Abstract
In this paper, we consider a class of singular fractional differential equations with infinite-point boundary conditions. The fractional orders are involved in the nonlinearity of the boundary value problem, and the nonlinearity is allowed to be singular with respect to not only the time variable but also to the space variable. Firstly, we give Green’s function and establish its properties. Then, we utilize the sequential technique and regularization to investigate the existence of positive solutions. Finally, we give an example of application of our result.
Similar content being viewed by others
1 Introduction
In this paper, we consider the following class of nonlinear singular fractional differential equations:
where \(\alpha,\mu_{i}\in\mathbb{R}^{1}_{+}=[0, +\infty)\), \(n\in \mathbb{N}\) (the set of natural numbers), and \(n-1<\alpha\leq n\), \(n\geq4\), \(i-1<\mu_{i}\leq i\) (\(i=1,2,\ldots,n-2\)), \(f(t,x_{1},x_{2},\ldots ,x_{n-1})\) may have singularity at \(t=0,1\), \(x_{i}=0\) (\(i=1,2,\ldots ,n-1\)), \(0<\xi_{1}<\xi_{2}<\cdots<\xi_{j}<\cdots<1\), \(\eta_{j}>0\) (\(j=1,2,\ldots\)), \(\sum_{j=1}^{\infty}\eta_{j}\xi_{j}^{\alpha-1}<\Theta\), \(\Theta =(\alpha-1)(\alpha-2)\cdots(\alpha-n+2)\), and \(D_{0^{+}}^{\alpha}u\), \(D^{\mu _{i}}_{0^{+}}u\) (\(i=1,2,\ldots,n-2\)) are the Riemann-Liouville derivatives.
Boundary value problems for nonlinear fractional differential equations arise from the studies of complex problems in many disciplinary areas such as fluid flows, electrical networks, rheology, biology chemical physics, and so on. Fractional-order models have been shown to be more accurate than integer-order models, and in applications of these models, it is important to theoretically establish conditions for the existence of positive solutions. In recent years, many authors investigated the existence of positive solutions for fractional equation boundary value problems (see [1–24] and the references therein), and a great deal of results have been developed for differential and integral boundary value problems. In [14], the author considered the following fractional differential equation:
where \(\alpha\in\mathbb{R}^{1}_{+}\), \(n\in\mathbb{N}\), \(n-1<\alpha \leq n\), \(n\geq3\), \(\alpha_{j}\geq0\), \(0<\xi_{1}<\xi_{2}<\cdots<\xi _{j-1}<\xi _{j}<\cdots<1\) (\(j=1,2,\ldots\)), and \(D_{0^{ +}}^{\alpha}\) is the standard Riemann-Liouville derivative, and \(f\in C((0, 1)\times(0,+\infty), \mathbb{R}^{1}_{+})\) allows singularities with respect to both time (\(t=0,1\)) and space variables (\(u=0\)). The author established the existence and multiplicity of positive solutions. In [15], the authors investigated the fractional differential equation
where \(\alpha,v,\mu\in\mathbb{R}^{1}_{+}\), \(3<\alpha\leq4\), \(0< v\leq1\), \(0<\mu\leq1\) are real numbers, \(D^{\alpha}_{0+}\), \(D^{v}_{0^{+}}\), and \(D^{\mu}_{0^{+}}\) are the Riemann-Liouville fractional derivatives, \(f(t,x,y,z)\) is a Carathéodory function singular at \(x,y,z=0\). The authors obtained the existence and multiplicity of positive solutions by means of Krasnoselskii’s fixed point theorem. As there do not exists \(0 < L < M\) such that (3.13) of [15] holds, the results of the multiple solutions are not correct in [15]. In [16], the authors investigated the fractional differential equation
where \(\alpha,\mu\in\mathbb{R}^{1}_{+}\), \(1<\alpha\leq2\), \(\mu >0\) are real numbers, \(\alpha-\mu\geq1\), \(D^{\alpha}_{0+}\) and \(D^{\mu}_{0^{+}}\) are the Riemann-Liouville fractional derivatives, f is a Carathéodory function, and \(f(t,x,y)\) is singular at \(x=0\). The authors obtained the existence of positive solutions by means of the Krasnoselskii’s fixed point theorem. In [17], the author investigated the fractional differential equation
where \(\alpha,\mu\in\mathbb{R}^{1}_{+}\), \(2<\alpha\leq3\), \(0<\mu \leq1\) are real numbers, \(D^{\alpha}_{0+}\) and \(D^{\mu}_{0^{+}}\) are the Riemann-Liouville fractional derivatives, f is a Carathéodory function, and \(f(t,x,y,z)\) is singular at \(x,y,z=0\). The authors obtained the existence of positive solutions by means of the Krasnoselskii’s fixed point theorem. In [18], the author investigated the singular problem
where \(\alpha\in\mathbb{R}^{1}_{+}\), \(n-1<\alpha\leq n\), \(n\geq2\), the nonlinear function \(f(t,x_{1},x_{2},\ldots,x_{n-1})\) may be singular at \(x_{i}=0\) (\(i=1,2,\ldots,n-1\)), and \(q(t)\) may be singular at \(t=0\). The existence results of positive solutions are obtained by a fixed point theorem for a mixed monotone operator.
