Solvability of Singular Fractional-Order Differential Equations with a Perturbation Term

Abstract

In this article, we study singular fractional-order differential equations with a variable coefficient, namely the linear operator of the differential equation containing a linear term with a variable coefficient. The coefficient $a\left(s\right)$ permits singularity at $s=0, 1$, and the nonlinearity $\mathfrak{f}(s,\chi )$ may be singular at $s=0, 1$ and $\chi =0$. By utilizing the fixed-point index theory, the existence of positive solutions are derived under sharp conditions concerning spectral radius.

1. Introduction

Differential equations serve as an excellent tool for modeling some of the anomalous phenomena of nature. For instance, the predator–prey model (i.e., Lotka-Volterra equations) can be used to depict the dynamics of a biological system. As for some recent developments on this issue, we mention [1,2,3]. Fractional differential equations (FDEs) have been a research focus during the last few decades. The main reason is that fractional-order models can depict the dynamic behavior in heredity and the memory of complex systems in nature, which can not be depicted by classical derivative models. FDEs involving various types of fractional derivatives have occurred widely in the field of engineering and natural science, such as anomalous diffusion [4], visco-elasticity [5,6], and control theory [7,8]. For example, FDEs with constant coefficients can be used to model visco-elastic damping [5,6]. Elshehawey et al. [6] studied the Endolymph equation:

$$D^{2}y\left(s\right)+a_{1}Dy\left(s\right)+a_2{D}^{{\displaystyle \frac{1}{2}}}y\left(s\right)+a_{3}y\left(s\right)=-g\left(s\right),$$

which can depict the dynamics of semicircular canals. In recent years, many papers focusing on fractional boundary value problems (FBVPs) have appeared (see [9,10,11,12,13,14,15,16,17,18]). One essential issue of these problems is the existence of (positive) solutions. Considering the practical meaning of a solution, positive solutions attract wider attention. Many researchers have been devoted to considering FBVPs consisting of certain boundary conditions and FDEs, in which the linear operator $\mathcal{L}\chi$ contains only one term: $D^{\alpha }\chi$. The usual approach to solving this problem is to convert the FBVP into a Fredholm operator by using Laplace transformation and to seek its fixed points. To gain the existence of fixed points, various nonlinear analysis methods are utilized, such as the nonlinear alternative of Leray-Schauder [9,10,11] fixed-point index theory [12,13], the Guo-Krasnosel’skii fixed-point theorem [14,15,16,17], and the Avery-Peterson fixed-point theorem [18]. The fixed-point index theory can be applied to estimate the number of solutions; for example, the Leggett-Williams fixed-point theorem considering multiple solutions is proved by calculating the fixed-point index.

Singular boundary value problems come from the field of engineering and natural science, such as coal slurry transport in non-Newtonian fluid theory and boundary layer theory [19]. In [20], the authors studied an FBVP as follows:

$$\left\{\begin{array}{c}-{\mathcal{D}}_{0+}^{\beta }\chi \left(s\right)=g(s,\chi \left(s\right)),0<s<1,n-1<\beta \le n,n>3\hfill \\ \chi \left(0\right)={\chi }^{\prime }\left(0\right)=\cdots ={\chi }^{(n-2)}\left(0\right)=0,{\mathcal{D}}_{0+}^{\alpha }\chi \left(1\right)={\displaystyle \sum _{i=1}^m}{D}_{0+}^{\gamma }{a}_i\chi \left({\eta }_i\right),\hfill \end{array}\right.$$

in which ${\mathcal{D}}_{0+}^{\alpha }$, ${\mathcal{D}}_{0+}^{\beta }$, and ${\mathcal{D}}_{0+}^{\gamma }$ are the Riemann-Liouville fractional derivatives (RLFDs). Based on the Leggett-Williams and Guo-Krasnosel’skii fixed-point theorems, they obtained the existence of multiple positive solutions under singular conditions.

In [21], according to the theory of $u_0$-positive operator, Cui considered a unique solution to FBVPs as follows:

$$\left\{\begin{array}{c}{\mathcal{D}}_{0+}^{\gamma }\chi \left(s\right)+a_1\left(s\right)g(s,\chi \left(s\right))+a_2\left(s\right)=0,0<s<1,\gamma \in (2,3]\hfill \\ \chi \left(0\right)={\chi }^{\prime }\left(0\right)=0,\chi \left(1\right)=0,\hfill \end{array}\right.$$

where $a_i\in L^1[0,1]\cap C(0,1)$, and ${\mathcal{D}}_{0+}^{\gamma }$ denotes RLFD.

He [10] discussed singular FBVPs as follows:

$$\left\{\begin{array}{c}{}^{C}D^{\beta }\chi \left(s\right)+g(s,\chi \left(s\right))=0,0<s<1,3<\beta <4,\hfill \\ {\chi }^{″}\left(0\right)={\chi }^{‴}\left(0\right)=0,\chi \left(1\right)={\chi }^{\prime }\left(0\right)=\xi {\int }_0^1\chi \left(t\right)dt,\hfill \end{array}\right.$$

where ${}^{C}D^{\beta }$ is the Caputo fractional derivative, and $g(s,\chi )$ may be singular at $\chi =0$. According to the nonlinear alternative of the Leray-Schauder and Krasnosel’skii fixed-point theorems, the author obtained results of the existence of positive solutions. To overcome the difficulties in keeping the compactness of the operator caused by singularity, the author made hypotheses as follows:

(H¹)

$g:[0,1]\times (0,+\infty )\to [0,+\infty )$ is continuous. In addition,

$$g(s,\chi )=y_1\left(\chi \right)+y_2\left(\chi \right),\forall (s,\chi )\in [0,1]\times (0,\infty ),$$

with $y_1\left(\chi \right)>0$ decreasing and $y_2\left(\chi \right)/y_1\left(\chi \right)$ increasing in $(0,\infty )$;

(H²)

$\exists M_0>0$, such that $y_1\left({\chi }_1{\chi }_2\right)\le M_0{y}_1\left({\chi }_1\right)y_1\left({\chi }_2\right)$, $\forall {\chi }_i\ge 0$, $i=1,2$;

(H³)

${\int }_0^1{y}_1\left(t\right)dt<\infty$.

Guo et al. [12] utilized the reduced-order method and fixed-point index to study a singular FBVP:

$$\left\{\begin{array}{c}{\mathcal{D}}_{0+}^{\beta }\chi \left(s\right)+a\left(s\right)g(s,\chi \left(s\right),{\mathcal{D}}_{0+}^{{\gamma }_1}\chi \left(s\right),\cdots ,{\mathcal{D}}_{0+}^{{\gamma }_{n-2}}\chi \left(s\right))=0, 0<s<1, n-1<\beta \le n, n\ge 3\hfill \\ \chi \left(0\right)=0, {\mathcal{D}}_{0+}^{{\gamma }_i}\chi \left(0\right)=0, i=1,2,\cdots ,n-2,\hfill \\ {\mathcal{D}}_{0+}^{{\gamma }_{n-2}}\chi \left(1\right)=\eta {\int }_0^{\xi }h\left(t\right){\mathcal{D}}_{0+}^{{\gamma }_{n-2}}\chi \left(t\right)d\mathcal{B}\left(t\right).\hfill \end{array}\right.$$

The authors made the following hypotheses:

(H⁴)

$a:(0,1)\to [0,+\infty )$ is continuous, $a\left(s\right)\not\equiv 0$ and

$$\begin{array}{c}\hfill {\int }_0^1\omega \left(t\right)a\left(t\right)dt<\infty .\end{array}$$

(H⁵)

