Existence and asymptotic analysis of positive solutions for a singular fractional differential equation with nonlocal boundary conditions

Abstract

In this paper, we focus on the existence and asymptotic analysis of positive solutions for a class of singular fractional differential equations subject to nonlocal boundary conditions. By constructing suitable upper and lower solutions and employing Schauder’s fixed point theorem, the conditions for the existence of positive solutions are established and the asymptotic analysis for the obtained solution is carried out. In our work, the nonlinear function involved in the equation not only contains fractional derivatives of unknown functions but also has a stronger singularity at some points of the time and space variables.

1 Introduction

The purpose of this paper is to establish some new results on existence and asymptotic analysis of positive solutions for the following singular fractional differential equation with nonlocal boundary condition:

$$ \textstyle\begin{cases} -\pmb{\mathscr{D}_{t}}^{\alpha } x(t)= f(t,x(t), \pmb{\mathscr{D}_{t}} ^{\gamma }x(t)), \quad 0< t< 1, \\ \pmb{\mathscr{D}_{t}} ^{\gamma }x(0)= \pmb{\mathscr{D}_{t}} ^{\gamma +1}x(0)=0, \qquad \pmb{\mathscr{D}_{t}} ^{\mu }x(1)=\int ^{1}_{0}\pmb{\mathscr{D}_{t}} ^{\mu }x(s)\,d\mathcal{X}(s), \end{cases} $$

(1.1)

where $2<\alpha \le 3$ with $0<\gamma \le \mu <\alpha -2$, $\pmb{\mathscr{D}_{t}}^{\alpha }$ is defined as the Riemann–Liouville derivative, $\int ^{1}_{0}\pmb{\mathscr{D}_{t}} ^{\mu }x(s)\,d \mathcal{X}(s)$ denotes a linear functional involving the Riemann–Stieltjes integrals, $\mathcal{X}$ is a function of bounded variation with a changing-sign measure $d\mathcal{X}$, $f:(0,1)\times (0, +\infty )\times (0, +\infty )\rightarrow [0,+\infty )$ is continuous, and $f(t,x_{1},x_{2})$ may be singular at $t = 0, 1$ and $x_{1}={x_{2}=0}$.

In describing viscoelasticity, Heymans and Kitagawa [1] pointed out that the accuracy and success of the model are their abilities to describe natural phenomena including memory effects in polymers. However, in many dynamic process, the influence of memory is often persistent, even if the factors affecting the process have disappeared, such as observed in stress relaxation after a nonmonotonous loading program. Thus in order to improve the accuracy of the model, based on the non-locality of fractional order derivative, one can choose a noninteger order differential equation to describe this type of physical phenomena with memory effects. In addition, fractional calculus also has many other applications in various fields of science and engineering, such as a HIV model [2, 3] and a fluid model [4,5,6,7,8]. Recently, Heymans and Podlubny [9] gave some physical interpretation for the fractional spring–pot model, the Zener model, the Maxwell model and the Voigt model. In [10], Abdon introduced a new concept of differentiation and integration combining fractal differentiation and fractional differentiation, which can explain the memory effect of heterogeneity, and elasco-viscosity of the medium and also the fractal geometry of the dynamic system. Using the time-scale fractional calculus, Nadia et al. [11] gave some applications of the fractional derivatives with arbitrary time scales in white noise from signal processing.

In the aspect of mathematical theory and application, to obtain further information of the relative natural phenomena, many authors are interested in the existence and properties of solutions for fractional differential models [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and many analytical techniques and methods have been developed to solve various differential equations, such as iterative methods [28,29,30,31,32,33,34,35,36,37], the Mawhin continuation theorem for resonance [38,39,40], the topological degree method [41, 42], the fixed point theorem [43,44,45,46,47,48,49,50,51,52,53,54,55], the variational method [56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73] and the upper and lower solution method [74, 75].

Inspired by the above work, in this paper, we mainly focus on the analytic results for Eq. (1.1). Our strategy is firstly introducing an accurate cone of Banach space and then constructing a couple of suitable upper and lower solutions, and finally establishing some new results on existence and asymptotic behavior of positive solutions for the equation by using the fixed point theorem. It is noteworthy that our approach and technique can solve the singularity of nonlinear term f at the space variables without the need of the complicated supremum and limit condition such as

• (A) $f\in C((0,1)\times (0,+\infty )\times (0,+\infty ),[0,+ \infty ))$ and for any $0 < r < R < +\infty$,

  $$ \lim_{n\to +\infty } \sup_{{{x\in K_{\frac{R}{\varGamma (\beta +1)}}} \atop {y\in \overline{K}_{R}\backslash K_{r} } }} \int _{e(n)}\omega (s)f\bigl(s,x(s),y(s)\bigr)\,ds=0, $$

  where $e(n)=[0,\frac{1}{n}] \cup [\frac{n-1}{n},1]$.

This is applied by Zhang et al. [14] for the spectral and singularity analysis for a fractional differential equation with signed measure. The main contributions of this work are as follows:

• (i) We present exact cone and suitable growth condition to overcome the difficulty due to the singularity of the nonlinear term f at the space variables.

• (ii) We establish a sufficient condition for the existence of positive solutions and give the estimation of the positive solution and asymptotic behavior of the derivative of positive solutions at the

• (iii) Nonsingular cases for the nonlinear term f at the time and space variables are discussed and some new results are established.

The rest of this paper is organized as follows. In Sect. 2, some preliminaries and lemmas are presented for subsequent developments. The main results are presented in Sect. 3.

2 Preliminaries and lemmas

For the convenience of the reader, we only present here some necessary properties from fractional calculus theory in the sense of Riemann–Liouville, and the corresponding definitions can be found in [76] or [12,13,14,15,16,17,18,19,20,21,22,23,24,25].