Motivated by the results mentioned, in this paper, we utilize the sequential technique and regularization to investigate the existence of positive solutions of BVP (1.1), where \(u\in C^{n-2}[0,1]\cap C^{n-1}(0,1)\) is said to be a positive solution of BVP (1.1) if and only if u satisfies (1.1) and \(u(t)>0\) for any \(t\in(0, 1]\). By using the sequential technique and regularization on a cone, some new existence results are obtained for the case where the nonlinearity is allowed to be singular with respect to both time and space variables. We emphasize here that our work presented in this paper has various new features. Firstly, we study singular nonlinear differential equation boundary value problems, that is, \(f(t,x_{1},x_{2},\ldots,x_{n-1})\) may have singularity at \(t=0,1\) and \(x_{i}=0\) (\(i=1,2,\ldots,n-1\)), which leads to many difficulties in analysis. Secondly, compared with [15–17], we complete the proof without the need of imposing the third condition of the Carathéodory conditions, that is, the condition \(|f(t,x_{1},x_{2},\ldots,x_{n-1})|\leq\varphi_{H}(t)\) is successfully removed, and, at the same time, the condition
is extended to
Thirdly, a special cone in a special space is established to overcome the difficulties caused by the singularity. Finally, values at infinite points are involved in the boundary conditions, and the fractional orders are involved in the nonlinearity of the boundary value problem (1.1).
For convenience, we list some conditions to be used throughout the paper.
- (H0):
-
f satisfies the local Carathéodory condition on \([0,1]\times(0,\infty)^{n-1}\) if
-
(1)
\(f(t,x_{1},x_{2},\ldots,x_{n-1}):[0,1]\rightarrow\mathbb{R}^{1}_{+}\) is measurable for all \((x_{1},x_{2},\ldots,x_{n-1})\in(0,+\infty)^{n-1}\);
-
(2)
\(f(t,x_{1},x_{2},\ldots,x_{n-1}):(0,+\infty)^{n-1}\rightarrow \mathbb{R}^{1}_{+} \) is continuous for a.e. \(t\in[0,1]\).
-
(1)
- (H1):
-
There exists a constant \(C>0\) such that, for a.e. \(t\in[0,1]\) and for any \((x_{1},x_{2},\ldots ,x_{n-1})\in(0,\infty)^{n-1}\),
$$ f(t,x_{1},x_{2},\ldots,x_{n-1})\geq C. $$(1.2) - (H2):
-
For all \((x_{1},x_{2},\ldots,x_{n-1})\in(0,\infty)^{n-1}\) and a.e. \(t\in[0,1]\),
$$f(t,x_{1},x_{2},\ldots,x_{n-1})\leq\beta(t) p(x_{1},x_{2},\ldots ,x_{n-1})+ \gamma(t)h(x_{1},x_{2},\ldots,x_{n-1}), $$where \(\beta,\gamma\in L^{1} ((0,1),(0,+\infty) )\), \(p\in C((0,\infty)^{n-1},\mathbb{R}^{1}_{+})\) is nonincreasing with respect to all arguments, \(h\in(\mathbb{R}^{n-1}_{+},\mathbb{R}^{1}_{+})\) is nondecreasing with respect to all arguments, and
$$\begin{aligned}& \int^{1}_{0}\beta(s)p \biggl(Ms^{\alpha-1}, \frac{(n-2-\mu _{1})M}{n!}s^{n-1-\mu_{1}},\ldots, \frac{(n-2-\mu_{n-2})M}{3!}s^{n-1-\mu_{n-2}} \biggr)\,ds< \infty, \\& \quad M=\frac{ C}{(\alpha-n+2)\Gamma(\alpha+1)}. \end{aligned}$$ - (H3):
-
$$\limsup_{x\rightarrow\infty}\frac{h(x,x,\ldots,x)}{x}=\lambda < \frac {\Gamma(n-1-\mu_{n-2})}{e\|\gamma\|_{1}}, $$
where \(e=\frac{\Theta}{P(0)\Gamma(\alpha-n+2)}\).
