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Positive solutions for a singular fractional nonlocal boundary value problem

Abstract

We investigate a singular fractional differential equation with an infinite-point fractional boundary condition, where the nonlinearity \(f(t,x)\) may be singular at \(x = 0\), and \(g(t)\) may also have singularities at \(t= 0\) or \(t=1\). We establish the existence of positive solutions using the fixed point index theory in cones.

Introduction

We consider the existence of positive solutions for the following fractional nonlocal boundary value problem:

$$ \textstyle\begin{cases} D^{\alpha }_{0^{+}}x(t)+\lambda g(t)f(t,x(t))=0,& t\in (0,1), \\ x(0)=x'(0)=\cdots =x^{(n-2)}(0)=0,& D^{\beta }_{0^{+}}x(1)=\sum_{i=1}^{\infty }\alpha_{i}D^{\gamma }_{0^{+}}x(\xi_{i}), \end{cases} $$
(1.1)

where \(\lambda >0\) is a parameter, \(D^{\alpha }_{0^{+}}\), \(D^{\beta } _{0^{+}}\), and \(D^{\gamma }_{0^{+}}\) denote the Riemann–Liouville fractional derivatives, \(2\leq n-1<\alpha \leq n\), \(1\leq \beta \leq n-2, 0\leq \gamma \leq \beta \), \(\alpha_{i}\geq 0\), \(0<\xi_{1}<\xi_{2}< \cdots <\xi_{i-1}<\xi_{i}<\cdots <1\) (\(i=1,2,\ldots\)), and \(\Gamma (\alpha -\gamma )>\Gamma (\alpha -\beta )\sum_{i=1}^{\infty } \alpha_{i}\xi_{i}^{\alpha -\gamma -1}\). The function \(f(t,x)\) may have singularity at \(x = 0\), and \(g(t)\) may be singular at \(t= 0\) and/or \(t=1\).

Fractional differential equations describe many phenomena in various fields of science and engineering [14]. For the development of the fractional differential equations, see [523] and the references therein. Recently, the existence of positive solutions for fractional differential equation multipoint boundary value problems (BVPs) have been studied by many authors; see [2433]. Using the compression expansion fixed point theorem due to Krasnosel’skii, Henderson and Luca [27] studied the fractional BVP

$$ \textstyle\begin{cases} D^{\alpha }_{0^{+}}x(t)+\lambda f(t,x(t))=0,& 0< t< 1, \\ x(0)=x'(0)=\cdots =x^{(n-2)}(0)=0,& D^{\beta }_{0^{+}}x(1)=\sum_{i=1}^{m}\alpha_{i}D^{\gamma }_{0^{+}}x(\xi_{i}), \end{cases} $$
(1.2)

where \(\lambda >0\), \(2\leq n-1<\alpha \leq n\), \(\alpha_{i}\geq 0\), \(0<\xi _{1}<\xi_{2}<\cdots <\xi_{m}<1\) (\(i=1,2,\ldots ,m\)), \(1\leq \beta \leq n-2\), \(0\leq \gamma \leq \beta \), and \(f(t,x)\) may be singular at \(t= 0,1\) and may change sign. In [28], for \(\lambda =1\), the authors investigated the existence and multiplicity of positive solutions for BVP (1.2). In [29, 30], the authors discussed the following infinite-point BVP:

$$ \textstyle\begin{cases} D^{\alpha }_{0^{+}}x(t)+f(t,x(t))=0,& 0< t< 1, \\ x(0)=x'(0)=\cdots =x^{(n-2)}(0)=0,& x^{(i)}(1)=\sum_{j=1}^{\infty }\alpha _{j}x(\xi_{j}), \end{cases} $$
(1.3)

where \(i\in \{1,2,\ldots ,n-2\}\), and \(\sum_{j=1}^{\infty }\alpha_{j} \xi_{j}^{\alpha -1}<(\alpha -1)\cdots (\alpha -i)\). The existence, uniqueness, and multiplicity of positive solutions for BVP (1.3) are established. Qiao and Zhou [31] discussed the singular BVP

$$ \textstyle\begin{cases} D^{\alpha }_{0^{+}}x(t)+g(t) f(t,x(t))=0, & 0< t< 1, \\ x(0)=x'(0)= \cdots =x^{(n-2)}(0)=0,& D^{\beta }_{0^{+}}x(1)=\sum_{i=1}^{\infty }\alpha_{i} x(\xi_{i}), \end{cases} $$
(1.4)

where \(\beta \in [1,\alpha -1]\), and \(\Gamma (\alpha )>\Gamma (\alpha -\beta )\sum_{i=1}^{\infty }\alpha_{i}\xi_{i}^{\alpha -1}\). For more results on the fractional infinite-point BVPs, see [24, 25, 32, 33] and the references therein.