Proposition 2.1

([76])

• (1) If $x,y:(0,+\infty )\rightarrow \mathbb{R}$ with order $\alpha > 0$, then

  $$ \pmb{\mathscr{D}_{t}}^{\alpha }\bigl(x(t)+y(t)\bigr)=\pmb{ \mathscr{D}_{t}}^{ \alpha }x(t)+\pmb{\mathscr{D}_{t}}^{\alpha }y(t). $$

• (2) If $x\in L^{1}(0, 1)$, $\nu >\gamma > 0$ and m is a positive integer, then

  $$ \begin{gathered} I^{\nu }I^{\gamma }x(t)=I^{\nu +\gamma }x(t), \qquad \pmb{ \mathscr{D}_{t}}^{\gamma }I^{\nu } x(t)=I^{\nu -\gamma } x(t), \\ \pmb{\mathscr{D}_{t}}^{\gamma }I^{\gamma } x(t)=x(t), \qquad \pmb{\mathscr{D}_{t}}^{m} \bigl(\pmb{ \mathscr{D}_{t}}^{\gamma }x(t) \bigr)= \pmb{\mathscr{D}_{t}}^{\gamma +m}x(t). \end{gathered} $$

• (3) If $\alpha >0$, $\gamma >0$, then

  $$ \pmb{\mathscr{D}_{t}}^{\alpha } t^{\gamma -1}= \frac{\varGamma (\gamma )}{ \varGamma (\gamma -\alpha )}t^{\gamma -\alpha -1}. $$

• (4) Suppose $\gamma > 0$, and $g(x)$ is integrable, then

  $$ I^{\gamma }\pmb{\mathscr{D}_{t}}^{\gamma }g(x)=g(x)+c_{1}x^{\gamma -1}+c _{2}x^{\gamma -2}+\cdots +c_{n}x^{\gamma -n}, $$

  where $c_{i}\in \mathbb{R}$ ($i=1,2,\ldots ,n$), n is the smallest integer greater than or equal to α.

In the rest of this paper, all discussions are based on the assumption $2<\alpha -\gamma \le 3$. We first give the following lemma.

Lemma 2.1

Let $x(t)=I^{\gamma }z(t)$, $z(t)\in C[0,1]$, then Eq. (1.1) is equivalent to the following boundary value problem:

$$ \textstyle\begin{cases} -\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } z(t)= f(t,I^{\gamma }z(t), z(t)), \\ z(0)=z'(0)=0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(1)=\int ^{1}_{0} \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(s)\,d\mathcal{X}(s). \end{cases} $$

(2.1)

Proof

Firstly, let $x(t)=I^{\gamma }z(t)$ and $z(t)\in C[0,1]$. It follows from Proposition 2.1(2) that

$$ \begin{aligned}[b] \pmb{\mathscr{D}_{t}}^{\gamma }x(t)= \pmb{\mathscr{D}_{t}}^{\gamma }I ^{\gamma }z(t)= z(t). \end{aligned} $$

(2.2)

On the other hand, $2<\alpha \le 3$ and $0<\gamma \le \mu <\alpha -2$ yield $\alpha -\gamma ,\alpha -\mu \in (2,3)$. Consequently, by the definition of the Riemann–Liouville derivative and integral and Proposition 2.1(2), one has

$$ \begin{aligned} & \pmb{\mathscr{D}_{t}}^{\gamma +1}x(t)= \pmb{\mathscr{D}_{t}}^{ \gamma +1}I^{\gamma }z(t)= z'(t), \\ &\pmb{\mathscr{D}_{t}}^{\alpha }x(t)=\frac{d ^{3}}{dt^{3}} \bigl(I^{3-\alpha }x(t) \bigr) =\frac{d^{3}}{dt^{3}} \bigl(I ^{3-\alpha }I^{\gamma }z(t) \bigr) =\frac{d^{3}}{dt^{3}} \bigl(I^{3- \alpha +\gamma }z(t) \bigr) \\ & \hphantom{\pmb{\mathscr{D}_{t}}^{\alpha }x(t)} =\pmb{\mathscr{D}_{t}}^{\alpha -\gamma }z(t).\end{aligned} $$

(2.3)

It follows from (1.1), (2.2) and (2.3) that $-\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } z(t)= f(t,I^{\gamma }z(t), z(t))$ with boundary conditions

$$ z(0)=\pmb{\mathscr{D}_{t}}^{\gamma }x(0)=0,\qquad z'(0)= \pmb{\mathscr{D}_{t}}^{\gamma +1}x(0)=0,\qquad \pmb{\mathscr{D}_{t}} ^{ \mu -\gamma }z(1)= \int ^{1}_{0}\pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(s)\,d \mathcal{X}(s). $$

Thus, Eq. (1.1) is turned into the boundary value problem (2.1).

Conversely, if $z\in C([0,1],[0,+\infty ))$ is a solution for the problem (2.1). Then letting $x(t)=I^{\gamma }z(t)$ and using (2.2) and (2.3), we get

$$ \begin{aligned}[b] -\pmb{\mathscr{D}_{t}}^{\alpha }x(t) &=-\pmb{\mathscr{D}_{t}}^{\alpha -\gamma }z(t) =f\bigl(t,I^{\gamma }z(t), z(t)\bigr) =f\bigl(t,x(t), \pmb{\mathscr{D}_{t}}^{\gamma }x(t) \bigr),\quad 0< t< 1,\end{aligned} $$

with boundary conditions

$$ \pmb{\mathscr{D}_{t}}^{\gamma }x(0)=z(0)=0,\qquad \pmb{ \mathscr{D}_{t}} ^{\gamma +1}x(0)=z'(0)=0,\qquad \pmb{ \mathscr{D}_{t}} ^{\mu }x(1)= \int ^{1} _{0}\pmb{\mathscr{D}_{t}} ^{\mu }x(s)\,d\mathcal{X}(s). $$

Consequently, the boundary value problem (2.1) is turned into Eq. (1.1). □

The following lemma is standard according to Proposition 2.1, and we omit the proof.