The main result of this paper is as follows.
Theorem 1.1
If (H0)-(H3) hold, then problem (1.1) has a positive solution x, and for \(t\in[0,1]\), we have
where M is defined by (H2).
In order to overcome the singularity, we utilize the sequential technique and regularization to testify the existence of positive solutions for problem (1.1). Next, for any \(m\in\mathbb {N}\), we define \(X_{m}:\mathbb{R}^{1}\rightarrow \mathbb{R}^{1}\) and \(f_{m}:[0,1]\times\mathbb{R}^{n-1}_{+}\rightarrow\mathbb {R}^{1}\) by the following formulas:
for all \((x_{1},x_{2},\ldots,x_{n-1})\in\mathbb{R}^{n-1}\) and a.e. \(t\in[0,1]\),
Then condition (H1) gives that \(f_{m}\) satisfies the local Carathéodory condition on \([0,1]\times\mathbb{R}^{n-1}_{+}\) and \(f_{m}(t,x_{1},x_{2},\ldots,x_{n-1})\geq C\) for a.e. \(t\in[0,1]\) and all \((x_{1},x_{2},\ldots,x_{n-1})\in\mathbb{R}^{n-1}_{+}\). Condition (H2) provides that
for a.e. \(t\in[0,1]\) and all \((x_{1},x_{2},\ldots,x_{n-1})\in\mathbb {R}^{n-1}_{+}\).
Next, we discuss the regular fractional differential equation
2 Preliminaries and lemmas
For convenience of the reader, we first present some basic definitions and lemmas. These definitions can be found in the recent literature such as [19, 20]. In this paper, \(\|x\|_{1}=\int^{1}_{0}|x(t)|\, dt\) is the norm in \(L^{1}[0,1]\), \(\|x\|=\max\{|x(t)|:t\in[0,1]\}\) is the norm in \(C[0,1]\), and
is the norm in \(E=C^{n-2}[0,1]\); \(\mathit{AC}^{k}[0,1]\) (\(k=0,1,2,\ldots\)) is the space of absolutely continuous functions having absolutely continuous kth-order derivatives on \([0,1]\).
Definition 2.1
The Riemann-Liouville fractional integral of order \(\alpha>0\) of a function \(y:(0,\infty)\rightarrow \mathbb{R}^{1}\) is given by
provided that the right-hand side is pointwise defined on \((0,\infty)\).
Definition 2.2
The Riemann-Liouville fractional derivative of order \(\alpha>0\) of a continuous function \(y:(0,\infty)\rightarrow\mathbb{R}^{1}\) is given by
with \(n=[\alpha]+1\), where \([\alpha]\) denotes the integer part of α, provided that the right-hand side is pointwise defined on \((0,\infty)\).
Lemma 2.1
[17]
We have
where \([\alpha]\) is the least integer greater than or equal to α, and \(\mathit{AC}^{0}[0,1]=\mathit{AC}[0,1]\).
Lemma 2.2
[21]
If \(x\in L^{1}[0,1]\) and \(\alpha+\beta\geq1\), then \((I^{\alpha}_{0^{+}}I^{\beta}_{0^{+}}x )(t)= (I^{\alpha+\beta}_{0^{+}}x )(t)\) for all \(t\in[0,1]\), that is,
Lemma 2.3
[22]
Suppose that \(\alpha>0\). If \(x\in C(0,1]\) and \(D^{\alpha}_{0^{+}}x\in L^{1}[0,1]\), then
for \(t\in(0,1]\), where \(n=[\alpha]+1\) and \(c_{k}\in\mathbb{R}^{1}\) (\(k=1,2,\ldots,n\)).
