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Positive solutions for a class of fractional 3-point boundary value problems at resonance
Advances in Difference Equations volume 2017, Article number: 7 (2017)
Abstract
In this paper, we study the nonlocal fractional differential equation:
where \(1 < \alpha< 2\), \(0 < \xi< 1\), \(\eta\xi^{\alpha-1}= 1\), \(D^{\alpha}_{0+}\) is the standard Riemann-Liouville derivative, \(f:[0,1]\times[0,+\infty)\rightarrow\mathbb{R}\) is continuous. The existence and uniqueness of positive solutions are obtained by means of the fixed point index theory and iterative technique.
1 Introduction
In this paper, we consider the following fractional differential equation:
where \(1 < \alpha< 2\), \(0 < \xi< 1\), \(\eta\xi^{\alpha-1}= 1\), \(D^{\alpha}_{0+}\) is the standard Riemann-Liouville derivative, \(f:[0,1]\times[0,+\infty)\rightarrow\mathbb{R}\) is continuous. Problem (1.1) happens to be at resonance, since \(\lambda=0\) is an eigenvalue of the linear problem
and \(ct^{\alpha-1},c\in\mathbb{R,}\) is the corresponding eigenfunction.
Fractional differential equations occur frequently in various fields such as physics, chemistry, engineering and control of dynamical systems, etc. During the last few decades, many papers and books on fractional calculus and fractional differential equations have appeared (see [1–22] and the references therein).
When \(0<\eta\xi^{\alpha-1}< 1\), problem (1.1) is non-resonant. In [9], the author studied the existence of positive solutions for the non-resonant case by means of the fixed point index theory under sublinear conditions.
In [18], the authors investigated the existence and multiplicity results of positive solutions by using of the fixed point theorem for the fractional differential equation given by
where \(1 <\alpha\leq2\), \(0 \leq\beta\leq1\), \(0 < \xi< 1\), \(0 \leq a\leq1\) with \(a\xi^{\alpha-\beta-2}< 1-\beta\), \(0\leq \alpha-\beta-1\).
Recently, there are some papers dealing with the existence of solutions of fractional boundary value problem at resonance by using the coincidence degree theory due to Mawhin (see [19–22]). In [22], the authors investigated the following fractional three-point boundary value problem (BVP for short) at resonance:
where \(1 < \alpha\leq2\), \(0 < \xi< 1\), \(\eta\xi^{\alpha-1}= 1\), \(D^{\alpha}_{0+}\) is the standard Riemann-Liouville derivative, \(f:[0,1]\times\mathbb{R}^{2}\rightarrow\mathbb{R}\) is continuous. By using the coincidence degree theory, the existence of solutions for BVP (1.4) are obtained under certain growth conditions.
To the best of our knowledge, there are only very few papers dealing with the existence of positive solutions for resonant boundary value problems since the corresponding linear operator is non-reversible. For the case that α is an integer, some work has been done dealing with the existence of positive solutions for resonant boundary value problems by using Leggett-Williams norm-type theorem for coincidence (see [23–25]). Webb [26] established existence of positive solutions for second order boundary value problems at resonance by considering equivalent non-resonant perturbed problems with the same boundary conditions.
Inspired by the work mentioned above, in this paper we aim to establish the existence of positive solutions for resonant problem (1.1). The paper is organized as follows. Firstly, we reduce non-perturbed boundary value problems at resonance to equivalent non-resonant perturbed problems with the same boundary conditions. Then we derive the corresponding Green’s function and argue its properties. Finally, the existence and uniqueness results of positive solutions are obtained by using of the fixed point index and iterative technique.
2 Basic definitions and preliminaries
In this section, we present some preliminaries and lemmas. The definitions and properties of fractional derivative can be found in the literature [1–22].
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 \((0,+\infty)\).
Denote
It is easy to check that \(g'(t)>0\) on \((0,+\infty)\), and
Therefore, there exists a unique \(b^{\ast}>0\) such that
For the convenience in presentation, we here list the assumptions to be used throughout the paper.
- (H1):
-
\(b\in(0, b^{\ast}]\) is a constant.
- (H2):
-
\(f:[0,1]\times[0,+\infty)\rightarrow\mathbb{R}\) is continuous and
$$ f(t,x)+bx\geq0. $$(2.3)
Set
where
is the Mittag-Leffler function (see [1, 2]).
Next we consider the following boundary value problem:
It is clear that (1.1) is equivalent to (2.6).
Set
Lemma 2.1
Suppose that (H1) holds, and \(y\in L[0,1]\). Then the unique solution of the problem
is
Proof
By [1, 2], we know that the solution of (2.10) can be expressed by
By \(u(0)=0\), we have \(c_{2}=0\).
