On the oscillation and asymptotic behavior for a kind of fractional differential equations
© Wang et al.; licensee Springer. 2014
Received: 29 September 2013
Accepted: 13 January 2014
Published: 31 January 2014
In this paper, we discuss the oscillations of the fractional order differential equation , , , where q is a positive real-valued function and f is a continuous functional; denotes the Riemann-Liouville differential operator of order α, . We use the Riccati transformation technique to obtain some sufficient conditions which guarantee that every solution of the equation is oscillatory or the limit inferior converges to zero. Two examples are given to show the applications of our main results.
The theory of fractional calculus goes back to Leibniz’s note in his list to L’Hospital , dated 30 September 1695, in which the meaning of the derivative of order 1/2 is discussed. After that in pure mathematics field the foundation of the fractional differential equations had been established. However, in recent years, many researchers found that the fractional differential equations are more accurate in describing some practical models, e.g. polymers. Today it has been used widely in physics, electrochemistry, control theory, and electromagnetic fields [2–7]. Furthermore, the fractional calculus can also provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a ‘memory’ term in the model [8–13]. Since these studies there has been much research actively concerned with the fractional differential equations and many useful achievements have been obtained [14–18].
From the 1960s, a lot of books and theses about the oscillatory behavior for first, second, and higher order differential equations are presented, see [19–21]. The study of the oscillatory problem with a view on fractional differential equation is just being initiated. As a new cross-cutting area, recently some attention has been paid to oscillations of fractional differential equations [22–29].
By the Riccati transformation technique the authors obtained some sufficient conditions, which guarantee that every solution of the equation is oscillatory.
where is the Liouville right-sided fractional derivative of order of y.
where denotes the Liouville right-sided fractional derivative of order α of x. Using a generalized Riccati function and the inequality technique, he established some new oscillation criteria.
where is a real number, is the Liouville right-sided fractional derivative of order α of y. By a generalized Riccati transformation technique, oscillation criteria for the nonlinear fractional differential equation are obtained.
where also denotes the Liouville right-sided fractional derivative and some sufficient conditions for the oscillation of the equation have been given.
The above works on the oscillation are all concerned with fractional equations with Liouville right-sided fractional derivative by the Riccati transformation technique.
We notice that very little attention is paid to oscillations of fractional differential equations with a Riemann-Liouville derivative. For work studying the oscillatory behavior of fractional differential equations with the Riemann-Liouville derivative we refer to [27, 28], and .
where denotes the Riemann-Liouville differential operator of order q with , and the operator is the Rieman-Liouville fractional integral operator. The authors obtained some new oscillation criteria by reducing the fractional differential equation to the equivalent Volterra fractional integral equation and by applying the inequality technique.
where is a Riemann-Liouville like discrete fractional difference operator of order α, and some oscillation criteria are established by the same method in .
where denotes the Riemann-Liouville or Caputo differential operator of order q with , , and the operator is the Rieman-Liouville fractional integral operator. The authors obtained some new oscillation criteria by the same method as .
and denotes the Riemann-Liouville integral operator.
We will use the method of the Riccati transformation technique to study the oscillatory behavior of the fractional differential equation (1.1). To the best of our knowledge, there is not any result on the oscillation of the fractional differential equation involving the Riemann-Liouville derivative by the method of the Riccati transformation technique.
A solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros on and otherwise it is non-oscillatory. An equation is said to be oscillatory if all its solutions are oscillatory.
The paper is organized as follows. In the next section, we present some basic definitions of the fractional differential and integral operators, and provide some necessary lemmas. In Section 3, we mainly use the Riccati transformation technique to get some sufficient conditions which guarantee that every solution of (1.1) is oscillatory or the limit inferior converges to zero. Our results are essential new. Finally we provide some examples to show applications of our criteria.
2 Some preliminary lemmas
Lemma 2.1 
for some , , where n is the smallest integer greater than or equal to α.
Before stating our main results, we begin with the following lemmas which are crucial in the proofs of the main results.
Proof Let x be an eventually positive solution of (1.1), which means that there exists a such that for .
From Lemma 2.1 we know that if exists, and this means for any , especially .
Obviously there exists a sufficient large such that . So for . The proof is complete. □
Lemma 2.3 
where equality holds if and only if .
3 Main results
where , then every solution x of (1.1) is oscillatory or .
If , from Lemma 2.2, there exists a such that for . Furthermore, using the same measure in Lemma 2.2, we can easily obtain the result that there exists a such that for . So we get for .
which is a contradiction to the condition (3.1) and the proof is complete. □
Then every solution x of (1.1) is either oscillatory or .
Proof This follows from Theorem 3.1 by taking . □
Then every solution x of (1.1) is either oscillatory or .
Proof Taking , then the condition (3.1) in Theorem 3.1 is reduced to (3.5). Hence the result is obtained from Theorem 3.1. □
then every solution x of (1.1) is oscillatory or .
