Oscillation Criteria for Nonlinear Fractional Differential Equations
Oscillation Criteria for Nonlinear Fractional Differential Equations
Run Xu
Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China
Received 5 October 2013; Accepted 3 December 2013
Academic Editor: Nazim Idrisoglu Mahmudov
Copyright © 2013 Run Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Several oscillation criteria are established for nonlinear fractional differential equations of the form where is the Liouville right-side fractional derivative of order of and is a quotient of two odd positive integers. We also give some examples to illustrate the main results. To the best of our knowledge, the results are initiation for the oscillatory behavior of the equations.
1. Introduction
In this paper, we are dealing with the oscillation problem of nonlinear fractional differential equations of the form where is a constant, is a ratio of two odd positive integers. is the Liouville right-side fractional derivative of order of defined by
Throughout this paper, we will suppose that the following conditions hold:, , , ;, there exists a function such that for and , .
By a solution of (1), we mean a function such that , and satisfies (1) on . A nontrivial solution of (1) is called oscillatory if it has arbitrary large zero. Otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all of its solutions are oscillatory.
The theory of fractional derivatives goes back to Leibniz’s note in [1], and this led to the appearance of the theory of derivatives and integrals of arbitrary order. The theory had taken a more or less finished form due primarily to Liouville, Grünwald, Letnikov, and Riemann by the end of nineteenth century. We can see some of the books such as [2, 3] on the subject of fractional derivatives and fractional integrals.
Fractional differential equations are generalizations of classical differential equations of integer order. In the last few decades, many researchers found that fractional derivatives and fractional integrals were applied in widespread fields of science and engineering, especially in mathematical modeling and simulation of systems and processes, instead of simply being applied in pure theoretical fields of mathematics. Nowadays, many articles have investigated some aspects of fractional differential equations, such as the existence, the uniqueness and stability of solutions, the methods for explicit and numerical solutions, and the stability of solutions (we refer the reader to see [4–9] and the references cited in there). In very recent days, the research on oscillation of various fractional differential equations is being a hot topic; see [10–16].
In [10], Grace et al. discussed the oscillation of the following question: where denotes the Riemann-Liouville differential operator of order with , and the functions , , and are continuous functions.
In [11], Chen et al. established several oscillation criteria for (3) with some additional initial conditions and , is an integer. They improved and extended some results of [10].
In [12], Chen considered the oscillation of the fractional differential equation with .
In [13], Han et al. have established some oscillation criteria for the equation
In [14], Qi and Cheng studied the following equation: with and established some new interval oscillation criteria by using a generalized Riccati transformation and inequality technique.
In [15], Feng and Meng paid attention to the oscillation of the fractional differential equation
In [16], Chen considered the oscillation of the fractional differential equation
The purpose of this paper is to establish some oscillation criteria for (1) by generalized Riccati function and present some applications for our results.
In order to prove our theories, we use the general weighted functions from the class . We say that a function belongs to the function class , if , where , which satisfies , for , and has nonpositive continuous partial derivative on .
2. Main Results
First, we set then it follows that
We give the following lemmas for our results.
Lemma 1 (see [17]). Let and be nonnegative; then
Lemma 2. Assume that is an eventually positive solution of (1), and then there exists a sufficiently large such that on , and one of the following two conditions hold:(i) on ,(ii) on and .
Proof. From the hypothesis, there exists a such that on , so that on , and we have Then is strictly increasing on , and we can conclude that is eventually of one sign. We claim that on , where is sufficiently large. Otherwise, if there exists a such that , then we can get on ; from those conditions we get that is, Integrating two sides of the previous inequality from to leads to Then from we have , which implies that for a certain constant (...truncated)