On the other hand, we have
Noting that \(\eta\xi^{\alpha-1}= 1\), and \(0 < \xi< 1\), we have
Equations (2.11) and (2.12) yield
Therefore, the solution of (2.10) is
This completes the proof. □
Lemma 2.2
Suppose that (H1) holds. The function \(K(t,s)\) has the following properties:
-
(1)
\(K(t,s) > 0, \forall t,s\in(0,1)\);
-
(2)
\(\omega_{2}(s)t^{\alpha-1}\leq K(t,s) \leq\omega_{1}(s)t^{\alpha-1}, \forall t,s\in[0,1]\), where
$$ \omega_{1}(s)=G_{b}(1-s)+G_{b}(1)q(s), \qquad \omega_{2}(s)=\frac {q(s)}{\Gamma(\alpha)}. $$(2.14)
Proof
It is clear that we just need to prove that (2) holds.
By (2.4), we can get
and
which implies \(G_{b}(t)\) is strictly increasing on \([0,1]\), and \(G'_{b}(t)\) is strictly decreasing on \((0,1]\).
By (2.15), we have
On the other hand, when \(0< t\leq s< 1\), noticing \(G_{b}(0)=0\), and the monotonicity of \(G_{b}(t)\), it is clear that
When \(0< s< t< 1\), we have
Integrating (2.20) with respect to s, we obtain
By (2.7), (2.19), and (2.21), we get \(K_{0}(t,s) > 0, \forall t,s\in(0,1)\). Then
This completes the proof. □
Let \(E=C[0,1]\) be endowed with the maximum norm \(\| u\| = \max_{0\leq t\leq1}| u(t)|\), θ is the zero element of E, \(B_{r}=\{u\in E : \| u\|< r\}\). Define a cone P by
Let
By means of Lemma 2.2 and the Arzela-Ascoli theorem, we can get \(A: P\rightarrow P\) is completely continuous, \(T: P\rightarrow P\) is completely continuous linear operator. By virtue of the Krein-Rutmann theorem and Lemma 2.2, we have the spectral radius \(r(T)> 0\) and T has a positive eigenfunction corresponding to its first eigenvalue \(\lambda_{1}=(r(T))^{-1}\). Since \(\lambda=0\) is the eigenvalue of the linear problems (1.2), and \(t^{\alpha-1}\) is the corresponding eigenfunction, we have the following lemma.
Lemma 2.3
Suppose that (H1) holds, then the first eigenvalue of T is \(\lambda_{1}=b\), and \(\varphi_{1}(t)=t^{\alpha-1}\) is the positive eigenfunction corresponding to \(\lambda_{1}\), that is, \(\varphi_{1}=bT\varphi_{1}\).
Lemma 2.4
[27]
Let P be a cone in a Banach space E, and Ω be a bounded open set in E. Suppose that A: \(\overline{\Omega}\cap P\rightarrow P\) is a completely continuous operator. If there exists \(u_{0}\in P\) with \(u_{0}\neq\theta\) such that
then \(i(A,\Omega\cap P,P)=0\).
Lemma 2.5
[27]
Let P be a cone in a Banach space E, and Ω be a bounded open set in E. Suppose that A: \(\overline{\Omega}\cap P\rightarrow P\) is a completely continuous operator. If
then \(i(A,\Omega\cap P,P)=1\).
3 The uniqueness result
Theorem 3.1
Assume that there exists \(\lambda\in(0,b)\) such that
then (1.1) has a unique nonnegative solution.
Proof
Firstly, we will prove A has fixed point in P.
Set
For any \(u \in P\setminus \{\theta\}\), let
By Lemma 2.2, it is obvious that \(l_{1}(u), l_{2}(u)> 0\), and
that is,
For any \(u_{0}\in P\setminus\{\theta\}\), let
We may suppose that \(u_{1}-u_{0} \neq\theta\) (otherwise, the proof is finished). Then there exists \(l_{2}(|u_{1}-u_{0}|)> 0\), such that
Thus
By induction, we can get
Then, for any \(n, m \in\mathbb{N}\), we have
So,
which implies \(\{u_{n}\}\) is a Cauchy sequence. Therefore, there exists a \(u^{\ast}\in P\), such that \(\{u_{n}\}\) converges to \(u^{\ast}\). Clearly, \(u^{\ast}\) is a fixed point of A.
In the following, we will prove the fixed point of A is unique.
Suppose \(v\neq u^{\ast}\) is a fixed point of A. Then there exists \(l_{2}(|u^{\ast}-v|)> 0\), such that
Then
By induction, we can get
So,
Consequently, the fixed point of A is unique.
This completes the proof. □
Remark 3.1
The unique nonnegative solution \(u^{\ast}\) of (1.1) can be approximated by the iterative schemes: for any \(u_{0}\in P\setminus\{\theta\}\), let
then \(u_{n}\rightarrow u^{\ast}\).
Remark 3.2
If \(f(t,0)\equiv0\) on \([0,1]\), then θ is the unique solution of (1.1) in P; If \(f(t,0)\not\equiv0\) on \([0,1]\), then the unique solution \(u^{\ast }\) is a positive solution.
4 Existence of positive solutions
Theorem 4.1
Assume that (H1), (H2), and the following assumptions hold:
Then (1.1) has at least one positive solution.