In the present paper, we investigate the existence of positive solutions for the singular fractional infinite-point BVP (1.1) using the fixed point index theory in cones. Note that \(f(t,x)\) may be singular at \(x = 0\) and \(g(t)\) may be singular at \(t= 0\) or \(t=1\).

Preliminaries and lemmas

Definition 2.1

([14])

The Riemann–Liouville fractional integral of order \(\alpha > 0\) of a function \(h: (0,+ \infty )\rightarrow \mathbb{R}\) is given by

$$ I_{0^{+}}^{\alpha }h(t)=\frac{1}{\Gamma (\alpha )} \int_{0}^{t} (t-s)^{ \alpha -1}h(s)\,ds, $$

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

Definition 2.2

([14])

The Riemann–Liouville fractional derivative of order \(\alpha > 0\) of a function \(h: (0,+ \infty )\rightarrow \mathbb{R}\) is given by

$$ D^{\alpha }_{0^{+}}h(t)=\frac{1}{\Gamma (n-\alpha )} \biggl( \frac{d}{dt} \biggr) ^{n} \int_{0}^{t} {\frac{h(s)}{(t-s)^{\alpha -n+1}}}\,ds, $$

where n is the smallest integer not less than α, provided that the right-hand side is defined pointwise on \((0,+\infty )\).

By arguments similar to those in [30, 31], we have the following two lemmas.

Lemma 2.1

Given \(y\in C(0,1)\cap L^{1}(0,1)\), the solution of the BVP

$$ \textstyle\begin{cases} D^{\alpha }_{0^{+}}x(t)+y(t)=0,& t\in (0,1), \\ x(0)=x'(0)=\cdots =x^{(n-2)}(0)=0,& D^{\beta }_{0^{+}}x(1)=\sum_{i=1}^{\infty } \alpha_{i}D^{\gamma }_{0^{+}}x(\xi_{i}), \end{cases} $$

is

$$ x(t)= \int_{0}^{1}G(t,s)y(s)\,ds, $$

where \(G(t,s)\) is the Green’s function given by

$$ G(t,s)=\frac{1}{\Gamma (\alpha )q(0)} \textstyle\begin{cases} q(s)(1-s)^{\alpha -\beta -1}t^{\alpha -1}-q(0)(t-s)^{\alpha -1},& 0\leq s\leq t\leq 1, \\ q(s)(1-s)^{\alpha -\beta -1}t^{\alpha -1},& 0\leq t\leq s\leq 1, \end{cases} $$

and

$$ q(s)=\frac{1}{\Gamma (\alpha -\beta )}-\frac{1}{\Gamma (\alpha - \gamma )}\sum_{s\leq \xi_{i}} \alpha_{i} \biggl( \frac{\xi_{i}-s}{1-s} \biggr) ^{\alpha -\gamma -1}(1-s)^{\beta -\gamma }. $$

Lemma 2.2

The functions q and G given in Lemma 2.1 have the following properties:

  1. (i)

    \(q\in C([0,1],(0,+\infty ))\) is nondecreasing;

  2. (ii)

    \(G(t,s)\in C([0,1]\times [0,1],[0,+\infty ))\);

  3. (iii)

    \(p(t)G(1,s)\leq G(t,s)\leq G(1,s), t,s\in [0,1]\), where \(p(t)=t^{\alpha -1}\).

