Further the result can be extended to more general equations. The coe cients of this equation are allowed to have negative and positive values, which is an expansion of the results in the references. The qualitative study of such equations has, besides its theoretical interest, signi. In this paper, we establish some new oscillation criteria for plaplacian delay dynamic equations with damping a t. We present new sufficient conditions for the oscillation of all solutions of such equations.
Speci c delay di erential equations are stood out by to be a rst approach of that generalization of ode theory. Not only does this theory unify those of differential equations and difference equations, but it also extends these. Oscillation of fourthorder delay differential equations. Stability and oscillations in delay differential equations of.
Keywordsoscillation, parabolic partial difference equations with delays, hyperbolic partial difference equations with. Some new oscillation criteria are given for first order neutral delay differential equations with variable coefficients. Nonoscillation in linear delay differential systems. Oscillation of a class of the fourthorder nonlinear difference equations. There he showed that the nth eigenfunction of a sturmliouville problem has precisely n1 roots. Tang, oscillations of delay differential equations with variable coefficients, chinese, j. Thus delay di erential equations with a constant delay. Some properties of oscillation of second order neutral. Oscillation of nonlinear neutral type second order delay.
Notes on oscillation of linear delay differential equations. This has resulted in hundreds of research papers in every major mathematical journal, and several books. Oscillation criteria for delay and advanced difference. In this paper, we consider the neutral delay partial difference equation 1,2,, 1,0 mn m n i m k n lii i aca pmna. Chapter 1 deals with an analysis of the dynamical characteristics of the.
Oscillation of neutral delay partial difference equation. Oscillation theorem for second order neutral delay. The results improve and complement some earlier ones in the literature. Oscillation of delay difference equation sciencedirect. Research article oscillation theorems for secondorder nonlinear neutral delay differential equations tongxingli 1 andyuriyv. However, inspite ofthe large number investigations of impulsive differential equations, their oscillation theory has not yet been fully elaborated, unlike the case of oscillation theory for delay differential equations. Oscillation properties of third order neutral delay. In a neutral delay differential equation, the highestorder derivative of the unknown function appears both with and without delay. Oscillation, delay dynamic equations, time scale, riccati technique. In 1994 koplatadze and kvinikadze 14 improved c 6, while in 1996 philos and s cas 24 and jaro s and stavroulakis 11 derived the conditions. Central south university of technology 29 3, 287288, 1998. Dec 28, 2012 oscillation of a class of the fourthorder nonlinear difference equations. Jul 19, 2010 some sufficient conditions are established for the oscillation of secondorder neutral differential equation, where. Them on o graphs by erbeetal, gyori and ladas, and laddeetal 3, 5.
Oscillation results for third order nonlinear neutral delay. Oscillation of nonlinear partial difference equations with delays. This monograph provides a definitive overview of recent advances in the stability and oscillation of autonomous delay differential equations. Recently the oscillatory behavior of impulsive delay differential equations has attracted the attention of many researchers. The results complement and improve those of grammatikopoulos et al. Our results generalize and extend some of the wellknown results in the literature. Greece introduction and statement of results a delay di. This article deals with the oscillation of a certain class of fourthorder delay differential equations. Solving second order delay differential equations by. In recent years there has been a resurgence of interest in the study of delay differential equations motivated largely by new applications in physics, biology, ecology, and physiology. Oscillation theory of delay differential equations. Rogovchenko 2 department of mathematics, linyi university, linyi, shandong, china. Delaydifference equations with periodic coefficients.
Result 1 see levin and may, 1976 for rr 0 and t odd, all roots of the characteristic equation are complex. Oscillation criteria for secondorder nonlinear neutral. Our results also improve some wellknown results in the literature even when applying them to. The most interesting oscillation criteria for the secondorder linear delay differential equation, the secondorder difference equation and the secondorder. Throughout, the main topics of study are shown in action. Little prior knowledge of the subject is required other than a firm grounding in the main techniques of differential equation theory. Oscillation theorems for certain delay difference inequalities pon sundar1, b. Oscillation of delay dynamic equations with oscillating coefficients author.
Oscillation of delay dynamic equations with oscillating. We state a new oscillation theorem for the sublinear case and we complete the existing. Ddes are also called timedelay systems, systems with aftereffect or deadtime, hereditary systems, equations with deviating argument, or differentialdifference equations. Topics include linear and nonlinear delay and integrodifferential equations, which have potential applications to both biological and physical dynamic processes.
Keywords oscillation, delay terms, bounded solutions, linear and nonlinear, difference inequalities. In mathematics, delay differential equations ddes are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. In this paper, we consider the nonoscillatory problems of linear delay differential systems of odddimension. Some new oscillation criteria are presented by transforming this equation to the firstorder delayed and advanced differential equations.