Lemma 2.2

Given $h\in L^{1}(0, 1)$, then the boundary value problem

$$ \textstyle\begin{cases} \pmb{\mathscr{D}_{t}}^{\alpha -\gamma }z(t)+h(t)=0 , \quad 0< t< 1 ,\\ z(0)=z'(0)=0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(1)=0, \end{cases} $$

(2.4)

has the unique solution

$$ z(t)= \int _{0}^{1}G(t,s)h(s)\,ds, $$

where $G(t,s)$ is the Green function of the boundary value problem (2.4) and

$$ G(t,s)= \textstyle\begin{cases} \frac{t^{\alpha -\gamma -1}(1-s)^{\alpha -\mu -1}-(t-s)^{\alpha - \gamma -1}}{\varGamma (\alpha -\gamma )}, & 0\leq s\leq t\leq 1, \\ \frac{t^{\alpha -\gamma -1}(1-s)^{\alpha - \mu -1}}{\varGamma (\alpha -\gamma )}, & 0\leq t\leq s\leq 1. \end{cases} $$

(2.5)

On the other hand, by Proposition 2.1, we know that the unique solution of the boundary value problem

$$ \textstyle\begin{cases} \pmb{\mathscr{D}_{t}}^{\alpha -\gamma }z(t)=0, \quad 0< t< 1,\\ z(0)=z'(0)=0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(1)=1, \end{cases} $$

is $\frac{\varGamma (\alpha -\mu )}{\varGamma (\alpha -\gamma )}t^{\alpha - \gamma -1}$. Let

$$ \mathcal{C}= \int _{0}^{1}\frac{\varGamma (\alpha -\mu )}{\varGamma (\alpha - \gamma )}t^{\alpha -\gamma -1}\,d \mathcal{X}(t), \qquad \mathcal{B}=\frac{\varGamma (\alpha -\mu )}{(1-\mathcal{C})\varGamma ( \alpha -\gamma )}, $$

(2.6)

and define

$$ \mathcal{G}_{\mathcal{X}}(s)= \int _{0}^{1}G(t, s)\,d\mathcal{X}(t). $$

Following the strategy in [24], the Green function for the boundary value problem (2.1) is

$$ W(t,s)={\mathcal{B}t^{\alpha -\gamma -1}} \mathcal{G}_{\mathcal{X}}(s)+G(t,s). $$

(2.7)

Thus we have the following lemma.

Lemma 2.3

Let $p\in L^{1}[0,1]$ and $2<\alpha \le 3$ with $0<\gamma \le \mu < \alpha -2$. Then the fractional differential equation

$$ \textstyle\begin{cases} -\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } z(t)= p(t), \\ z(0)=z'(0)=0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(1)=\int ^{1}_{0} \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(s)\,d\mathcal{X}(s), \end{cases} $$

has the unique solution

$$ \begin{aligned}[b] z(t)= \int _{0}^{1} W(t,s)p(s)\,ds, \end{aligned} $$

where $W(t,s)$ is defined by (2.7).

In order to guarantee the nonnegativity of the Green function, the following condition is necessary.

• (F0) $\mathcal{X}$ is a function of bounded variation satisfying $\mathcal{G}_{\mathcal{X}}(s)\ge 0$, $s\in [0, 1]$ and $\mathcal{C}\in [0,1)$.

Lemma 2.4

Assume (F0) is satisfied, then for the Green function in (2.7) one has the following estimation:

• (1) $W(t,s) > 0$ for all $0< t$, $s<1$.

• (2)
  $$ \mathcal{B}t^{\alpha -\gamma -1}\mathcal{G}_{\mathcal{X}}(s) \le W(t,s) \le c(s)t^{\alpha -\gamma -1}, $$

  (2.8)

  where

  $$ c(s)=\frac{(1-s)^{\alpha -\mu -1}}{\varGamma (\alpha -\gamma )}+ \mathcal{B}\mathcal{G}_{\mathcal{X}}(s). $$

Proof

The conclusion of (1) is clear. In what follows, we prove the conclusion (2). Using (2.5) and (2.7), one gets

$$ \begin{aligned}[b] W(t,s) &={\mathcal{B}t^{\alpha -\gamma -1}} \mathcal{G}_{\mathcal{X}}(s)+G(t,s) \le \mathcal{B}\mathcal{G}_{\mathcal{X}}(s)+G(t,s) \\ &\le \frac{t^{ \alpha -\gamma -1}(1-s)^{\alpha -\mu -1}}{\varGamma (\alpha -\gamma )}+ \mathcal{B}t^{\alpha -\gamma -1}\mathcal{G}_{\mathcal{X}}(s)= c(s)t ^{\alpha -\gamma -1} \end{aligned} $$

and

$$ \begin{aligned}[b] W(t,s) &={\mathcal{B}t^{\alpha -\gamma -1}} \mathcal{G}_{\mathcal{X}}(s)+G(t,s) \ge {\mathcal{B}t^{\alpha -\gamma -1}} \mathcal{G}_{\mathcal{X}}(s). \end{aligned} $$

 □

It follows from Lemma 2.3 that we have the following.

Lemma 2.5

If $z \in C([0,1], \mathbb{R})$ satisfies

$$ z(0)=z'(0)=0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(1)= \int ^{1}_{0} \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(s)\,d\mathcal{X}(s), $$

and $\pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(t)\le 0$ for all $t\in (0,1)$, then $z(t)\ge 0$, $t\in [0,1]$.

3 Singular cases

In this section, we first give the definition of upper and lower solution on the boundary value problem (2.1), and then introduce some theories of function space and give our main results.

Definition 3.1

We call a continuous function $\psi (t)$ as a lower solution for the boundary value problem (2.1), if

$$ \textstyle\begin{cases} -\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } \psi (t)\le f(t,I^{\gamma } \psi (t), \psi (t)), \\ \psi (0)\ge 0, \psi '(0)\ge 0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }\psi (1)\ge \int ^{1}_{0} \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }\psi (s)\,d\mathcal{X}(s). \end{cases} $$

Definition 3.2

We call a continuous function $\phi (t)$ as a upper solution for the boundary value problem (2.1), if

$$ \textstyle\begin{cases} -\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } \phi (t)\ge f(t,I^{\gamma } \phi (t), \phi (t)), \\ \phi (0)\le 0, \phi '(0)\le 0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }\phi (1)\le \int ^{1}_{0} \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }\phi (s)\,d\mathcal{X}(s). \end{cases} $$

Let

$$ e(t)=t^{\alpha -\gamma -1},\qquad \kappa (t)=\frac{\varGamma ( {\alpha - \gamma })}{\varGamma (\alpha )}t^{\alpha -1},\quad t\in [0,1], $$

and define our work space $E= C[0,1]$ and a subset $P_{e}$ of E,

$$ \begin{aligned}[b] P_{e}&=\bigl\{ z\in E: \text{ there exist two positive numbers } 0< l_{z}< 1< L_{z} \text{ such that} \\ &\quad l_{z} e(t)\le z(t)\le L_{z} e(t), t\in [0,1]\bigr\} .\end{aligned} $$

(3.1)

Clearly, $P_{e}$ is nonempty since $e(t)\in P_{e}$. For any $z \in P_{e}$, define an operator B by

$$ \begin{aligned}[b] (B z) (t)= \int _{0}^{1} W(t,s)f\bigl(s,I^{\gamma }z(s), z(s)\bigr)\,ds. \end{aligned} $$

(3.2)

To overcome the difficulties of the singularity at the space variables, we introduce the following growth conditions for f:

• (F1) $f\in C((0,1)\times (0,\infty )\times (0,\infty ),[0,+ \infty ))$, and $f(t,x_{1},x_{2})$ is decreasing in $x_{i}>0$ for $i=1,2$.

