Oscillation of second order neutral type emdenfowler. Introduction this article is concerned with the oscillatory behavior of neutral difference equations of the form, di. In a neutral delay differential equation, the highestorder derivative of the unknown function appears both with and without delay. Necessary and sufficient condition for oscillations of neutral differential equations volume 28 issue 3 m.
This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives. In this paper some necessary and sufficient conditions are obtained to guarantee the oscillation for bounded and all solutions of second order nonlinear neutral delay difference equations. The world is too rich and complex for our minds to grasp it whole, for our minds are but a small part of the richness of the world. Some riccati type difference inequalities are established for the secondorder nonlinear difference equations with negative neutral term. Pdf oscillation and asymptotic behaviour of solutions of. We investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form. Recently, there has been a lot of activity concerning the oscillation and nonoscillation behavior of neutral differential and difference equations see 1, 2, 3, 4. Properties of the solutions of linear difference equations with constant coefficients property 10. Furthermore, by using,, the stability, oscillatory, controllability and other behaviors of solutions of delay or neutral differential equations can be achieved. Difference equations differential equations to section 1. The neutral difference equations are seen as the discrete analogue of the neutral differential equations the oscillatory and asymptotic behavior of delay difference equations and neutral difference equations have been intensively studied in recent years due to its various application in. Examples are presented to illustrate the importance of the results. Convergence of the solutions for a neutral difference.
Linear di erence equations in this chapter we discuss how to solve linear di erence equations and give some applications. Pdf convergence and divergence of the solutions of a. For example, by using, a necessary and sufficient condition was obtained for the relative controllability of the neutral differential equations with delay see. Compared with some existing ones, our results are derived without the use of the mode transformation method and the. Introduction second order neutral delay difference equation is gaining. Oscillation criteria of second order neutral difference equations. The series focuses on the latest achievements in functional differential and difference equations, publishing contributions presented at the conferences as well as other high quality papers that outline the recent progress in functional differential and difference equations. Eventually positive and bounded solutions of evenorder. The semicycles of solutions of neutral difference equations.
In this paper, we establish some oscillation criteria of the secondorder delay difference equation. In simple cases, a di erence equation gives rise to an associated auxiliary equation rst explained in 7. In this section we will consider the simplest cases. In this paper we consider the existence of a positive sequence and utilize it in the capacity of integrating factor to. Pdf on oscillation of solutions to second order neutral.
In the last few decades several studies on quantitative and qualitative properties of nsfdes were carried out see 4, 5, 20 and the references therein. Almost oscillation criteria for secondorder neutral difference. On neutral impulsive stochastic differential equations. Nonoscillation of firstorder neutral impulsive difference. The time delays can be constant, timedependent, or statedependent, and the choice of the solver function dde23, ddesd, or ddensd depends on the type of delays in the equation.
In this paper some criteria for the oscillation of mixed type third order neutral difference equation of the form a n d n xn b n xn 1 cn xn 2 q x n n. In theorem 5 and theorem 8, we have studied the oscillation criteria as well as the asymptotic behavior, where was established some sufficient conditions to ensure that every solution are either oscillates. This paper is concerned with the general solution of linear fractional neutral differential difference equations. View pdf asymptotic stability analysis of fractionalorder neutral. Pdf asymptotic stability in totally nonlinear neutral. Request pdf oscillation of third order mixed type neutral difference equations in this paper, we obtain some sufficient conditions for the oscillation of all. Chapter 6 deals with nonoscillation and oscillation properties of scalar linear neutral differential equations. Necessary and sufficient conditions for oscillation of second order neutral difference equations. The present di erence equation would be presented as.
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. Necessary and sufficient condition for oscillations of. Oscillatory behavior of second order nonlinear difference. A neutral delay differential equation is a differential equation in which the highest order derivative of the unknown function appears both with and without delay. Neutral difference equations zhan zhou, jianshe yu and zhicheng wang department of appl.
Di erence equations relate to di erential equations as discrete mathematics relates to continuous mathematics. Asymptotic behavior of higherorder neutral difference equations with general arguments. Some properties of oscillation of second order neutral. Convergence and divergence of the solutions of a neutral hindawi. The discrete counterparts of neutral differential equations are called neutral.
