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Positive solutions of conformable fractional differential equations with integral boundary conditions
- Wenyong Zhong^{1}Email authorView ORCID ID profile and
- Lanfang Wang^{1}
- Received: 14 March 2018
- Accepted: 2 September 2018
- Published: 12 September 2018
Abstract
In this paper, we discuss the existence of positive solutions of the conformable fractional differential equation \(T_{\alpha }x(t)+f(t,x(t))=0\), \(t\in [0,1]\), subject to the boundary conditions \(x(0)=0\) and \(x(1)= \lambda \int_{0}^{1}x(t)\,\mathrm{d}t\), where the order α belongs to \((1,2]\), \(T_{\alpha }x(t)\) denotes the conformable fractional derivative of a function \(x(t)\) of order α, and \(f:[0,1]\times [0,\infty)\mapsto [0,\infty)\) is continuous. By use of the fixed point theorem in a cone, some criteria for the existence of at least one positive solution are established. The obtained conditions are generally weaker than those derived by using the classical norm-type expansion and compression theorem. A concrete example is given to illustrate the possible application of the obtained results.
Keywords
- Conformable fractional derivatives
- Integral boundary value problems
- Positive solutions
- Fixed point theorems
1 Introduction
The fractional derivative is a generalization of the classical one to an arbitrary order, and the question of what is a fractional derivative was first raised by L’Hôpital in a letter to Leibniz in 1695. Since then, fractional calculus has been extensively studied, and it has been applied to almost every field of science, engineering, and mathematics in the last four decades [1–10]. It is worth emphasizing that there exist a number of definitions of fractional derivatives in the literature, and the different definitions are constructed to satisfy various constraints.
Recently, in [11] Khalil et al. introduced a new well-behaved definition of a fractional derivative termed the conformable fractional derivative. The new definition has drawn much interest from many researchers. And some results have been obtained on the properties of the conformable fractional derivative [11–13]. Several applications and generalizations of the definition were also discussed in [14–20], among which [14] indicated that several specific conformable fractional models are consistent with experimental date, and which [15] interpreted the physical and geometrical meaning of the conformable fractional derivative. Although the definite meaning indicates potential applications of the conformable fractional derivative in physics and engineering, it is worth noting that the investigation of the theory of conformable fractional differential equations has only entered an initial stage.
Initial value problems of conformable fractional differential equations were discussed in [21–23]; and analytical solutions to some specific conformable fractional partial differential equations were studied in [24–30]. For the discussion of boundary value problems (BVPs for short) of conformable fractional differential equations, some theoretical developments have also been achieved. In particular, Lyapunov type inequalities for some conformable boundary value problems were established in [31, 32]; a regular conformable fractional Sturm–Liouville eigenvalue problem was considered in [33]; solvability of some two-point fractional BVP was considered in [34–36] by using topological transversality theorem; a type of three-point fractional BVP was studied in [37] by means of fixed point theorems; and a class of periodic BVP was discussed in [38] by virtue of methods of lower and upper solutions. Applying approximation methods of operators and fixed point theorems, Xiaoyu Dong et al. [39] investigated the existence of positive solutions to a specific type of two-point BVP of p-Laplacian.
In the context of the conformable fractional derivatives, to the best of our knowledge, there have been very few results in the literature for the existence of positive solution to the conformable fractional differential equations with integral boundary conditions. It is worth pointing out that the obtained Green function in this work is singular, while the Green functions of BVPs of some new fractional derivatives with nonsingular kernels are nonsingular [40, 41].
The rest of paper is organized as follows. Section 2 preliminarily provides some definitions and lemmas which are crucial to the following discussion. In Sect. 3, we establish some criteria for the existence of at least one positive solution to the BVP (1.1) by means of the fixed point theorem in a cone. The obtained conditions are generally weaker than those derived by using the classical norm-type expansion and compression theorem [42]. Finally, an example is given to illustrate the possible application of the obtained results.
2 Preliminaries
In this section, we preliminarily provide some definitions and lemmas which play a key role in the following discussion.
Definition 2.1
Definition 2.2
Definition 2.3
([12])
Lemma 2.1
Let α be in \((n,n+1]\). If f is a continuous function on \([0,\infty)\), then, for all \(t>0\), \(T_{\alpha }I_{\alpha }f(t)=f(t)\).
Lemma 2.2
Let α be in \((n,n+1]\). Then \(T_{\alpha }t^{k}=0\) for t in \([0,1]\) and \(k=1,2,\dots,n\).