Lemma 2.4
Suppose that \(i-1<\mu_{i}\leq i\) (\(i=1,2,\ldots ,n-2\)) and \(u\in C^{n-2}[0,1]\), \(u^{(i)}(0)=0\) (\(i=0,1,2,\ldots,n-3\)). Then \(D^{\mu_{i}}_{0^{+}}u\in C[0,1]\) (\(i=1,2,\ldots,n-2\)), and
Proof
By integration by parts we get
So
Hence, we have
Further, by the continuity of
we get that \(D^{\mu_{i}}_{0^{+}}u(t)\) (\(i=1,2,\ldots,n-2\)) is continuous on \([0,1]\). □
Lemma 2.5
Given \(g\in C(0,1)\cap L^{1}(0,1)\),
is the unique solution in \(C^{n-2}[0,1]\cap C^{n-1}(0,1)\) of the equation
where
and
Proof
Applying Lemma 2.3, we can reduce (2.2) to the equivalent integral equation
for \(C_{1}, C_{2},\ldots, C_{n}\in\mathbb{R}^{1}\). By Lemma 2.3 we have that
is a solution of (2.2) in \(C(0,1]\). Since \(u(0)=u'(0)=\cdots =u^{(n-2)}(0)=0\), we have \(C_{2}=C_{3}=\cdots=C_{n}=0\), but \(C_{1}\neq0\), and thus
By Lemma 2.1 we have \(I_{0^{+}}^{\alpha}g\in \mathit{AC}^{n-2}[0,1]\), so that
is a solution of (2.2) in the space \(\mathit{AC}^{n-2}[0,1]\). Taking the derivative step by step for (2.4), we have
On the other hand, the equality \(u^{(n-2)}(1)=\sum_{j=1}^{\infty}\eta _{j}u(\xi_{j})\), combined with
gives
where
Hence,
where
and
□
Lemma 2.6
The Green function \(G(t,s)\) defined in Lemma 2.5 has the following properties:
-
(1)
$$\frac{\partial^{j}}{\partial t^{j}}G(t,s)\in C \bigl([0,1]\times [0,1] \bigr),\quad j=0,1,2, \ldots,n-2; $$
-
(2)
$$0\leq\frac{\partial^{j}}{\partial t^{j}}G(t,s)\leq\frac{\Theta }{\Gamma (\alpha-j)P(0)},\quad (t,s)\in[0,1] \times[0,1], j=0,1,2,\ldots,n-2; $$
-
(3)
$$\int_{0}^{1}\frac{\partial^{j}}{\partial t^{j}}G(t,s)\,ds\geq \frac{ t^{\alpha-j-1}}{(\alpha-n+2)\Gamma(\alpha-j+1)}, \quad t\in[0,1], j=0,1,2,\ldots,n-3 $$
and
$$\int_{0}^{1}\frac{\partial^{n-2}}{\partial t^{n-2}}G(t,s)\,ds\geq \frac{ t(1-t)}{\Gamma(\alpha-n+3)}, \quad t\in[0,1]. $$
Proof
By calculating the derivative we get
for \(j=0,1,2,\ldots,n-2\).
(1) For \(n-1<\alpha\leq n\), \(j\leq n-2\), we have \(\alpha-j>1\). Hence, by (2.5) we have that \(\frac{\partial^{j}}{\partial t^{j}}G(t,s)\) (\(j=0,1,2,\ldots,n-2\)) are continuous on \([0,1]\times[0,1]\), and so (1) holds.
(2) By direct calculation we get \(P'(s)\geq0\), \(s\in[0,1]\). Thus, \(P(s)\) is nondecreasing with respect to \(s\in[0,1]\), and we easily get
On the other hand, \(P(s)\) is nondecreasing on \([0,1]\), so we have
Hence,
Next, we will prove that
Since \(\frac{1-s}{t-s}\) is increasing with respect to s on \((0,t)\), we get that \(\frac{1-s}{t-s}>\frac{1}{t}\). For \(0\leq s\leq t\leq1\), we get
On the other hand, for \(0\leq t\leq s\leq1\), we get
Hence,
so the proof of (2) is completed. We further prove (3).
(3) For \(j=0,1,2,\ldots,n-3\), \(n-j\geq3\), we have
Changing j of (2.5) by \(n-2\), we have
Hence, for \(t\in[0,1]\), we have
It is clear that
The proof of Lemma 2.6 is completed. □
3 Auxiliary regular problem
Let \(E=C^{n-2}[0,1]\) and define the cone K in E as
By Lemma 2.4 and (2.1) we have
For any \(m\in\mathbb{N}\), define the operator \(Q_{m}:K\rightarrow E\) as follows:
Lemma 3.1
Let (H0)-(H3) hold. Then, for any \(m\in\mathbb{N}\), \(Q_{m}: K\rightarrow K\) is a completely continuous operator.