$g:[0,1]\times (0,\infty )\times (0,+\infty )\to [0,\infty )$ is continuous. For any $0<r_1<r_2<\infty ,$

$$\begin{array}{cc}\hfill \underset{j\to \infty }{lim\; sup}& \{sup{\int }_{e\left(j\right)}\omega \left(t\right)a\left(t\right)g(t,{\chi }_1\left(t\right),...,{\chi }_n\left(t\right))dt:\hfill \\ & {\chi }_i\in P_{r_3}\setminus P_{r_1}(i=1,...,n-1),{\chi }_n\in {\overline{P}}_{r_2}\setminus P_{r_1}\}=0,\hfill \end{array}$$

where $e\left(j\right)=[0,{\displaystyle \frac{1}{j}}]\cup [{\displaystyle \frac{j-1}{j}},1]$, and $r_3={\displaystyle \frac{r_2}{\mathrm{\Gamma }({\gamma }_{n-1}-{\gamma }_{n-2}+1)}}.$

It should be noted that Green’s function plays a crucial role when some nonlinear analysis approaches are used on cones. In general, the cone is closely related to the properties of Green’s function. However, it is very difficult to study the properties of Green’s function for the FDEs containing linear operators with variable coefficients:

$$\mathfrak{L}\chi =-{\mathcal{D}}_{0+}^{\beta }\chi +a\left(t\right)\chi$$

since the Laplace transform cannot be utilized. For some recent developments on this problem, we mention [22,23,24] and the references therein. Graef [22] and Zou [24] studied an FBVP:

$$\left\{\begin{array}{c}-{\mathcal{D}}_{0+}^{\beta }\chi \left(s\right)+a\left(s\right)\chi \left(s\right)=g(s,\chi \left(s\right)),0<s<1,2<\beta <3,\hfill \\ \chi \left(0\right)={\chi }^{\prime }\left(0\right)=0,{\chi }^{\prime }\left(1\right)=0,\hfill \end{array}\right.$$

where $a\in C[0,1]$. In Ref. [22], based on spectral theory, the authors derived Green’s function and obtained some positive properties of it. Then, they obtained the uniqueness of positive solutions by using the fixed-point theorem of a mixed monotone operator. Zou [24] obtained some new upper and lower estimates for the same Green function and obtained results of the existence of positive solutions to the FBVP. In the case that $a\left(s\right)\equiv a$ is a constant, by means of the Laplace transform, the present author [25] derived the associated Green function of an FBVP:

$$\left\{\begin{array}{c}-{\mathcal{D}}_{0+}^{\beta }\chi \left(s\right)+a\chi \left(s\right)=g(s,\chi \left(s\right)),0<s<1,2<\beta <3,\hfill \\ \chi \left(0\right)={\chi }^{\prime }\left(0\right)=0,\chi \left(1\right)=0,\hfill \end{array}\right.$$

as a series of functions concerning the Mittag-Leffler function. Then, we obtained some positive properties of it. However, the method fails to work in the case where $a\left(s\right)$ is not a constant.

Motivated by the aforementioned work, this article considers the existence of positive solutions for the following singular FBVPs:

$$\left\{\begin{array}{c}-{\mathcal{D}}^{\alpha }\chi \left(s\right)+a\left(s\right)\chi \left(s\right)=\mathfrak{f}(s,\chi \left(s\right)),s\in (0,1), n-1<\alpha <n, n\ge 3,\hfill \\ \chi \left(0\right)={\chi }^{\prime }\left(0\right)=\cdots ={\chi }^{(n-2)}\left(0\right)=0,\chi \left(1\right)=0,\hfill \end{array}\right.$$

(1)

where $a\in L^1[0, 1]$, $\mathfrak{f}:(0, 1)\times (0,+\infty )\to {\mathbb{R}}^{+}$ is continuous and may be singular at $s=0, 1$ and $\chi =0$.

Our work presented in this paper has some features. First, the significant difference from the existing results [10,20,21] lies in that the linear operator of the FDE contains a linear term with a variable coefficient; that is, $a\left(s\right)\chi$. Second, compared with [22,24], the coefficient of the linear term and the nonlinearity possess singularity, meaning a permits singularity at $s=0, 1$, and $\mathfrak{f}(s,\chi )$ may be singular at $s=0, 1$ and $\chi =0$. Third, based on spectral theory, this paper utilizes the fixed-point index theorem to study positive solutions of an FBVP (1). To overcome the difficulties in keeping the compactness of the operator caused by singularity, the extension theorem of a completely continuous operator is considered. These are different from [10,22,24]. The paper is organized as follows. In Section 2, we introduce some definitions and lemmas from fractional calculus theory and deduce some of the properties of Green’s function. In Section 3, we establish the results of the existence of positive solutions to the FBVP (1) by using the fixed-point index theory. Finally, we illustrate an example to show the application of our theoretical results.

2. Preliminaries and Lemmas

Definition 1

([26]). The RLFD of order $\alpha >0$ for a function χ is defined by

$${\mathcal{D}}_{0+}^{\alpha }\chi \left(s\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}{\left({\displaystyle \frac{d}{dt}}\right)}^n{\int }_0^s{(s-t)}^{n-\alpha -1}\chi \left(t\right)dt,$$

where $\left[\alpha \right]=n-1$ denotes the integer part of α.

Definition 2

([26]). The Riemann-Liouville integral of order $\alpha >0$ for a function, χ, is defined by

$$I_{0+}^{\alpha }\chi \left(s\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^s{(s-t)}^{\alpha -1}\chi \left(t\right)dt.$$

Lemma 1

([27]). Let $\mathfrak{h}\left(s\right)\in C(0,1)\cap L^1[0,1]$; then, the unique solution of FBVP

$$\left\{\begin{array}{c}-{\mathcal{D}}^{\alpha }\chi \left(s\right)=\mathfrak{h}\left(s\right),0<s<1,\hfill \\ \chi \left(0\right)={\chi }^{\prime }\left(0\right)=\cdots ={\chi }^{(n-2)}\left(0\right)=0,\chi \left(1\right)=0,\hfill \end{array}\right.$$

is

$$\chi \left(s\right)={\int }_0^1{H}_0(s,t)\mathfrak{h}\left(t\right)dt,$$

here,

$$H_0(s,t)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\left\{\begin{array}{cc}{s}^{\alpha -1}{(1-t)}^{\alpha -1}-{(s-t)}^{\alpha -1},\hfill & 0\le t\le s\le 1,\hfill \\ s^{\alpha -1}{(1-t)}^{\alpha -1},\hfill & 0\le s\le t\le 1.\hfill \end{array}\right.$$

(2)

Lemma 2

([27]). $H_0(s,t)$ satisfies the properties as follows:

(1)

$H_0(s,t)=H_0(1-t,1-s)$, $fors,t\in [0, 1]$;

(2)

$H_0(s,t)\le (\alpha -1)\phi (1-s)$, $fors,t\in [0, 1]$;

(3)

$\mathrm{\Gamma }\left(\alpha \right)\phi (1-s)\phi \left(t\right)\le H_0(s,t)\le (\alpha -1)\phi \left(t\right)$, $fors,t\in [0, 1]$.

where

$$\phi \left(s\right)={\displaystyle \frac{{(1-s)}^{\alpha -1}s}{\mathrm{\Gamma }\left(\alpha \right)}}.$$

(3)

Denote $\mathcal{E}=C[0,1]$ utilizing the norm $\parallel x\parallel ={max}_{t\in [0,1]}\left|x\left(t\right)\right|$. We also denote

$$B_r=\{\chi \in \mathcal{E}: \parallel \chi \parallel <r\}.$$

Then, $\mathcal{E}$ is a Banach space. Define the function $H:[0, 1]\times [0, 1]\to \mathbb{R}$ as follows:

$$\begin{array}{c}\hfill H\left(s,t\right)=\sum _{n=0}^{\infty }{(-1)}^n{H}_n\left(s,t\right),\end{array}$$

here, $H_0\left(s,t\right)$ is defined by (2), and

$$H_n\left(s,t\right)={\int }_0^{1}a\left(\tau \right)H_0(s,\tau )H_{n-1}(\tau ,t)d\tau ,n\ge 1.$$

(4)

Lemma 3

([28]). Assume that E is a Banach space, $\mathcal{L}:E\to E$ is a linear operator, and $r\left(\mathcal{L}\right)$ is the spectral radius of $\mathcal{L}$; then, the following holds:

(i)

$r\left(\mathcal{L}\right)\le \parallel \mathcal{L}\parallel$;

(ii)

If $r\left(\mathcal{L}\right)<1$; then, $(\mathcal{I}-\mathcal{L})$ is invertable and

$$\begin{array}{c}\hfill {(\mathcal{I}-\mathcal{L})}^{-1}=\sum _{n=0}^{\infty }{\mathcal{L}}^n,\end{array}$$

in which $\mathcal{I}$ denotes the identity operator.