Set \(E=C[0,1]\) and \(\Vert x \Vert =\sup_{t\in [0,1]}\vert x(t) \vert \). We define the cones

$$ P= \bigl\{ x\in E: x(t)\geq 0, t\in [0,1] \bigr\} \quad \mbox{and}\quad K= \bigl\{ x \in P: x(t)\geq p(t)\Vert x \Vert , t\in [0,1] \bigr\} . $$

For \(0< r<+\infty \), denote \(K_{r}=\{x\in K:\Vert x \Vert < r\}\), \(\partial K_{r}= \{x\in K:\Vert x \Vert =r\}\) and \(\overline{K}_{r}=\{x\in K:\Vert x \Vert \leq r\}\). Define the operators \(A:\overline{K}_{R}\backslash K_{r}\rightarrow P\) and \(L: E\rightarrow E\) by

$$\begin{aligned}& Ax(t)=\lambda \int_{0}^{1}G(t,s)g(s)f \bigl(s,x(s) \bigr)\,ds,\quad t\in [0,1], \\& Lx(t)=\int_{0}^{1}G(t,s)g(s)x(s)\,ds,\quad t\in [0,1]. \end{aligned}$$

Clearly, \(L: K\rightarrow K \) is a completely continuous linear operator. Moreover, if x is a fixed point of A, then x is a solution of BVP (1.1).

We further assume that:

\((H_{1})\) :

\(g\in C((0,1), [0,\infty ))\) and \(0<\int_{0}^{1} G(1,s)g(s)\,ds <+\infty \).

\((H_{2})\) :

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

$$ \lim_{m\rightarrow \infty }\sup_{u\in \overline{K}_{R}\backslash K _{r}} \int_{D(m)}g(s)f \bigl(s,x(s) \bigr)\,ds=0, $$

where \(D(m)=[0,\frac{1}{m}] \cup [\frac{m-1}{m},1]\).

We obtain the following lemma using proofs similar to those in [34, 35].

Lemma 2.3

Suppose that \((H_{1})\) and \((H_{2})\) hold. Then \(A: \overline{K}_{R}\backslash K_{r}\rightarrow K\) is completely continuous.

By Lemma 2.2 we can show that the spectral radius \(r(L)>0\); see, for example, Lemma 2.5 of [36]. Using the Krein–Rutman theorem (see Theorem 19.2 on p. 226 of [37]), we have the following result.

Lemma 2.4

Suppose that \((H_{1})\) and \((H_{2})\) are satisfied. Then the first eigenvalue of L is \(\lambda_{1}=(r(L))^{-1}>0\), and there exists a positive eigenfunction \(\varphi_{1}\) such that \(\varphi_{1}=\lambda_{1} L \varphi_{1}\).

The main tool in the paper is the following fixed point index theorem.

Lemma 2.5

([38])

Let K be a cone in a Banach space E, and let \(T:\overline{K}_{r} \rightarrow K\) be a completely continuous operator.

  1. (i)

    If there exists \(u_{0}\in K\backslash \{\theta \}\) such that \(u-Tu\neq\mu u_{0}\) for any \(u\in \partial K_{r}\) and \(\mu \geq 0\), then \(i(T,K_{r},K)=0\).

  2. (ii)

    If \(Tu\neq\mu u\) for any \(u\in \partial K_{r}\) and \(\mu \geq 1\), then \(i(T,K_{r},K)=1\).

Main results

Theorem 3.1

Suppose that \((H_{1})\) and \((H_{2})\) are satisfied. If

$$ 0\leq f^{\infty }:=\limsup_{x\to +\infty }\max_{t\in [0,1]} \frac{f(t,x)}{x} < \lambda_{1}< f_{0}:=\liminf _{x\to 0}\min_{t\in [0, 1]} \frac{f(t,x)}{x}\leq + \infty , $$

then BVP (1.1) has at least one positive solution for any

$$ \lambda \in \biggl( \frac{\lambda_{1}}{ f_{0}}, \frac{\lambda_{1}}{ f ^{\infty }} \biggr) . $$
(3.1)

Proof

By (3.1) we have \(f_{0}>\frac{\lambda_{1}}{\lambda }\), and there exists \(r_{1}>0\) such that \(f(t,x) \geq \frac{\lambda_{1}}{\lambda }x\) for \(0< x\leq r_{1}\) and \(0 \leq t \leq 1\). For any \(x \in \partial K_{r _{1}}\), we obtain

$$ (Ax) (t)=\lambda \int_{0}^{1} G(t,s)g(s)f \bigl(s,x(s) \bigr)\,ds \geq \lambda_{1}(Lx) (t),\quad t\in [0, 1]. $$

