Positive solutions for a class of nonlinear Hadamard fractional differential equations with a parameter
 Haisong Huang^{1}Email author and
 Weihua Liu^{2}
https://doi.org/10.1186/s1366201815519
© The Author(s) 2018
Received: 27 December 2017
Accepted: 8 March 2018
Published: 15 March 2018
Abstract
In this paper, we investigate a class of boundary value problem of nonlinear Hadamard fractional differential equations with a parameter. By means of the properties of the Green function and Guo–Krasnosel’skii fixedpoint theorem on cones, the existence and nonexistence of positive solutions are obtained. Finally, some examples are presented to show the effectiveness of our main results.
Keywords
MSC
1 Introduction
Fractional differential equations have given rise to abroad attention of many researchers by the intensive development of the theory of fractional calculus itself. On the other hand, fractional differential equations can better describe many phenomena than ordinary differential equations in many diverse and widespread fields of science and engineering. For the development of fractional calculus and applications, we refer the reader to [1–10] and the references therein. For example, by employing the Avery–Henderson fixedpoint theorem, Li [11] obtained the existence of positive solutions as considered for a fractional differential equation with pLaplacian operator. In [12], existence and uniqueness results for a new class of boundary value problems of sequential fractional differential equations with nonlocal nonseparated boundary conditions involving lowerorder fractional derivatives were given by some standard fixedpoint theorems. The existence and multiplicity of solutions or positive solutions for nonlinear boundary value problems involving fractional differential equations with kinds of boundary value conditions were studied by some wellknown fixedpoint theorems, the lower and upper solutions method and the monotone iterative technique; see [13, 14] and the references therein. For example, the authors of [15] investigated the solutions of fractional integrodifferential equations with boundary value conditions, respectively. In [16], the existence and multiplicity of positive solutions were obtained for nonlinear Caputo fractional differential equations with integral boundary conditions. Henderson and Luca investigated the positive solutions of nonlinear boundary value problems for systems of fractional differential equations in the book [17]. In [18], by applying the fixedpoint theorem due to Leggett–Williams, the authors considered the existence of positive solutions for a system of fractional multipoint boundary value problem with pLaplacian operator.
In the past ten years, most of the work on the topic is based on Riemann–Liouville and Caputotype fractional differential equations. Recently, more and more scholars paid attention to the boundary value problems of nonlinear Hadamard fractional differential equations [19–24]. By applying some standard fixedpoint theorems, Ahmad and Ntouyas [25, 26] studied the existence and uniqueness of solutions for Hadamardtype fractional differential equations for boundary value problems and systems with integral boundary conditions, respectively. Based on standard fixedpoint theorems for multivalued maps, Ahmad et al. [27] investigated the existence of solutions for fractional boundary value problems involving Hadamardtype fractional differential inclusions and integral boundary conditions. Aljoudi et al. [28] studied a nonlocal boundary value problem of Hadamardtype coupled sequential fractional differential equations supplemented with coupled strip conditions. By discussing a continuity, integrable estimation, and the asymptotic property on Mittag–Leffler functions, Li and Wang [29] investigated the existence of solutions and finitetime stability for a class of nonlinear Hadamard fractional differential equations with constant coefficient. In [30, 31], the existence of positive solutions for nonlinear Hadamard fractional differential equations with fourpoint coupled and coupled integral boundary conditions were given by the Guo–Krasnosel’skii fixedpoint theorems, respectively.
2 Preliminaries
For convenience of the reader, we present some necessary definitions and lemmas from Hadamard fractional calculus theory in this section.
Definition 2.1
([32])
Definition 2.2
([32])
Now we will give the Green function of linear equation and some properties of the Green function.
Lemma 2.1
Proof
Lemma 2.2
Proof
For \(1\leq t\leq s\leq e\), It is easy to see that \(G(t,s)\geq0\) and \(g(t)G(e,s)= G(t,s)\leq G(e,s)\).
Our main results are based on the following Guo–Krasnosel’skii fixedpoint theorem on cones.
Lemma 2.3
 \((B_{1})\) :

\(\Vert Sw \Vert \leq \Vert w \Vert \), \(w\in \mathscr{P}\cap\partial\Omega_{1}\), and \(\Vert Sw \Vert \geq \Vert w \Vert \), \(w\in \mathscr{P}\cap\partial\Omega_{2}\),
 \((B_{2})\) :

\(\Vert Sw \Vert \geq \Vert w \Vert \), \(w\in \mathscr{P}\cap\partial\Omega_{1}\), and \(\Vert Sw \Vert \leq \Vert w \Vert \), \(w\in \mathscr{P}\cap\partial\Omega_{2}\).
Lemma 2.4
\(S_{\lambda}:\mathscr{P}\rightarrow \mathscr{P}\) is completely continuous.
Proof
The operator \(S_{\lambda}:\mathscr{P}\rightarrow \mathscr{P}\) is continuous in view of continuity of \(G(t,s)\), \(a(t)\) and \(f(x(t))\). By means of the Arzela–Ascoli theorem, \(S_{\lambda}:\mathscr{P}\rightarrow \mathscr{P}\) is completely continuous. □
3 Main results
In this section, we establish some sufficient conditions for the existence and nonexistence of positive solutions for boundary value problem (1).