Proof
First, we show that \(Q_{m}:K\rightarrow K\). Given \(u\in K\), by Lemma 2.6 we get that
are nonnegative and continuous on \([0,1]\times[0,1]\) and \(G(0,s)=0\) for \(s\in[0,1]\). So we have
As a result, \(Q_{m}:K\rightarrow K\).
In order to prove that \(Q_{m}\) is a continuous operator, let \(\{u_{k}\} \subset K\) be a convergent sequence. Suppose that \(\lim_{k\rightarrow \infty}u_{k}=u\in K\). Then
uniformly for \(t\in[0,1]\). For \(i-1<\mu_{i}\leq i\) (\(i=1,2,\ldots,n-2\)) and \(t\in[0,1]\), we get
so we get
uniformly for \(t\in[0,1]\). Moreover, \(\{u_{k}\}\subset K\) is a convergent sequence. There exists \(r>0\) such that \(\|u_{k}\|_{2}\leq r\) (\(k\in\mathbb{N}\)). Then \(\| u^{(j)}_{k}\|\leq r\) (\(j=0,1,2,\ldots,n-2\); \(k\in\mathbb{N}\)). For \(i-1<\mu_{i}\leq i\) (\(i=1,2,\ldots,n-2\)), by (2.1), for any \(t\in [0,1]\), we have
Let
For \(s\in[0,1]\setminus\Gamma\), where \(\operatorname{mes}(\Gamma)=0\), \(f_{m}(s,x_{1},x_{2},\ldots,x_{n-1})\) is continuous on \(\mathbb {R}^{n-1}_{+} \) with respect to \(x_{i}\), so \(f_{m}(s,x_{1},x_{2},\ldots ,x_{n-1})\) is uniformly continuous with respect to \(x_{i}\) on
Hence, for any \(\varepsilon>0\), there exists \(\delta>0\) such that, for any \(x_{1}^{1}, x_{1}^{2}\in[0,r], x_{2}^{1}, x_{2}^{2}\in [0,\frac{r}{\Gamma(n-1-\mu_{1})}], \ldots, x_{n-1}^{1}, x_{n-1}^{2}\in[0,\frac{r}{\Gamma(n-1-\mu_{n-2})}]\), \(|x_{1}^{1}-x_{1}^{2}|<\delta, |x_{2}^{1}-x_{2}^{2}|<\delta, \ldots , |x_{n-1}^{1}-x_{n-1}^{2}|<\delta\), we have
Since \(\|u_{k}-u\|_{2}\rightarrow0\), for the above \(\delta>0\), there exists \(N\in\mathbb{N}\) such that, for \(k>N\),
Therefore, for \(k>N\), by (3.5) we have
It follows from (3.6) that
By (1.4) we have
By the integrability of \(\beta(t)\), \(\gamma(t)\) on \([0,1]\) we get that \(\varphi_{r}\in L^{1}(0,1)\), and by (3.7) and (3.8) we have
It follows from the relations in Lemma 2.6 and the Lebesgue dominated convergence theorem that, for any \(m\in\mathbb{N}\),
Hence, for any \(m\in\mathbb{N}\), we have
uniformly for \(t\in[0,1]\). Therefore, for any \(m\in\mathbb{N}\), \(Q_{m}\) is a continuous operator.
Now, for any bounded set \(D\subset K\), we need to prove that \(\{ Q_{m}(D)\}\) is relatively compact in E. In order to apply the Arzelà-Ascoli theorem, we have to prove that \(\{Q_{m}(D)\}\) is bounded in E and that,for any \(m\in\mathbb{N}\), \(\{ (Q_{m}(D))^{(n-2)}(t) \}\) is equicontinuous on \([0,1]\). By the boundedness of \(D\subset K\) there exists a positive number \(R>0\) such that
Then (3.3) means that
Put ρ as in (3.4) and \(0\leq\rho(t)\leq\varphi_{R}(t)\). Then, for \(u\in D\), we have
which shows that, for any \(m\in\mathbb{N}\), \(\{Q_{m}(D)\}\) is bounded in E. Moreover, for \(0\leq t_{1}\leq t_{2}\leq1\) and \(u\in D\), we have
where e is from (H3). Since \((t-s)^{\alpha-n+1}\) is uniformly continuous on \([0,1]\times[0,1]\) and \(t^{\alpha-n+1}\) is uniformly continuous on \([0,1]\), for any \(\varepsilon>0\), there exists \(\delta>0\) such that, for \(0\leq t_{1}\leq t_{2}\leq1\), \(t_{2}-t_{1}<\delta\), \(0< s\leq t_{1}\),
Consequently, for all \(u\in D\), \(0\leq t_{1}\leq t_{2}\leq1\), and \(t_{2}-t_{1}<\min \{\delta,\sqrt[\alpha-n+1]{\varepsilon } \}\), we have the inequality
Hence, for any \(m\in\mathbb{N}\), \(\{(Q_{m}(D))^{(n-2)}(t) \}\) is equicontinuous on \([0,1]\). Therefore, for any \(m\in\mathbb{N}\), \(Q_{m}: K\rightarrow K\) is a completely continuous operator. □
To prove the main results, we need the following well-known fixed point theorem.
