philos type oscillation criteria for linear differential equations with impulsive effects abstract the present paper deals with the problem of oscillation for a second order linear differential equation in which an impulsive effect is considered this equation can be regarded as an equation of motion in which the moving speed of a mass point changes sharply by some influence it is proved that there is a case that the mass point may oscillate due to the influence of the impulsive effect even if the mass point does not oscillate in the model removing the impulsive effect it is also shown that the obtained results extend some previous ones through the use of an example

new oscillation criteria for forced second order half linear differential equations with damping abstract using variational principle and riccati technique new oscillation criteria for forced second order half linear differential equation with damped term are established which improve a recent paper by li and cheng w t li s s cheng an oscillation criterion for nonhomogenuous half linear differential equations appl math lett 15 2002 259 263 and generalized some of the results in the literature

interval criteria for oscillation of nonlinear second order dynamic equations on time scales interval oscillation criteria are established for a second order nonlinear dynamic equation on time scales by utilizing a generalized riccati technique and an integral averaging technique the theory can be applied to second order dynamic equations regardless of the choice of delta or nabla derivatives

nonlinear functionals and a theorem of sun we use a recently developed theory of nonlinear functionals in the study of oscillations of second order symmetric vector differential systems to extend a number of theorems of sun new kamenev type theorems for second order linear matrix differential systems appl math lett 2004 in press under a common theme the criteria presented here are of the form the integral of the coefficient matrix is bounded at infinity in a sense to be made explicit in the paper and bounded away from a positive absolute constant implies oscillation at infinity

on the oscillation of certain second order nonlinear differential equations abstract in this paper oscillation criteria for the nonlinear second order ordinary differential equation r t x t x t p 2 x t c t f x t 0 are given the results extend the integral averaging technique and include earlier results our methodology is somewhat different from that of previous authors

second order oscillation of forced functional differential equations with oscillatory potentials new oscillation criteria are established for second order differential equations containing both delay and advanced arguments of the form k t x t p t x t t a 1x t t q t x s t b 1x s t e t t 0 where a 1 and b 1 k p q e t s are continuous real valued functions k t 0 is nondecreasing t and o are nondecreasing t t t s t t and lim t t t the potentials p q and e are allowed to change sign and the information on the whole half line is not required as opposed to the usual case in most articles among others as an application of the results we are able to deduce that every solution of x t m 1sintx t p12 m 2costx t p6 cos2t m 1 m 2 0 is oscillatory provided that either m 1 orm 2 is sufficiently large

oscillation of second order neutral equations with distributed deviating argument oscillation criteria are established for the second order neutral delay differential equation with distributed deviating argument r t j x t z t a bq t x f x g t x d s x 0 t t 0 where z t x t p t x t t these results are extensions of the integral averaging techniques due to coles and kamenev and improve some known oscillation criteria in the existing literature

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

oscillation theorems for self adjoint matrix hamiltonian systems involving general means by use of monotone functionals and positive linear functionals a generalized riccati transformation and the general means technique some new oscillation criteria for the following self adjoint hamiltonian matrix system e x t a t x t b t y t y t c t x t a t y t are obtained the results obtained improve and complement that of kumari et al 2000 on kamenev type theorems moreover these results generalize and improve earlier results due to meng 2002 for e erbe et al 1993 meng et al 1998 and wang 2001 for p t x t q t x t 0 or its special cases and wong 2001 for the scalar system x t q t x t 0

oscillation of even order partial differential equations with distributed deviating arguments some kamenev type oscillation criteria are established for a class of boundary value problems associated with even order partial differential equations with distributed deviating arguments our approach is to reduce the high dimensional oscillation problem to a one dimensional oscillation one and the general means developed by philos and wong is used as the main tool the results obtained here extend and improve some known results in the literature

oscillation criteria for half linear elliptic inequalities with p x laplacians via riccati method abstract riccati inequality for half linear elliptic inequalities with p x laplacians is established and oscillation criteria are derived by using the riccati inequality our method is to reduce oscillation problems for half linear elliptic inequalities with p x laplacians to one dimensional riccati inequality with variable exponents generalizations to more general elliptic inequalities with p x laplacians are also discussed

a new razumikhin theorem for delay difference equations abstract in this paper we established a new razumikhin type theorem on stability for delay difference equations

oscillation criteria for second order nonlinear differential equations involving general means consider the second order nonlinear differential equation e y a t f y 0 where a t c 0 f y c1 f y 0 and yf y 0 for y 0 furthermore f y also satisfies either a superlinear or a sublinear condition which covers the nonlinear function f y y y 1 with 1 and 0 1 respectively commonly known as the emden fowler case here the coefficient function a t is allowed to be negative for arbitrarily large values of t kamenev type oscillation criteria involving integral averages for the linear equations l y a t y 0 are extended to the nonlinear equation e by using more general means the results extend similar results on general means by philos for the linear equation l and also results based upon kamenev s integral averaging method concerning the nonlinear equation e

oscillation criteria of kamenev type for pde with p laplacian some oscillation criteria are given for the partial differential equation with p laplacian and the nonlinearity of the emden fowler typediv email protected email protected p email protected u c x u p 2u 0 p 1 the results are extension of integral averaging technique of kamenev and improve the result of usami