Lemma 4.

Assume that $(\alpha -1){\int }_0^1\left|a\left(\tau \right)\right|\phi (1-\tau )d\tau <1$ and $y\in L^1[0, 1]\cap C(0, 1)$. Then, the solution of the FBVP

$$\left\{\begin{array}{cc}\hfill & -{\mathcal{D}}^{\alpha }\chi \left(s\right)+a\left(s\right)\chi \left(t\right)=y\left(s\right), s\in (0,1),\hfill \\ \hfill & \chi \left(0\right)={\chi }^{\prime }\left(0\right)=\cdots ={\chi }^{(n-2)}\left(0\right)=0, \chi \left(1\right)=0.\hfill \end{array}\right.$$

(5)

is

$$\chi \left(t\right)={\int }_0^{1}H(s,t)y\left(t\right)dt.$$

Proof. 

First, we will prove that the series of functions defined by (4) is uniformly convergent on $[0, 1]\times [0, 1]$. According to Lemma 2, we have

$$\begin{array}{cc}\hfill \mid H_1\left(s,t\right)|& =|{\int }_0^{1}a\left(\tau \right)H_0(s,\tau )H_0(\tau ,t)d\tau |\le {\int }_0^1\left|a\left(\tau \right)\right|{(\alpha -1)}^2\phi (1-s)\phi \left(t\right)d\tau \hfill \\ \hfill & ={(\alpha -1)}^2\left({\int }_0^1\left|a\left(\tau \right)\right|d\tau \right)\phi (1-s)\phi \left(t\right).\hfill \end{array}$$

(6)

In view of the notation of the function $H_2$ and (6), we have

$$\begin{array}{cc}\hfill |H_2\left(s,t\right)|& \le {\int }_0^1\left|a\left(\tau \right)\right|H_0(s,\tau )H_1(\tau ,t)d\tau \hfill \\ & \le {\int }_0^1{(\alpha -1)}^3\left|a\left(\tau \right)\right|\phi (1-s)\left({\int }_0^{1}a\left(\tau \right)d\tau \right)\phi (1-\tau )\phi \left(t\right)d\tau \hfill \\ \hfill & ={(\alpha -1)}^3\left({\int }_0^1\left|a\left(\tau \right)\right|d\tau \right)\left({\int }_0^1\left|a\left(\tau \right)\right|\phi (1-\tau )d\tau \right)\phi (1-s)\phi \left(t\right).\hfill \end{array}$$

Then, by induction, we obtain

$$|H_n{\left(s,t\right)|\le (\alpha -1)}^{n+1}\left({\int }_0^1\left|a\left(\tau \right)\right|d\tau \right){\left({\int }_0^1\left|a\left(\tau \right)\right|\phi (1-\tau )d\tau \right)}^{n-1}\phi (1-s)\phi \left(t\right),n\ge 1.$$

(7)

Note that $(\alpha -1){\int }_0^1\left|a\left(\tau \right)\right|\phi (1-\tau )d\tau <1$. Hence, the series of functions

$$\sum _{n=0}^{+\infty }{(-1)}^n{H}_n\left(s,t\right)$$

is uniformly convergent on $[0, 1]\times [0, 1]$. It is clear that $H\left(s,t\right)$ is continuous on $[0, 1]\times [0, 1]$ since $H_n(s,t)$$(n=0,1,2,\cdots )$ is continuous.

Now, let $\chi \in \mathcal{E}$ be a solution of the FBVP (5). According to Lemma 1,

$$\chi \left(s\right)={\int }_0^1{H}_0(s,t)\left(y\right(t)-a(t\left)\chi \right(t\left)\right)dt,$$

i.e.,

$$\chi \left(s\right)+{\int }_0^1{H}_0(s,t)a\left(t\right)\chi \left(t\right)dt={\int }_0^1{H}_0(s,t)y\left(t\right)dt.$$

(8)

Denote an operator $\mathcal{L}:\mathcal{E}\to \mathcal{E}$ by using

$$\mathcal{L}\chi \left(s\right)={\int }_0^{1}a\left(t\right)H_0(s,t)\chi \left(t\right)dt.$$

Set

$$\left(Ty\right)\left(s\right)={\int }_0^1{H}_0(s,t)y\left(t\right)dt.$$

Notice that

$$\parallel \mathcal{L}\parallel =\underset{\parallel \chi \parallel =1}{max}\parallel \mathcal{L}\chi \parallel \le (\alpha -1){\int }_0^1\left|a\left(t\right)\right|\phi (1-t)dt<1;$$

it follows from Lemma 3 that

$$({{\mathcal{I}+\mathcal{L})}^{-1}=\mathcal{I}-\mathcal{L}+\mathcal{L}}^2+···+{(-1)}^n{\mathcal{L}}^n+···,$$

in which $\mathcal{I}$ denotes the identity operator. Therefore, (8) can be expressed as

$$\chi =\sum _{n=0}^{+\infty }{(-\mathcal{L})}^{n}Ty.$$

By induction, we can verify that

$$\left({(-\mathcal{L})}^{n}Ty\right)\left(s\right)={\int }_0^1{(-1)}^n{H}_n(s,t)y\left(t\right)dt,n=0,1,2,\cdots .$$

According to the above discussion, we obtain

$$\chi \left(s\right)={\int }_0^{1}H(s,t)y\left(t\right)dt.$$

□

For convenience, we give the notations as follows:

$$\begin{array}{cc}\hfill & {\mathrm{\Delta }}_1={\int }_0^1\left|a\left(\tau \right)\right|d\tau ,{\mathrm{\Delta }}_2=(\alpha -1){\int }_0^1\left|a\left(\tau \right)\right|\phi (1-\tau )d\tau ,\hfill \\ \hfill & \mathrm{\Delta }={\displaystyle \frac{{(\alpha -1)}^2{\mathrm{\Delta }}_1}{\mathrm{\Gamma }\left(\alpha \right)(1-{\mathrm{\Delta }}_2)}},\hfill \\ \hfill & {\mathrm{\Lambda }}_1=1+\mathrm{\Delta },{\mathrm{\Lambda }}_2=1-\mathrm{\Delta },\hfill \\ \hfill & \mathrm{\Lambda }={\displaystyle \frac{\mathrm{\Gamma }(\alpha -1){\mathrm{\Lambda }}_2}{{\mathrm{\Lambda }}_1}}.\hfill \end{array}$$

Lemma 5.

Assume that ${\mathrm{\Delta }}_2<1$ and ${\mathrm{\Lambda }}_2>0$ hold; then, the function $H(t,s)$ satisfies the following properties:

(P1)

$H\left(s,t\right)\ge \mathrm{\Gamma }\left(\alpha \right){\mathrm{\Lambda }}_2\phi (1-s)\phi \left(t\right),fors,t\in [0,1]$;

(P2)

$H\left(s,t\right)\le (\alpha -1){\mathrm{\Lambda }}_1\phi \left(t\right),fors,t\in [0,1]$;

(P3)

$H\left(s,t\right)\le (\alpha -1){\mathrm{\Lambda }}_1\phi (1-s),fors,t\in [0,1]$,

in which φ is defined by (3).

Proof. 