Suppose that \(\varphi_{1}\) is the positive eigenfunction corresponding to \(\lambda_{1}\) and that A has no fixed points on \(\partial K_{r_{1}}\). We claim that

$$ x-Ax\neq\mu \varphi_{1},\quad x\in \partial K_{r_{1}}, \mu\geq 0. $$
(3.2)

Otherwise, there would exist \(x_{1}\in \partial K_{r_{1}}\) and \(\mu_{1}\geq 0\) such that \(x_{1}-Ax_{1}=\mu_{1} \varphi_{1}\). Then \(\mu_{1}> 0\) and \(x_{1}=Ax_{1}+\mu_{1} \varphi_{1}\geq \mu_{1} \varphi _{1}\). Denote \(\overline{\mu }=\sup \{\mu \mid x_{1}\geq \mu \varphi_{1}\}\). Then \(\overline{\mu }\geq \mu_{1}\), \(x_{1}\geq \overline{\mu } \varphi_{1}\), and \(A x_{1}\geq \lambda_{1} \overline{\mu } L\varphi _{1}=\overline{\mu } \varphi_{1}\). Thus

$$ x_{1}=Ax_{1}+\mu_{1} \varphi_{1} \geq \overline{\mu } \varphi_{1}+\mu _{1} \varphi_{1}=(\overline{\mu }+\mu_{1}) \varphi_{1}, $$

which contradicts to the definition of μ̅. It follows from (3.2) and Lemma 2.5(i) that

$$ i(A, K_{r_{1}},K)=0. $$
(3.3)

On the other hand, by (3.1) we have \(f^{\infty }<\frac{\lambda_{1}}{ \lambda }\), and there exist \(r_{2}>r_{1}\) and \(0< \sigma <1\) such that \(f(t,x)\leq \sigma \frac{\lambda_{1}}{\lambda }x\) for \(x\geq r_{2}\) and \(0 \leq t \leq 1\). We define \(L_{1}u= \sigma \lambda_{1}Lu\). Obviously, the linear operator \(L_{1}:E\rightarrow E\) is bounded, and \(L_{1}(K) \subset K\). From the definition of \(\lambda_{1}\) and \(0< \sigma <1\) it follows that

$$ \bigl(r(L_{1}) \bigr)^{-1}=(\sigma \lambda_{1})^{-1} \bigl(r(L) \bigr)^{-1}=\sigma^{-1}>1. $$
(3.4)

Choose \(\varepsilon_{0}=\frac{1}{2}(1-r(L_{1}))\). Then by Gelfand’s formula there exists a natural number \(N\geq 1\) such that \(\Vert L^{k}_{1} \Vert \leq [r(L_{1})+\varepsilon_{0}]^{k}\) for \(k\geq N\). We now define

$$ \Vert x \Vert ^{*}=\sum_{i=1}^{N} \bigl[r(L_{1})+\varepsilon_{0} \bigr]^{N-i} \bigl\Vert L^{i-1}_{1}x \bigr\Vert ,\quad x\in E, $$

where \(L^{0}_{1}=I\) is the identity operator. Since \(L_{1}\) is linear, it is easy to verify that \(\Vert x \Vert ^{*}\) is a norm in E. Let \(M_{0}=\sup_{x\in \partial K_{r_{2}}}\lambda \int_{0}^{1}G(1,s)g(s)f(s,x(s))\,ds\). Then \(M_{0} < + \infty \). We define \(M_{0}^{*}=\Vert M_{0} \Vert ^{*}\) and take \(r_{3}>\max \{r_{2}, 2M_{0}^{*}\varepsilon^{-1}_{0}\}\). Noting that \(\Vert x \Vert ^{*}>[r(L_{1})+\varepsilon_{0}]^{N-1}\Vert x \Vert \), we can find \(r_{4}>r_{3}\) large enough such that \(\Vert x \Vert \geq r_{4}\)and thus \(\Vert x \Vert ^{*}>r_{3}\).

