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Positive solutions for a class of fractional infinite-point boundary value problems
Boundary Value Problems volume 2018, Article number: 118 (2018)
Abstract
In this paper, we consider a class of fractional differential equations with infinite-point boundary value conditions. Under some conditions concerning the spectral radius with respect to the relevant linear operator, both the existence of uniqueness and the nonexistence of positive solution are obtained by means of the iterative technique.
1 Introduction
In this paper, we consider the following fractional differential equations (FDE for short) with infinite-point boundary value conditions:
where \(\alpha \geq 2\), \(n-1 < \alpha \leq n\), \(0< \beta < \alpha -1\), \(\eta_{i}\in \mathbb{R}\), \(0 < \xi_{1}< \cdots < \xi_{i}<\xi_{i+1}< \cdots < 1\) (\(i=1,2,\ldots \)), \(\sum^{\infty }_{i=1}\eta_{i}\xi_{i}^{ \alpha -\beta -1}\neq 1\), \(D^{\alpha }_{0+}\) is the standard Riemann–Liouville derivative, \(f:[0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous, \(a(t)\in C((0,1),[0,+\infty ))\) may be singular at \(t=0,1\).
FDE serve as an excellent instrument for the description of memory and hereditary properties of various materials and processes. During the last few decades, much attention has been paid to the study of boundary value problems (BVP for short) of fractional differential equation, such as the nonlocal BVP [1, 3, 7, 13, 18], singular BVP [6, 8, 11, 19, 20, 25], semipositone BVP [14–16, 23], resonant BVP [2, 12], and impulsive BVP [10, 27]. Since only positive solutions are meaningful in most practical problems, some work has been done to study the existence of positive solutions for fractional boundary value problems by using the techniques of nonlinear analysis. In [16], the authors discussed the following fractional m-point BVP:
where \(0< \beta < \alpha -1\), \(0 < \xi_{1}< \cdots < \xi_{m-2} < 1\) with \(\sum^{m-2}_{i=1}\eta_{i}\xi_{i}^{\alpha -\beta -1}< 1\). Some new positive properties of the corresponding Green function were established, and a number of theorems on the existence of positive solutions were obtained. However, there are few works on the uniqueness of solution for fractional boundary value problems [4, 5, 9, 17, 21, 22, 24, 26]. To obtain the uniqueness of solution, the main tool used in most of the works in literature is the Banach contraction map principle provided the nonlinearity satisfies the Lipschitz condition. Liang and Zhang [9] studied the uniqueness of positive solution for the following fractional three-point BVP:
where \(0 < \eta < 1\) with \(0< \beta \eta^{\alpha -3}< 1\), \(f:[0,1] \times [0,+\infty )\rightarrow [0,+\infty )\) is continuous and nondecreasing with respect to the second variable, and there exists
such that, for \(u,v \in [0,+\infty )\) with \(u\geq v\) and \(t\in [0,1]\),
It should be noted that (1.2) implies \(f(t,u)-f(t,v)\leq \lambda (u-v)\) since \(\ln (1+x)\leq x, \forall x \geq 0\), that is, f satisfies the Lipschitz condition.
Motivated by the above works, in this paper we aim to establish the existence of uniqueness and the nonexistence of positive solution to BVPÂ (1.1). It is well known that the cone plays a very important role in seeking positive solution. Our analysis relies on the iterative technique on the cone derived from the properties of the Green function. Our work presented in this paper has the following features. Firstly, the uniqueness results are obtained under some conditions concerning the spectral radius with respect to the relevant linear operator. Secondly, the Lipschitz condition is generalized. In addition, the error estimation of iterative sequences is also given. Thirdly, the nonexistence results of positive solution are obtained under conditions concerning the spectral radius of the relevant linear operator. Finally, we impose weaker positivity conditions on the nonlocal boundary term, that is, some of the coefficients \(\eta_{i}\) can be negative.
2 Preliminaries and lemmas
In this section, we present some notations and lemmas that will be used in the proof of our main results.
Definition 2.1
The fractional integral of order \(\alpha > 0\) of a function \(u:(0,+\infty )\rightarrow R\) is given by
provided that the right-hand side is point-wise defined on \((0,+ \infty )\).
Definition 2.2
The Riemann–Liouville fractional derivative of order \(\alpha > 0\) of a function \(u:(0,+\infty )\rightarrow R\) is given by
where \(n=[\alpha ]+1\), \([\alpha ]\) denotes the integer part of number α, provided that the right-hand side is point-wise defined on \((0,+\infty )\).
Set
where
Remark 2.1
Assume that \(\sum^{\infty }_{i=1}\eta_{i}\xi_{i} ^{\alpha -\beta -1}\) is convergent. Then \(p\in C[0,1]\).
Remark 2.2
If \(\eta_{i}= 0\) (\(i=1,2,\ldots \)), we have \(p(s)\equiv 0\) and \(q(s)\equiv 0\); if \(\eta_{i}\geq 0\) (\(i=1,2,\ldots \)) and \(p(0)<1\), we have \(q(s)\geq 0\) on \([0,1]\).