In this paper, we study the semicycles of solutions of neutral delay difference equation. Certain third order mixed neutral difference equations b. Read pdf basic electrical engineering equations basic electrical engineering equations youve tried the rest, now try the best see the reallife story of. Some properties of oscillation of second order neutral delay difference equations. Anyone who has made a study of di erential equations will know that even supposedly elementary examples can be hard to solve. General solution of linear fractional neutral differential. In this paper, the authors using summation averaging method and an inequality present some new oscillation criteria for the second order neutral type emdenfowler delay difference equation. Some new criteria for the oscillation of certain neutral difference equations of the form, di. There are two kinds of neutral equations, one of them can be integrated leading to a term with a concentrated delay and an integral term. Upper bound of numbers of terms of semicycles are determined. Second order neutral delay difference equation is gaining interest because they are the discrete analogue of differential equations. Recent progress in differential and difference equations. In the case of nonlinear problems, whether in differential or difference equations, it is difficult and in some cases impossible to invert the problem and obtain a suitable mapping that can be effectively used in fixed point theory to qualitatively analyze its solutions. Existence of a nonoscillatory solution of a secondorder linear.
Oscillatory and asymptotic behavior of a firstorder. By a solution of the neutral type difference equation e, we mean a sequence of. Existence of uncountably many bounded positive solutions for second order nonlinear neutral delay difference equations. Some properties of oscillation of second order neutral delay. Oscillation criteria for secondorder neutral delay. In this paper, a collocation method based on hermite polynomials is presented for the numerical solution of the neutral functionaldifferential equations nfdes with proportional delays. Oscillation of nonlinear thirdorder delay difference. That is, we have looked mainly at sequences for which we could write the nth term as a n fn for some known function f. Pdf oscillation of a higher order neutral difference equation with a. Pdes are used to formulate problems involving functions of several variables, and are either solved in closed form, or used to. Typically the time delay relates the current value of the derivative to the value of the solution at some prior time, but.
Research article new oscillation criteria for thirdorder nonlinear mixed neutral difference equations elmetwallymohammedelabbasy, 1 magdyyosephbarsom, 1 andfaisalsalehaldheleai 2 department of mathematics, faculty of science, mansoura university, mansoura, egypt. Neutral stochastic functional differential equations nsfdes have been initiated in and their usage in aeroelasticity was pointed out. Exponential stability of mild solutions to impulsive. Global attractivity of neutral difference equations. Oscillation of third order mixed type neutral difference equations. An application of measures of noncompactness in the investigation of boundedness of solutions of secondorder neutral difference equations, advances in difference equations, 20, pp. Second order neutral delay differential equations have applications in problems. The qualitative study of such equations has, besides its theoretical interest, signi. In this article we consider the existence of positive solutions of a system of periodic neutral difference equations. Oscillations of neutral difference equations of second. The difference between neutral and ground on the electric panel this one gives a detailed description of how the ground. Pdf the authors obtain oscillation results for the even order forced neutral difference equation. One can think of time as a continuous variable, or one can think of time as a discrete variable.
The established criteria significantly improve and extend related results in the literature. Eventually positive and bounded solutions of evenorder nonlinear neutral differential equations. Oscillatory and asymptotic behavior of solutions of. Stability of linear impulsive neutral delay differential.
Delay differential equations contain terms whose value depends on the solution at prior times. Ocillation of first order neutral delay 3 difference equations, electric journal of qualitative of differential equations,proc. In this equation, if 1 0, it is no longer an differential equation and so 1 cannot be 0. This paper is concerned with the oscillation of the bounded solutions of neutral difference equation where. A partial differential equation pde is a differential equation that contains unknown multivariable functions and their partial derivatives. Free fulltext pdf articles from hundreds of disciplines, all in one place. Oscillation criteria for secondorder neutral delay difference equations. Characterization of higherorder neutral difference. In this paper we present some new oscillation criteria for second order nonlinear neutral type delay difference equation of the form where, is a forward difference operator defined by.
Some improved criteria on exponential stability of neutral. An application of measures of noncompactness in the. Gomathi jawahar department of mathematics, karunya university, coimbatore641114, tamil nadu, india. Stability and boundedness in nonlinear and neutral. Existence, uniqueness, and stability for nonlinear differentialdifference equations in the neutral case, n. Research article new oscillation criteria for thirdorder. Oscillation of solutions of non linear neutral difference equations with nonlinear neutral term. Characterization of higherorder neutral difference equations with several delays n. To cope with the complexity, we reason hierarchically.
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