Oscillation of solutions for certain fractional partial differential equations
- Wei Nian Li^{1}Email author
Received: 23 April 2015
Accepted: 11 January 2016
Published: 22 January 2016
Abstract
Keywords
MSC
1 Introduction
The fractional calculus may be considered an old and yet novel topic. On 30 September 1695, Leibniz wrote a letter to L’Hôspital to discuss the meaning of the derivative of order \(\frac{1}{2}\) [1]. After that in pure mathematics field the foundation of the fractional differential equations had been established. At the same time, many researchers found that the fractional differential equations play increasingly important roles in the modeling of engineering and science problems. It has been established that, in many situations, these models provide more suitable results than analogous models with integer derivatives. In the past few years, the fractional calculus and the theory of fractional differential equations have been investigated extensively. For example, see [2–22] and the references cited therein.
In recent years, the research on the oscillatory behavior of solutions of fractional differential equation has been a hot topic and some results have been established. For example, see [11–19]. However, to the best of the author’s knowledge, very little is known regarding the oscillatory behavior of fractional partial differential equations which involve the Riemann-Liouville fractional partial derivative up to now [20–22].
2 Formulation of the problems
- (A1)
\(a\in C(\mathbb{R}_{+};(0,\infty))\);
- (A2)\(m\in C(\overline{G}\times\mathbb{R};\mathbb{R})\), and$$m(x,t,\xi) \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \geq0,&\mbox{if } \xi\in(0,\infty),\\ \leq0,& \mbox{if } \xi\in(-\infty,0); \end{array}\displaystyle \right . $$
- (A3)
\(f\in C(\overline{G};\mathbb{R})\).
By a solution of the problem (1)-(2) (or (1)-(3)) we mean a function \(u(x,t)\) which satisfies (1) on G̅ and the boundary condition (2) (or (3)).
A solution \(u(x,t)\) of the problem (1)-(2) (or (1)-(3)) is said to be oscillatory in G if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.
Definition 2.1
Definition 2.2
Definition 2.3
Lemma 2.1
[20]
3 Main results
First, we introduce the following fact [23]:
Next, we establish some useful lemmas.
Lemma 3.1
Proof
The proof of the following lemma is similar to that of Lemma 3.1 and we omit it.
Lemma 3.2
Lemma 3.3
Proof
Similarly we can obtain the following lemma.
Lemma 3.4
Finally, we give our main results.
Theorem 3.1
If the inequality (8) has no eventually positive solutions and the inequality (11) has no eventually negative solutions, then every solution of the problem (1)-(2) is oscillatory in G.
Proof
Suppose to the contrary that there is a nonoscillatory solution \(u(x,t)\) of the problem (1)-(2). It is obvious that there exist \(t_{0}\geq0\) such that \(\mid(x,t)\mid> 0\) for \(t\geq t_{0}\). Therefore \(u(x,t)>0\) or \(u(x,t)<0\), \(t\geq t_{0}\).
If \(u (x,t)>0\), \(t\geq t_{0}\), using Lemma 3.1 we see that \(U (t)>0\) is a solution of the inequality (8), which is a contradiction.
If \(u (x,t)<0\), \(t\geq t_{0}\), using Lemma 3.2, it is easy to see that \(U (t)<0\) is a solution of the inequality (11), which is a contradiction. This completes the proof. □
Theorem 3.2
Proof
We prove that the (8) has no eventually positive solutions and the inequality (11) has no eventually negative solutions.
Assume that (11) has a negative solution \(\widetilde{V}(t)\). Noting that condition (17) holds and using the above mentioned method, we easily obtain a contradiction. This completes the proof of Theorem 3.2. □
Using Lemma 3.3 and Lemma 3.4, we easily establish the following theorems.
Theorem 3.3
If the inequality (12) has no eventually positive solutions and the inequality (15) has no eventually negative solutions, then every solution of the problem (1)-(3) is oscillatory in G.
Theorem 3.4
4 Examples
In this section, we give two illustrative examples.
Example 4.1
Example 4.2
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (10971018). The author is very grateful to the reviewers for their valuable suggestions and useful comments on this paper.
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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