• (F2) For any $\tau >0$, $f(t,\frac{ \tau }{\varGamma (\gamma +1)} t^{\gamma },\tau )\not \equiv 0$, and

  $$ 0< \int _{0}^{1} c(s)f\bigl(s,\tau \kappa (s), \tau e(s)\bigr)\,ds< +\infty . $$

Lemma 3.1

Assume (F0) (F1) and (F2) are satisfied, then $B (P_{e})\subset P_{e}$ and B is well defined.

Proof

For any $z\in P_{e}$, it follows from the definition of $P_{e}$ that there exist two numbers $0< l_{z}<1<L_{z}$ such that $l_{z} e(t)\le z(t) \le L_{z} e(t)$ for any $t\in [0,1]$. Notice that $\kappa (t)=I^{ \gamma } e(t)$, then by (2.8) and (F1)–(F2), one gets

$$ \begin{aligned}[b] (Bz) (t) &= \int _{0}^{1} W(t,s)f\bigl(s,I^{\gamma }z(s), z(s)\bigr)\,ds \\ &\le \int _{0}^{1} c(s)f\bigl(s,I^{\gamma } \bigl(l_{z} e(s)\bigr), l_{z} e(s)\bigr)\,ds \\ &\le \int _{0}^{1} c(s)f\bigl(s,l_{z}\kappa (s),l_{z}e(s)\bigr)\,ds \\ &< +\infty .\end{aligned} $$

(3.3)

Take $\tau =\max_{t\in [0,1]} z(t)$, it follows from (F2) that, for any $s\in [0,1]$,

$$ \mathcal{G}_{\mathcal{X}}(s)f \biggl(s,\frac{ \tau s^{\gamma }}{\varGamma (\gamma +1)},\tau \biggr)\not \equiv 0. $$

Consequently, from the continuity of f, one has

$$ \int _{0}^{1}\mathcal{G}_{\mathcal{X}}(s) f \biggl(s,\frac{ \tau s^{ \gamma }}{\varGamma (\gamma +1)},\tau \biggr)\,ds>0. $$

This yields

$$ \begin{aligned}[b] & \int _{0}^{1} \mathcal{G}_{\mathcal{X}}(s)f \bigl(s,I^{\gamma }\tau , \tau \bigr)\,ds= \int _{0}^{1} \mathcal{G}_{\mathcal{X}}(s)f \biggl(s,\frac{ \tau s^{\gamma }}{\varGamma (\gamma +1)},\tau \biggr)\,ds>0.\end{aligned} $$

(3.4)

By (2.8), (3.3) and (3.4), we have

$$ (Bz) (t) = \int _{0}^{1} W(t,s)f\bigl(s,I^{\gamma }z(s), z(s)\bigr)\,ds\ge \mathcal{B} e(t) \int _{0}^{1} \mathcal{G}_{\mathcal{X}}(s)f \bigl(s,I^{ \gamma }\tau , \tau \bigr)\,ds\ge l'_{z} e(t), $$

(3.5)

where

$$ l'_{z}=\min \biggl\{ \frac{1}{2}, \mathcal{B} \int _{0}^{1} \mathcal{G} _{\mathcal{X}}(s)f \bigl(s,I^{\gamma }\tau , \tau \bigr)\,ds \biggr\} . $$

On the other hand, in view of (2.8), we also have

$$ (Bz) (t) = \int _{0}^{1} W(t,s)f\bigl(s,I^{\gamma }z(s), z(s)\bigr)\,ds\le e(t) \int _{0}^{1} c(s)f\bigl(s,l_{z}\kappa (s),l_{z}e(s)\bigr)\,ds\le L'_{z} e(t), $$

(3.6)

where

$$ \begin{aligned}[b] L'_{z}&= \max \biggl\{ 2, \int _{0}^{1} c(s)f\bigl(s,l_{z}\kappa (s),l_{z}e(s)\bigr)\,ds \biggr\} .\end{aligned} $$

Thus it follows from (3.3)–(3.6) that $B (P_{e})\subset P_{e}$ and B is well defined. □

Theorem 3.1

(Existence)

Suppose (F0)–(F2) hold. Then Eq. (1.1) has at least one positive solution.

Proof

Firstly by Lemma 2.3 and (3.2), we have

$$ \textstyle\begin{cases} -\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } (Bz)(t)= f(t,I^{\gamma }z(t), z(t)), \\ (Bz)(0)=(Bz)'(0)=0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }(Bz)(1)=\int ^{1}_{0} \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }(Bz)(s)\,d\mathcal{X}(s). \end{cases} $$

(3.7)

Next we seek for a couple of lower and upper solutions of the boundary value problem (2.1). To do this, take

$$ \eta (t) =\min \bigl\{ e(t), Be(t)\bigr\} , \qquad \xi (t)=\max \bigl\{ e(t), Be(t)\bigr\} . $$

(3.8)

Obviously, if $e(t)=Be(t)$, then $e(t)$ is a positive solution of Eq. (1.1). If $e(t)\neq Be(t)$, then we have $\xi (t), \eta (t) \in P_{e}$ and

$$ \eta (t)\le e(t)\le \xi (t). $$

(3.9)

Letting

$$ \psi (t) = B\xi (t), \qquad \phi (t) = B\eta (t), $$

we claim that the functions $\psi (t)$, $\phi (t)$ shall be the lower solution and upper solution of the boundary value problem (2.1), respectively.