Lemma 3.2
[20]
Let K be a positive cone in a Banach space E, \(\Omega_{1}\) and \(\Omega_{2}\) be two bounded open sets in E such that \(\theta\in\Omega_{1}\) and \(\overline{\Omega}_{1}\subset \Omega _{2}\), and \(A:K\cap(\overline{\Omega}_{2}\setminus\Omega _{1})\rightarrow K\) be a completely continuous operator, where θ denotes the zero element of E. Suppose that one of the following two conditions holds:
-
(i)
\(\|Au\|\leq\|u\|\), \(\forall u\in K\cap\partial\Omega_{1}\); \(\|Au\| \geq\| u\|\), \(\forall u\in K\cap\partial\Omega_{2}\);
-
(ii)
\(\|Au\|\geq\|u\|\), \(\forall u\in K\cap\partial\Omega_{1}\); \(\|Au\| \leq \|u\|\), \(\forall u\in K\cap\partial\Omega_{2} \).
Then A has a fixed point in \(P\cap(\overline{\Omega}_{2}\setminus \Omega_{1})\).
Theorem 3.1
Let (H0)-(H3) hold. Then problem (1.5) has a solution \(x_{m}\in K\), and
where M is defined by (H2).
Proof
By Lemma 3.1, \(Q_{m}:K\rightarrow K\) is a completely continuous operator. Then by (1.2) and Lemma 2.6 we have
and hence, \(\|Q_{m}u\|\geq M\) and \(\|Q_{m}u\|_{2}\geq M\) for \(u\in K\). Let \(\Omega_{1}=\{u\in E:\|u\|_{2}< M\}\). Then
Let \(W_{m}=p (\frac{1}{m},\frac{1}{m},\ldots,\frac{1}{m} )\). For any \(u\in K\) and \(t\in[0,1]\), by Lemma 2.6 and (1.4) we have
where e is from (H3).
For any \(u\in K\) such that \(\|u\|_{2}\leq S\) (\(S>M\)), by (3.3) and (3.11) we have
By (H3), taking \(\overline{\lambda}>0\) such that
we have that there exists \(G>M+1\) such that, for any \(x>G\),
Taking
we have \(\frac{S}{\Gamma(n-1-\mu_{n-2})}+1>G\) and \(S>M\). Let \(\Omega _{2}= \{u\in E:\|u\|_{2}< S \}\). Then, for any \(u\in K\cap \partial\Omega_{2}\), by (3.12) and (3.13) we get
Hence,
By Lemma 3.2 we get that the operator \(Q_{m}\) has a fixed point in \(K\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\), and, as a result, \(x_{m}\) is a solution of problem (1.5). Since \(x_{m}\) is a solution of problem (1.5),
and \(x_{m}\) satisfies \(x_{m}(t)\geq Mt^{\alpha-1}\). In addition, Lemma 2.6 and (1.2) imply
By (2.1) we get
Further, since
we get
Furthermore,
and then it follows from \(x_{m}(t)\geq Mt^{\alpha-1}\), (3.16) and (3.17) that, for \(t\in[0,1]\) and \(m\in\mathbb{N}\), we have
where M is defined by (H2). □
In order to finish the main result, we also need the following lemma.
Lemma 3.3
Let \(x_{m}\) be a solution of problem (1.5). If (H0)-(H3) hold, then the sequence \(\{x_{m}\}\) is relatively compact in E.