Lemma 2.3
([12])
Lemma 2.4
Proof
The functions \(\mathcal{G}\), \(\mathcal{H}\) and \(\mathcal{K}\) have several important properties as follows.
Lemma 2.5
Proof
The key tool in our approach is the following well-known fixed point theorem in a cone [42, 43].
Lemma 2.6
- (C1)
\(\|\Phi x\|\leqslant \|x\|\), \(x\in \mathcal{P}\cap \partial \Omega_{1}\).
- (C2)
There exists \(\psi \in \mathcal{P}\setminus \{0\}\) such that \(x\neq \Phi x+\lambda \psi \) for \(x\in \mathcal{P}\cap \partial \Omega_{2}\) and \(\lambda >0\).
3 Main results
- (H)
The function f is nonnegative, and continuous on \([0,1]\times [0, \infty)\), and the parameter λ belongs to \([0,2)\).
Under the hypothesis (H), the operator is well defined and has the following property.
Lemma 3.1
If the hypothesis (H) holds, then \(\Phi (\mathcal{P})\subset \mathcal{P}\).
Proof
Furthermore, observe that the kernel \(\mathcal{G}(t,s)\) of \(\Phi_{1}\) is singular on \([0,1]\times [0,1]\), and that the complete continuity of the operator \(\Phi_{1}\) was verified in [34, 39] by using approximations of the operator. As for the operator \(\Phi_{2}\), its kernel \(\mathcal{H}(t,s)\) is continuous on \([0,1]\times [0,1]\), and using the standard argument, we can easily check that it is also completely continuous. Thus we obtain the following lemma.
Lemma 3.2
If the hypothesis (H) holds, then the operator \(\Phi:\mathcal{P} \mapsto \mathcal{P}\) completely continuous.
The next lemma transforms the BVP (1.1) into an equivalent fixed point problem.
Lemma 3.3
If the hypothesis (H) holds, then a function x in \(C[0,1]\) is a positive solution of the BVP (1.1) if and only if it is a fixed point of Φ in \(\mathcal{P}\).
Proof
On the other hand, if x is a positive solution of the BVP (1.1), then Lemma 2.4 implies \(\Phi x=x\). Moreover, by the same type of argument as for the proof of Lemma 3.1, we also get \(x(t)\geqslant q(t)\|x\|\) for \(t\in [0,1]\). Hence x is a fixed point of Φ in \(\mathcal{P}\). We consequently complete the proof. □
Now we are in a position to give and show the main results.
Theorem 3.1
Assume that the hypothesis (H) holds. If \(f_{0}>\Lambda_{1}\) and \(f^{\infty }< \frac{\Lambda_{2}}{2}\), then the BVP (1.1) has at least one positive solution.
Proof
The assertion will be proven by Lemma 2.6. Observe that Lemma 3.2 ensures that the operator \(\Phi:\mathcal{P}\rightarrow \mathcal{P}\) is completely continuous.
Theorem 3.2
Assume that the hypothesis (H) holds. If \(f^{0}<\Lambda_{2}\) and \(f_{\infty }>\Lambda_{1}\), then the BVP (1.1) has at least one positive solution.
Proof
The assertion will be shown by Lemma 2.6. Note that the complete continuity of the operator Φ is guaranteed by Lemma 3.2. We only need to prove that the operator Φ satisfies the conditions (C1) and (C2) in Lemma 2.6.
Remark 3.1
The conditions in Lemma 2.6 are weaker than those in the classical norm-type expansion and compression theorem [42], and accordingly, it is generally difficult to utilize the latter to prove Theorem 3.1 and 3.2.
By Theorem 3.1 and 3.2, we directly obtain the following corollary.
Corollary 3.1
If \(f_{0}=\infty \) and \(f^{\infty }=0\), or if \(f^{0}=0\) and \(f_{\infty }=\infty\), then the BVP (1.1) has at least one positive solution.
3.1 An illustrative example
4 Conclusion
By using the fixed point theorem in a cone, we establish some criteria for the existence of at least one positive solution to the conformable fractional differential equations with integral boundary conditions. The obtained conditions are generally weaker than those derived by using the classical norm-type expansion and compression theorem and are easy to satisfy and check. We will further investigate boundary value problems of fractional differential equations with nonsingular kernel in the future.
Declarations
Acknowledgements
The authors are grateful to the referees for carefully reading the paper and for their comments and suggestions.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
The paper is supported by the Natural Science Foundation of Hunan Province of China (Grant no. 11JJ3007).
Authors’ contributions
All authors contributed equally and significantly in writing this article. 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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