If condition cg is satisfied and sw, 15 then all solutions of 10. T patulaa secondorder nonlinear difference equation. Cooke received june 8, 1988 consider secondorder difference equations of the form ufn,u0 e and 2mfly. Oscillatory behaviour of a class of nonlinear systems of. Mohankumar 2 assistant professor, department of mathematics, aarupadaiveedu institute of technology, vinayaka missions university, paiyanoor, kancheepuram, tamilnadu, india. Numerical methods for oscillatory solutions to hyperbolic. The approach to the conditional oscillation requires the calculation of the critical oscillation constant depending on coe cients of the treated equations. Application of first order linear homogeneous difference equations to the real life and its oscillatory behavior a.
The overflow blog defending yourself against coronavirus scams. Theorems on oscillatory and asymptotic behavior of second order nonlinear neutral difference equations a. Grace initiated the study of oscillatory theory of fde, and he considered the equations of the form where denotes the riemannliouville differential. In the discussion of the oscillatory solutions of 1. In superlinear and sublinear cases, necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions. Application of first order linear homogeneous difference.
On the oscillatory behavior of solutions of a class of nonlinear. This paper is concerned with the oscillatory behavior of firstorder nonlinear difference equations with variable deviating arguments. Pdf oscillation of thirdorder difference equations researchgate. This paper considers a class of fourth order nonlinear difference equations. They may also have further applications in the analysis. In recent years, many authors interested in studying the oscillatory and nonoscilla tory behavior of various classes of difference equations, see for examples 1, 2. By means of the averaging technique and the generalized riccati transformation technique, we establish some oscillation criteria for the secondorder quasilinear neutral delay dynamic equations, where, and the time scale interval is. Existence of nonoscillatory solutions to second order neutral type. Oscillatory behavior of delay partial difference equations. Oscillatory behavior of first order neutral delay di erence equations in the proof of our main results. Popenda 2 were explained several new fundamental concepts in this fast developing area of research. Oscillation behavior of certain third order nonlinear.
Oscillatory behavior of thirdorder nonlinear neutral delay. Besides the purely mathematical problem, the interest in the behavior of the solutions to difference equations with retarded arguments is. In several recent papers 2, 581 oscillation and asymptotic behavior of solutions of secondorder linear difference equations have been. A note on the oscillatory property for nonlinear difference. Oscillation behavior for a class of differential equation. In here, we consider the difference methods for the follow ing model equations which share some similarity with the general linear equations with highly oscillatory coefficients, initial values, and forcing terms 4.
We study the oscillation of a fourth order nonlinear differential equation with a middle term. On asymptotic behavior of solutions of first order difference. Oscillation of difference, differential, and dynamic equations. Moreover, i am not proposing the equations simply as a model of behavior nor. Also the behavior of solutions of the equations y111 x px y11 x qx y1 x rx yox 0 1. Delay equations are used to study phenomena in which some part of the history of the system determines its evolution.
Li lefschetz center for dynamical systems, division of applied mathematics, brown university, providence, rhode island 02912 submitted by k. Equation 1 is said to possess property if all its solutions defined in a neighbourhood of are oscillating when is even. Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation article pdf available in mathematica slovaca 572. Jun 30, 2016 the question regarding the analysis of oscillatory behavior of solutions to with other methods that do not require these assumptions remains open at the moment. Oscillatory behavior of difference equations of second. In this paper, some sufficient conditions are obtained by discussing the oscillatory behavior of solutions for a class of evenorder nonlinear functional differential equations, and our results generalize and improve some known results. Cooke criteria are given to determine the oscillatory property of solutions of the nonlinear difference equation. Oscillation conditions for difference equations with a monotone or.
Classifying nonoscillatory solutions and oscillation of a. Oscillatory and asymptotic behavior of thirdorder nonlinear. The oscillations of higherand fourthorder differential equations have been studied by several authors, and several techniques have been proposed for obtaining oscillatory criteria for higherand. Oscillatory behavior of first order neutral delay di erence. Illustrative examples are also given to support the validity of the method. Oscillatory and asymptotic behavior of solutions of nonlinear neutraltype difference equations volume 38 issue 2 john r. This paper is concerned with oscillatory and asymptotic behavior of solutions of a class of thirdorder nonlinear functional differential equations. Oscillating behavior and limits larson calculus calculus 10e. Oscillatory and non oscillatory properties of fourth order difference.
Oscillating behavior and limits contact us if you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below. Research article oscillation behavior for a class of. Difference equation, sequence, oscillation and non oscillation, double summation, riccatis. Among equations which are oscillatory at the equations which possess the properties or, i. Hence, the coe cients have to be measurable in a certain sense. This paper is concerned with oscillatory behavior of a certain class of secondorder neutral delay. But the fractional difference equations are studied by very few authors, see 912. Oscillatory behavior of the secondorder nonlinear neutral difference equations. China and department of mathematics, binzhou normal college shandong, binzhou 256604, p. This is an investigation of the causes of oscillatory behaviour in solutions of stochastic delay differential equations. New theorems do not need several restrictive assumptions required in related results reported in the literature. The difference equation 1 is non oscillatory if and only if there exists. Pandian department of mathematics madras university p.