We next prove that

$$ Ax\neq\mu x,\quad x\in \partial K_{r_{4}}, \mu \geq 1. $$
(3.5)

Arguing indirectly, we get that there exist \(x_{2}\in \partial K_{r _{4}}\) and \(\mu_{2} \geq 1\) such that \(Ax_{2}=\mu_{2}x_{2}\). We define \(\widetilde{x}(t)=\min \{x_{2}(t), r_{2}\}\) for \(t\in [0,1]\) and \(H(x_{2})=\{t\in [0,1]: x_{2}(t)>r_{2}\}\). It is easy to see that \(\Vert \widetilde{x} \Vert =r_{2}\). We have \(\widetilde{x}\in \partial K_{r _{2}}\) since \(\widetilde{x}(t)=\min \{x_{2}(t), r_{2}\}\geq \min \{p(t)r _{4}, r_{2}\}\geq p(t)r_{2}\), \(t\in [0,1]\). It follows that

$$\begin{aligned} \mu_{2}x_{2}(t)&=(Ax_{2}) (t) \\ &=\lambda \int_{0}^{1}G(t,s)g(s)f \bigl(s,x_{2}(s) \bigr)\,ds \\ & \leq \lambda \int_{H(x_{2})}G(t,s)g(s)f \bigl(s,x_{2}(s) \bigr)\,ds + \lambda \int_{[0,1]\backslash H(x_{2})}G(1,s)g(s)f \bigl(s,x _{2}(s) \bigr)\,ds \\ & \leq \sigma \lambda_{1} \int_{0}^{1}G(t,s)g(s)x_{2}(s)\,ds + \lambda \int_{0}^{1}G(1,s)g(s)f \bigl(s,\widetilde{x}(s) \bigr)\,ds \\ & \leq (L_{1}x_{2}) (t)+M_{0},\quad t\in [0,1]. \end{aligned}$$

Since \(L_{1}(K)\subset K\), we have \(0\leq (L^{j}_{1}(Ax_{2})(t)) \leq (L^{j}_{1}(L_{1}x_{2}+M_{0})(t))\), \(j=0,1,2,\ldots , N-1\). Then \(\Vert L^{j}_{1}(Ax_{2}) \Vert \leq \Vert L^{j}_{1}(L_{1}x_{2}+M_{0}) \Vert \), \(j=0,1,2, \ldots , N-1\), and hence

$$ \Vert Ax_{2} \Vert ^{*}\leq \sum _{i=1}^{N} \bigl[r(L_{1})+ \varepsilon_{0} \bigr]^{N-i}\bigl\Vert L^{i-1}_{1}(L_{1}x_{2}+M_{0}) \bigr\Vert = \Vert L_{1}x_{2}+M_{0} \Vert ^{*}. $$

Therefore we obtain

$$\begin{aligned} \mu_{2}\Vert x_{2} \Vert ^{*} &=\Vert Ax_{2} \Vert ^{*} \\ &\leq \Vert L_{1}x_{2} \Vert ^{*}+M_{0}^{*} \\ &= \sum _{i=1}^{N} \bigl[r(L_{1})+ \varepsilon_{0} \bigr]^{N-i}\bigl\Vert L^{i}_{1}x_{2} \bigr\Vert +M _{0}^{*} \\ & \leq \bigl[r(L_{1})+\varepsilon_{0} \bigr]\sum _{i=1}^{N-1} \bigl[r(L_{1})+ \varepsilon_{0} \bigr]^{N-i-1}\bigl\Vert L^{i}_{1}x_{2} \bigr\Vert + \bigl[r(L_{1})+\varepsilon_{0} \bigr]^{N} \Vert x_{2} \Vert +M_{0}^{*} \\ & = \bigl[r(L_{1})+\varepsilon_{0} \bigr]\sum _{i=1}^{N} \bigl[r(L _{1})+ \varepsilon_{0} \bigr]^{N-i}\bigl\Vert L^{i-1}_{1}x_{2} \bigr\Vert +M_{0}^{*} \\ & = \bigl[r(L _{1})+\varepsilon_{0} \bigr] \Vert x_{2} \Vert ^{*}+M_{0}^{*} \\ &\leq \bigl[r(L_{1})+ \varepsilon_{0} \bigr]\Vert x_{2} \Vert ^{*}+\frac{\varepsilon_{0}}{2}r_{3} \\ & < \bigl[r(L _{1})+\varepsilon_{0} \bigr] \Vert x_{2} \Vert ^{*}+\frac{\varepsilon_{0}}{2}\Vert x_{2} \Vert ^{*} \\ &= \biggl[ \frac{1}{4}r(L_{1})+ \frac{3}{4} \biggr] \Vert x_{2} \Vert ^{*}. \end{aligned}$$

