left(\xi \right)\in L^1[0,1]$. The associated GF and some estimates of it were derived under a general condition on $a\left(\xi \right)$.

Graef et al. [13] considered the lower-order FBVP

$$\left\{\begin{array}{c}-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)+a\left(\xi \right){\rm Y}\left(\xi \right)=g(\xi ,{\rm Y}\left(\xi \right)),0<\xi <1,1<\alpha <2,\hfill \\ {\rm Y}\left(0\right)={\rm Y}\left(1\right)=0,\hfill \end{array}\right.$$

in which $a\in C[0,1]$. The authors derived the GF under the condition $\overline{a}:=\max \left\{\right|a\left(t\right)|:t\in [0,1]\}<4^{\alpha -1}\Gamma \left(\alpha \right)$, but they failed to obtain its positivity. When $\mathrm{a}\left(t\right)\equiv a\in \mathbb{R}$, the present author [19] deduced the GF of FBVPs (1)–(2) as a series concerning Mittag–Leffler functions and established some positive properties for it. However, the approach fails to handle cases where $a\left(t\right)\not\equiv a$. As for discrete delta FDEs with Dirichlet-type BCs and a perturbation term, we mention the articles [20,21]. To the best of our knowledge, to date, no-one has obtained the positive properties of the GF for lower-order FBVPs (1)–(2). We will establish some positive properties for the GF of FBVPs (1)–(2) to fill this gap. The key difficulty is that the properties of GF corresponding to lower-order FBVPs without a perturbation term are not good enough. To overcome this difficulty, we rewrite the expression of GF and utilize a special transformation.

The features of the present article are as follows. Firstly, the FDE we considered contains a perturbation term with a variable coefficient, which is different from that considered in [6,7,8,9]. Secondly, compared with [18], FDE (1) considered herein is a lower-order FDE. Compared with [14,15,16,17], FDE (1) considered herein is a lower-order FDE, and the coefficient of the perturbation term $\mathrm{a}\left(t\right)$ permits singularity at $t=0,1$, that is, we consider more general conditions. Thirdly, compared with [13], we consider positive properties for the GF of FBVPs (1)–(2).

2. Preliminaries

Definition 1

([2]). The R-L fractional integral of order $\varkappa (\varkappa >0)$ of a function Ψ is given by

$$I_{0+}^{\varkappa }\Psi \left(\xi \right)={\displaystyle \frac{1}{\Gamma \left(\varkappa \right)}}{\int }_0^{\xi }{(\xi -\zeta )}^{\varkappa -1}\Psi \left(\zeta \right)d\zeta ,$$

provided that the right-hand side is point-wise defined on $(0,+\infty )$.

Definition 2

([2]). The R-L derivative of order ϰ$(n-1<\varkappa <n)$ of a function Ψ is given by

$$D_{0+}^{\varkappa }\Psi \left(\xi \right)={\displaystyle \frac{1}{\Gamma (n-\varkappa )}}{\left({\displaystyle \frac{d}{d\xi }}\right)}^n{\int }_0^{\xi }{(\xi -\zeta )}^{n-\varkappa -1}\Psi \left(\zeta \right)d\zeta ,$$

where $n=\left[\varkappa \right]+1$, $\left[\varkappa \right]$ is the integer part of ϰ.

For simplicity, we use some notations:

$$\begin{array}{cc}\hfill & G_0(\xi ,\zeta )={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left\{\begin{array}{cc}{\left[\xi (1-\zeta )\right]}^{\alpha -1},\hfill & 0\le \xi \le \zeta \le 1,\hfill \\ {\xi (1-\zeta )]}^{\alpha -1}-{(\xi -\zeta )}^{\alpha -1},\hfill & 0\le \zeta \le \xi \le 1;\hfill \end{array}\right.\hfill \\ \hfill & K_0(\xi ,\zeta )={\xi }^{2-\alpha }{G}_0(\xi ,\zeta );\hfill \\ \hfill & K_n(\xi ,\zeta )={\int }_0^1\mathrm{a}\left(\eta \right){\eta }^{\alpha -2}{K}_0(\xi ,\eta )K_{n-1}(\eta ,\zeta )d\eta ,n=1,2,\cdots ;\hfill \\ \hfill & {\Lambda }_1={\int }_0^1{\left|\mathrm{a}\left(\xi \right)\right|\left[\xi (1-\xi )\right]}^{\alpha -1}d\xi ,{\Lambda }_2={\int }_0^1\left|\mathrm{a}\left(\xi \right)\right|{\left[\xi (1-\xi )\right]}^{\alpha -2}d\xi ;\hfill \\ \hfill & \Lambda ={\displaystyle \frac{{\Lambda }_2}{(\alpha -1)(\Gamma \left(\alpha \right)-{\Lambda }_1)}},\Delta ={\displaystyle \frac{(\alpha -1)(1-\Lambda )}{1+\Lambda }};\hfill \\ \hfill & p\left(\xi \right)=\xi (1-\xi ),q\left(\xi \right)=\xi {(1-\xi )}^{\alpha -1}.\hfill \end{array}$$

Throughout this paper, we suppose that the function $\mathrm{a}\left(\xi \right)$ in FDE (1) always satisfies the following:

$\left(H_1\right)$$\mathrm{a}\left(\xi \right)\in L^1[0,1]$ with ${\Lambda }_1<\Gamma \left(\alpha \right)$, ${\Lambda }_2<+\infty$ and $\Lambda <1$.

It is known that $G_0(\xi ,\zeta )$ is the GF of FBVPs (1)–(2) with $\mathrm{a}\left(t\right)\equiv 0$.

Lemma 1

([12]). The function $G_0(\xi ,\zeta )$ possesses the following properties:

(i) 

$G_0(\xi ,\zeta )=G_0(1-\zeta ,1-\xi ),\forall \xi ,\zeta \in [0,1]$;

(ii) 

${\displaystyle \frac{q(1-\xi )q\left(\zeta \right)}{\Gamma (\alpha -1)}}\le G_0(\xi ,\zeta )\le {\displaystyle \frac{q\left(\zeta \right)q(1-\xi )}{\Gamma \left(\alpha \right)p\left(\zeta \right)}},\forall \xi ,\zeta \in (0,1)$;

(iii) 

${\displaystyle \frac{q(1-\xi )q\left(\zeta \right)}{\Gamma (\alpha -1)}}\le G_0(\xi ,\zeta )\le {\displaystyle \frac{q\left(\zeta \right)q(1-\xi )}{\Gamma \left(\alpha \right)p\left(\xi \right)}},\forall \xi ,\zeta \in (0,1)$.

According to Lemma 1 and the notation of $K_0(\xi ,\zeta )$, we have the following Lemma:

Lemma 2.