In fact, it follows from (F1) that B is nonincreasing relative to z. By (3.8)–(3.9), we have

$$ \begin{aligned} &\psi (t) = B\xi (t)\le B\eta (t)=\phi (t),\\ &\psi (t) = B\xi (t)\le B e(t) \le \xi (t),\\ &\phi (t) = B\eta (t)\ge B e(t)\ge \eta (t), \end{aligned} $$

(3.10)

and $\psi (t), \phi (t)\in P_{e}$. Thus (3.7) and (3.10) yield

$$ \begin{aligned} &\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } \psi (t) +f\bigl(t,I^{\gamma } \psi (t), \psi (t)\bigr)\\ &\quad =\pmb{ \mathscr{D}_{t}}^{\alpha -\gamma }(B\xi ) (t) +f\bigl(t,I^{\gamma }(B \xi ) (t), (B\xi ) (t)\bigr) \\ &\quad \ge \pmb{\mathscr{D}_{t}} ^{\alpha -\gamma }(B\xi ) (t) +f \bigl(t,I^{\gamma }\xi (t), \xi (t)\bigr)=0, \quad t\in (0,1), \\ &\psi (0)=\psi '(0)=0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }\psi (1)= \int ^{1}_{0} \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }\psi (s)\,d\mathcal{X}(s),\end{aligned} $$

(3.11)

and

$$ \begin{aligned} &\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } \phi (t) +f\bigl(t,I^{\gamma } \phi (t), \phi (t)\bigr)\\ &\quad =\pmb{ \mathscr{D}_{t}}^{\alpha -\gamma } (B\eta ) (t)+f\bigl(t,I ^{\gamma }(B\eta ) (t), (B\eta ) (t)\bigr) \\ &\quad \le \pmb{\mathscr{D}_{t}}^{ \alpha -\gamma } (B\eta ) (t)+f \bigl(t,I^{\gamma }\eta (t), \eta (t)\bigr)=0, \quad t\in (0,1), \\ &\phi (0)=\phi '(0)=0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }\phi (1)= \int ^{1}_{0} \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }\phi (s)\,d\mathcal{X}(s).\end{aligned} $$

(3.12)

Thus (3.10)–(3.12) show that $\phi (t)$ and $\psi (t)$ are the lower and upper solutions of the boundary value problem (2.1), respectively, and $\psi (t), \phi (t)\in P_{e}$.

Now define the function

$$ F(t,z)= \textstyle\begin{cases} f(t,I^{\gamma }\psi (t), \psi (t)), & z< \psi (t),\\ f(t,I^{\gamma }z(t),z(t)), & \psi (t)\le z\le \phi (t),\\ f(t,I^{\gamma }\phi (t),\phi (t)), & z>\phi (t), \end{cases} $$

(3.13)

then from (3.13), $F[0,1]\times [0,+\infty )\to [0,+\infty )$ is a continuous function.

Next let us consider the following auxiliary boundary value problem:

$$ \textstyle\begin{cases} -\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } z(t)=F(t,z), \quad 0< t< 1 ,\\ z(0)=z'(0)=0, \quad \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(1)=\int ^{1}_{0} \pmb{\mathscr{D}_{t}}^{\mu -\gamma }z(s)\,d\mathcal{X}(s). \end{cases} $$

(3.14)

Define an operator A in E by

$$ (Az) (t)= \int _{0}^{1} W(t,s)F\bigl(s,z(s)\bigr)\,ds, \quad \forall z\in E. $$

Obviously, from Lemma 2.3, a fixed point of A is a solution of the boundary value problem (3.14).

For all $z\in E$, as $\psi \in P_{e}$, there exists a constant $0< l_{\psi }<1$ such that $\psi (t)\ge l_{\psi }e(t)$, $t\in [0,1]$. Thus by Lemma 2.4, we have

$$ \begin{aligned}[b] (Az) (t) &\le \int _{0}^{1} c(s)F\bigl(s,z(s)\bigr)\,ds \le \int _{0}^{1} c(s)f\bigl(s,I ^{\gamma }\psi (s), \psi (s)\bigr)\,ds \\ &\le \int _{0}^{1} c(s)f(s,f\bigl(s,l_{ \psi }I^{\gamma } e(s), l_{\psi }e(s)\bigr)\,ds \\ & = \int _{0}^{1} c(s)f\bigl(s,l _{\psi }\kappa (s),l_{\psi }e(s)\bigr)\,ds \\ &< +\infty .\end{aligned} $$

So A is bounded. In addition, according to the continuity of F and K, we find that $A:E \to E$ is continuous.

Let Ω be a bounded subset of E, then we have $\Vert z\Vert\le N$ for some positive constant $N>0$ and all $z\in \varOmega$. Let $L = \max_{0\le t\le 1, 0\le z\le N}|F(t,z)| + 1$. It follows from the uniform continuity of $W(t,s)$ that, for any $\epsilon > 0$ and $s\in [0,1]$, there exists $\sigma > 0$ such that

$$ \bigl\vert W(t_{1},s)-W(t_{2},s) \bigr\vert < \frac{\epsilon }{L}, $$

for $\vert t_{1}-t_{2}\vert < \sigma$. Then

$$ \bigl\vert Az(t_{1})-Az(t_{2}) \bigr\vert \leq \int _{0}^{1} \bigl\vert W(t_{1},s)-W (t _{2},s) \bigr\vert \bigl\vert F\bigl(s,z(s)\bigr) \bigr\vert \,ds< \epsilon . $$

This implies that $A(\varOmega )$ is equicontinuous.

Thus according to the Arzelà–Ascoli theorem, $A:E\to E$ is a completely continuous operator. Consequently it follows from the Schauder fixed point theorem that A has a fixed point w such that $w=Aw$.