Thus \(\frac{1}{4}r(L_{1})+\frac{3}{4}\geq 1\), that is, \(r(L_{1}) \geq 1\), which contradicts (3.4). It follows from (3.5) and Lemma 2.5(ii) that

$$ i(A,K_{r_{4}},K)=1. $$
(3.6)

By (3.3), (3.6), and the additivity of the fixed point index we have

$$ i(A,K_{r_{4}}\backslash \overline{K}_{r_{1}},K)=i(A,K_{r_{4}},K)-i(A, K_{r_{1}},K)=1. $$

Therefore A has at least one fixed point \(x^{*}\in K_{r_{4}}\backslash \overline{K}_{r_{1}}\), which is a positive solution of BVP (1.1). □

An example

Let \(\alpha =\frac{7}{2}\), \(\beta =\frac{3}{2}\), \(\gamma =\frac{1}{2}, \alpha_{i}=\frac{2}{i^{2}}\), \(\xi_{i}=1-\frac{1}{i+1} (i=1,2,\ldots )\), \(g(t)=\frac{1}{\sqrt[4]{t(1-t)}}\),\(f(t,x)=\sqrt{2-t+\vert \ln x \vert }\). Consider the following fractional BVP:

$$ \textstyle\begin{cases} D^{\frac{7}{2}}_{0^{+}}x(t)+\lambda \frac{1}{\sqrt[4]{t(1-t)}}\sqrt{2-t+\vert \ln x(t) \vert } =0, & t\in (0,1), \\ x(0)=x'(0)=x''(0)=0,& D^{\frac{3}{2}}_{0^{+}}x(1)=\sum_{i=1}^{\infty }\frac{2}{i^{2}}D^{ \frac{1}{2}}_{0^{+}}x ( 1-\frac{1}{i+1} ) . \end{cases} $$
(4.1)

Direct computation shows that \(\Gamma (\alpha -\beta )=1, \Gamma (\alpha -\gamma )=2\), \(\sum_{i=1}^{\infty }\alpha_{i}\xi_{i} ^{\alpha -\gamma -1}=2 ( \frac{\pi^{2}}{6}-1 ) \), and \(\frac{1}{\Gamma (\alpha -\beta )}-\frac{1}{\Gamma (\alpha -\gamma )} \sum_{i=1}^{\infty }\alpha_{i}\xi_{i}^{\alpha -\gamma -1}\approx 0.355>0\).

Let \(K=\{ x\in C[0,1]: x(t)\geq p(t)\Vert x \Vert , t\in [0,1] \}\), where \(p(t)=t^{\frac{5}{2}}\). For \(x\in \overline{K}_{R}\backslash K_{r}\), we obtain \(\vert \ln x(t) \vert \leq \vert \ln rp(t) \vert +\vert \ln R \vert \). Due to \(\int_{0}^{1}\vert \ln p(t) \vert dt =\frac{5}{2}\), we have \(\lim_{m\rightarrow \infty }\int_{D(m)}\vert \ln p(t) \vert \,dt=0\). Since \(0\leq G(t,s)\leq G(1,s) \leq \frac{1}{\Gamma (\frac{7}{2})(2-\frac{\pi^{2}}{6})}\), it follows that \(\int_{0}^{1}G(1,s)g(s)\,ds\leq \frac{1}{\Gamma (\frac{7}{2})(2-\frac{\pi^{2}}{6})} \int_{0}^{1}g(s)\,ds=\frac{2[\Gamma (\frac{3}{4})]^{2}}{\Gamma (\frac{7}{2})(2-\frac{\pi^{2}}{6})\sqrt{\pi }}\). For \(x\in \overline{K}_{R}\backslash K_{r}\), we have

$$ \int_{0}^{1}f^{2} \bigl(s,x(s) \bigr) \,ds \leq \int_{0}^{1} \bigl(2-s+\vert \ln r \vert + \vert \ln R \vert +\bigl\vert \ln p(s) \bigr\vert \bigr)\,ds=4+\vert \ln r \vert +\vert \ln R \vert . $$

