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.
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1 Introduction
We consider the existence of positive solutions for the following fractional nonlocal boundary value problem:
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 [1–4]. For the development of the fractional differential equations, see [5–23] 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 [24–33]. Using the compression expansion fixed point theorem due to Krasnosel’skii, Henderson and Luca [27] studied the fractional BVP
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:
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
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 \(t=1\).
2 Preliminaries and lemmas
Definition 2.1
The Riemann–Liouville fractional integral of order \(\alpha > 0\) of a function \(h: (0,+ \infty )\rightarrow \mathbb{R}\) is given by
provided that the right-hand side is defined pointwise on \((0,+\infty )\).
Definition 2.2
The Riemann–Liouville fractional derivative of order \(\alpha > 0\) of a function \(h: (0,+ \infty )\rightarrow \mathbb{R}\) is given by
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
is
where \(G(t,s)\) is the Green’s function given by
and
Lemma 2.2
The functions q and G given in Lemma 2.1 have the following properties:
-
(i)
\(q\in C([0,1],(0,+\infty ))\) is nondecreasing;
-
(ii)
\(G(t,s)\in C([0,1]\times [0,1],[0,+\infty ))\);
-
(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
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
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.
-
(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 \(i(T,K_{r},K)=0\).
-
(ii)
If \(Tu\neq\mu u\) for any \(u\in \partial K_{r}\) and \(\mu \geq 1\), then \(i(T,K_{r},K)=1\).
3 Main results
Theorem 3.1
Suppose that \((H_{1})\) and \((H_{2})\) are satisfied. If
then BVP (1.1) has at least one positive solution for any
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
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
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
which contradicts to the definition of μ̅. It follows from (3.2) and Lemma 2.5(i) that
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
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
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
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
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
Therefore we obtain
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
By (3.3), (3.6), and the additivity of the fixed point index we have
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). □
4 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:
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
Therefore
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 \(\lambda \in (0,+\infty )\).
5 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 \(t=1\).
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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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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