- Research
- Open Access
Existence and asymptotic analysis of positive solutions for a singular fractional differential equation with nonlocal boundary conditions
- Jianxin He^{1}Email author,
- Xinguang Zhang^{2, 3},
- Lishan Liu^{3, 4},
- Yonghong Wu^{3} and
- Yujun Cui^{5}
- Received: 8 March 2018
- Accepted: 3 December 2018
- Published: 12 December 2018
Abstract
In this paper, we focus on the existence and asymptotic analysis of positive solutions for a class of singular fractional differential equations subject to nonlocal boundary conditions. By constructing suitable upper and lower solutions and employing Schauder’s fixed point theorem, the conditions for the existence of positive solutions are established and the asymptotic analysis for the obtained solution is carried out. In our work, the nonlinear function involved in the equation not only contains fractional derivatives of unknown functions but also has a stronger singularity at some points of the time and space variables.
Keywords
- Asymptotic analysis
- Nonlocal boundary conditions
- Upper and lower solutions method
- Fractional differential equation
1 Introduction
In describing viscoelasticity, Heymans and Kitagawa [1] pointed out that the accuracy and success of the model are their abilities to describe natural phenomena including memory effects in polymers. However, in many dynamic process, the influence of memory is often persistent, even if the factors affecting the process have disappeared, such as observed in stress relaxation after a nonmonotonous loading program. Thus in order to improve the accuracy of the model, based on the non-locality of fractional order derivative, one can choose a noninteger order differential equation to describe this type of physical phenomena with memory effects. In addition, fractional calculus also has many other applications in various fields of science and engineering, such as a HIV model [2, 3] and a fluid model [4–8]. Recently, Heymans and Podlubny [9] gave some physical interpretation for the fractional spring–pot model, the Zener model, the Maxwell model and the Voigt model. In [10], Abdon introduced a new concept of differentiation and integration combining fractal differentiation and fractional differentiation, which can explain the memory effect of heterogeneity, and elasco-viscosity of the medium and also the fractal geometry of the dynamic system. Using the time-scale fractional calculus, Nadia et al. [11] gave some applications of the fractional derivatives with arbitrary time scales in white noise from signal processing.
In the aspect of mathematical theory and application, to obtain further information of the relative natural phenomena, many authors are interested in the existence and properties of solutions for fractional differential models [12–27] and many analytical techniques and methods have been developed to solve various differential equations, such as iterative methods [28–37], the Mawhin continuation theorem for resonance [38–40], the topological degree method [41, 42], the fixed point theorem [43–55], the variational method [56–73] and the upper and lower solution method [74, 75].
- (A)\(f\in C((0,1)\times (0,+\infty )\times (0,+\infty ),[0,+ \infty ))\) and for any \(0 < r < R < +\infty \),where \(e(n)=[0,\frac{1}{n}] \cup [\frac{n-1}{n},1]\).$$ \lim_{n\to +\infty } \sup_{{{x\in K_{\frac{R}{\varGamma (\beta +1)}}} \atop {y\in \overline{K}_{R}\backslash K_{r} } }} \int _{e(n)}\omega (s)f\bigl(s,x(s),y(s)\bigr)\,ds=0, $$
- (i)
We present exact cone and suitable growth condition to overcome the difficulty due to the singularity of the nonlinear term f at the space variables.
- (ii)
We establish a sufficient condition for the existence of positive solutions and give the estimation of the positive solution and asymptotic behavior of the derivative of positive solutions at the
- (iii)
Nonsingular cases for the nonlinear term f at the time and space variables are discussed and some new results are established.
The rest of this paper is organized as follows. In Sect. 2, some preliminaries and lemmas are presented for subsequent developments. The main results are presented in Sect. 3.
2 Preliminaries and lemmas
For the convenience of the reader, we only present here some necessary properties from fractional calculus theory in the sense of Riemann–Liouville, and the corresponding definitions can be found in [76] or [12–25].
Proposition 2.1
([76])
- (1)If \(x,y:(0,+\infty )\rightarrow \mathbb{R}\) with order \(\alpha > 0\), then$$ \pmb{\mathscr{D}_{t}}^{\alpha }\bigl(x(t)+y(t)\bigr)=\pmb{ \mathscr{D}_{t}}^{ \alpha }x(t)+\pmb{\mathscr{D}_{t}}^{\alpha }y(t). $$
- (2)If \(x\in L^{1}(0, 1)\), \(\nu >\gamma > 0\) and m is a positive integer, then$$ \begin{gathered} I^{\nu }I^{\gamma }x(t)=I^{\nu +\gamma }x(t), \qquad \pmb{ \mathscr{D}_{t}}^{\gamma }I^{\nu } x(t)=I^{\nu -\gamma } x(t), \\ \pmb{\mathscr{D}_{t}}^{\gamma }I^{\gamma } x(t)=x(t), \qquad \pmb{\mathscr{D}_{t}}^{m} \bigl(\pmb{ \mathscr{D}_{t}}^{\gamma }x(t) \bigr)= \pmb{\mathscr{D}_{t}}^{\gamma +m}x(t). \end{gathered} $$
- (3)If \(\alpha >0\), \(\gamma >0\), then$$ \pmb{\mathscr{D}_{t}}^{\alpha } t^{\gamma -1}= \frac{\varGamma (\gamma )}{ \varGamma (\gamma -\alpha )}t^{\gamma -\alpha -1}. $$
- (4)Suppose \(\gamma > 0\), and \(g(x)\) is integrable, thenwhere \(c_{i}\in \mathbb{R}\) (\(i=1,2,\ldots ,n\)), n is the smallest integer greater than or equal to α.$$ I^{\gamma }\pmb{\mathscr{D}_{t}}^{\gamma }g(x)=g(x)+c_{1}x^{\gamma -1}+c _{2}x^{\gamma -2}+\cdots +c_{n}x^{\gamma -n}, $$
In the rest of this paper, all discussions are based on the assumption \(2<\alpha -\gamma \le 3\). We first give the following lemma.