The function $K_0(\xi ,\zeta )$ possesses the following properties:

(i) 

${\displaystyle \frac{p\left(\xi \right)q\left(\zeta \right)}{\Gamma (\alpha -1)}}\le K_0(\xi ,\zeta )\le {\displaystyle \frac{q\left(\zeta \right)p\left(\xi \right)}{\Gamma \left(\alpha \right)p\left(\zeta \right)}},\forall \xi ,\zeta \in (0,1)$;

(ii) 

${\displaystyle \frac{p\left(\xi \right)q\left(\zeta \right)}{\Gamma (\alpha -1)}}\le K_0(\xi ,\zeta )\le {\displaystyle \frac{q\left(\zeta \right)}{\Gamma \left(\alpha \right)}},\forall \xi ,\zeta \in [0,1]$.

Lemma 3.

Assume that $\left(H_1\right)$ holds. Then,

$$\begin{array}{c}\hfill \sum _{n=0}^{+\infty }{(-1)}^n{K}_n(\xi ,\zeta )\end{array}$$

is a uniformly convergent series of functions on $[0,1]\times [0,1]$.

Proof. 

According to Lemma 2 and the notation of $K_n$, we have

$$\begin{array}{cc}\hfill |K_1(\xi ,\zeta )|& \le {\int }_0^1\left|\mathrm{a}\left(\eta \right)\right|{\eta }^{\alpha -2}\times {\displaystyle \frac{\xi (1-\xi ){(1-\eta )}^{\alpha -2}}{\Gamma \left(\alpha \right)}}\times {\displaystyle \frac{\zeta {(1-\zeta )}^{\alpha -1}}{\Gamma \left(\alpha \right)}}d\eta \hfill \\ \hfill & ={\displaystyle \frac{{\Lambda }_2}{{(\Gamma \left(\alpha \right))}^2}}\xi (1-\xi )\zeta {(1-\zeta )}^{\alpha -1}\hfill \\ & ={\displaystyle \frac{{\Lambda }_{2}p\left(\xi \right)q\left(\zeta \right)}{{(\Gamma \left(\alpha \right))}^2}}.\hfill \end{array}$$

Suppose that

$$|K_m(\xi ,\zeta )|\le {\displaystyle \frac{{\Lambda }_2{\Lambda }_1^{m-1}p\left(\xi \right)q\left(\zeta \right)}{{(\Gamma \left(\alpha \right))}^{m+1}}}.$$

Then,

$$\begin{array}{cc}\hfill |K_{m+1}(\xi ,\zeta )|& =|{\int }_0^1\mathrm{a}\left(\eta \right){\eta }^{\alpha -2}{K}_0(\xi ,\eta )K_m(\eta ,\zeta )d\eta |\hfill \\ & \le {\displaystyle \frac{{\Lambda }_2{\Lambda }_1^{m-1}}{{(\Gamma \left(\alpha \right))}^{m+2}}}{\int }_0^1\left|\mathrm{a}\left(\eta \right)\right|{\eta }^{\alpha -2}{\displaystyle \frac{q\left(\eta \right)p\left(\xi \right)}{p\left(\eta \right)}}p\left(\eta \right)q\left(\zeta \right)d\eta \hfill \\ & ={\displaystyle \frac{{\Lambda }_2{\Lambda }_1^{m-1}}{{(\Gamma \left(\alpha \right))}^{m+2}}}{\int }_0^1\left|\mathrm{a}\left(\eta \right)\right|{\left[\eta (1-\eta )\right]}^{\alpha -1}p\left(\xi \right)q\left(\zeta \right)d\eta \hfill \\ \hfill & ={\displaystyle \frac{{\Lambda }_2{\Lambda }_1^{m}p\left(\xi \right)q\left(\zeta \right)}{{(\Gamma \left(\alpha \right))}^{m+2}}}.\hfill \end{array}$$

Through induction, we can obtain

$$|K_n(\xi ,\zeta )|\le {\displaystyle \frac{{\Lambda }_2{\Lambda }_1^{n-1}p\left(\xi \right)q\left(\zeta \right)}{{(\Gamma \left(\alpha \right))}^{n+1}}},n=1,2,\cdots .$$

(3)

Note that since ${\Lambda }_1<\Gamma \left(\alpha \right)$, we determine that

$$\begin{array}{c}\hfill \sum _{n=0}^{+\infty }{(-1)}^n{K}_n(\xi ,\zeta )\end{array}$$

is a uniformly convergent series of functions on $[0,1]\times [0,1]$. □

3. Main Results

Let $\mathrm{E}=C[0,1]$ and $\parallel {\rm Y}\parallel ={\max }_{0\le \xi \le 1}|{\rm Y}\left(\xi \right)|$; then, $(\mathrm{E},\parallel ·\parallel )$ is a Banach space.

Theorem 1.

Assume that $\left(H_1\right)$ holds and $\mathrm{y}\in L^1[0,1]$; then, the FBVP

$$\left\{\begin{array}{c}-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)+\mathrm{a}\left(\xi \right){\rm Y}\left(\xi \right)=\mathrm{y}\left(\xi \right),0<\xi <1,\hfill \\ {\rm Y}\left(0\right)={\rm Y}\left(1\right)=0.\hfill \end{array}\right.$$

(4)

has a unique solution:

$${\rm Y}\left(\xi \right)={\int }_0^1{\xi }^{\alpha -2}{\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )\mathrm{y}\left(\zeta \right)d\zeta ,$$

where

$$\begin{array}{c}\hfill {\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )=\sum _{n=0}^{+\infty }{(-1)}^n{K}_n(\xi ,\zeta ).\end{array}$$

In addition, ${\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )$ satisfies the following:

(i) 

${\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )\ge {\displaystyle \frac{(1-\Lambda )p\left(\xi \right)q\left(\zeta \right)}{\Gamma (\alpha -1)}},\forall \xi ,\zeta \in [0,1];$

(ii) 

${\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )\le {\displaystyle \frac{(1+\Lambda )p\left(\xi \right)q\left(\zeta \right)}{\Gamma \left(\alpha \right)p\left(\zeta \right)}},\forall t,s\in (0,1);$

(iii) 

${\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )\le {\displaystyle \frac{(1+\Lambda )q\left(\zeta \right)}{\Gamma \left(\alpha \right)}},\forall \xi ,\zeta \in [0,1].$

Proof. 