Theorem 3.1
Proof
Theorem 3.2
Proof
Theorem 3.3
Proof
For the remainder of this section, we will need the following assumption.
Assumption 1
\((\min_{x\in[g(l)r,r]}f(x))/r>0\), where \(l\in(1,e)\).
In view of the continuity of \(f(x)\) and Assumption 1, we have \(0<\lambda_{1}\leq+\infty\) and \(0\leq\lambda_{2}<+\infty\).
Theorem 3.4
Suppose Assumption 1 holds. If \(f_{0}=f_{\infty}=+\infty\), then boundary value problem (1) has at least two positive solutions for each \(\lambda\in(0,\lambda_{1})\).
Proof
Corollary 3.1
Suppose Assumption 1 holds. If \(f_{0}=+\infty\) or \(f_{\infty}=+\infty\), then boundary value problem (1) has at least one positive solution for each \(\lambda\in(0,\lambda_{1})\).
Theorem 3.5
Suppose Assumption 1 holds. If \(f_{0}=f_{\infty}=0\), then boundary value problem (1) has at least two positive solutions for each \(\lambda\in(\lambda_{2},+\infty)\).
Proof
Corollary 3.2
Suppose Assumption 1 holds. If \(f_{0}=0\) or \(f_{\infty}=0\), then boundary value problem (1) has at least one positive solution for each \(\lambda\in(\lambda_{2},+\infty)\).
By the above theorems, we can obtain the following results.
Corollary 3.3
Suppose Assumption 1 holds. If \(f_{0}=+\infty\), \(f_{\infty}=d\) or \(f_{\infty}=+\infty\), \(f_{0}=d\), then boundary value problem (1) has at least one positive solution for each \(\lambda\in(0,(dC_{1})^{1})\).
Corollary 3.4
Suppose Assumption 1 holds. If \(f_{0}=0\), \(f_{\infty}=d\) or \(f_{\infty}=0\), \(f_{0}=d\), then boundary value problem (1) has at least one positive solution for each \(\lambda\in((g(l)dC_{2})^{1},+\infty)\).
Theorem 3.6
Suppose Assumption 1 holds. If \(F_{0}<+\infty\) and \(F_{\infty}<+\infty\), then there exists a \(\lambda_{0}>0\) such that, for all \(0<\lambda<\lambda_{0}\), boundary value problem (1) has no positive solution.
Proof
Theorem 3.7
Suppose Assumption 1 holds. If \(f_{0}>0\) and \(f_{\infty}>0\), then there exists a \(\lambda_{0}>0\) such that, for all \(\lambda>\lambda_{0}\), boundary value problem (1) has no positive solution.
Proof
4 Some examples
In this section, we will present some examples to illustrate the main results.
Example 4.1
Example 4.2
Since \(\alpha=2.5\) and \(a(t)=\ln t\), we have \(C_{1}\leq0.2006\) and \(C_{2}\geq0.1146\). Let \(f(x)=x^{\vartheta}\), \(0<\vartheta<1\). Then from [35], we have \(F_{\infty}=0\) and \(f_{0}=+\infty\). Choose \(l=e^{0.5}\). Then \(g(e^{0.5})=0.5^{1.5}\approx0.3536\). So \(g(l)C_{2}f_{0}>F_{\infty}C_{1}\) holds. Thus, by Theorem 3.2, the boundary value problem (29) has a positive solution for each \(\lambda\in(0,+\infty)\).
Example 4.3
 (i)
Choose \(l=e^{0.5}\). Then \(g(e^{0.5})=0.5^{1.5}\approx0.3536\). So \(g(l)C_{2}f_{\infty}>F_{0} C_{1}\) holds. Thus, by Theorem 3.1, the boundary value problem (30) has a positive solution for each \(\lambda\in(0.1234,2.4925)\).
 (ii)
By Theorem 3.6, the boundary value problem (30) has no positive solution for all \(\lambda\in(0,0.0083)\).
 (iii)
By Theorem 3.7, the boundary value problem (30) has no positive solution for all \(\lambda\in(12.3388,+\infty)\).
5 Conclusions
By means of the properties of the Green function and the Guo–Krasnosel’skii fixedpoint theorem on cones, we have investigated the existence and nonexistence of positive solutions for a class of boundary value problems of nonlinear Hadamard fractional differential equations with a parameter. Three examples are given to show the effectiveness of the obtained results. Furthermore, by using similarly the method in this paper, we can also obtain the existence and nonexistence of positive solutions for nonlinear Hadamard fractional boundary value problems as follows: \((D^{\alpha}x)(t)+\lambda a(t)f(x(t))=0\), \(x(1)=x(e)=(\delta x)(1)=0\), \(\alpha\in(2,3]\), \(t\in[1,e]\).
Declarations
Acknowledgements
The authors would like to thank the anonymous referees and the editor for their constructive suggestions for improving the presentation of the paper.
Authors’ contributions
All authors contributed equally in this article. They 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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