Proof
For \(t\in[0,1]\) and \(m\in\mathbb{N}\), since p is nondecreasing, by (3.18) we have
and by Lemma 2.6, (1.4), (3.3), (3.15), and (3.19), for \(t\in [0,1]\) and \(m\in\mathbb{N}\), we get
where
By (H2) and (3.20) we have
and \(\Upsilon<\infty\).
By (H3), taking \(\lambda_{1}>0\) such that
we have that there exists \(A>M+1\) such that, for any \(x>A\),
In order to prove that \(\{x_{m}\}\subset K\) is relatively compact in \(E=C^{(n-2)}[0,1]\), we need to prove that \(\{x_{m}\}\) is bounded in E and \(\{x_{m}\}\) is equicontinuous on \([0,1]\). First, we prove that \(\{ x_{m}\}\) is bounded in E. If \(\{x_{m}\}\) is unbounded, then there exists a subsequence \(\{x_{m_{j}}\}\subset\{x_{m}\}\) such that \(\| x_{m_{j}}\|_{2}\rightarrow+\infty\), and then there exists \(j_{0}\) such that
and then \(\frac{\|x_{m_{j_{0}}}\|_{2}}{\Gamma(n-1-\mu_{n-2})}+1>A\), and by (3.21) and (3.22) we get
This is a contradiction, which means that \(\{x_{m}\}\) is bounded in E. Next, we will prove that \(\{x_{m}^{(n-2)}(t)\}\) is equicontinuous on \([0,1]\). Since \(\{x_{m}\}\) is bounded in E, there exists \(\Lambda >0\) such that \(\|x_{m}\|\leq\Lambda\). Let
Then \(\Upsilon=\int_{0}^{1}\phi(t)\, dt\), and for any \(m\in\mathbb{N}\) and a.e. \(t\in[0,1]\),
Assume that \(0\leq t_{1}< t_{2}\leq1\). Then by (3.24), for any \(m\in \mathbb{N}\), we have
Hence, we can prove that \(\{x_{m}^{(n-2)}(t)|m=1,2,\ldots \}\) is equicontinuous on \([0,1]\). □
4 Proof of Theorem 1.1
Proof of Theorem 1.1
According to Theorem 3.1, we know that (1.5) has a solution \(x_{m}\in K\) for any \(m\in\mathbb{N}\). Moreover, Lemma 3.3 implies that \(\{x_{m}\}\) is relatively compact in E and satisfies inequality (3.16) for \(t\in[0,1]\) and \(m\in\mathbb{N}\). The sequence \(\{x_{m}\}\) has a subsequence converging to \(x^{\star}\subset K\). Without loss of generality, we still assume that \(\{x_{m}\}\) itself uniformly converges to \(x^{\star}\). So \(x^{\star}\in K\) satisfies the boundary conditions of (1.1), and according to (2.1), we get
in \(C[0,1]\). Take the limit in (3.16) as \(m\rightarrow\infty\). Then \(x^{\star}\) satisfies (1.3). Moreover, for a.e. \(t\in[0,1]\),
By (3.3) we get
Therefore, for all \(m\in\mathbb{N}\) and a.e. \((t,s)\in[0,1]\times[0,1]\), we get
where ϕ is defined by (3.23). Taking \(m\rightarrow\infty\) in (3.15) and combining with (4.1), we obtain
by the Lebesgue dominated convergence theorem. Hence, \(x^{\star}\) is a positive solution of problem (1.1) and satisfies inequality (1.3). □
5 Example
We consider the following nonlinear singular fractional differential equation:
where \(\alpha=\frac{9}{2}\), \(\mu_{1}=\frac{1}{2}\), \(\mu_{2}=\frac {3}{2}\), \(\mu _{3}=\frac{5}{2}\), \(\eta_{j}=\frac{14}{j^{2}}\), \(\xi_{j}=\frac {1}{j^{\frac {9}{7}}}\), \(a\in (0,\frac{2}{7} )\), \(b\in (0,\frac {1}{9} )\), \(c\in (0,\frac{2}{7} )\), \(d\in (0,\frac {2}{5} )\), \(b_{1},c_{1},d_{1}\in(0,1)\). Letting \(\gamma (t)=\frac {3+t^{\frac{1}{2}}}{2\text{,}100t^{\frac{1}{2}}}\) and \(\beta(t)=\frac {1+t^{\frac {1}{2}}}{t^{\frac{1}{2}}}\), we easily get
Then the function
satisfies the hypotheses (H0), (H1), and (H2) for
and
so hypothesis (H3) is also satisfied. Therefore, Theorem 1.1 guarantees that the fractional differential equation (5.1) has one positive solution u satisfying inequality (1.3) for \(t\in [0,1]\), where \(M=\frac{ 7}{(\alpha-4)\Gamma(\alpha+1)}\).