From (7), it follows that

$$|H_n{(s,t)|\le (\alpha -1)}^2{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2^{n-1}\phi (1-s)\phi \left(t\right),n\ge 1.$$

Note that ${\mathrm{\Delta }}_2<1$; then, we obtain

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }\left|H_n\left(s,t\right)\right|& \le \sum _{n=1}^{\infty }{(\alpha -1)}^2{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2^{n-1}\phi (1-s)\phi \left(t\right)\hfill \\ & ={\displaystyle \frac{{(\alpha -1)}^2{\mathrm{\Delta }}_1}{1-{\mathrm{\Delta }}_2}}·\phi (1-s)\phi \left(t\right).\hfill \end{array}$$

(9)

From Lemma 2, we obtain

$$\phi (1-s)\phi \left(t\right)\le {\displaystyle \frac{H_0(s,t)}{\mathrm{\Gamma }\left(\alpha \right)}}.$$

By combining this inequality with (9), we have

$$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }\left|H_n\left(s,t\right)\right|& \le {\displaystyle \frac{{(\alpha -1)}^2{\mathrm{\Delta }}_1}{1-{\mathrm{\Delta }}_2}}·\phi (1-s)\phi \left(t\right)\hfill \\ & \le {\displaystyle \frac{{(\alpha -1)}^2{\mathrm{\Delta }}_1}{1-{\mathrm{\Delta }}_2}}·{\displaystyle \frac{H_0\left(s,t\right)}{\mathrm{\Gamma }\left(\alpha \right)}}\hfill \\ & =\mathrm{\Delta }{H}_0\left(s,t\right).\hfill \end{array}$$

Then, we obtain

$$\begin{array}{cc}\hfill H\left(s,t\right)& =H_0\left(s,t\right)+\sum _{n=1}^{\infty }{(-1)}^n{H}_n\left(s,t\right)\hfill \\ & \le (1+\mathrm{\Delta })H_0\left(s,t\right)={\mathrm{\Lambda }}_1{H}_0\left(s,t\right),\hfill \end{array}$$

(10)

and

$$\begin{array}{cc}\hfill H\left(s,t\right)& \ge (1-\mathrm{\Delta })H_0\left(s,t\right)={\mathrm{\Lambda }}_2{H}_0\left(s,t\right).\hfill \end{array}$$

(11)

Consequently, from the inequalities (10), (11), and Lemma 2, we see that $H(s,t)$ satisfies (P1), (P2), and (P3). □

In this article, we suppose that the hypothesis listed below always holds:

(H1)

$a\in L^1[0,1]$ with ${\mathrm{\Delta }}_2<1$ and ${\mathrm{\Lambda }}_2>0$.

Here, we also list some hypotheses that will be used in the remainder of this article.

(H2)

$\mathfrak{f}\in C\left(\right(0, 1)\times (0,+\infty \left)\right)$ is non-negative. In addition, for any $r_0>r>0$, $\mathrm{\exists } {\mathrm{\Psi }}_{r,r_0}\in L^1[0, 1]\bigcap C(0, 1)$ such that

$$\begin{array}{c}\hfill \mathfrak{f}(s,\chi )\le {\mathrm{\Psi }}_{r,r_0}\left(s\right),\forall s\in (0,1),\chi \in \left[r\mathrm{\Lambda }\phi (1-t),r_0\right].\end{array}$$

(H3)

∃ $r_1>0$ and a non-negative function $b_1\in L^1[0, 1]$ with ${\int }_0^1{b}_1\left(t\right)dt>0$ such that

$$\begin{array}{c}\hfill \mathfrak{f}(s,\chi )\ge b_1\left(s\right)\chi ,\forall (s,\chi )\in (0, 1)\times (0, r_1];\end{array}$$

(H4)

∃ $r_2>0$ and a non-negative function $b_2\in L^1[0, 1]$ with ${\int }_0^1{b}_2\left(t\right)dt>0$ such that

$$\begin{array}{c}\hfill \mathfrak{f}(s,\chi )\le b_2\left(s\right)\chi ,\forall (s,\chi )\in (0, 1)\times [r_2,+\infty );\end{array}$$

(H5)

There exist $0<\eta <\zeta <1$ such that

$$\begin{array}{c}\hfill \underset{\chi \to 0+}{lim\; inf}\underset{s\in [\eta ,\zeta ]}{min}\mathfrak{f}(s,\chi )=+\infty .\end{array}$$

Define the operators A and $L_i(i=1,2)$ by using

$$\begin{array}{cc}\hfill & \left(A\chi \left)\right(s\right)={\int }_0^{1}H(s,t)\mathfrak{f}(t,\chi (t\left)\right)dt,\hfill \\ \hfill & \left(L_i\chi \left)\right(s\right)={\int }_0^{1}H(s,t)b_i\left(t\right)\chi \left(t\right)dt, i=1,2.\hfill \end{array}$$

(12)

Lemma 6

([29]). If $\mathcal{E}$ is a Banach space, $\mathcal{P}\subset \mathcal{E}$ is a total cone, and the linear operators $L:\mathcal{E}\to \mathcal{E}$ are continuous, and $L\left(\mathcal{P}\right)\subset \mathcal{P}$; in addition, $\mathrm{\exists } \varrho \notin (-\mathcal{P})$ and $\kappa >0$ such that $\varrho \le \kappa L\left(\varrho \right)$; then, the spectral radius $r\left(L\right)>0$, and $\mathrm{\exists } \varphi \in \mathcal{P}$ such that $L\varphi =r\left(L\right)\varphi$.

Denote a cone $\mathcal{P}$ by

$$\mathcal{P}=\left\{\chi \in \mathcal{E}:\chi \left(s\right)\ge \mathrm{\Lambda }\parallel \chi \parallel \phi (1-s),s\in [0,1]\right\}.$$

From Lemmas 5 and 6, we have the lemma as follows:

Lemma 7.

The linear operators $L_i:\mathcal{P}\to \mathcal{P}$$(i=0,1)$ defined in (12) are completely continuous. In addition, $r\left(L_i\right)>0$, and $\mathrm{\exists } {\phi }_i\in \mathcal{P}$$(i=1,2)$ such that $L_i{\phi }_i=r\left(L_i\right){\phi }_i$$(i=1,2)$.

Lemma 8.

Assume that the hypothesis (H2)t holds; then, $A:\mathcal{P}\setminus B_r\to \mathcal{P}$ is completely continuous for any $r>0$.

Proof. 

First, for any $\chi \in \mathcal{P}\setminus B_r$, the denotation of $\mathcal{P}$ yields

$$\begin{array}{c}\hfill \chi \left(s\right)\ge \mathrm{\Lambda }\parallel \chi \parallel \phi (1-s)\ge \mathrm{\Lambda }r\phi (1-s).\end{array}$$

According to the hypothesis (H2), there exists ${\mathrm{\Psi }}_{r,\parallel \chi \parallel }\in L^1[0, 1]\bigcap C(0, 1)$ such that

$$\begin{array}{c}\hfill \mathfrak{f}(s,\chi \left(s\right))\le {\mathrm{\Psi }}_{r,\parallel \chi \parallel }\left(s\right),\forall s\in (0,1).\end{array}$$

According to Lemma 2, we have

$$\begin{array}{c}\hfill \left(A\chi \left)\right(s\right)={\int }_0^{1}H(s,t)\mathfrak{f}(t,\chi (t\left)\right)dt\le (\alpha -1){\mathrm{\Lambda }}_1\phi (1-s){\int }_0^1{\mathrm{\Psi }}_{r,\parallel \chi \parallel }\left(t\right)dt,\end{array}$$

$$\begin{array}{c}\hfill \left(A\chi \left)\right(s\right)={\int }_0^{1}H(s,t)\mathfrak{f}(t,\chi (t\left)\right)dt\le (\alpha -1){\mathrm{\Lambda }}_1{\int }_0^1\phi \left(t\right){\mathrm{\Psi }}_{r,\parallel \chi \parallel }\left(t\right)dt,\end{array}$$

(13)

and

$$\begin{array}{c}\hfill A\chi \left(s\right)\ge \mathrm{\Gamma }\left(\alpha \right){\mathrm{\Lambda }}_2\phi (1-s){\int }_0^1\phi \left(t\right)\mathfrak{f}(t,\chi (t\left)\right)dt\ge \mathrm{\Lambda }\phi (1-s)∥A\chi ∥.\end{array}$$

The above inequalities imply that $A:\mathcal{P}\setminus B_r\to \mathcal{P}$ is well-defined.