It is easy to see that ${\rm Y}\in \mathrm{E}$ is a solution of FBVP (4) if and only if

$${\rm Y}\left(\xi \right)={\int }_0^1{G}_0(\xi ,\zeta )(\mathrm{y}\left(\zeta \right)-\mathrm{a}\left(\zeta \right){\rm Y}\left(\zeta \right))d\zeta .$$

(5)

Assume that $\mathrm{v}\left(\xi \right)={\xi }^{2-\alpha }{\rm Y}\left(\xi \right),\xi \in [0,1]$. Then, Equation (5) can be written as

$$\mathrm{v}\left(\xi \right)={\int }_0^1{K}_0(\xi ,\zeta )(\mathrm{y}\left(\zeta \right)-\mathrm{a}\left(\zeta \right){\zeta }^{\alpha -2}\mathrm{v}\left(\zeta \right))d\zeta ;$$

that is,

$$\mathrm{v}\left(\xi \right)+{\int }_0^1{K}_0(\xi ,\zeta )\mathrm{a}\left(\zeta \right){\zeta }^{\alpha -2}\mathrm{v}\left(\zeta \right)d\zeta ={\int }_0^1{K}_0(\xi ,\zeta )\mathrm{y}\left(\zeta \right)d\zeta .$$

(6)

Let $\mathrm{I}$ denote the identity operator. Define the linear operators $\mathrm{L}:\mathrm{E}\to \mathrm{E}$ as

$$\mathrm{Lv}\left(\xi \right)={\int }_0^1{K}_0(\xi ,\zeta )\mathrm{a}\left(\zeta \right){\zeta }^{\alpha -2}\mathrm{v}\left(\zeta \right)d\zeta .$$

According to Lemma 2 and ${\Lambda }_1<\Gamma \left(\alpha \right)$, we can obtain $\parallel \mathrm{L}\parallel <1.$ Thus, the linear operator of $(\mathrm{L}+\mathrm{I})$ is invertible, and ${(\mathrm{L}+\mathrm{I})}^{-1}=\mathrm{I}-\mathrm{L}+{\mathrm{L}}^2+···+{(-1)}^n{\mathrm{L}}^n+···.$ From (6) and Lemma 3, we have

$$\begin{array}{cc}\hfill \mathrm{v}\left(\xi \right)& ={(\mathrm{L}+\mathrm{I})}^{-1}\left({\int }_0^1{K}_0(\xi ,\zeta )\mathrm{y}\left(\zeta \right)d\zeta \right)=\sum _{n=0}^{+\infty }{(-\mathrm{L})}^n\left({\int }_0^1{K}_0(\xi ,\zeta )\mathrm{y}\left(\zeta \right)d\zeta \right)\hfill \\ \hfill & =\sum _{n=0}^{+\infty }{\int }_0^1{(-1)}^n{K}_n(\xi ,\zeta )\mathrm{y}\left(\zeta \right)d\zeta ={\int }_0^1{\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )\mathrm{y}\left(\zeta \right)d\zeta .\hfill \end{array}$$

Thus, we determine that

$${\rm Y}\left(\xi \right)={\int }_0^1{\xi }^{\alpha -2}{\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )\mathrm{y}\left(\zeta \right)d\zeta .$$

In addition, from Lemmas 2 and (3), it follows that

$$|K_n(\xi ,\zeta )|\le {\displaystyle \frac{{\Lambda }_2{\Lambda }_1^{n-1}{K}_0(\xi ,\zeta )}{(\alpha -1){(\Gamma \left(\alpha \right))}^n}},n=1,2,\cdots .$$

Therefore,

$$\sum _{n=1}^{+\infty }\left|K_n(\xi ,\zeta )\right|\le {\displaystyle \frac{{\Lambda }_2{K}_0(\xi ,\zeta )}{(\alpha -1)(\Gamma \left(\alpha \right)-{\Lambda }_1)}}=\Lambda K_0(\xi ,\zeta ).$$

Then, we have

$$(1-\Lambda )K_0(\xi ,\zeta )\le {\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )\le (1+\Lambda )K_0(\xi ,\zeta ).$$

Combining the above inequality and Lemma 2, we conclude that the properties $\left(\mathrm{i}\right)$, $\left(\mathrm{ii}\right)$, and $\left(\mathrm{iii}\right)$ hold. □

Remark 1.

It is easy to verify that ${\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )=t^{2-\alpha }G(\xi ,\zeta )$, in which $G(\xi ,\zeta )$ is the GF derived by the authors in [13] for the case of $\mathrm{a}\left(\xi \right)\in C[0,1]$. Moreover, the function $G(\xi ,\zeta )$ satisfies the following:

(i) 

$G(\xi ,\zeta )\ge {\displaystyle \frac{(1-\Lambda )q(1-\xi )q\left(\zeta \right)}{\Gamma (\alpha -1)}},\forall \xi ,\zeta \in [0,1];$

(ii) 

$G(\xi ,\zeta )\le {\displaystyle \frac{(1+\Lambda )q(1-\xi )q\left(\zeta \right)}{\Gamma \left(\alpha \right)p\left(\zeta \right)}},\forall \xi ,\zeta \in (0,1);$

(iii) 

$G(\xi ,\zeta )\le {\displaystyle \frac{(1+\Lambda )q(1-\xi )q\left(\zeta \right)}{\Gamma \left(\alpha \right)p\left(\xi \right)}},\forall \xi ,\zeta \in (0,1).$

Remark 2.

If $\mathrm{a}\in C[0,1]$, then ${\Lambda }_2<+\infty$ holds naturally. In this case, $\left(H_1\right)$ is equivalent to ${\Lambda }_1<\Gamma \left(\alpha \right)$, and $\Lambda <1$. Moreover, $\overline{a}:=\max \left\{\right|\mathrm{a}\left(t\right)|:t\in [0,1]\}<4^{\alpha -1}\Gamma \left(\alpha \right)$ implies that ${\Lambda }_1<\Gamma \left(\alpha \right)$.

4. Applications

4.1. Singular Problems

In this section, we study a singular FBVP as follows:

$$\left\{\begin{array}{c}-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)+\mathrm{a}\left(\xi \right){\rm Y}\left(\xi \right)=\mathfrak{g}(\xi ,{\rm Y}\left(\xi \right),{\rm Y}\left(\xi \right)),0<\xi <1,1<\alpha <2,\hfill \\ {\rm Y}\left(0\right)={\rm Y}\left(1\right)=0,\hfill \end{array}\right.$$

(7)

We list the hypotheses used in this section below:

$\left(H_2\right)$ $\mathfrak{g}:(0,1)\times (0,+\infty )\times (0,+\infty )\to [0,+\infty )$ is continuous, and $\mathfrak{g}(\xi ,x_1,x_2)$ is nondecreasing on $x_1$ and nonincreasing on $x_2$. ∃ $\sigma \in (0,1)$, such that

$$\mathfrak{g}(\xi ,\tau x_1,{\displaystyle \frac{x_2}{\tau }})\ge {\tau }^{\sigma }\mathfrak{g}(\xi ,x_1,x_2),\forall x_1,x_2>0,\tau \in (0,1);$$

$\left(H_3\right)$ $0<{\int }_0^1{(1-\zeta )}^{\alpha -2}\mathfrak{g}(\zeta ,(1-\zeta ){\zeta }^{\alpha -1},(1-\zeta ){\zeta }^{\alpha -1})d\zeta <+\infty .$

Define a cone $\mathcal{P}$ as

$$\mathcal{P}=\left\{{\rm Y}\in \mathrm{E}:\exists l_{{\rm Y}}>0,suchthat,\Delta ·\parallel {\rm Y}\parallel ·p\left(\xi \right)\le {\rm Y}\left(\xi \right)\le l_{{\rm Y}}·p\left(\xi \right),\xi \in [0,1]\right\}.$$

Let

$$\mathrm{A}({{\rm Y}}_1,{{\rm Y}}_2)\left(\xi \right)={\int }_0^1{\mathrm{K}}_{\mathrm{a}}(\xi ,\zeta )\mathfrak{g}(\zeta ,{\zeta }^{\alpha -2}{{\rm Y}}_1\left(\zeta \right),{\zeta }^{\alpha -2}{{\rm Y}}_2\left(\zeta \right))d\zeta .$$

Theorem 2.