Therefore

$$\begin{aligned} &\lim_{m\rightarrow \infty }\sup_{x\in \overline{K}_{R}\backslash K _{r}} \int_{D(m)}g(s)f \bigl(s,x(s) \bigr)\,ds \\ & \quad \leq \lim_{m\rightarrow \infty }\sup_{x\in \overline{K}_{R}\backslash K _{r}} \biggl( \int_{D(m)}g^{2}(s)\,ds \biggr) ^{\frac{1}{2}} \biggl( \int_{D(m)}f ^{2} \bigl(s,x(s) \bigr)\,ds \biggr) ^{\frac{1}{2}} \\ & \quad \leq \lim_{m\rightarrow \infty }\sqrt{\pi } \biggl( \int_{D(m)} \bigl(2-s+\vert \ln r \vert +\vert \ln R \vert +\bigl\vert \ln p(s) \bigr\vert \bigr)\,ds \biggr) ^{\frac{1}{2}}=0. \end{aligned}$$

Direct computation yields \(f^{\infty }=0\) and \(f_{0}=+\infty \). Using Theorem 3.1, we can conclude that the BVP (4.1) has at least one positive solution for any \(\lambda \in (0,+\infty )\).

Conclusions

We established the existence of positive solutions for the singular fractional differential equation infinite-point BVP (1.1) using the fixed point index theory in cones. Note that the nonlinearity may possess singularities, that is, \(f(t,x)\) may have a singularity at \(x = 0\), and \(g(t)\) may be singular at \(t= 0\) or \(t=1\).

References

  1. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  3. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  4. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

    Book  Google Scholar 

  5. Goodrich, C.S.: Coercive nonlocal elements in fractional differential equations. Positivity 21, 377–394 (2017)

    MathSciNet  Article  Google Scholar 

  6. Shabibi, M., Postolache, M., Rezapour, Sh.: Positive solutions for a singular sum fractional differential system. Int. J. Anal. Appl. 13, 108–118 (2017)

    MATH  Google Scholar 

  7. Shabibi, M., Postolache, M., Rezapour, Sh., Vaezpour, S.M.: Investigation of a multi-singular pointwise defined fractional integro-differential equation. J. Math. Anal. 7, 61–77 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Zhang, X., Liu, L., Wu, Y.: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26–33 (2014)

    MathSciNet  Article  Google Scholar 

  9. Zhang, X., Mao, C., Liu, L., Wu, Y.: Exact iterative solution for an abstract fractional dynamic system model for bioprocess. Qual. Theory Dyn. Syst. 16, 205–222 (2017)

    MathSciNet  Article  Google Scholar 

  10. Zhang, X., Liu, L., Wu, Y., Wiwatanapataphee, B.: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252–263 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Zhang, X., Liu, L., Wu, Y., Lu, Y.: The iterative solutions of nonlinear fractional differential equations. Appl. Math. Comput. 219, 4680–4691 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Cui, Y.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)

    MathSciNet  Article  Google Scholar 

  13. Cui, Y., Ma, W., Sun, Q., Su, X.: New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal., Model. Control 23, 31–39 (2018)

    MathSciNet  Article  Google Scholar 

  14. Yan, F., Zuo, M., Hao, X.: Positive solution for a fractional singular boundary value problem with p-Laplacian operator. Bound. Value Probl. 2018, Article ID 51 (2018)

    MathSciNet  Article  Google Scholar 

  15. Zou, Y., He, G.: On the uniqueness of solutions for a class of fractional differential equations. Appl. Math. Lett. 74, 68–73 (2017)

    MathSciNet  Article  Google Scholar 

  16. Hao, X.: Positive solution for singular fractional differential equations involving derivatives. Adv. Differ. Equ. 2016, Article ID 139 (2016)

    MathSciNet  Article  Google Scholar 

  17. Hao, X., Wang, H., Liu, L., Cui, Y.: Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator. Bound. Value Probl. 2017, Article ID 182 (2017)

    MathSciNet  Article  Google Scholar 

  18. Zuo, M., Hao, X., Liu, L., Cui, Y.: Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Bound. Value Probl. 2017, Article ID 161 (2017)

    MathSciNet  Article  Google Scholar 

  19. Hao, X., Zuo, M., Liu, L.: Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities. Appl. Math. Lett. 82, 24–31 (2018)

    MathSciNet  Article  Google Scholar 

  20. Zhang, X., Zhong, Q.: Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions. Fract. Calc. Appl. Anal. 20, 1471–1484 (2017)