Proof
It follows from (4.1) that there exists \(r_{1} > 0\) such that
Thus, for any \(u\in\partial B_{r_{1}}\cap P\), we have
We may suppose that A has no fixed points on \(\partial B_{r_{1}}\cap P\) (otherwise, the proof is finished). Now we show that
If otherwise, there exist \(u_{1}\in\partial B_{r_{1}}\cap P\) and \(\mu_{0}> 0\) such that
Then
Denote
It is clear that \(\mu^{\ast}\geq \mu_{0}\) and \(u_{1}\geq\mu^{\ast}\varphi_{1}\). Since \(T(P)\subset P\), we have \(b Tu_{1}\geq\mu^{\ast}b T\varphi_{1}=\mu^{\ast}\varphi_{1}\). Then
contradicts the definition of \(\mu^{\ast}\). Hence (4.5) holds and we see from Lemma 2.4 that
On the other hand, it follows from (4.2) that there exist \(0 < \sigma< 1\) and \(r_{2} > r_{1}\) such that
Let \(T_{1}u=\sigma b Tu\). Then \(T_{1}\) is a bounded linear operator and \(T_{1}(P)\subset P\). Set
In the following, we will prove that W is bounded.
For any \(u \in W\), set \(\tilde{u}(t)=\min\{u(t),r_{2}\}\). Then
Therefore,
where
Thus \((I-T_{1})u(t)\leq M,t\in[0,1]\). Noticing b is the first eigenvalue of T and \(0 < \sigma< 1\), we have \((r(T_{1}))^{-1}=\sigma^{-1}>1\). So the inverse operator of \(I-T_{1}\) exists, and
It follows from \(T_{1}(P)\subset P\) that \((I-T_{1})^{-1}(P)\subset P\). We have
which implies W is bounded.
Select \(r_{3}> \max\{r_{2},M\| (I-T_{1})^{-1} \|\}\). Then by Lemma 2.5, we have
which implies that A has at least one fixed point on \((B_{r_{3}}\backslash\bar{B}_{r_{1}})\cap P\). This means that BVP (1.1) has at least one positive solution.
This completes the proof. □
Theorem 4.2
Assume that (H1), (H2), and the following assumptions hold:
and \(f(t,0)\not\equiv0\) on \([0,1]\). Then (1.1) has at least one positive solution.
Proof
It follows from (4.12) that there exists \(r_{1} > 0\) such that
Denote \(T_{2}u=bTu\). Obviously, \(r(T_{2})=1\).
We may suppose that A has no fixed points on \(\partial B_{r_{1}}\cap P\) (otherwise, the proof is finished). In the following, we prove that
If otherwise, there exist \(u_{1}\in \partial B_{r_{1}}\cap P,\mu_{0}> 1\), such that \(Au_{1}= \mu_{0} u_{1}\). It is clear that \(\mu_{0} u_{1}=Au_{1}\leq T_{2}u_{1}\), and
Therefore,
Thus \(r(T_{2})=\lim_{n\rightarrow+\infty}\sqrt[n]{\| T_{2}^{n}\|}\geq \mu_{0} > 1\), which contradicts \(r(T_{2})=1\). We have from Lemma 2.5
Since \(f(t,0)\not\equiv0\) on \([0,1]\), clearly we have \(A\theta\neq \theta\), here θ is the zero element of E. So (4.15) implies that the problem (1.1) has at least one positive solution. □
Remark 4.1
Suppose u is a positive solution of (1.1), then there exist \(l_{1}, l_{2}>0\), such that
Example 4.1
A 3-point boundary value problem at resonance
Consider the following problem:
Since \(\Gamma(\cdot)\) is strictly increasing on \([2,+\infty)\), for any \(t\in[0,+\infty)\), we have
Noticing \(\frac{1}{2\sqrt{\pi}}\thickapprox0.282\), \(\frac{1}{5}e^{\frac {1}{5}}\thickapprox0.243\), we have \(g(\frac{1}{5})< 0\). Therefore \(b^{\ast}> b_{1}:=\frac{1}{5}\).
Let
where \(b\in(0,b_{1}]\). It is clear that (H1) and (H2) hold. Moreover,
Therefore the assumptions of Theorem 4.1 are satisfied. Thus Theorem 4.1 ensures that (4.16) has at least one positive solution.
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Acknowledgements
The authors would like to thank the referee for his/her very important comments, which improved the results and the quality of the paper. This work was supported financially by the National Natural Science Foundation of China (11371221, 11571296), the Natural Science Foundation of Shandong Province of China (ZR2013AQ014, ZR2015AL002), the Specialized Research Fund for the Doctoral Program of Higher Education (20133705120003), and the Project of Shandong Province Higher Educational Science and Technology Program (J13LI08, J15LI16).
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Wang, Y., Liu, L. Positive solutions for a class of fractional 3-point boundary value problems at resonance. Adv Differ Equ 2017, 7 (2017). https://doi.org/10.1186/s13662-016-1062-5
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DOI: https://doi.org/10.1186/s13662-016-1062-5
Keywords
- fractional differential equation
- positive solution
- resonance
- fixed point index