In order to show that w is also a fixed point of the operator B, we only need to prove

$$ \psi (t)\le w(t)\le \phi (t), \quad t\in [0,1]. $$

We firstly verify that $w(t)\le \phi (t)$. Let $z(t)=\phi (t)-w(t)$, $t\in [0,1]$. Noticing that w is a fixed point of A and (3.12), we have

$$ z(0)=z'(0)=0, \quad \pmb{ \mathscr{D}_{t}} ^{\mu -\gamma }z(1)= \int ^{1}_{0} \pmb{\mathscr{D}_{t}} ^{\mu -\gamma }z(s)\,d\mathcal{X}(s). $$

(3.15)

On the other hand, it follows from (3.11) and (F1) that

$$ f\bigl(t,I^{\gamma }\phi (t), \phi (t)\bigr)\le f \bigl(t,I^{\gamma }\eta (t), \eta (t)\bigr). $$

(3.16)

Thus by the definition of F and (3.16), one gets

$$ \begin{aligned}[b] f\bigl(t,I^{\gamma }\psi (t), \psi (t)\bigr)&\le F \bigl(t,u(t)\bigr)\le f\bigl(t,I^{\gamma } \phi (t), \phi (t)\bigr)\\ &\le f \bigl(t,I^{\gamma }\eta (t), \eta (t)\bigr),\quad \forall u \in E, \forall t\in [0,1]. \end{aligned} $$

(3.17)

It follows from (3.7) and (3.17) that

$$ \begin{aligned}[b] \pmb{\mathscr{D}_{t}}^{\alpha -\gamma }z(t) &=\pmb{\mathscr{D}_{t}} ^{\alpha -\gamma } \phi (t)-\pmb{ \mathscr{D}_{t}}^{\alpha -\gamma } w(t)= \pmb{\mathscr{D}_{t}}^{\alpha -\gamma } \bigl(B\eta (t)\bigr)+F\bigl(w(t)\bigr) \\ &=- f\bigl(t,I ^{\gamma }\eta (t), \eta (t)\bigr)+F(w(t) \le 0. \end{aligned} $$

(3.18)

Thus Lemma 2.5, (3.15) and (3.18) imply that

$$ z(t)=\phi (t)-w(t)\ge 0, \quad t\in [0, 1]. $$

Similarly, we also have $w(t)-\psi (t)\ge 0$ on $[0, 1]$. Thus the following estimation is valid:

$$ \phi (t)\ge w(t)\ge \psi (t), \quad t\in [0,1], $$

(3.19)

which also implies $F(t,w(t))=f(t,I^{\gamma }w(t),w(t))$, $t\in [0,1]$.

Combined with the above facts, we get that the fixed point of A is also the fixed point of B. So $w(t)$ is a positive solution of the boundary value problem (2.1), and consequently $x(t)=I^{\gamma }w(t)$ is a positive solution of Eq. (1.1). □

Theorem 3.2

(Estimation and asymptotic behavior)

Assume (F0)–(F2) are satisfied. Then there exist two positive constants m, n such that the solution $x(t)$ to Eq. (1.1) satisfies

$$ mt^{\alpha -1}\le x(t)\le nt^{\alpha -1} \quad \mathit{and} \quad \pmb{ \mathscr{D}_{t}} ^{\gamma }x(t)=o\bigl(t^{\alpha -2}\bigr). $$

Proof

It follows from $\psi \in P_{e}$ and (3.19) that there exists $0< l_{\psi }<1$ such that

$$ w(t)\ge \psi (t)\ge l_{\psi }e(t). $$

(3.20)

Thus, from (3.20) and (2.8), we have

$$ \begin{aligned}[b] w(t) &= \int _{0}^{1} W(t,s)f\bigl(s,I^{\gamma }w(s),w(s) \bigr)\,ds \\ &\le e(t) \int _{0}^{1} c(s)f\bigl(s,l_{\psi }I^{\gamma } e(s),l_{\psi }e(s)\bigr)\,ds \\ &=e(t) \int _{0}^{1} c(s)f\bigl(s, l_{\psi }\kappa (s), l_{\psi }e(s)\bigr)\,ds \\ &\le Le(t).\end{aligned} $$

(3.21)

Consequently, one gets

$$ l_{\psi }e(t)\le w(t)\le Le(t). $$

We have

$$ I^{\gamma }e(t)=\frac{1}{\varGamma (\gamma )} \int _{0}^{t}(t-s)^{\gamma -1} e(s)\,ds= \frac{\varGamma (\alpha -\gamma )}{\varGamma (\alpha )}t^{\alpha -1}. $$

(3.22)

By (3.22), we have

$$ m t^{\alpha -1}=l_{\psi }\frac{\varGamma (\alpha -\gamma )}{\varGamma ( \alpha )}t^{\alpha -1}\le I^{\gamma }w(t)=x(t)\le L\frac{\varGamma ( \alpha -\gamma )}{\varGamma (\alpha )}t^{\alpha -1}=nt^{\alpha -1}. $$

In the end, by the l’Hospital rule,

$$ \lim_{t\to 0+} \frac{\pmb{\mathscr{D}_{t}} ^{\gamma }x(t)}{t^{\alpha -2}}=\lim_{t \to 0+} \frac{\pmb{\mathscr{D}_{t}} ^{\gamma +1}x(t)}{(\alpha -2)t^{ \alpha -3}}=0, $$

that is, $\pmb{\mathscr{D}_{t}} ^{\gamma }x(t)=o(t^{\alpha -2})$. □

4 Nonsingular cases

In this section, we are interested in some nonsingular cases of the nonlinear term f at time and space variables.

Case 1: f may be singular at $t=0$ and (or) $t=1$, but f is nonsingular at $x_{1}=x_{2}=0$:

Theorem 4.1

Suppose (F0) and the following assumptions are satisfied:

• (B1) $f\in C((0,1)\times [0,\infty )\times [0,\infty ),[0,+ \infty ))$, and $f(t,x_{1},x_{2})$ is decreasing in $x_{i}>0$ for $i=1,2$.