Lemma 2.1
Proof
The following lemma is standard according to Proposition 2.1, and we omit the proof.
Lemma 2.2
Lemma 2.3
- (F0)
\(\mathcal{X}\) is a function of bounded variation satisfying \(\mathcal{G}_{\mathcal{X}}(s)\ge 0\), \(s\in [0, 1]\) and \(\mathcal{C}\in [0,1)\).
Lemma 2.4
- (1)
\(W(t,s) > 0\) for all \(0< t\), \(s<1\).
- (2)where$$ \mathcal{B}t^{\alpha -\gamma -1}\mathcal{G}_{\mathcal{X}}(s) \le W(t,s) \le c(s)t^{\alpha -\gamma -1}, $$(2.8)$$ c(s)=\frac{(1-s)^{\alpha -\mu -1}}{\varGamma (\alpha -\gamma )}+ \mathcal{B}\mathcal{G}_{\mathcal{X}}(s). $$
Proof
It follows from Lemma 2.3 that we have the following.
Lemma 2.5
3 Singular cases
In this section, we first give the definition of upper and lower solution on the boundary value problem (2.1), and then introduce some theories of function space and give our main results.
Definition 3.1
Definition 3.2
- (F1)
\(f\in C((0,1)\times (0,\infty )\times (0,\infty ),[0,+ \infty ))\), and \(f(t,x_{1},x_{2})\) is decreasing in \(x_{i}>0\) for \(i=1,2\).
- (F2)For any \(\tau >0\), \(f(t,\frac{ \tau }{\varGamma (\gamma +1)} t^{\gamma },\tau )\not \equiv 0\), and$$ 0< \int _{0}^{1} c(s)f\bigl(s,\tau \kappa (s), \tau e(s)\bigr)\,ds< +\infty . $$
Lemma 3.1
Assume (F0) (F1) and (F2) are satisfied, then \(B (P_{e})\subset P_{e}\) and B is well defined.
Proof
Proof
Thus according to the Arzelà–Ascoli theorem, \(A:E\to E\) is a completely continuous operator. Consequently it follows from the Schauder fixed point theorem that A has a fixed point w such that \(w=Aw\).
Combined with the above facts, we get that the fixed point of A is also the fixed point of B. So \(w(t)\) is a positive solution of the boundary value problem (2.1), and consequently \(x(t)=I^{\gamma }w(t)\) is a positive solution of Eq. (1.1). □
Theorem 3.2
(Estimation and asymptotic behavior)
Proof
4 Nonsingular cases
In this section, we are interested in some nonsingular cases of the nonlinear term f at time and space variables.
Case 1: f may be singular at \(t=0\) and (or) \(t=1\), but f is nonsingular at \(x_{1}=x_{2}=0\):
Theorem 4.1
- (B1)
\(f\in C((0,1)\times [0,\infty )\times [0,\infty ),[0,+ \infty ))\), and \(f(t,x_{1},x_{2})\) is decreasing in \(x_{i}>0\) for \(i=1,2\).
- (B2)\(f(t,0,0)\not \equiv 0\) for any \(t\in (0,1)\), and$$ 0< \int _{0}^{1} c(s)f(s,0,0)\,ds< +\infty . $$
Proof
Case 2: \(f(t,x_{1},x_{2})\) is nonsingular at both \(t=0,1\) and \(x_{i} =0\), \(i=1,2\). Then, by Theorem 4.1, the following conclusion is valid.
Theorem 4.2
Proof
In fact, if \(f(t,x_{1},x_{2}): [0,1]\times [0,\infty )\times [0, \infty ) \to [0,+\infty )\) is continuous and \(f(t,0,0)\not \equiv 0\), then the condition (B2) holds naturally. □
5 Numerical examples
Example 1
Proof
Let \(\alpha =\frac{5}{2}\), \(\gamma =\frac{1}{4}\), \(\mu =\frac{1}{3}\), \(f(t,x_{1},x_{2})=10 t^{-\frac{1}{4}} [x_{1}^{-\frac{1}{3}}+x_{2} ^{-\frac{1}{8}} ]\). Then \(2<\alpha \le 3\) satisfying \(0<\gamma \le \mu <\alpha -2\) and f is singular at \(t=0\) and \(x_{1}=x_{2}=0\).
Declarations
Acknowledgements
The authors would like to thank the referees for their useful suggestions.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
The authors are supported financially by the National Natural Science Foundation of China (11571296).
Authors’ contributions
The study was carried out in collaboration among all authors. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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.
Authors’ Affiliations
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