Remark 5.1
In the examples of [15], the index of the independent variables of h cannot be 1, but the index of the independent variables of h can be 1 in this paper because \(\lim_{x\rightarrow \infty}\frac{h(x,x,\ldots,x)}{x}=0\) is replaced by \(\limsup_{x\rightarrow\infty}\frac{h(x,x,\ldots,x)}{x}=\lambda<\frac {\Gamma (n-1-\mu_{n-2})}{e\|\gamma\|_{1}}\).
6 Conclusions
In this paper, some existence results are obtained successfully for the boundary value problem (1.1) for the case where the nonlinearity is allowed to be singular with respect to not only the time variable but also the space variable and also the boundary conditions may involve infinite number of points. Compared with previous work [15–17], we complete the proof without imposing the third Carathéodory condition, that is, the condition \(|f(t,x_{1},x_{2},\ldots,x_{n-1})|\leq\varphi_{H}(t)\) is successfully removed, and, at the same time, the condition
is extended to
which leads to more general results. Moreover, the results of [15] seem to be wrong when \(\lim_{x\rightarrow0}\frac{h(x,x,\ldots,x)}{x}=0\). So we have improved the result of [15–17]. The method we utilized for the analysis is the sequential technique and regularization, and the existence of positive solutions is obtained by the fixed point theorem.
References
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)
Liang, S, Zhang, J: Existence and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem. Comput. Math. Appl. 62, 1333-1340 (2011)
Jiang, J, Liu, L, Wu, Y: Positive solutions for nonlinear fractional differential equations with boundary conditions involving Riemann-Stieltjes integrals. Abstr. Appl. Anal. 2012, Article ID 708192 (2012)
Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677-2682 (2008)
Zhang, X, Liu, L, Wiwatanapataphee, B, Wu, Y: The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition. Appl. Math. Comput. 235, 412-422 (2014)
Xu, X, Jiang, D, Yuan, C: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Anal. 71, 4676-4688 (2009)
Zhang, X, Liu, L, Wu, Y, Lu, Y: The iterative solutions of nonlinear fractional differential equations. Appl. Math. Comput. 219, 4680-4691 (2013)
Kou, C, Zhou, H, Yan, Y: Existence of solutions of initial value problems for nonlinear fractional functional differential equations on the half-axis. Nonlinear Anal. 74, 5975-5986 (2011)
Li, X, Liu, S, Jiang, W: Positive solutions for boundary value problem of nonlinear fractional functional differential equations. Appl. Math. Comput. 217, 9278-9285 (2011)
Liu, L, Hao, X, Wu, Y: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model. 57, 836-847 (2013)
Jing, W, Huang, X, Guo, W, Zhang, Q: The existence of positive solutions for the singular fractional differential equation. Appl. Math. Comput. 41, 171-182 (2013)
Sun, Y, Zhao, M: Positive solutions for a class of fractional differential equations with integral boundary conditions. Appl. Math. Lett. 34, 17-21 (2014)
Jiang, J, Liu, L, Wu, Y: Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions. Abstr. Appl. Anal. 2012, Article ID 696283 (2012)
Zhang, X: Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions. Appl. Math. Lett. 39, 22-27 (2015)
Bai, Z, Sun, W: Existence and multiplicity of positive solutions for singular fractional boundary value problems. Comput. Math. Appl. 63, 1369-1381 (2012)
Agarwal, RP, O’Regan, D, Stanĕk, S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57-68 (2010)
Stanĕk, S: The existence of positive solutions of singular fractional boundary value problems. Comput. Math. Appl. 59, 1379-1388 (2011)
Zhang, S: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 59, 1300-1309 (2010)
Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999)
Kibas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Bai, Z, Lv, H: Positive solutions of boundary value problems of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 2761-2767 (2005)
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. 27, 26-33 (2014)
Zhang, X, Liu, L, Wu, Y: The eigenvalue problem for a singular higher fractional differential equation involving fractional derivatives. Appl. Math. Comput. 218, 8526-8536 (2012)
Acknowledgements
The authors were supported financially by the National Natural Science Foundation of China (11371221, 11571296).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
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, 114 (2016). https://doi.org/10.1186/s13661-016-0621-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-016-0621-8