Now, we prove that $A:\mathcal{P}\setminus B_r\to \mathcal{P}$ is compact.

Let $S\subset \mathcal{P}\setminus B_r$ be a bounded set; then, there is a $r_0>r$ such that $r\le \parallel \chi \parallel \le r_0,\forall \chi \in S$. Inequality (13) implies

$$\begin{array}{c}\hfill A\chi \left(s\right)\le (\alpha -1){\mathrm{\Lambda }}_1{\int }_0^1\phi \left(t\right){\mathrm{\Psi }}_{r,r_0}\left(t\right)dt<+\infty .\end{array}$$

Thus, $A\left(S\right)$ is uniformly bounded.

On the other hand, $H(s,t)$ is continuous on $[0, 1]\times [0, 1]$. For ∀$\epsilon >0$, ∃$\sigma \in (0, 1)$, when $s_1,s_2\in [0, 1]$ and $|s_2-s_1|<\sigma$, one has

$$\begin{array}{c}\hfill \left|H(s_2,t)-H(s_1,t)\right|<{\displaystyle \frac{\epsilon }{{\int }_0^1{\mathrm{\Psi }}_{r,r_0}\left(t\right)dt+1}}.\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill \left|A\chi \left(s_2\right)-A\chi \left(s_1\right)\right|& \le {\int }_0^1\left|H(s_2,t)-H(s_1,t)\right|\mathfrak{f}(t,\chi (t\left)\right)dt\hfill \\ & \le {\int }_0^1\left|H(s_2,t)-H(s_1,t)\right|{\mathrm{\Psi }}_{r,r_0}\left(t\right)dt\hfill \\ & \le {\int }_0^1{\displaystyle \frac{\epsilon }{{\int }_0^1{\mathrm{\Psi }}_{r,r_0}\left(t\right)dt+1}}{\mathrm{\Psi }}_{r,r_0}\left(t\right)dt<\epsilon ,\hfill \end{array}$$

which implies $A\left(S\right)$ is equicontinuous. According to the Arzela-Ascoli theorem, we see that $A:P\setminus B_r\to P$ is compact.

Finally, we shall show the continuity of A.

Let $\left\{{\chi }_n\right\}\subset \mathcal{P}\setminus B_r$ and $\parallel {\chi }_n-{\chi }_0\parallel \to 0$$(n\to +\infty )$; then, there is a number, $R>r$, such that $r\le ∥{\chi }_n∥\le R, n=0,1,2,\cdots$. Considering the absolute continuity of the integral, for ∀$ϵ>0$, there is a $\delta \in (0,{\displaystyle \frac{1}{2}})$ such that

$$\begin{array}{c}\hfill {\int }_0^{\delta }\phi \left(t\right){\mathrm{\Psi }}_{r,R}\left(t\right)dt\le {\displaystyle \frac{\epsilon }{6(\alpha -1){\mathrm{\Lambda }}_1}},{\int }_{1-\delta }^1\phi \left(t\right){\mathrm{\Psi }}_{r,R}\left(t\right)dt\le {\displaystyle \frac{\epsilon }{6(\alpha -1){\mathrm{\Lambda }}_1}}.\end{array}$$

Note that $\mathfrak{f}(t,x)$ is continuous on $[\delta , 1-\delta ]\times [r\mathrm{\Lambda }{m}_{\delta }, R]$, where

$$\begin{array}{c}\hfill m_{\delta }=min\left\{\phi \right(s\left)\right|s\in [\delta , 1-\delta ]\},\end{array}$$

there is a positive integer N; when $n>N$, we obtain

$$\begin{array}{c}\hfill |\mathfrak{f}(s,{\chi }_n\left(s\right))-\mathfrak{f}(s,{\chi }_0\left(s\right))| <{\displaystyle \frac{\epsilon }{3(\alpha -1){\mathrm{\Lambda }}_1{\int }_0^1\phi \left(t\right)dt}},\forall s\in [\delta ,1-\delta ].\end{array}$$

Then,

$$\begin{array}{cc}\hfill \parallel A{\chi }_n-A{\chi }_0\parallel \le & \underset{s\in [0,1]}{max}{\int }_0^{1}H(s,t)|\mathfrak{f}(t,{\chi }_n\left(t\right))-\mathfrak{f}(t,{\chi }_0\left(t\right))|dt\hfill \\ \hfill \le & (\alpha -1){\mathrm{\Lambda }}_1{\int }_0^1\phi \left(t\right)|\mathfrak{f}(t,{\chi }_n\left(t\right))-\mathfrak{f}(t,{\chi }_0\left(t\right))t|dt\hfill \\ \hfill =& (\alpha -1){\mathrm{\Lambda }}_1{\int }_0^{\delta }\phi \left(t\right)|f(t,{\chi }_n\left(t\right))-f(t,{\chi }_0\left(t\right))|dt\hfill \\ & +(\alpha -1){\mathrm{\Lambda }}_1{\int }_{\delta }^{1-\delta }\phi \left(t\right)|\mathfrak{f}(t,x_n\left(t\right))-\mathfrak{f}(t,x_0\left(t\right))|dt\hfill \\ \hfill & +(\alpha -1){\mathrm{\Lambda }}_1{\int }_{1-\delta }^1\phi \left(t\right)|\mathfrak{f}(t,{\chi }_n\left(t\right))-\mathfrak{f}(t,{\chi }_0\left(t\right))|dt\hfill \\ \hfill \le & 2(\alpha -1){\mathrm{\Lambda }}_1\left({\int }_0^{\delta }\phi \left(t\right){\mathrm{\Psi }}_{r,R}\left(t\right)dt+{\int }_{1-\delta }^1\phi \left(t\right){\mathrm{\Psi }}_{r,R}\left(t\right)dt\right)\hfill \\ \hfill & +(\alpha -1){\mathrm{\Lambda }}_1{\int }_{\delta }^{1-\delta }\phi \left(t\right)|\mathfrak{f}(t,{\chi }_n\left(t\right))-\mathfrak{f}(t,{\chi }_0\left(t\right))|dt\hfill \\ \hfill <& {\displaystyle \frac{\epsilon }{3}}+{\displaystyle \frac{\epsilon }{3}}+{\displaystyle \frac{\epsilon }{3}}=\epsilon .\hfill \end{array}$$

Therefore, A is continuous. This completes the proof. □

Remark 1.

For any $r>0$, the extension theorem of a completely continuous operator [30] yields that $A:\mathcal{P}\setminus B_r\to \mathcal{P}$ can be extended to a completely continuous operator $\overline{A}:\mathcal{P}\to \mathcal{P}$. Clearly, if $\mathcal{V}\in \mathcal{P}$ with $\parallel \mathcal{V}\parallel >r$ is a fixed point of $\overline{A}$, then $\mathcal{V}$ is also a fixed point of A. For convenience, we still write the extension operator as A.

Lemma 9

([30]). Suppose that $\mathcal{P}$ is a cone in the Banach space $\mathcal{E}$, $\mathrm{\Omega }\subset \mathcal{E}$ is a bounded open set, and $A:\overline{\mathrm{\Omega }}\cap \mathcal{P}\to \mathcal{P}$ is completely continuous. If there is a ${\mathcal{V}}_0\in \mathcal{P}$ with ${\mathcal{V}}_0\ne \theta$ such that

$$\mathcal{V}-A\mathcal{V}\ne \lambda {\mathcal{V}}_0,\forall \lambda \ge 0, \mathcal{V}\in \partial \mathrm{\Omega }\cap \mathcal{P},$$

then, $i(A,\mathrm{\Omega }\cap \mathcal{P},\mathcal{P})=0$.