    MathSciNet  Article  Google Scholar 

  21. Zhang, X., Zhong, Q.: Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables. Appl. Math. Lett. 80, 12–19 (2018)

    MathSciNet  Article  Google Scholar 

  22. Hao, X., Sun, H., Liu, L.: Existence results for fractional integral boundary value problem involving fractional derivatives on an infinite interval. Math. Meth. Appl. Sci. (2018). https://doi.org/10.1002/mma.5210

  23. Hao, X., Wang, H.: Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions. Open Math. 16, 581–596 (2018)

    MathSciNet  Article  Google Scholar 

  24. Guo, L., Liu, L., Wu, Y.: Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions. Nonlinear Anal., Model. Control 21, 635–650 (2016)

    MathSciNet  Article  Google Scholar 

  25. Guo, L., Liu, L., Wu, Y.: Existence of positive solutions for singular higher-order fractional differential equations with infinite-point boundary conditions. Bound. Value Probl. 2016, Article ID 114 (2016)

    MathSciNet  Article  Google Scholar 

  26. Salen, H.A.H.: On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies. J. Comput. Appl. Math. 224, 565–572 (2009)

    MathSciNet  Article  Google Scholar 

  27. Henderson, J., Luca, R.: Existence of positive solutions for a singular fractional boundary value problem. Nonlinear Anal., Model. Control 22, 99–114 (2017)

    MathSciNet  Article  Google Scholar 

  28. Pu, R., Zhang, X., Cui, Y., Li, P., Wang, W.: Positive solutions for singular semipositone fractional differential equation subject to multipoint boundary conditions. J. Funct. Spaces 2017, Article ID 5892616 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Zhai, C., Wang, L.: Some existence, uniqueness results on positive solutions for a fractional differential equation with infinite-point boundary conditions. Nonlinear Anal., Model. Control 22, 566–577 (2017)

    MathSciNet  Article  Google Scholar 

  30. Zhang, X.: Positive solutions for a class of singular fractional differential equations with infinite-point boundary value conditions. Appl. Math. Lett. 39, 22–27 (2015)

    MathSciNet  Article  Google Scholar 

  31. Qiao, Y., Zhou, Z.: Existence of positive solutions of singular fractional differential equations with infinite-point boundary conditions. Adv. Differ. Equ. 2017, Article ID 8 (2017)

    MathSciNet  Article  Google Scholar 

  32. Zhong, Q., Zhang, X.: Positive solution for higher-order singular infinite-point fractional differential equation with p-Laplacian. Adv. Differ. Equ. 2016, Article ID 11 (2016)

    MathSciNet  Article  Google Scholar 

  33. Li, B., Sun, S., Sun, Y.: Existence of solutions for fractional Langevin equation with infinite-point boundary conditions. J. Appl. Math. Comput. 53, 683–692 (2017)

    MathSciNet  Article  Google Scholar 

  34. Hao, X., Liu, L., Wu, Y., Sun, Q.: Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions. Nonlinear Anal. 73, 1653–1662 (2010)

    MathSciNet  Article  Google Scholar 

  35. Wang, Y., Liu, L., Wu, Y.: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 74, 3599–3605 (2011)

    MathSciNet  Article  Google Scholar 

  36. Webb, J.R.L., Lan, K.Q.: Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topol. Methods Nonlinear Anal. 27, 91–115 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Deimling, K.: Nonlinear Functional Analysis. Spring, Berlin (1985)

    Book  Google Scholar 

  38. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)

    MATH  Google Scholar 

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Supported financially by the National Natural Science Foundation of China (11501318, 11871302), the China Postdoctoral Science Foundation (2017M612230), and the Natural Science Foundation of Shandong Province of China (ZR2017MA036).

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Correspondence to Xinan Hao.

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Zhang, L., Sun, Z. & Hao, X. Positive solutions for a singular fractional nonlocal boundary value problem. Adv Differ Equ 2018, 381 (2018). https://doi.org/10.1186/s13662-018-1844-z

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  • DOI: https://doi.org/10.1186/s13662-018-1844-z

MSC

  • 26A33
  • 34B08
  • 34B10
  • 34B16
  • 34B18

Keywords

  • Positive solution
  • Singular
  • Infinite-point fractional boundary condition
  • Fixed point index