If $\left(H_1\right)-\left(H_3\right)$ hold, then FBVP (7) has exactly one positive solution $\mathcal{V}\left(t\right)\in \mathcal{P}$. In addition, for any $w_0\in \mathcal{P}\setminus \left\{\theta \right\}$, there is a ${\tau }_0\in (0,1)$ such that

$${\tau }_0^{1-\sigma }{w}_0\le \mathrm{A}(w_0,w_0)\le {\tau }_0^{-(1-\sigma )}{w}_0.$$

Set

$${{\rm Y}}_0={\tau }_0{w}_0,{\Psi }_0={\tau }_0^{-1}{w}_0,$$

and ${{\rm Y}}_n=\mathrm{A}({{\rm Y}}_{n-1},{\Psi }_{n-1})$, ${\Psi }_n=\mathrm{A}({\Psi }_{n-1},{{\rm Y}}_{n-1})$, $n=1,2,\cdots$; then, ${\xi }^{\alpha -2}{{\rm Y}}_n\left(\xi \right)\to \mathcal{V}\left(\xi \right)$.

Proof. 

The proof is similar to Theorem 4.5 of [19], so we omit it. □

Example 1.

Consider the following BVP:

$$\left\{\begin{array}{l}-D_{0+}^{1.5}{\rm Y}\left(\xi \right)+{\displaystyle \frac{{\rm Y}\left(\xi \right)}{20{\xi }^{\frac{1}{4}}{(1-\xi )}^{\frac{1}{4}}}}={\xi }^{-{\displaystyle \frac{1}{6}}}{(1-\xi )}^{-{\displaystyle \frac{1}{6}}}\left({[{\rm Y}\left(\xi \right)]}^{{\displaystyle \frac{1}{6}}}+{[{\rm Y}\left(\xi \right)]}^{-{\displaystyle \frac{1}{6}}}\right),0<\xi <1,\\ {\rm Y}\left(0\right)=0,{\rm Y}\left(1\right)=0.\end{array}\right.$$

(8)

Set $\mathrm{a}\left(\xi \right)={\displaystyle \frac{1}{20}}{\xi }^{-{\displaystyle \frac{1}{4}}}{(1-\xi )}^{-{\displaystyle \frac{1}{4}}}$; then,

$$\Gamma (1.5)\approx 0.88623;$$

$${\Lambda }_1={\displaystyle \frac{1}{20}}{\int }_0^1{\left[\xi (1-\xi )\right]}^{{\displaystyle \frac{1}{4}}}d\xi \approx 0.03090;$$

$${\Lambda }_2={\displaystyle \frac{1}{20}}{\int }_0^1{\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{3}{4}}}d\xi \approx 0.37077;$$

$$\Lambda ={\displaystyle \frac{{\Lambda }_2}{\frac{1}{2}(\Gamma \left(\frac{3}{2}\right)-{\Lambda }_1)}}\approx 0.87328.$$

Thus, $\left(H_1\right)$ holds. Let $\sigma ={\displaystyle \frac{1}{6}}$, and

$$\mathfrak{g}(\xi ,x_1,x_2)={\xi }^{-{\displaystyle \frac{1}{6}}}{(1-\xi )}^{-{\displaystyle \frac{1}{6}}}\left[x_1^{{\displaystyle \frac{1}{6}}}+x_2^{-{\displaystyle \frac{1}{6}}}\right].$$

It is clear that $\left(H_2\right)$ holds. Through calculation, we obtain

$${\int }_0^1{(1-\zeta )}^{-{\displaystyle \frac{1}{2}}}\mathfrak{g}\left(\zeta ,(1-\zeta ){\zeta }^{{\displaystyle \frac{1}{2}}},(1-\zeta ){\zeta }^{{\displaystyle \frac{1}{2}}}\right)d\zeta \approx 8.56694.$$

So, Theorem 2 guarantees that the singular FBVP (8) has only one positive solution $\mathcal{V}\left(t\right)\in \mathcal{P}$.

Set $w_0\left(\xi \right)=p\left(\xi \right)\in \mathcal{P}\setminus \left\{\theta \right\}$; then, $w_0\in \mathcal{P}\setminus \left\{\theta \right\}$. According to $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$ of Theorem 1, we have

$$0.053064\times w_0\le \mathrm{A}(w_0,w_0)\le 18.10854\times w_0.$$

Let ${\tau }_0={\displaystyle \frac{1}{50}}$; then,

$${\tau }_0^{1-\sigma }{w}_0\le \mathrm{A}(w_0,w_0)\le {\tau }_0^{-(1-\sigma )}{w}_0.$$

Set

$${{\rm Y}}_0={\tau }_0{w}_0,{\Psi }_0={\tau }_0^{-1}{w}_0,$$

and ${{\rm Y}}_n=\mathrm{A}({{\rm Y}}_{n-1},{\Psi }_{n-1})$, ${\Psi }_n=\mathrm{A}({\Psi }_{n-1},{{\rm Y}}_{n-1})$, $n=1,2,\cdots$; then, ${\xi }^{\alpha -2}{{\rm Y}}_n\left(\xi \right)\to \mathcal{V}\left(\xi \right)$.

4.2. Semipositone Problem

In this section, we study positive solutions for a semipositone problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)=\mu \mathcal{F}(\xi ,{\rm Y}\left(\xi \right)),\xi \in (0,1),\hfill \\ {\rm Y}\left(0\right)={\rm Y}\left(1\right)=0,\hfill \end{array}\right.\end{array}$$

(9)

where $\mu >0$, $\mathcal{F}$ permits singularity at $t=0,1$, and $\mathcal{F}$ may change its sign.

We list the hypotheses used in this section below:

$\left(H_1^{\prime }\right)$ There are non-negative functions, $\mathrm{b},\mathrm{e}\in L^1(0,1)\cap C(0,1)$, such that ${\int }_0^1\mathrm{e}\left(\xi \right)d\xi >0$, ${\int }_0^1\mathrm{b}\left(\xi \right)d\xi >0$, ${\int }_0^1{(1-\xi )}^{\alpha -2}\mathrm{e}\left(\xi \right)d\xi <+\infty$ and ${\int }_0^1\mathrm{b}\left(\xi \right){\left[\xi (1-\xi )\right]}^{\alpha -2}d\xi <+\infty$.