Lemma 10

([30]). Suppose that $\mathcal{P}$ is a cone in the Banach space $\mathcal{E}$; the zero element $\theta \in \Omega$ and $\Omega \subset \mathcal{E}$ represent a bounded openset, and $A:\overline{\Omega }\cap \mathcal{P}\to \mathcal{P}$ is completely continuous. If

$$A\mathcal{V}\ne \lambda \mathcal{V},\forall \lambda \ge 1,\mathcal{V}\in \partial \Omega \cap \mathcal{P},$$

then $i(A,\Omega \cap \mathcal{P},\mathcal{P})=1$.

3. Main Results

Theorem 1.

Suppose that the hypotheses (H2), (H3), and (H4) hold. Moreover, $0<r\left(L_2\right)<1\le r\left(L_1\right)$. Then, there is at least one positive solution to the FBVP (1).

Proof. 

For ∀$\chi \in \partial B_{r_1}\cap \mathcal{P}$, in view of (H3), we have

$$\begin{array}{c}\hfill A\chi \left(s\right)={\int }_0^{1}H(s,t)\mathfrak{f}(t,\chi (t\left)\right)dt\ge {\int }_0^{1}H(s,t)b_1\left(t\right)\chi \left(t\right)dt=L_1\chi \left(s\right).\end{array}$$

(14)

If A has fixed points on $\partial B_{r_1}\cap \mathcal{P}$, the proof is completed. Therefore, we suppose that there are no fixed points of A on $\partial B_{r_1}\cap P$. According to Lemma 7, there exists ${\phi }_1\in \mathcal{P}$ such that $L_1{\phi }_1=r\left(L_1\right){\phi }_1$. Next, we will prove that

$$\begin{array}{c}\hfill \chi -A\chi \ne \lambda {\phi }_1,\forall \chi \in \partial P\cap B_{r_1}, \lambda \ge 0.\end{array}$$

(15)

On the contrary, assue that (15) does not hold. Then, there are ${\chi }_0\in \partial B_{r_1}\cap \mathcal{P}$ and ${\lambda }_0>0$ such that ${\chi }_0-A{\chi }_0={\lambda }_0{\phi }_1$. Thus,

$${\chi }_0=A{\chi }_0+{\lambda }_0{\phi }_1\ge {\lambda }_0{\phi }_1.$$

Set

$${\lambda }^{*}=sup\{\lambda :{\chi }_0\ge \lambda {\phi }_1\}.$$

Then, we can obtain ${\lambda }^{*}\ge {\lambda }_0$ and $x_0\ge {\lambda }^{*}{\phi }_1$. Note that $L_1$ is a positive linear operator. Consequently,

$$L_1{\chi }_0\ge {\lambda }^{*}{L}_1{\phi }_1={\lambda }^{*}r\left(L_1\right){\phi }_1\ge {\lambda }^{*}{\phi }_1.$$

This and (14) yield the following:

$${\chi }_0=A{\chi }_0+{\lambda }_0{\phi }_1\ge L_1{\chi }_0+{\lambda }_0{\phi }_1\ge ({\lambda }^{*}+{\lambda }_0){\phi }_1.$$

This contradicts the definition of ${\lambda }^{*}$. According to Lemma 9, we obtain

$$\begin{array}{c}\hfill i(A,B_{r_1}\cap \mathcal{P},\mathcal{P})=0.\end{array}$$

(16)

Now, we set

$$\mathfrak{D}=\{\chi \in \mathcal{P}\setminus B_{r_1}:\chi =\lambda A\chi ,\lambda \in [0,1]\}.$$

In the sequel, we shall show that $\mathfrak{D}$ is bounded.

For $\chi \in \mathfrak{D}$, according to (H4), we have $\parallel \chi \parallel \ge r_1$ and

$$\mathfrak{f}(s,\chi \left(s\right))\le b_2\left(s\right)\chi \left(s\right)+\mathfrak{f}(s,\tilde{\chi }\left(s\right)),$$

in which $\tilde{\chi }\left(s\right)=min\{\chi \left(s\right),r_2\}$. Clearly, $\chi \left(s\right)\ge \mathrm{\Lambda }{r}_1\phi (1-s)$. Note that

$$\begin{array}{c}\hfill \mathrm{\Lambda }{r}_1\phi (1-s)={\displaystyle \frac{(1-\mathrm{\Delta })(1-s)s^{\alpha -1}{r}_1}{(\alpha -1)(1+\mathrm{\Delta })}}<r_1.\end{array}$$

Thus,

$$\begin{array}{c}\hfill r_2\ge \tilde{\chi }\left(s\right)\ge \mathrm{\Lambda }{r}_1\phi (1-s).\end{array}$$

Denote $K=(\alpha -1){\mathrm{\Lambda }}_1{\int }_0^1\phi \left(s\right){\mathrm{\Psi }}_{r_1,r_2}\left(s\right)ds$. Then,

$$\chi \left(s\right)=\lambda A\chi \left(s\right)\le A\chi \left(s\right)\le L_2\chi \left(s\right)+A\tilde{\chi }\left(s\right)\le L_2\chi \left(s\right)+K,$$

which implies $\left((\mathcal{I}-L_2)\chi \right)\left(s\right)\le K, s\in [0, 1]$. Since $r\left(L_2\right)<1$, it follows from Lemma 3 that $(\mathcal{I}-L_2)$ is invertible and

$${(\mathcal{I}-L_2)}^{-1}=\mathcal{I}+L_2+L_2^2+\cdots +L_2^n+\cdots .$$

Hence, $\parallel \chi \parallel \le K∥{(\mathcal{I}-L_2)}^{-1}∥$. So, $\mathfrak{D}$ is bounded.

Let $R=r_2+K\parallel {(\mathcal{I}-L_2)}^{-1}\parallel$. Then,

$$A\chi \ne \lambda \chi ,\forall \lambda \ge 1,\chi \in \partial B_R\cap \mathcal{P}.$$

According to Lemma 10, we have

$$\begin{array}{c}\hfill i(A,B_R\cap \mathcal{P},\mathcal{P})=1.\end{array}$$

(17)

From (16) and (17), it follows that

$$\begin{array}{c}\hfill i(A,(B_R\setminus \overline{B_{r_1}})\cap \mathcal{P},\mathcal{P})=1.\end{array}$$

This means that there exists $\mathcal{V}\in (B_R\setminus \overline{B_{r_1}})\cap P$, which is a fixed point of A. Clearly, $\mathcal{V}$ is also a positive solution of the FBVP (1). □

Corollary 1.

Theorem 1 still holds if $0<r\left(L_2\right)<1\le r\left(L_1\right)$ is replaced by the hypothesis as follows:

$$\begin{array}{c}\hfill \underset{t\in [\varsigma ,1-\varsigma ]}{inf}{\int }_{\varsigma }^{1-\varsigma }H(s,t)b_1\left(t\right)dt\ge 1>\underset{t\in [0,1]}{sup}{\int }_0^{1}H(s,t)b_2\left(t\right)dt,forsome\varsigma \in (0,{\displaystyle \frac{1}{2}}).\end{array}$$

Theorem 2.

Assume that there exists $r_2>0$ such that the hypotheses (H2), (H4), and (H5) hold. Moreover, $0<r\left(L_2\right)<1$. Then, there is at least one positive solution to the FBVP (1).

Proof. 

