Periodic solutions for a kind of second order neutral functional differential equations☆
Introduction
Forced second order differential equations such as Duffing and Rayleigh equations have been the subject of many investigations since they arise in many physical model. In particular, periodic solutions of such equations with delays were studied by using Mawhin’s continuation theorem by Iannacci and Nkashama [1], Wang and Cheng [2], and many others (see for example [3], [4], [5], [6], [7], [8]). But the works to study the existence of periodic solutions for neutral functional differential equations (NFDE) by using Mawhin’s continuation theorem rarely appeared [9], [10]. The reason for it lies in the following two respects. The first is that the criterion of L-compact of nonlinear operator N on the set is difficulty to establish; the second is that a priori bound of solutions is not easy to estimate. In paper [9], Enrice Serra studied a kind of first order NFDE in the following form:under the condition andwhere is Caraéodory function of 2π period in the first variable, P is a subset of with positive measure. The author obtained that the above equation has at least one periodic solution (see Theorem 3.1 in [9] ).
In this paper, we investigate the existence of periodic solutions to the second order NFDEwhere are constants and the forcing function is a T periodic continuous function with zero mean over , i.e.,
In this paper, we will establish criteria for the existence of T periodic solutions of (1) under a sublinear condition on the function g. Our criteria, as will be explained in the last section, are compatible with the existent ones for the well known Duffing equations. The techniques for obtaining the existence criteria are based on a priori estimation and Mawhin’s continuation theorem.
Section snippets
Preparatory results
Let with the norm , and with the norm . Clearly, CT and are two Banach spaces. We also define operator A in the following form Lemma 1 LetIf , then B satisfies the following conditions . .
The conclusion of case 1 is obviously true. The conclusion of case 2 follows from
Main results
In order to use Lemma 3, throughout this paper, we assume that p is real T-periodic function with zero mean over , g is real continuous function and , . We denote . Theorem 1 If there are constants and such thatand eitherorThen (1) has a T-periodic solution. Proof It is easy to see that Eq. (1) has a T-periodic solution if and only if the following operator equationhas a T-periodic solution, where L is defined in Section 2
Remarks and examples
Remark 1 It was proved in [3] that if there exist constants such that for and for , then (1) has a 2π periodic solution when c = 0. In Theorem 1 we only assume sublinear g and hence is an improvement. Moreover, we extend the results in [11] to neutral differential equations. Example 1 Consider the following equation:Corresponding to Eq. (1), we have andSo the conditions of
References (13)
- et al.
A priori bounds for periodic solutions of a delay Rayleigh equation
Appl. Math. Lett.
(1999) - et al.
periodic solutions for a kind of Rayleigh equation with a deviating argument
Appl. Math. Lett.
(2004) - et al.
Periodic solutions for a kind of second order differential equations with multiple deviating argument
Appl. Math. Comput.
(2003) Periodic solutions for some nonlinear differential equations of neutral type
Nonlinear Anal. TMA
(1991)- et al.
Periodic solutions to neutral differential equations with deviating argument
Appl. Math. Comput.
(2004) - et al.
Periodic solutions of a second order forced sublinear differential equations with delay
Appl. Math. Lett.
(2005)
Cited by (8)
Positive periodic solutions of third-order neutral differential equations with delayed derivative terms
2024, Bulletin des Sciences MathematiquesInteraction between periodic elastic waves and two contact nonlinearities
2012, Mathematical Models and Methods in Applied SciencesPositive periodic solutions for singular high-order neutral functional differential equations
2018, Mathematica SlovacaPositive periodic solutions of second-order neutral functional differential equations with delayed derivative terms
2014, Acta Mathematica Sinica, Chinese SeriesAntiperiodic solutions for a generalized high-order (p, q)-Laplacian neutral differential system with delays in the critical case
2013, Abstract and Applied Analysis
- ☆
Supported by the NNSF of China (No. 10571050) and Key Project of Hunan Province Department of Education of China.