$\left(H_4\right)$ $\mathcal{F}(\xi ,\mathrm{x}):(0,1)\times [0,+\infty )\to \mathbb{R}$ is continuous, and

$$\begin{array}{c}\hfill \mathcal{F}(\xi ,\mathrm{x})\ge -\mathrm{b}\left(\xi \right)\mathrm{x}-\mathrm{e}\left(\xi \right),\forall (\xi ,\mathrm{x})\in (0,1)\times [0,+\infty ).\end{array}$$

$\left(H_5\right)$ There are non-negative functions, $\mathrm{z}\in C(0,1)$ and $\mathrm{h}\in C[0,+\infty )$, such that ${\int }_0^1\mathrm{z}\left(\xi \right){(1-\xi )}^{\alpha -2}d\xi <+\infty$ and

$$\begin{array}{c}\hfill \mathcal{F}(\xi ,t^{\alpha -2}\mathrm{x})\le \mathrm{z}\left(\xi \right)\mathrm{h}\left(\mathrm{x}\right),\forall (\xi ,\mathrm{x})\in (0,1)\times [0,+\infty ).\end{array}$$

$\left(H_6\right)$ There is $[\mathrm{c},\mathrm{d}]\subset (0,1)$ such that

$$\begin{array}{c}\hfill \underset{\mathrm{x}\to +\infty }{\lim \inf }\underset{\xi \in [\mathrm{c},\mathrm{d}]}{\min }{\displaystyle \frac{\mathcal{F}(\xi ,\mathrm{x})}{\mathrm{x}}}=+\infty \end{array}$$

Denote

$${\mu }_1={\displaystyle \frac{(\alpha -1)\Gamma \left(\alpha \right)}{2{\int }_0^1\mathrm{b}\left(\xi \right){\left[\xi (1-\xi )\right]}^{\alpha -2}d\xi +(\alpha -1){\int }_0^1\mathrm{b}\left(\xi \right){\left[\xi (1-\xi )\right]}^{\alpha -1}d\xi }}.$$

Lemma 4.

Assume that the hypothesis $\left(H_1^{\prime }\right)$ holds and $0<\mu <{\mu }_1$. Then, the function ${\mathrm{K}}_{\mu \mathrm{b}}(t,s)$ satisfies the following:

(i) 

${\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )\ge {\displaystyle \frac{p\left(\xi \right)q\left(\zeta \right)}{2\Gamma (\alpha -1)}},\forall \xi ,\zeta \in [0,1];$

(ii) 

${\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )\le {\displaystyle \frac{3p\left(\xi \right)q\left(\zeta \right)}{2\Gamma \left(\alpha \right)p\left(\zeta \right)}},\forall \xi ,\zeta \in (0,1);$

(iii) 

${\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )\le {\displaystyle \frac{3q\left(\zeta \right)}{2\Gamma \left(\alpha \right)}},\forall \xi ,\zeta \in [0,1].$

Proof. 

Let us replace the function $\mathrm{a}\left(\xi \right)$ in Theorem 1 with $\mu \mathrm{b}\left(\xi \right)$. It is easy to verify that $\mu \mathrm{b}$ satisfies the hypothesis $\left(H_1\right)$. Moreover, we can obtain

$$\begin{array}{cc}\hfill 0<\Lambda & ={\displaystyle \frac{{\int }_0^1\mu \mathrm{b}\left(\xi \right){\left[\xi (1-\xi )\right]}^{\alpha -2}d\xi }{(\alpha -1)(\Gamma \left(\alpha \right)-{\int }_0^1\mu \mathrm{b}\left(\xi \right){\left[\xi (1-\xi )\right]}^{\alpha -1}d\xi )}}\hfill \\ & <{\displaystyle \frac{{\int }_0^1\mathrm{b}\left(\xi \right){\left[\xi (1-\xi )\right]}^{\alpha -2}d\xi }{(\alpha -1)\left(\frac{\Gamma \left(\alpha \right)}{{\mu }_1}-{\int }_0^1\mathrm{b}\left(\xi \right){\left[\xi (1-\xi )\right]}^{\alpha -1}d\xi \right)}}\hfill \\ & ={\displaystyle \frac{1}{2}}.\hfill \end{array}$$

Then, from Theorem 1, we can determine that the properties $\left(i\right)$, $\left(ii\right)$, and $\left(iii\right)$ hold. □

According to Lemma 4 and Theorem 1, we obtain the following Lemma:

Lemma 5.

Assume that the hypothesis $\left(H_1^{\prime }\right)$ holds and $0<\mu <{\mu }_1$. Then, the solution to FBVP

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-D_{0+}^{\alpha }{\rm Y}\left(\xi \right)+\mu \mathrm{b}\left(\xi \right){\rm Y}\left(t\right)=\mu \mathrm{e}\left(\xi \right),0<\xi <1,\hfill \\ {\rm Y}\left(0\right)={\rm Y}\left(1\right)=0,\hfill \end{array}\right.\end{array}$$

is

$$\begin{array}{c}\hfill {\varphi }_{\mu }\left(\xi \right)=\mu {\int }_0^1{\xi }^{\alpha -2}{\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )\mathrm{e}\left(\zeta \right)d\zeta .\end{array}$$

Moreover,

$$\begin{array}{c}\hfill {\varphi }_{\mu }\left(\xi \right)\le {\displaystyle \frac{3\mu q(1-\xi )}{2\Gamma \left(\alpha \right)}}{\int }_0^1{(1-\zeta )}^{\alpha -2}\mathrm{e}\left(\zeta \right)d\zeta .\end{array}$$

(10)

Denote

$$\begin{array}{cc}\hfill & g(\xi ,\mathrm{x})=\mathcal{F}(\xi ,\mathrm{x})+\mathrm{b}\left(\xi \right)\mathrm{x}+\mathrm{e}\left(\xi \right);\hfill \\ \hfill & {[\mathrm{u}\left(\xi \right)-{\varphi }_{\mu }\left(\xi \right)]}^{+}=\max \{\mathrm{u}\left(\xi \right)-{\varphi }_{\mu }\left(\xi \right),0\}.\hfill \end{array}$$

For the rest of this section, we suppose that the hypothesis $\left(H_1^{\prime }\right)$ always holds and $0<\mu <{\mu }_1$.