For the convenience in presentation, we here list the assumption to be used throughout the paper.
- (\(A_{1}\)):
-
\(p(0)\neq 1\), \(q(s)\geq 0\) on \([0,1]\);
- (\(A_{2}\)):
-
\(f:[0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous, \(a(t)\in C((0,1),[0,+\infty ))\), and
$$ 0< \int^{1}_{0}a(s)\,ds< +\infty . $$
Lemma 2.1
Assume that \(y\in L[0,1]\) and \(p(0)\neq 1\). Then the unique solution of the problem
is
Proof
The proof is similar to Lemma 2.3 in [17], so we omit it. □
Lemma 2.2
The function \(K(t,s)\) has the following properties:
-
(1)
\(K(t,s) > 0\), \(\forall t,s\in (0,1)\);
-
(2)
\(K(t,s) \leq \frac{t^{\alpha -1}(1-s)^{\alpha -\beta -1}}{\Gamma (\alpha )}\), \(\forall t,s\in [0,1]\);
-
(3)
\(\frac{t^{\alpha -1}}{\Gamma (\alpha )} h(s)\leq K(t,s) \leq \frac{M _{1}}{\Gamma (\alpha )}h(s)\), \(\forall t,s\in [0,1]\), where
$$ h(s)= \bigl[ 1-(1-s)^{\beta } \bigr] (1-s)^{\alpha -\beta -1}, \qquad M_{1}=\max \bigl\{ 1,\beta^{-1} \bigr\} . $$
Proof
It is obvious that \((1)\), \((2)\) hold. In the following, we will prove (3).
Case (i): \(0< s\leq t< 1\).
If \(0 < \beta < 1\), we have
It follows from
that
It follows from (2.1) and (2.2) that
If \(\beta \geq 1\), we have
Therefore,
On the other hand, noticing \(0< \beta < \alpha -1\), we have
Case (ii): \(0\leq t \leq s \leq 1\).
If \(\beta \geq 1\), we have
If \(0 < \beta < 1\). It follows from (2.2) that
On the other hand, noticing \(0< \beta < \alpha -1\), we have
It follows from (2.3)–(2.8) that (3) holds. □
Lemma 2.3
The function \(G(t,s)\) has the following properties:
-
(1)
\(G(t,s) > 0\), \(\forall t,s\in (0,1)\);
-
(2)
\(G(t,s) \leq t^{\alpha -1}\Phi_{1}(s)\), \(\forall t,s \in [0,1]\);
-
(3)
\(t^{\alpha -1}\Phi_{2}(s)\leq G(t,s) \leq M_{1}\Phi_{2}(s)\), \(\forall t,s\in [0,1]\),
where
Proof
It can be directly deduced from Lemma 2.2 and the definition of \(G(t,s)\), so we omit the proof. □
Let \(E=C[0,1]\) be endowed with the maximum norm \(\Vert u \Vert = \max_{0\leq t\leq 1}\vert u(t) \vert \), \(B_{r}=\{u\in E : \Vert u \Vert < r\}\). Define cones P, Q by
Lemma 2.4
Define \(A:P\rightarrow E\) as follows:
then \(A: P\rightarrow Q\) is completely continuous.
Proof
By Lemma 2.3, it is easy to check that \(A: P\rightarrow Q\). By means of the Arzela–Ascoli theorem, \(A: P\rightarrow Q\) is completely continuous. □
Remark 2.3
Lemma 2.4 is valid if \((A_{2})\) is replaced by \((A'_{2})\) \(f:[0,1]\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous, \(a(t)\in C((0,1),[0,+\infty ))\) and
Let
It is clear that T is a completely continuous linear operator and \(T(P)\subset Q\). By virtue of the Krein–Rutmann theorem and Lemma 2.3, we have the following lemma.
Lemma 2.5
The spectral radius \(r(T)> 0\) and T has a positive eigenfunction \(\varphi_{1}\) corresponding to its first eigenvalue \((r(T))^{-1}\), that is, \(T\varphi_{1}=r(T)\varphi_{1}\).
3 Main results
3.1 Uniqueness results
Theorem 3.1
Suppose that \(f(t,0)\not \equiv 0\) on \([0,1]\), and there exists \(\lambda \in C(0,1)\cap L[0,1]\) satisfying
such that
Then (1.1) has a unique positive solution if the spectral radius \(r(L_{\lambda })\in (0,1)\), where
Proof
It follows from \(f(t,0)\not \equiv 0\) that θ is not a fixed point of A. Then we only need to prove that A has a unique fixed point in Q.
Firstly, we will prove that A has a fixed point in Q.