Proof Suppose x is a non-oscillatory solution of (1.1). We only consider the case that is eventually positive, since the case that is eventually negative is similar. Assume that for all with large enough .
which contradicts (3.7). The proof is complete. □
where . Then every solution of (1.1) is either oscillatory or .
which is a contradiction of (3.10). So the proof is complete. □
In this section, we will show applications of our main results.
which implies that all conditions in Theorem 3.1 hold. So by Theorem 3.1 every solution of (4.1) is oscillatory or .
which yields the result that all conditions on Theorem 3.3 hold. Therefore, by Theorem 3.3 every solution of (4.2) is oscillatory or .
This research is supported by the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).
- Leibniz GW: Mathematische Schriften. Georg Olms Verlagsbuchhandlung, Hildesheim; 1962.Google Scholar
- Diethelm K: The Analysis of Fractional Differential Equations. Springer, Berlin; 2010.View ArticleGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Das S: Functional Fractional Calculus for System Identification and Controls. Springer, New York; 2008.Google Scholar
- Oldham KB, Spanier J: The Fractional Calculus. Academic Press, San Diego; 1974.Google Scholar
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.Google Scholar
- Metzler R, Schick S, Kilian HG, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 1995, 103: 7180-7186. 10.1063/1.470346View ArticleGoogle Scholar
- Glöckle WG, Nonnenmacher TF: A fractional calculus approach to self similar protein dynamics. Biophys. J. 1995, 68: 46-53. 10.1016/S0006-3495(95)80157-8View ArticleGoogle Scholar
- Diethelm K, Freed AD: Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional-derivative kernels with application to the human calcaneal fat pad. Biomech. Model. Mechanobiol. 2006, 5: 203-215. 10.1007/s10237-005-0011-0View ArticleGoogle Scholar
- Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, River Edge; 2000.View ArticleGoogle Scholar
- Magin RL: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 2004, 32: 1-377. 10.1615/CritRevBiomedEng.v32.10View ArticleGoogle Scholar
- Baillie RT: Long memory processes and fractional integration in econometrics. J. Econ. 1996, 73: 5-59. 10.1016/0304-4076(95)01732-1MathSciNetView ArticleGoogle Scholar
- Diethelm K, Ford NJ: Analysis of fractional differential equations. J. Math. Anal. Appl. 2002, 265: 229-248. 10.1006/jmaa.2000.7194MathSciNetView ArticleGoogle Scholar
- Cabada A, Wang GT: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 2012, 389: 403-411. 10.1016/j.jmaa.2011.11.065MathSciNetView ArticleGoogle Scholar
- Sun S, Zhao Y, Han Z, Li Y: The existence of solutions for boundary value problem of fractional hybrid differential equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 4961-4967. 10.1016/j.cnsns.2012.06.001MathSciNetView ArticleGoogle Scholar
- Galeone L, Garrappa R: Explicit methods for fractional differential equations and their stability properties. J. Comput. Appl. Math. 2009, 288: 548-560.MathSciNetView ArticleGoogle Scholar
- Zhang X, Liu L, Wu Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 2012, 55: 1263-1274. 10.1016/j.mcm.2011.10.006MathSciNetView ArticleGoogle Scholar
- Han Z, Sun S, Shi B: Oscillation criteria for a class of second order Emden-Fowler delay dynamic equations on time scales. J. Math. Anal. Appl. 2007, 334: 847-858. 10.1016/j.jmaa.2007.01.004MathSciNetView ArticleGoogle Scholar
- Bohner M, Erbe L, Peterson A: Oscillation for nonlinear second order dynamic equations on a time scale. J. Math. Anal. Appl. 2005, 301: 491-507. 10.1016/j.jmaa.2004.07.038MathSciNetView ArticleGoogle Scholar
- Erbe L, Peterson A: Boundedness and oscillation for nonlinear dynamic equations on a time scale. Proc. Am. Math. Soc. 2004, 132: 735-744. 10.1090/S0002-9939-03-07061-8MathSciNetView ArticleGoogle Scholar
- Chen D: Oscillation criteria of fractional differential equations. Adv. Differ. Equ. 2012., 2012: Article ID 33Google Scholar
- Chen D: Oscillatory behavior of a class of fractional differential equations with damping. Univ. Politeh. Bucharest Sci. Bull. 2013, 75: 107-118.Google Scholar
- Zheng B: Oscillation for a class of nonlinear fractional differential equations with damping term. J. Adv. Math. Stud. 2013, 6: 107-115.MathSciNetGoogle Scholar
- Han Z, Zhao Y, Sun Y, Zhang C: Oscillation for a class of fractional differential equation. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 390282Google Scholar
- Qi C, Huang S: Interval oscillation criteria for a class of fractional differential equations with damping term. Math. Probl. Eng. 2013., 2013: Article ID 301085 10.1155/2013/301085Google Scholar
- Grace SR, Agarwal RP, Wong JY, Zafer A: On the oscillation of fractional differential equations. Fract. Calc. Appl. Anal. 2012, 15: 222-231.MathSciNetView ArticleGoogle Scholar
- Marian SL: Oscillation of fractional nonlinear difference equations. Math. Æterna 2012, 2: 805-813.MathSciNetGoogle Scholar
- Chen D, Qu P, Lan Y: Forced oscillation of certain fractional differential equations. Adv. Differ. Equ. 2013., 2013: Article ID 125Google Scholar
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1959.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.