Obviously, FBVP (9) is equivalent to

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-D_{0+}^{\alpha }({\rm Y}\left(\xi \right)+{\varphi }_{\mu }\left(\xi \right))+\mu \mathrm{b}\left(\xi \right)({\rm Y}\left(\xi \right)+{\varphi }_{\mu }\left(\xi \right))=\mu g(\xi ,{\rm Y}\left(\xi \right)),\xi \in (0,1),\hfill \\ {\rm Y}\left(0\right)={\rm Y}\left(1\right)=0,\hfill \end{array}\right.\end{array}$$

(11)

Set $\Psi \left(\xi \right)={\rm Y}\left(\xi \right)+{\varphi }_{\mu }\left(\xi \right)$; then, (11) can be written as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-D_{0+}^{\alpha }\Psi \left(\xi \right)+\mu \mathrm{b}\left(\xi \right)\Psi \left(\xi \right)=\mu g(\xi ,\Psi \left(\xi \right)-{\varphi }_{\mu }\left(\xi \right)),\xi \in (0,1),\hfill \\ \Psi \left(0\right)=\Psi \left(1\right)=0,\hfill \end{array}\right.\end{array}$$

Now we consider the auxiliary FBVP:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-D_{0+}^{\alpha }\Psi \left(\xi \right)+\mu \mathrm{b}\left(\xi \right)\Psi \left(\xi \right)=\mu g(\xi ,{[\Psi \left(\xi \right)-{\varphi }_{\mu }\left(\xi \right)]}^{+}),\xi \in (0,1),\hfill \\ \Psi \left(0\right)=\Psi \left(1\right)=0,\hfill \end{array}\right.\end{array}$$

(12)

According to Theorem 1, $\Psi \in \mathrm{E}$ is a solution of (12), where $\Psi$ is a fixed point of the operator $\mathrm{T}$ defined by

$$\mathrm{Tv}\left(\xi \right)=\mu {\int }_0^1{\xi }^{\alpha -2}{\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )g(\zeta ,{[\mathrm{v}\left(\zeta \right)-{\varphi }_{\mu }\left(\zeta \right)]}^{+})d\zeta .$$

Denote

$$\mathrm{Q}=\left\{\mathrm{w}\in \mathrm{E}:\exists {\mathrm{l}}_{\mathrm{w}}>0,s.t.{\displaystyle \frac{(\alpha -1)\parallel \mathrm{w}\parallel p\left(\xi \right)}{3}}\le \mathrm{w}\left(\xi \right)\le {\mathrm{l}}_{\mathrm{w}}p\left(\xi \right),\xi \in [0,1]\right\},$$

$${\mathrm{B}}_{\mathrm{r}}=\{\mathrm{w}\left(\xi \right)\in \mathrm{E}:\parallel \mathrm{w}\parallel <\mathrm{r}\},{\mathrm{Q}}_{\mathrm{r}}=\mathrm{Q}\cap {\mathrm{B}}_{\mathrm{r}},\partial {\mathrm{Q}}_{\mathrm{r}}=\mathrm{Q}\cap \partial {\mathrm{B}}_{\mathrm{r}}.$$

Let

$$\mathcal{A}\mathrm{w}\left(\xi \right)=\mu {\int }_0^1{\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )g(\zeta ,{\zeta }^{\alpha -2}{[\mathrm{w}\left(\zeta \right)-{\zeta }^{2-\alpha }{\varphi }_{\mu }\left(\zeta \right)]}^{+})d\zeta .$$

By using the Ascoli–Arzela theorem, we can obtain the following Lemma:

Lemma 6.

If $\left(H_4\right)$ and $\left(H_5\right)$ hold and $0<\mu <{\mu }_1$, then $\mathcal{A}$: $\mathrm{Q}\to \mathrm{Q}$ is completely continuous.

Lemma 7

([22]). Assume that $\mathrm{Q}$ is a cone of a Banach space $\mathrm{E}$, ${\Theta }_i\subset \mathrm{E}$$(i=1,2)$ are bounded open sets with $\theta \in {\Theta }_1$ and $\overline{{\Theta }_1}\subset {\Theta }_2$, and $\mathcal{A}:$$\mathrm{Q}\cap (\overline{{\Theta }_2}\setminus {\Theta }_1)\to \mathrm{Q}$ is completely continuous. If one of the following assumptions holds, then there is a fixed point of $\mathcal{A}$ in $\mathrm{Q}\cap (\overline{{\Theta }_2}\setminus {\Theta }_1)$:

(1) 

$\parallel \mathcal{A}\Psi \parallel \ge \parallel \Psi \parallel ,\Psi \in \mathrm{Q}\cap \partial {\Theta }_1,\parallel \mathcal{A}\Psi \parallel \le \parallel \Psi \parallel ,\Psi \in \mathrm{Q}\cap \partial {\Theta }_2$;

(2) 

$\parallel \mathcal{A}\Psi \parallel \le \parallel \Psi \parallel ,\Psi \in \mathrm{Q}\cap \partial {\Theta }_1,\parallel \mathcal{A}\Psi \parallel \ge \parallel \Psi \parallel ,\Psi \in \mathrm{Q}\cap \partial {\Theta }_2$,

Theorem 3.

Suppose that $\left(H_4\right)$–$\left(H_6\right)$ hold. Then, there exists ${\mu }^{*}>0$ such that the FBVP (9) has positive solutions for any $\mu \in (0,{\mu }^{*})$.

Proof. 

It is clear that if $\mathrm{w}\in \mathrm{E}$ with $\mathrm{w}\left(t\right)\ge t^{2-\alpha }{\varphi }_{\mu }\left(t\right)$, and $\mathrm{w}$ is a fixed point of the operator $\mathcal{A}$, then $t^{\alpha -2}\mathrm{w}-{\varphi }_{\mu }$ is a positive solution of FBVP (9).

Denote

$${\mathrm{r}}_1={\displaystyle \frac{9{\int }_0^1\mathrm{e}\left(\zeta \right){(1-\zeta )}^{\alpha -2}d\zeta }{4{\int }_0^1{\mathrm{b}\left(\zeta \right)[\zeta (1-\zeta ]}^{\alpha -2}d\zeta }},$$

and

$${\mu }^{*}=\min \left\{{\mu }_1,{\displaystyle \frac{2\Gamma \left(\alpha \right){\mathrm{r}}_1}{3{\int }_0^{1}q\left(\zeta \right)[\mathrm{z}\left(\zeta \right){\mathrm{h}}^{★}\left({\mathrm{r}}_1\right)+{\mathrm{r}}_1\mathrm{b}\left(\zeta \right){\zeta }^{\alpha -2}+\mathrm{e}\left(\zeta \right)]d\zeta }}\right\},$$

where

$${\mathrm{h}}^{★}\left(\mathrm{x}\right)=\max \{\mathrm{h}\left(\tau \right):0\le \tau \le \mathrm{x}\}.$$

For any $\mathrm{w}\in \mathrm{Q}\setminus {\mathrm{B}}_{{\mathrm{r}}_1}$, we have

$$\mathrm{w}\left(\xi \right)\ge {\displaystyle \frac{(\alpha -1)\parallel \mathrm{w}\parallel p\left(\xi \right)}{3}}.$$