Set
For any \(u \in Q\setminus \{\theta \}\), let
It follows from Lemma 2.3 that \(l_{1}(u)\), \(l_{2}(u)> 0\) and
Therefore,
Similar to Lemma 2.5, we have that the spectral radius \(r(L_{\lambda })>0\) and \(L_{\lambda }\) has a positive eigenfunction \(\psi_{1}\), that is, \(L_{\lambda }\psi_{1}=r(L_{\lambda })\psi_{1}\). It is easy to see that
For any \(u_{0}\in Q\setminus \{\theta \}\), let
We may suppose that \(u_{1}-u_{0} \neq \theta \) (otherwise, the proof is finished). It follows from (3.1) and (3.2) that
Then
By induction, we can get
Then, for any \(n, m \in \mathbb{N}\), we have
It follows from \(r(L_{\lambda })<1\) that
which implies \(\{u_{n}\}\) is a Cauchy sequence. Therefore, there exists \(u^{\ast }\in Q\) such that \(\{u_{n}\}\) converges to \(u^{\ast }\). Clearly \(u^{\ast }\) is a fixed point of A.
In the following, we will prove that the fixed point of A is unique.
Suppose that \(v \neq u^{\ast }\) is a positive fixed point of A. Then there exists \(l_{1}(\vert u^{\ast }-v \vert )> 0\) such that
Therefore,
Similar to the proof of (3.7), we can get
It follows from \(r(L_{\lambda })<1\) that
which implies that the positive fixed point of A is unique. □
Remark 3.1
The unique positive solution \(u^{\ast }\) of (1.1) can be approximated by the iterative schemes: for any \(u_{0}\in Q\setminus \{\theta \}\), let
then \(u_{n}\rightarrow u^{\ast }\). Furthermore, we have error estimation
and with the rate of convergence
Remark 3.2
The spectral radius satisfies \(r(L_{\lambda })= \lim_{n\rightarrow \infty }\Vert L_{\lambda }^{n} \Vert ^{\frac{1}{n}}\) and \(r(L_{\lambda })\leq \Vert L_{\lambda }^{n} \Vert ^{\frac{1}{n}}\). Particularly,
3.2 Nonexistence results
Theorem 3.2
Suppose that there exists \(b_{1} \in C(0,1)\cap L[0,1]\) satisfying
such that
Then (1.1) has no positive solution if the spectral radius \(r(L_{b _{1}})\in (0,1)\), where
Proof
We only need to prove that A has no fixed point in \(Q_{1}\). If otherwise, there exists \(v\in Q_{1}\) such that \(Av= v\). Similar to Lemma 2.5, we have that the spectral radius \(r(L_{b_{1}})>0\) and \(L_{b_{1}}\) has a positive eigenfunction \(\varphi_{b_{1}}\) satisfying
It is clear that \(v, \varphi_{b_{1}}\in Q_{1}\). Therefore, there exists \(c_{1}>0\) such that
By \(f(t,x)\leq b_{1} (t)x\), we have \(v=Av\leq L_{b_{1}}v\). It is obvious that \(L_{b_{1}}\) is increasing in \(Q_{1}\). By induction, we can get \(v\leq L^{n}_{b_{1}}v\), \(\forall n=1,2,3,\ldots \) . Then
Noticing \(r(L_{b_{1}})<1\), we have \(v=\theta \), which contradicts \(v\in Q_{1}\). □
Theorem 3.3
Suppose that there exists \(b_{2}\in C(0,1)\cap L[0,1]\) satisfying
such that
Then (1.1) has no positive solution if the spectral radius \(r(L_{b_{2}})> 1\), where
Proof
The proof is just similar to Theorem 3.2, so we omit it. □
4 Example
Example 4.1
Consider the following problem:
where
Let \(\lambda (t)=\sqrt{t}\), we have
It is clear that
where \(\Delta :=\Gamma (\frac{5}{2})[1-p(0)]\).
Let
Then
By direct calculations, we have
and
It is clear that
On the other hand, we have
Noticing that
we have
which implies that
It follows from Lemma 2.5 and Remark 3.2 that
Therefore the assumptions of Theorem 3.1 are satisfied. Thus Theorem 3.1 ensures that BVP (4.1) has a unique positive solution.
5 Conclusions
In this paper, we consider both the existence of uniqueness and the nonexistence of positive solution for fractional differential equations with infinite-point boundary value conditions. The interesting point lies in that both the existence of uniqueness and the nonexistence of positive solution are closely associated with the spectral radius with respect to the relevant linear operator.
Abbreviations
- FDE:
-
Fractional differential equations
- BVP:
-
Boundary value problems
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Acknowledgements
The authors would like to thank the referees for their pertinent comments and valuable suggestions.
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Funding
The authors are supported financially by the Natural Science Foundation of Shandong Province of China (ZR2017MA036, ZR2014AM034), the National Natural Science Foundation of China (11571296).
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Wang, Y., Liu, L. Positive solutions for a class of fractional infinite-point boundary value problems. Bound Value Probl 2018, 118 (2018). https://doi.org/10.1186/s13661-018-1035-6
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DOI: https://doi.org/10.1186/s13661-018-1035-6