On the oscillatory behaviour of stochastic delay equations. During the last several years, oscillatory, nonoscillatory and asymptotic behaviour of solutions of nonlinearlinear systems of. Oscillatory behavior of second order neutral differential. Oscillatory and asymptotic behavior of solutions of. On the oscillatory behavior of solutions of second order. Oscillation, second order, nonlinear neutral delay differential equations. However, to the best of the authors knowledge very little is known regarding the oscillatory behavior of differential equation with fractionalorder derivatives up to now except for 1727. Oscillation criteria for first order nonlinear neutral delay. Such a solution is called oscillatory if it has arbitrarily large zeros, otherwise it is called nonoscillatory. In this paper, and motivated by the above mentioned work, we investigate the oscillatory behavior of the nonlinear fractional difference equation with damping term of the form.
Browse other questions tagged ordinarydifferentialequations or ask your own question. Monotone and oscillatory solutions of a rational di. In this paper, the oscillatory behavior of solutions of a general class of nonlinear neutral delay differential equations is discussed. Oscillatory behavior for evenorder nonlinear functional. Introduction in the last few years, there has been an increasing interest in the study of oscillatory behavior of solutions of first order neutral delay differential equations with positive and negative coefficients. In this paper, we consider the oscillation behavior of solutions of the following fractional difference equation. The oscillatory behivor is described as the vehicle bounces around the center of the lane as it moves down the road. On oscillatory behavior of solutions of thirdorder nonlinear. The scorepenalty function for the oscillating behavior is defined as follows. Journal of mathematical analysis and applications 150, 414424 1990 oscillatory behavior of difference equations of second order zdzislaw szafranski and blazej szmanda institute of mathematics, technical university, piotrowo 3a, 60965 poznan, poland submitted by kenneth l. We study oscillatory behavior of solutions to a class of secondorder nonlinear neutral differential equations under the assumptions that allow applications to differential equations with delayed and advanced arguments. Papers written in english should be submitted as tex and pdf files using. Accompanied with the development of the theory on fractional differential equations, fractional difference equations have also been studied more inten sively of.
Oscillatory behavior of secondorder nonlinear neutral. Several examples are provided to show that the results obtained. We are here concerned with the oscillatory behavior of solutions of the following second order ordinary differential equation. Using a certain energy function, we describe the properties of oscillatory solutions.
Monotonic behavior on the invariant ray may or may not be representative of other solutions. Oscillatory difference equations 421 from theorems 2 and 1 we have, respectively, corollary 1. In most cases in the next section, the behavior on the invariant ray is in fact representative of all solutions but in section 4 this is not the case. Oscillatory behavior of difference equations of second order.
Lalli department of mathematics university of saskatchewan saskatoon, canada, stn owo transmitted by melvin scott abstract the asymptotic and oscillatory behavior of solutions. However, to the best of the author s knowledge very little is known regarding the oscillatory behavior of di erential equation with fractionalorder derivatives up to now except for. These equations are perfectly general and are consonant with research findings in decision theory and psychology, 4 and in form similar to the eigenvector, eigenvalue equations or characteristic functions basic to much contemporary physical science and engineering. Oscillation and asymptotic behavior of solutions of oddorder. North oscillatory and nonoscillatory behavior of secondorder functional difference equations e. Oscillatory and asymptotic behaviour of fourth order. Oscillating differential equation encyclopedia of mathematics. Elaydi 4 was given some basic introduction about difference equations and briefly explained their oscillatory behaviors of solutions of difference equations. Pdf oscillatory and asymptotic behaviour of a nonlinear. By using the generalized riccati transformation and the integral averaging technique, two new sufficient conditions which insure that the solution is oscillatory or converges to zero are established. The authors examine the oscillatory and nonoscillatory behavior of solutions of a class of second order difference equations of neutral type that includes halflinear equations as a special case. Nonlinear, neutral, delay difference equation, oscillatory behavior, variable coefficients.
Oscillatory behavior of quasilinear neutral delay dynamic. Pdf oscillatory solutions of nonlinear fourth order. Oscillatory and nonoscillatory behavior of secondorder. Introduction neutral delay difference equations contain the difference of the unknown sequence both with and without delays.
Nn 0 wherefn, y may be classified as superlinear, sublinear, strongly superlinear and strongly sublinear. Theorems on oscillatory and asymptotic behavior of second. The aim of this paper is the study of the boundedness, asymptotic stability and oscillatory behavior of the following rational difference equation. Oscillatory behavior of the secondorder nonlinear neutral. Oscillatory behavior of delay partial difference equations with positive and negative coefficients shu tang liu department of automatic control engineering south china university of technology guangzhou 510641, p. Oscillations of nonlinear difference equations with deviating.