This inequality, together with (10), leads to

$$\begin{array}{cc}\hfill & \mathrm{w}\left(\xi \right)-{\xi }^{2-\alpha }{\varphi }_{\mu }\left(\xi \right)\hfill \\ \hfill & \ge p\left(\xi \right)\left({\displaystyle \frac{(\alpha -1)\parallel \mathrm{w}\parallel }{3}}-{\displaystyle \frac{3\mu {\int }_0^1{(1-\zeta )}^{\alpha -2}\mathrm{e}\left(\zeta \right)d\zeta }{2\Gamma \left(\alpha \right)}}\right)\hfill \\ \hfill & \ge p\left(\xi \right)\left({\displaystyle \frac{(\alpha -1)\parallel \mathrm{w}\parallel }{3}}-{\displaystyle \frac{3{\mu }_1{\int }_0^1{(1-\zeta )}^{\alpha -2}\mathrm{e}\left(\zeta \right)d\zeta }{2\Gamma \left(\alpha \right)}}\right)\hfill \\ \hfill & \ge p\left(\xi \right)\left({\displaystyle \frac{(\alpha -1)\parallel \mathrm{w}\parallel }{3}}-{\displaystyle \frac{3(\alpha -1){\int }_0^1{(1-\zeta )}^{\alpha -2}\mathrm{e}\left(\zeta \right)d\zeta }{4{\int }_0^1\mathrm{b}\left(\zeta \right){\left[\zeta (1-\zeta )\right]}^{\alpha -2}d\zeta }}\right)\hfill \\ \hfill & ={\displaystyle \frac{(\alpha -1)(\parallel \mathrm{w}\parallel -{\mathrm{r}}_1)p\left(\xi \right)}{3}}\hfill \\ & \ge 0.\hfill \end{array}$$

(13)

Then, for any ${\mathrm{w}}_0\in \partial {\mathrm{Q}}_{{\mathrm{r}}_1}$, in view of Lemma 4 and $\left(H_5\right)$, we have

$$\begin{array}{cc}\hfill & \mathcal{A}{\mathrm{w}}_0\left(\xi \right)=\mu {\int }_0^1{\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )g(\zeta ,{\zeta }^{\alpha -2}{[{\mathrm{w}}_0\left(\zeta \right)-{\zeta }^{2-\alpha }{\varphi }_{\mu }\left(\zeta \right)]}^{+})d\zeta \hfill \\ \hfill & \le {\displaystyle \frac{3\mu }{2\Gamma \left(\alpha \right)}}{\int }_0^{1}q\left(\zeta \right)\left[\mathrm{z}\left(\zeta \right)\mathrm{h}\left({[{\mathrm{w}}_0\left(\zeta \right)-{\zeta }^{2-\alpha }{\varphi }_{\mu }\left(\zeta \right)]}^{+}\right)+\mathrm{b}\left(\zeta \right){\zeta }^{\alpha -2}\parallel {\mathrm{w}}_0\parallel +\mathrm{e}\left(\zeta \right)\right]d\zeta \hfill \\ \hfill & <{\displaystyle \frac{3{\mu }^{*}}{2\Gamma \left(\alpha \right)}}{\int }_0^{1}q\left(\zeta \right)\left[\mathrm{z}\left(\zeta \right){\mathrm{h}}^{★}\left({\mathrm{r}}_1\right)+{\mathrm{r}}_1\mathrm{b}\left(\zeta \right){\zeta }^{\alpha -2}+\mathrm{e}\left(\zeta \right)\right]d\zeta \hfill \\ \hfill & \le {\mathrm{r}}_1=\parallel {\mathrm{w}}_0\parallel .\hfill \end{array}$$

(14)

On the other hand, denote

$$\mathcal{G}={\displaystyle \frac{12\Gamma (\alpha -1)}{\mu (\alpha -1)p^2\left(\mathrm{c}\right)p^2\left(\mathrm{d}\right){\int }_{\mathrm{c}}^{\mathrm{d}}{\zeta }^{\alpha -1}{(1-\zeta )}^{\alpha -1}d\zeta }}.$$

$\left(H_6\right)$ implies that there exists $\mathcal{X}>0$ such that

$$\mathcal{F}(\xi ,\mathrm{x})\ge \mathcal{G}\mathrm{x},\forall \mathrm{x}\ge \mathcal{X},\xi \in [\mathrm{c},\mathrm{d}].$$

Denote

$${\mathrm{r}}_2=\max \left\{2{\mathrm{r}}_1,{\displaystyle \frac{6\mathcal{X}}{(\alpha -1)p\left(\mathrm{c}\right)p\left(\mathrm{d}\right)}}\right\}.$$

Let $\mathrm{w}\in \partial {\mathrm{Q}}_{{\mathrm{r}}_2}$. The inequality (13) implies that

$$\mathrm{w}\left(\xi \right)-{\xi }^{2-\alpha }{\varphi }_{\mu }\left(\xi \right)\ge {\displaystyle \frac{(\alpha -1)({\mathrm{r}}_2-{\mathrm{r}}_1)p\left(\xi \right)}{3}},\xi \in [0,1].$$

Then,

$$\mathrm{w}\left(\xi \right)-{\xi }^{2-\alpha }{\varphi }_{\mu }\left(\xi \right)\ge 0,\xi \in [0,1],$$

and

$$\mathrm{w}\left(\xi \right)-{\xi }^{2-\alpha }{\varphi }_{\mu }\left(\xi \right)\ge {\displaystyle \frac{(\alpha -1){\mathrm{r}}_{2}p\left(\xi \right)}{6}}\ge {\displaystyle \frac{(\alpha -1)p\left(\mathrm{c}\right)p\left(\mathrm{d}\right){\mathrm{r}}_2}{6}},\xi \in [\mathrm{c},\mathrm{d}].$$

Hence,

$$\mathrm{w}\left(\xi \right)-{\xi }^{2-\alpha }{\varphi }_{\mu }\left(\xi \right)\ge \mathcal{X},\xi \in [\mathrm{c},\mathrm{d}].$$

Then, for any ${\mathrm{w}}_1\in \partial {\mathrm{Q}}_{{\mathrm{r}}_2}$, we have

$$\begin{array}{cc}\hfill & \mathcal{A}{\mathrm{w}}_1\left(\xi \right)=\mu {\int }_0^1{\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )g(\zeta ,{\zeta }^{\alpha -2}{[{\mathrm{w}}_1\left(\zeta \right)-{\zeta }^{2-\alpha }{\varphi }_{\mu }\left(\zeta \right)]}^{+})d\zeta \hfill \\ \hfill & \ge \mu {\int }_{\mathrm{c}}^{\mathrm{d}}{\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )g(\zeta ,{\zeta }^{\alpha -2}{[{\mathrm{w}}_1\left(\zeta \right)-{\zeta }^{2-\alpha }{\varphi }_{\mu }\left(\zeta \right)]}^{+})d\zeta \hfill \\ \hfill & \ge \mu {\int }_{\mathrm{c}}^{\mathrm{d}}{\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta )\mathcal{F}(\zeta ,{\zeta }^{\alpha -2}{[{\mathrm{w}}_1\left(\zeta \right)-{\zeta }^{2-\alpha }{\varphi }_{\mu }\left(\zeta \right)]}^{+})d\zeta \hfill \\ \hfill & \ge \mu \mathcal{G}{\int }_{\mathrm{c}}^{\mathrm{d}}{\mathrm{K}}_{\mu \mathrm{b}}(\xi ,\zeta ){\zeta }^{\alpha -2}({\mathrm{w}}_1\left(\zeta \right)-{\zeta }^{2-\alpha }{\varphi }_{\mu }\left(\zeta \right))d\zeta \hfill \\ \hfill & \ge {\displaystyle \frac{\mu \mathcal{G}(\alpha -1)p\left(\mathrm{c}\right)p\left(\mathrm{d}\right){\mathrm{r}}_2{\int }_{\mathrm{c}}^{\mathrm{d}}{\zeta }^{\alpha -1}{(1-\zeta )}^{\alpha -1}d\zeta }{12\Gamma (\alpha -1)}}p\left(\xi \right)\hfill \\ \hfill & ={\displaystyle \frac{{\mathrm{r}}_{2}p\left(\xi \right)}{p\left(\mathrm{c}\right)p\left(\mathrm{d}\right)}}.\hfill \end{array}$$