From the hypothesis (H5), for

$$\begin{array}{c}\hfill M={\displaystyle \frac{r_2}{\rho \mathrm{\Gamma }\left(\alpha \right){\mathrm{\Lambda }}_2{\int }_{\eta }^{\zeta }\phi \left(s\right)ds}},\end{array}$$

in which

$$\begin{array}{c}\hfill \rho =\underset{t\in [\eta ,\zeta ]}{min}\phi (1-t),\end{array}$$

there is $r_3\in (0, r_2)$ such that

$$\begin{array}{c}\hfill \mathfrak{f}(s,\chi )\ge M,for(s,\chi )\in [\eta , \zeta ]\times (0, r_3].\end{array}$$

For any $\chi \in B_{r_3}\cap \mathcal{P}$,

$$\begin{array}{c}\hfill A\chi \left(s\right)={\int }_0^{1}H(s,t)\mathfrak{f}(t,\chi \left(t\right))dt\ge \mathrm{\Gamma }\left(\alpha \right){\mathrm{\Lambda }}_{2}M\phi (1-s){\int }_{\eta }^{\zeta }\phi \left(t\right)dt={\displaystyle \frac{r_2\phi (1-s)}{\rho }},\end{array}$$

which implies that $\parallel A\chi \parallel \ge r_2>r_3\ge \parallel \chi \parallel$. Then, we see that any element of $B_{r_3}\cap \mathcal{P}$ is not a fixed point of A. We may suppose that there is no fixed point of A on $\overline{B_{r_3}}\cap \mathcal{P}$. According to the Kronecker existence theorem, we have

$$\begin{array}{c}\hfill i(A,B_{r_3}\cap \mathcal{P},\mathcal{P})=0.\end{array}$$

(18)

According to the proof of Theorem 1, it follows from (H4) that there is $R>r_2$ such that

$$\begin{array}{c}\hfill i(A,B_R\cap \mathcal{P},\mathcal{P})=1.\end{array}$$

(19)

Then, in view of (18) and (19), we see that

$$\begin{array}{c}\hfill i(A,(B_R\setminus \overline{B_{r_3}})\cap \mathcal{P},\mathcal{P})=1.\end{array}$$

This yields the fact that the FBVP (1) has at least one positive solution. □

Corollary 2.

Theorem 2 still holds if $0<r\left(L_2\right)<1)$ is replaced by the hypothesis as follows:

$$\begin{array}{c}\hfill \underset{t\in [0,1]}{sup}{\int }_0^{1}H(s,t)b_2\left(t\right)dt<1.\end{array}$$

Theorem 3.

Assume that the hypotheses listed below hold:

(H6)

There are continuous functions $g,h:(0,+\infty )\to [0,+\infty )$ and non-negative functions $l\in L[0, 1]\cap C(0, 1)$ such that

$$\begin{array}{c}\hfill \mathfrak{f}(s,\chi )=l\left(s\right)\left[g\right(\chi )+h(\chi \left)\right],(s,\chi )\in (0, 1)\times (0,\infty ),\end{array}$$

in which $g\left(\chi \right)$ is non-increasing and ${\displaystyle \frac{h\left(\chi \right)}{g\left(\chi \right)}}$ is non-decreasing in $\chi \in (0,\infty )$. Moreover,

$$\begin{array}{c}\hfill \underset{\chi \to 0^{+}}{lim}g\left(\chi \right)=+\infty ;0<{\int }_0^{1}l\left(s\right)g\left(r\mathrm{\Lambda }\phi \right(1-s\left)\right)ds<+\infty ,\forall r>0.\end{array}$$

(H7)

There exists $r_4>0$ such that

$$\begin{array}{c}\hfill (\alpha -1){\mathrm{\Lambda }}_1\left(1+{\displaystyle \frac{h\left(r_4\right)}{g\left(r_4\right)}}\right){\int }_0^1\phi \left(s\right)l\left(s\right)g\left(r_4\mathrm{\Lambda }\phi (1-s)\right)ds<r_4.\end{array}$$

Then, BVP (1) has at least one positive solution.

Proof. 

It is easy to verify that hypothesis (H6) implies that (H2) and (H5) hold. According to the proof of Theorem 2, there exists $0<r_3<r_4$ such that

$$\begin{array}{c}\hfill i(A,B_{r_3}\cap \mathcal{P},\mathcal{P})=0.\end{array}$$

(20)

Otherwise, A has at least one positive fixed point on $\overline{B_{r_3}}\cap P$, which completes the proof.

If A has no fixed points on $\partial B_{r_4}\cap \mathcal{P}$, we shall prove that

$$\begin{array}{c}\hfill A\chi \ne \lambda \chi ,\forall \chi \in \partial B_{r_4}\cap \mathcal{P},\lambda \ge 1.\end{array}$$

Otherwise, there exist ${\chi }_1\in \partial B_{r_4}\cap \mathcal{P}$ and ${\lambda }_1>1$ such that $A{\chi }_1={\lambda }_1{\chi }_1$. It is clear that

$$\begin{array}{c}\hfill {\chi }_1\left(s\right)\ge \mathrm{\Lambda }\parallel {\chi }_1\parallel \phi (1-s)=\mathrm{\Lambda }{r}_4\phi (1-s).\end{array}$$

From (H6) and (H7), we have

$$\begin{array}{cc}\hfill {\lambda }_1{\chi }_1\left(s\right)& =A{\chi }_1\left(s\right)={\int }_0^{1}H(s,t)\mathfrak{f}(t,{\chi }_1\left(t\right))dt\hfill \\ \hfill \le & (\alpha -1){\mathrm{\Lambda }}_1{\int }_0^1\phi \left(t\right)l\left(t\right)g\left({\chi }_1\left(t\right)\right)\left[1+{\displaystyle \frac{h\left({\chi }_1\left(t\right)\right)}{g\left({\chi }_1\left(t\right)\right)}}\right]dt\hfill \\ \hfill \le & (\alpha -1){\mathrm{\Lambda }}_1\left[1+{\displaystyle \frac{h\left(r_4\right)}{g\left(r_4\right)}}\right]{\int }_0^1\phi \left(t\right)l\left(t\right)g\left({\chi }_1\left(t\right)\right)dt\hfill \\ \hfill \le & (\alpha -1){\mathrm{\Lambda }}_1\left(1+{\displaystyle \frac{h\left(r_4\right)}{g\left(r_4\right)}}\right){\int }_0^1\phi \left(t\right)l\left(t\right)g\left(r_4\mathrm{\Lambda }\phi (1-t)\right)dt<r_4.\hfill \end{array}$$

This contradicts ${\chi }_1\in \partial B_{r_4}\cap \mathcal{P}$. According to Lemma 10, we have

$$\begin{array}{c}\hfill i(A,B_{r_4}\cap \mathcal{P},\mathcal{P})=1.\end{array}$$

(21)

Then, in view of (20) and (21), we see that

$$\begin{array}{c}\hfill i(A,(B_{r_4}\setminus \overline{B_{r_3}})\cap \mathcal{P},\mathcal{P})=1.\end{array}$$

This yields that FBVP (1) has at least one positive solution. □

Corollary 3.

Theorem 3 still holds if (H7) is replaced by the hypothesis as follows:

$$\begin{array}{c}\hfill \underset{r\ge 1}{sup}{\displaystyle \frac{rg\left(r\right)}{g\left(r\right)+h\left(r\right)}}>(\alpha -1){\mathrm{\Lambda }}_1{\int }_0^1\phi \left(s\right)l\left(s\right)g\left(\mathrm{\Lambda }\phi (1-s)\right)ds.\end{array}$$

Corollary 4.

Theorem 3 still holds if (H7) is replaced by the hypothesis as follows:

$$\begin{array}{c}\hfill \underset{\chi \to +\infty }{lim}{\displaystyle \frac{\chi g\left(\chi \right)}{g\left(\chi \right)+h\left(\chi \right)}}=+\infty .\end{array}$$

Example 1.