• (B2) $f(t,0,0)\not \equiv 0$ for any $t\in (0,1)$, and

  $$ 0< \int _{0}^{1} c(s)f(s,0,0)\,ds< +\infty . $$

Then Eq. (1.1) has at least one positive solution $x(t)$ satisfying

$$ 0\le x(t)\le \mathcal{M}^{*}t^{\gamma } $$

for some constant $\mathcal{M}^{*}> 0$. Moreover, the positive solution $x(t)$ has boundary asymptotic behavior

$$ \pmb{\mathscr{D}_{t}} ^{\gamma }x(t)=o\bigl(t^{\alpha -2} \bigr). $$

Proof

In fact, we only replace the set $P_{e}$ in Theorem 3.1 by using

$$ P_{1} =\bigl\{ x\in E: x(t)\ge 0, t \in [0, 1]\bigr\} . $$

Let

$$ \eta (t) =\min \{0, B0\}=0, \qquad \xi (t)=\max \{0, B0\}=B0, $$

and set

$$ \psi (t) = B\xi (t)=B(B0), \qquad \phi (t) = B\eta (t)=B0. $$

Then we have $\phi (t), \psi (t)\in P_{1}$ and

$$ 0\le \phi (t)=B0 \quad \text{and} \quad 0\le \psi (t)=(B\phi ) (t)\le B0=\phi (t). $$

(4.1)

On the other hand, we also have

$$ \begin{aligned}[b] &\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } \psi (t)+f\bigl(t,I^{\gamma } \psi (t), \psi (t)\bigr) \\ &\quad =\pmb{\mathscr{D}_{t}}^{\alpha -\gamma }(B \xi ) (t) +f \bigl(t,I^{\gamma }\psi (t), \psi (t)\bigr) \\ &\quad =-f\bigl(t,I^{\gamma } \xi (t), \xi (t)\bigr)+f\bigl(t,I^{\gamma }\psi (t), \psi (t)\bigr) \\ &\quad =-f\bigl(t,I^{ \gamma }\phi (t), \phi (t)\bigr)+f\bigl(t,I^{\gamma }\psi (t), \psi (t)\bigr) \\ &\quad \ge 0, \quad t\in (0,1),\end{aligned} $$

(4.2)

and

$$ \begin{aligned}[b] &\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } \phi (t)+f\bigl(t,I^{\gamma } \phi (t), \phi (t)\bigr) \\ &\quad =\pmb{\mathscr{D}_{t}}^{\alpha -\gamma } (B \eta ) (t)+f \bigl(t,I^{\gamma }\phi (t), \phi (t)\bigr) \\ &\quad =-f\bigl(t,I^{\gamma } \eta (t), \eta (t)\bigr)+f\bigl(t,I^{\gamma } \phi (t), \phi (t)\bigr) \\ &\quad =-f(t,0, 0)+f\bigl(t,I ^{\gamma }\phi (t), \phi (t)\bigr) \\ &\quad \le 0, \quad t\in (0,1).\end{aligned} $$

(4.3)

Thus from (4.1)–(4.3), $\phi (t)$ and $\psi (t)$ are still the lower and upper solutions of the boundary value problem (2.1), respectively.

Finally, it follows from Lemma 2.3 that

$$ \begin{gathered} \phi (t)=B0= \int _{0}^{1}W(t,s)f(s,0,0)\,ds\le { \int _{0}^{1} c(s)f(s,0,0)\,ds}= \mathcal{N}^{*},\\ I^{\gamma }\mathcal{N}^{*}=\frac{\mathcal{N}^{*}}{ \varGamma (\gamma )} \int _{0}^{t}(t-s)^{\gamma -1}\,ds= \mathcal{M}^{*}t^{ \gamma }. \end{gathered} $$

Thus according to the proofs of Theorems 3.1–3.2, the conclusion of Theorem 4.1 is true. □

Case 2: $f(t,x_{1},x_{2})$ is nonsingular at both $t=0,1$ and $x_{i} =0$, $i=1,2$. Then, by Theorem 4.1, the following conclusion is valid.

Theorem 4.2

Assume that $f(t,x_{1},x_{2}): [0,1]\times [0,\infty )\times [0, \infty ) \to [0,+\infty )$ is a continuous and decreasing function in $x_{i}$, $i=1,2$ with $f(t,0,0)\not \equiv 0$ for any $t\in [0,1]$. If (F0) holds, then Eq. (1.1) has at least one positive solution $x(t)$ with the estimation

$$ 0\le x(t)\le \mathcal{M}^{*}t^{\gamma } $$

for some constant $\mathcal{M}^{*}> 0$ and boundary asymptotic behavior $\pmb{\mathscr{D}_{t}} ^{\gamma }x(t)=o(t^{\alpha -2})$.

Proof

In fact, if $f(t,x_{1},x_{2}): [0,1]\times [0,\infty )\times [0, \infty ) \to [0,+\infty )$ is continuous and $f(t,0,0)\not \equiv 0$, then the condition (B2) holds naturally. □

5 Numerical examples

Example 1

Consider the existence of positive solutions for the following singular fractional differential equation with nonlocal boundary condition:

$$ \textstyle\begin{cases} -\pmb{\mathscr{D}_{t}}^{\frac{5}{2}} x(t)=10 t^{-\frac{1}{4}} [( \pmb{\mathscr{D}_{t}} ^{\frac{1}{2}}x(t))^{-\frac{1}{8}}+x^{- \frac{1}{3}}(t) ],\quad 0< t< 1, \\ \pmb{\mathscr{D}_{t}}^{\frac{1}{2}}x(0)= \pmb{\mathscr{D}_{t}}^{\frac{3}{2}}x(0)=0, \qquad \pmb{\mathscr{D}_{t}} ^{\frac{1}{2}}x(1)=\int ^{1}_{0} \pmb{\mathscr{D}_{t}}^{\frac{1}{2}}x(s)\,d \mathcal{X}(s), \end{cases} $$

(5.1)

where $\mathcal{X}$ is a function of bounded variation such that

$$ \mathcal{X}(t)= \textstyle\begin{cases} 0, & t\in [0,\frac{1}{2} ) ,\\ \frac{3}{2}, & t\in [\frac{1}{2},\frac{3}{4} ), \\ 1, & t\in [\frac{3}{4},1 ]. \end{cases} $$

(5.2)

By simple calculation, Eq. (5.1) can be transformed to the following 4-point boundary value problem with coefficients of both signs in the boundary condition:

$$ \textstyle\begin{cases} -\pmb{\mathscr{D}_{t}}^{\frac{5}{2}} x(t)=10 t^{-\frac{1}{4}} [( \pmb{\mathscr{D}_{t}} ^{\frac{1}{4}}x(t))^{-\frac{1}{8}}+x^{- \frac{1}{3}}(t) ],\quad 0< t< 1, \\ \pmb{\mathscr{D}_{t}}^{\frac{1}{4}}x(0)= \pmb{\mathscr{D}_{t}}^{\frac{5}{4}}x(0)=0, \qquad \pmb{\mathscr{D}_{t}} ^{\frac{1}{3}}x(1)=\frac{3}{2}\pmb{\mathscr{D}_{t}}^{\frac{1}{3}}x (\frac{1}{2} )-\frac{1}{2}\pmb{\mathscr{D}_{t}}^{ \frac{1}{3}}x (\frac{3}{4} ). \end{cases} $$

(5.3)

Conclusion: The BVP (5.1) has at least one positive solution $x(t)$, and there exist two positive constants m, n such that

$$ mt^{\frac{3}{2}}\le x(t)\le nt^{\frac{3}{2}} $$

with boundary asymptotic behavior $\pmb{\mathscr{D}_{t}} ^{ \frac{1}{4}}x(t)=o(t^{\frac{1}{2}})$.