Thus,

$$\parallel \mathcal{A}{\mathrm{w}}_1\parallel \ge \underset{\xi \in [0,1]}{\max }{\displaystyle \frac{{\mathrm{r}}_{2}p\left(\xi \right)}{p\left(\mathrm{c}\right)p\left(\mathrm{d}\right)}}>{\mathrm{r}}_2=\parallel {\mathrm{w}}_1\parallel .$$

(15)

□

From Lemma 7, inequalities (14) and (15) imply that $\mathcal{A}$ has a fixed point $\mathrm{w}\in \mathrm{Q}\cap ({\mathrm{B}}_{{\mathrm{r}}_2}\setminus \overline{{\mathrm{B}}_{{\mathrm{r}}_1}})$. Clearly, ${\xi }^{\alpha -2}\mathrm{w}\left(\xi \right)-{\varphi }_{\mu }\left(\xi \right)$ is a positive solution of FBVP (9).

Example 2.

Consider the following semipositone BVP:

$$\left\{\begin{array}{c}-D_{0+}^{{\displaystyle \frac{7}{4}}}{\rm Y}\left(\xi \right)=\mu \mathcal{F}(\xi ,{\rm Y}\left(\xi \right)),0<t<1,\hfill \\ {\rm Y}\left(0\right)={\rm Y}\left(1\right)=0,\hfill \end{array}\right.$$

(16)

where

$$\mathcal{F}(\xi ,\mathrm{x})={\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{1}{2}}}{\mathrm{x}}^{{\displaystyle \frac{7}{4}}}-{\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{1}{4}}}{\mathrm{x}}^{{\displaystyle \frac{1}{4}}}.$$

Then,

$$\mathcal{F}(\xi ,\mathrm{x})\ge -{\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{1}{4}}}{\mathrm{x}}^{{\displaystyle \frac{1}{4}}}\ge -{\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{1}{4}}}(\mathrm{x}+1),$$

and

$$\mathcal{F}(\xi ,{\xi }^{-{\displaystyle \frac{1}{4}}}\mathrm{x})\le {\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{1}{2}}}{({\xi }^{-{\displaystyle \frac{1}{4}}}\mathrm{x})}^{{\displaystyle \frac{7}{4}}}\le {\xi }^{-{\displaystyle \frac{15}{16}}}{(1-\xi )}^{-{\displaystyle \frac{1}{2}}}{\mathrm{x}}^{{\displaystyle \frac{7}{4}}}.$$

Let $\mathrm{b}\left(\xi \right)=\mathrm{e}\left(\xi \right)={\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{1}{4}}}$, $\mathrm{z}\left(\xi \right)={\xi }^{-{\displaystyle \frac{15}{16}}}{(1-\xi )}^{-{\displaystyle \frac{1}{2}}}$, $\mathrm{h}\left(\mathrm{x}\right)={\mathrm{x}}^{{\displaystyle \frac{7}{4}}}$. Through calculation, we have

$${\int }_0^1\mathrm{b}\left(\xi \right)d\xi ={\int }_0^1\mathrm{e}\left(\xi \right)d\xi ={\int }_0^1{\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{1}{4}}}d\xi \approx 1.69443,$$

$${\int }_0^1\mathrm{b}\left(\xi \right){\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{1}{4}}}={\int }_0^1{\left[\xi (1-\xi )\right]}^{-{\displaystyle \frac{1}{2}}}d\xi \approx 3.14159,$$

$${\int }_0^1\mathrm{e}\left(\xi \right){(1-\xi )}^{-{\displaystyle \frac{1}{4}}}={\int }_0^1{\xi }^{-{\displaystyle \frac{1}{4}}}{(1-\xi )}^{-{\displaystyle \frac{1}{2}}}d\xi \approx 2.39628,$$

$${\int }_0^1\mathrm{z}\left(\xi \right){(1-\xi )}^{-{\displaystyle \frac{1}{4}}}={\int }_0^1{\xi }^{-{\displaystyle \frac{15}{16}}}{(1-\xi )}^{-{\displaystyle \frac{3}{4}}}d\xi \approx 19.58384.$$

So $\left(H_1^{\prime }\right)$, $\left(H_4\right)$ and $\left(H_5\right)$ hold.

On the other hand,

$$\begin{array}{c}\hfill \underset{\mathrm{x}\to +\infty }{\lim \inf }\underset{\xi \in [{\displaystyle \frac{1}{5}},{\displaystyle \frac{4}{5}}]}{\min }{\displaystyle \frac{\mathcal{F}(\xi ,\mathrm{x})}{\mathrm{x}}}=+\infty \end{array}$$

implies that $\left(H_6\right)$ holds. Thus, all the assumptions of Theorem 3 hold. Then, semipositone BVP (16) has positive solutions provided the positive parameter μ is small enough.

5. Conclusions

Graef et al. [13,14] developed an approach for converting the FDE

$$-D_{0+}^{\gamma }{\rm Y}\left(\xi \right)+\mathrm{a}\left(\xi \right){\rm Y}\left(\xi \right)=\mathrm{y}\left(\xi \right),\xi \in (0,1)$$

(17)

with certain BCs into integral equations. Based on this approach, many results on higher-order FBVPs with a perturbation term have been obtained. To discuss the positivity of the GF of FDE (17) with certain BCs, the GF of the FDE

$$-D_{0+}^{\gamma }{\rm Y}\left(\xi \right)=\mathrm{y}\left(\xi \right),\xi \in (0,1),$$

(18)

with the same BCs plays a key role. For FDE (18), usually, the GF of high-order FDEs has better properties than that of lower-order ones. So, the method of [14,18] cannot be directly used to study FBVPs (1)–(2). Therefore, Graef et al. [13] failed to establish the positivity of the GF of FBVPs (1)–(2). Graef et al. [14] pointed out that the lack of positivity seriously restricts the application of the GF. To overcome this difficulty, we utilize a special transformation and rewrite the expression of GF.

The method of reduction order is an important method to study FBVPs in which there is nonlinearity involving a fractional derivative of unknown function (see [10,11]). In addition, for lower-order FDEs (18) with nonlocal BCs, such as m-point BCs and integral BCs, the GF can be expressed by the GF of FBVPs (18)–(2). So the approach of this article can be used to study some other FBVPs.