On the oscillation of the solutions to linear difference equations with variable delay. Oscillation criteria for secondorder delay, difference. They belong to the class of systems with the functional state, i. Meimaridou, oscillation of secondorder neutral delay differential equations, rat. In this article, a class of fourthorder difference equations with quasidifferences and deviating argument is considered. These criteria extend and generalize those results in the literature. Oscillation criteria for secondorder neutral delay. Shanmugavalli, oscillation conditions for first order neutral difference equations with positive and negative variable coefficients, malaya journal of matematik communicated. In this paper, via comparison with first order oscillatory di. Oscillation theorems for certain delay difference inequalities pon sundar1. The aim of this paper is to obtain new conditions for the oscillation of solutions for equations with distributed delay.
Moreover, the theory can be applied to other different types of time scales. Oscillation results for third order nonlinear neutral. Keywords oscillation, parabolic partial difference equations with delays, hyperbolic partial difference equations with delays, neutral type, discrete gaussian formula. Moreover, some illustrating examples are also provided to show the importance of our results.
Oscillation results for secondorder delay dynamic equations. In fact, in the last 25 years oscillation theory of difference and functional differential equations has attracted many researchers. Oscillation theory was initiated by jacques charles francois sturm in his investigations of sturmliouville problems from 1836. Our results not only unify the oscillation of second order delay di erential and di erence equations but also are new for qdi erence equations and can be applied on any time scale. The frequency of the oscillation in hertz is, and the period is. A further result on the oscillation of delay difference equations. Oscillation theory for difference and functional differential.
The results obtained in the paper improve some results from c. On oscillation of solutions of differential equations with. Oscillation, delay differential problem, impulse, bounded solution. Oscillation of nonlinear delay difference equations. We establish some new sufficient conditions which insure that every solution of this equation either oscillates or converges to zero. Second order delay dynamic equations 3 the paper is organized as follows. Oscillation criteria for delay and advanced difference equations 353 as it has been mentioned above, it is an interesting problem to. Oscillation theorems for certain delay difference inequalities. The aim of this note is to give some new explicit conditions for oscillation and nonoscillation for 1. Oscillation criteria are established for thirdorder neutral delay differential equations with deviating arguments.
A new impetus to the investigation of oscillation was given by mahfoud who deduce oscillation of delay equations from that of ordinary equations. Oscillation criteria for secondorder delay, difference, and. Gomathi jawahar department of mathematics, karunya university, coimbatore641114, tamil nadu, india. Oscillation of secondorder nonlinear neutral delay. Employing suitable comparison theorems we establish new results on oscillation of the studied equation.
From the publication of this paper up to the present time, impulsive delay di. Some new oscillation criteria including hille and neharitype criteria are presented. By comparison with some first difference equations whose oscillatory characters are known and by means of a riccati transformation technique, we obtain several new sufficient conditions for the oscillation of all solutions of the nonlinear neutral difference equation of the form. Stability and oscillations in delay differential equations. Some examples are considered to illustrate the main results. Oscillation of difference, differential, and dynamic equations. This monograph is devoted to a rapidly developing area of research of the qualitative theory of difference and functional differential equations. Some properties of oscillation of second order neutral delay difference equations. In this paper, we are concerned with oscillation of a secondorder nonlinear neutral delay difference equation of the form. Oscillation properties of higher order impulsive delay. Oscillation of nonlinear partial difference equations with. Oscillations umd department of physics umd physics. Some sufficient conditions are established for the oscillation of secondorder neutral differential equation, where.
Oscillation of solutions to neutral delay and advanced. Introduction in this article, we study the oscillation of solutions to the rstorder linear delay di. Similar conditions for equations with distributed delay are far less known. Oscillation criteria for p laplacian delay dynamic equations. In the first chapter of this monograph, we address oscillation of solutions to difference equations of.
Abstractin this paper we present some new oscillation criteria for second order nonlinear neutral type delay difference equation of the form a n x n c n x n k q n x n 1 0, n n 0 0, where, is a forward difference operator defined by a nonnegative integer, k and are positive integers. Oscillation and nonoscillation of solutions of delay partial difference equations is receiving much attention 47. This paper deals with the oscillation criteria for the linear delay differential equations. The aim of this monograph is to present a reasonably selfcontained account of the advances in the oscillation theory of this class of equations. The frequency and period of the oscillation are both determined by the constant, which appears in the simple harmonic oscillator equation, whereas the amplitude, and phase angle, are. Consider the secondorder linear delay differential equation equations has been a very active research area in the past decades, and many references and. Period of oscillation is independent of the amplitude of the oscillation. A further result on the oscillation of delay difference. Ladas, recent developments in the oscillation of delay difference equations, in international conference on differential equations, stability and gontro dekker, new york, 1990. Based upon the corresponding characteristic equations, we get some criteria for nonoscillation by utilizing the matrix measures. Our results also improve some wellknown results in the literature even when applying them to the linear delay difference equation. Some properties of oscillation of second order neutral delay.