Proof

Let $\alpha =\frac{5}{2}$, $\gamma =\frac{1}{4}$, $\mu =\frac{1}{3}$, $f(t,x_{1},x_{2})=10 t^{-\frac{1}{4}} [x_{1}^{-\frac{1}{3}}+x_{2} ^{-\frac{1}{8}} ]$. Then $2<\alpha \le 3$ satisfying $0<\gamma \le \mu <\alpha -2$ and f is singular at $t=0$ and $x_{1}=x_{2}=0$.

Clearly,

$$ G(t,s)=\frac{1}{\varGamma (\frac{9}{4})} \textstyle\begin{cases} t^{\frac{5}{4}}(1-s)^{\frac{7}{6}}-(t-s)^{\frac{5}{4}}=:G_{1}(t,s), & 0\leq s\leq t\leq 1, \\ t^{\frac{5}{4}}(1-s)^{\frac{7}{6}}=:G_{2}(t,s), & 0\leq t\leq s\leq 1,\end{cases} $$

and

$$ \mathcal{G}_{\mathcal{X}}(s)=\frac{1}{\varGamma (\frac{9}{4})} \textstyle\begin{cases} \frac{3}{2}G_{1} (\frac{1}{2},s )-\frac{1}{2}G_{1} (\frac{3}{4},s )= (\frac{1}{2} )^{\frac{9}{4}} (3- (\frac{3}{2} ) ^{\frac{5}{4}} )(1-s)^{\frac{7}{6}}-\frac{3}{2} (\frac{1}{2}-s ) ^{\frac{5}{4}}+\frac{1}{2} (\frac{3}{4}-s )^{\frac{5}{4}}, \\\quad 0\leq s< \frac{1}{2}, \\ \frac{3}{2}G_{2} (\frac{1}{2},s )- \frac{1}{2}G_{1} (\frac{3}{4},s )= (\frac{1}{2} ) ^{\frac{9}{4}} (3- (\frac{3}{2} )^{\frac{5}{4}} )(1-s)^{ \frac{7}{6}}+\frac{1}{2} (\frac{3}{4}-s )^{\frac{5}{4}}, \\\quad \frac{1}{2}\leq s< \frac{3}{4},\\ \frac{3}{2}G_{2} (\frac{1}{2},s )- \frac{1}{2}G_{2} (\frac{3}{4},s )= (\frac{1}{2} ) ^{\frac{9}{4}} (3- (\frac{3}{2} )^{\frac{5}{4}} )(1-s)^{ \frac{7}{6}}, \\\quad \frac{3}{4}\leq s\le 1.\end{cases} $$

Thus

$$ \begin{gathered} \mathcal{G}_{\mathcal{X}}(s)\ge 0, \qquad \mathcal{C}=\frac{\varGamma (\frac{13}{6})}{\varGamma (\frac{9}{4})} \int _{0}^{1}t^{\frac{5}{4}}\,d\mathcal{X}(t)= \frac{\varGamma (\frac{13}{6})}{ \varGamma (\frac{9}{4})} \biggl(1- \int _{0}^{1}\mathcal{X}(t)\,dt^{ \frac{5}{4}} \biggr)=0.2691< 1, \\ c(s)=\frac{1}{\varGamma (\frac{9}{4})}(1-s)^{\frac{7}{6}}+ \frac{1}{1.3070}\mathcal{G}_{\mathcal{X}}(s). \end{gathered} $$

Consequently, (F0) and (F1) hold.

Since

$$ e(t)=t^{\frac{5}{4}}, \kappa (t)=\frac{\varGamma (\frac{9}{4})}{\varGamma (\frac{5}{2})}t^{\frac{3}{2}}, $$

for any $\tau >0$ and $t\in (0,1)$, we have $f(t,\frac{ \tau }{ \varGamma (\gamma +1)} t^{\gamma },\tau )= 10 t^{-\frac{1}{4}} [ (\frac{ \tau }{ \varGamma (\frac{5}{4})} )^{-\frac{1}{3}} t^{-\frac{1}{12}}+ \tau ^{-\frac{1}{8}} ]\not \equiv 0$ and

$$ \begin{aligned}[b] 0 &< \int _{0}^{1} c(s)f\bigl(s,\tau \kappa (s), \tau e(s)\bigr)\,ds \\ &=10 \int _{0} ^{1} s^{-\frac{1}{4}} \biggl[ \frac{1}{\varGamma (\frac{9}{4})}(1-s)^{ \frac{7}{6}}+ \frac{1}{1.3070}\mathcal{G}_{\mathcal{X}}(s) \biggr] \biggl[ \biggl(\frac{\varGamma (\frac{9}{4})\tau }{\varGamma (\frac{5}{2})} \biggr) ^{-\frac{1}{3}}s^{-\frac{1}{2}}+ \tau ^{-\frac{1}{8}} s^{-\frac{5}{32}} \biggr] \,ds < +\infty .\end{aligned} $$

Thus (F2) holds.

It follows from Theorem 3.1 that Eq. (5.1) has at least a positive solution $x(t)$ satisfying the estimation

$$ mt^{\frac{3}{2}}\le x(t)\le nt^{\frac{3}{2}} $$

for some positive constants m, n and boundary asymptotic behavior $\pmb{\mathscr{D}_{t}} ^{\frac{1}{4}}x(t)=o(t^{\frac{1}{2}})$. □

Remark 5.1

In [14], Zhang et al. use the condition (A) to overcome the singularity of the equation. Obviously, Example 1 indicates that (F1) and (F2) are easier to check than (A), thus the growth condition in this paper is more popular in handling a singularity in the space variable.