Consider the FBVP as follows:

$$\left\{\begin{array}{c}-{\mathcal{D}}^{{\displaystyle \frac{5}{2}}}\chi \left(s\right)+{\pi }^{-{\displaystyle \frac{5}{2}}}{s}^{-{\displaystyle \frac{1}{2}}}{(1-s)}^{-{\displaystyle \frac{1}{2}}}\chi \left(s\right)=\mathfrak{f}(s,\chi \left(s\right)), 0<s<1,\hfill \\ \chi \left(0\right)={\chi }^{\prime }\left(0\right)=\chi \left(1\right)=0,\hfill \end{array}\right.$$

(22)

where

$$\mathfrak{f}(s,\chi )=2s^{{\displaystyle \frac{1}{2}}}{(1-s)}^{-{\displaystyle \frac{1}{2}}}\left({\chi }^{-{\displaystyle \frac{1}{3}}}+(1+{sin}^2\chi ){\chi }^{{\displaystyle \frac{1}{3}}}\right).$$

According to the calculation, we have

$$\begin{array}{cc}\hfill & {\mathrm{\Delta }}_1={\pi }^{-{\displaystyle \frac{3}{2}}}\approx 0.1796,{\mathrm{\Delta }}_2={\displaystyle \frac{8}{15}}{\pi }^{-3}\approx 0.0172,\hfill \\ \hfill & \mathrm{\Delta }={\displaystyle \frac{3{\mathrm{\Delta }}_1}{\sqrt{\pi }(1-{\mathrm{\Delta }}_2)}}\approx 0.3093,\hfill \\ \hfill & {\mathrm{\Lambda }}_1=1+\mathrm{\Delta }\approx 1.3093,{\mathrm{\Lambda }}_2=1-\mathrm{\Delta }\approx 0.6904,\hfill \\ \hfill & \mathrm{\Lambda }={\displaystyle \frac{\sqrt{\pi }{\mathrm{\Lambda }}_2}{2{\mathrm{\Lambda }}_1}}\approx 0.4673.\hfill \end{array}$$

For $r_0>r>0$, denote

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_{r,r_0}\left(t\right)=2{\left[r\mathrm{\Lambda }\right]}^{-{\displaystyle \frac{1}{3}}}{\left(\mathrm{\Gamma }\left({\displaystyle \frac{5}{2}}\right)\right)}^{{\displaystyle \frac{1}{3}}}{(1-t)}^{-{\displaystyle \frac{5}{6}}}+4R^{{\displaystyle \frac{1}{3}}}{t}^{{\displaystyle \frac{1}{2}}}{(1-t)}^{-{\displaystyle \frac{1}{2}}}.\end{array}$$

It is clear that

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_{r,r_0}\in L^1[0, 1]\bigcap C(0, 1).\end{array}$$

For any $(s,\chi )\in \{(\tau ,y):r\mathrm{\Lambda }\phi (1-\tau )\le y\le r_0,0<\tau <1\}$, we have

$$\begin{array}{cc}\hfill \mathfrak{f}(s,\chi )& \le 2s^{{\displaystyle \frac{1}{2}}}{(1-s)}^{-{\displaystyle \frac{1}{2}}}\left({\left[r\mathrm{\Lambda }\phi (1-s)\right]}^{-{\displaystyle \frac{1}{3}}}+2R^{{\displaystyle \frac{1}{3}}}\right)\hfill \\ \hfill & =2t^{{\displaystyle \frac{1}{2}}}{(1-s)}^{-{\displaystyle \frac{1}{2}}}\left({\left[r\mathrm{\Lambda }\right]}^{-{\displaystyle \frac{1}{3}}}{\left(\mathrm{\Gamma }\left({\displaystyle \frac{5}{2}}\right)\right)}^{{\displaystyle \frac{1}{3}}}{(1-s)}^{-{\displaystyle \frac{1}{3}}}{s}^{-{\displaystyle \frac{1}{2}}}+2R^{{\displaystyle \frac{1}{3}}}\right)\hfill \\ \hfill & =2{\left[r\mathrm{\Lambda }\right]}^{-{\displaystyle \frac{1}{3}}}{\left(\mathrm{\Gamma }\left({\displaystyle \frac{5}{2}}\right)\right)}^{{\displaystyle \frac{1}{3}}}{(1-s)}^{-{\displaystyle \frac{5}{6}}}+4R^{{\displaystyle \frac{1}{3}}}{s}^{{\displaystyle \frac{1}{2}}}{(1-s)}^{-{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

that is,

$$\begin{array}{c}\hfill \mathfrak{f}(s,\chi )\le {\mathrm{\Psi }}_{r,R}\left(s\right),\forall s\in (0,1),\chi \in \left[r\mathrm{\Lambda }\phi (1-s),r_0\right].\end{array}$$

Let $r_2=8$, and

$$\begin{array}{c}\hfill b_2\left(t\right)=2t^{{\displaystyle \frac{1}{2}}}{(1-t)}^{-{\displaystyle \frac{1}{2}}}.\end{array}$$

Then, ${\int }_0^1{b}_2\left(s\right)ds=\pi$ and

$$\begin{array}{c}\hfill \mathfrak{f}(s,\chi )\le b_2\left(s\right)\chi ,\forall (s,\chi )\in (0, 1)\times [r_2,+\infty ).\end{array}$$

By direct calculation, we have

$$\begin{array}{c}\hfill \underset{\chi \to 0+}{lim\; inf}\underset{s\in [{\displaystyle \frac{1}{4}},{\displaystyle \frac{3}{4}}]}{min}\mathfrak{f}(s,\chi )=+\infty .\end{array}$$

According to the above discussion, we see that the hypotheses (H2), (H4), and (H5) hold. Moreover, we have

$$\begin{array}{cc}\hfill |\left(L_2\chi \right)\left(s\right)|& =|{\int }_0^{1}H(s,t)b_2\left(t\right)\chi \left(t\right)dt|\le {\int }_0^{1}H(s,t)b_2\left(t\right)dt\parallel \chi \parallel \hfill \\ \hfill & \le {\displaystyle \frac{4{\mathrm{\Lambda }}_1}{\sqrt{\pi }}}{\int }_0^1(1-t)t^{{\displaystyle \frac{3}{2}}}dt\parallel \chi \parallel ={\displaystyle \frac{16{\mathrm{\Lambda }}_1}{35\sqrt{\pi }}}\parallel \chi \parallel ,\hfill \end{array}$$

which implies that $r\left(L_2\right)\le {\displaystyle \frac{16{\mathrm{\Lambda }}_1}{35\sqrt{\pi }}}\approx 0.3377<1$. On the other hand, Lemma 8 yields $r\left(L_2\right)>0$.

So, the hypotheses of Theorem 2 are fulfilled. Then, Theorem 2 guarantees that the FBVP (22) has positive solutions.

4. Conclusions

This paper considers a class of singular FBVPs with a perturbation term, meaning the FDEs contain a linear operator with variable coefficients as follows:

$$-{\mathcal{D}}_{0+}^{\alpha }\chi +a\left(t\right)\chi .$$

It should be noted that the coefficient of the perturbation term a may be singular at $t=0, 1$. By virtue of spectral theory, the associated Green function $H(s,t)$ is constructed as a series of functions. We obtain some interesting properties of the Green function. Some results of the existence of positive solutions are derived under conditions that $\mathfrak{f}(s,\chi )$ possesses singularity at $s=0, 1$ and $\chi =0$. To overcome the difficulties in keeping the compactness of the operator caused by singularity, we made hypotheses (H1) and (H2), which are no stronger than the hypotheses used in [10] or [12]. Compared with the conditions used in Corollary 1 and 2, which are usually used in sublinear conditions, the conditions associated with the spectral radii are more sharp (see [31]). The main tool used in this paper is the fixed-point index theory, and some results of existence are closely associated with the spectral radii of the relevant linear operators. In addition, if some non-local conditions are added, we can obtain the multiplicity of a positive solution. The methods used in this paper can also be utilized for some other FBVPs.