new kamenev type oscillation criteria for second order nonlinear differential equations with damping abstract some new oscillation criteria are established for the nonlinear damped differential equation r t y p t y q t f y 0 that are different from most known ones in the sense that they are based on a class of new functions t s r defined in the sequel our results are sharper than some previous results which can be seen by the examples at the end of this paper

an oscillation criterion for second order sublinear differential equations abstract an oscillation criterion is given for the second order sublinear differential equation y a t f y 0 where a t is not assumed to be non negative and f y is non decreasing in y and yf y 0 for y 0 f y also satisfies a sublinear condition which includes the special case f y y y sgn y 0 this result extends earlier oscillation theorems of philos and the first author

integral inequalities and second order linear oscillation on the interval 0 a u 0 below by a solution we mean a non trivial solution a solution is oscillatory if it has an infinite number of zeros numerous oscillation criteria are known and diversified approaches have proved fruitful among these is the use of the associated riccati equation although the theory of differential and integral inequalities has been used by some investigators see e g hartman 4 71 the authors believe that it has not been fully exploited the non oscillation of 1 1 is equivalent to the existence of a solution of the associated riccati equation on a half line thus one technique of proving oscillation is to show that any solution of the riccati equation diverges to i co at a finite point the theory of differential and integral inequalities comes in handy in the estimation of the growth of such a solution below we describe some known results that are relevant to results established in this paper although these are stated here for the case i co in the rest of the paper we consider both cases a 0 and a 00 it is well known that if q is large in the mean then 1 1 is oscillatory a typical example of such a criterion is due to fite leighton and wintner if

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

oscillation criteria for second order neutral equations with distributed deviating argument abstract we present new oscillation criteria for the second order neutral delay differential equations with distributed deviating argument r t x t z t a b p t f x g t d 0 t t 0 where z t x t q t x t new oscillation criteria are established which are based on a class of new functions t s l defined in the sequel our results are sharper than some previous results which can be seen by the example at the end of this paper

discrete time approximations of stochastic delay equations the milstein scheme in this paper we develop a strong milstein approximation scheme for solving stochastic delay differential equations sddes the scheme has convergence order 1 in order to establish the scheme we prove an infinite dimensional ito formula for tame functions acting on the segment process of the solution of an sdde it is interesting to note that the presence of the memory in the sdde requires the use of the malliavin calculus and the anticipating stochastic analysis of nualart and pardoux given the nonanticipating nature of the sdde the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

oscillation criteria for certain forced second order nonlinear differential equations with delayed argument some new sufficient conditions for the oscillation criteria are given for the forced second order nonlinear differential equations with delayed argument in the form x t q t f x t t e t the results are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our methodology is somewhat different from that of previous authors our results generalize and extend some earlier results of sun 1 furthermore we are able to answer the question in recent paper 1 for the oscillation of bounded solutions of the above equation when f x x lsgn x 0 l 1

note on wong s paper using the method of interval integral average we establish some new oscillation criteria which improve the oscillation criteria due to james s w wong

a note on the lasalle type theorems for stochastic differential delay equations the main aim of this note is to improve some results obtained in the author s earlier paper 1999 j math anal appl 236 350 369 from the improved result follow some useful criteria on the stochastic asymptotic stability and boundedness

oscillation and nonoscillation of hill s equation with periodic damping we prove new results on the oscillation and nonoscillation of the hill s equation with periodic damping y p t y q t y 0 t 0 where p t and q t are continuous and periodic the results show that the equation y sint y cost y 0 is nonoscillatory whilst the equation y cost y sint y 0 is oscillatory

oscillation criterion for half linear differential equations with periodic coefficients abstract in this paper we present an oscillation criterion for second order half linear differential equations with periodic coefficients the method is based on the nonexistence of a proper solution of the related modified riccati equation our result can be regarded as an oscillatory counterpart to the nonoscillation criterion by sugie and matsumura 2008 these two theorems provide a complete half linear extension of the oscillation criterion of kwong and wong 2003 dealing with the hill s equation

oscillation criteria for second order nonlinear differential equations with damping abstract for a second order nonlinear differential equation with a damping term we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations and both sets of oscillation criteria established in the paper are of independent interest several examples are provided to illustrate the relevance of the new theorems

oscillation criteria of a certain class of third order nonlinear delay differential equations with damping abstract in this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form r 2 t r 1 t y p t y q t f y g t 0 by using a generalized riccati transformation and integral averaging technique we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero in particular several examples are given to illustrate the importance of our results

a comparison theorem for the iterative method with the preconditioner i s max in 1991 gunawardena et al linear algebra appl 154 156 1991 123 have reported the modified gauss seidel method with a preconditioner i s in this article we propose to use a preconditioner i smax instead of i s here smax is constructed by only the largest element at each row of the upper triangular part of a by using the lemma established neumann and plemmons linear algebra appl 88 89 1987 559 we get the comparison theorem for the proposed method simple numerical examples are also given

simple conditions for parametrically excited oscillations of generalized mathieu equations abstract the following equation is considered in this paper x cos t x 0 where and are real parameters and 0 this equation is referred to as mathieu s equation when 2 the parameters determine whether all solutions of this equation are oscillatory or nonoscillatory our results provide parametric conditions for oscillation and nonoscillation there is a feature in which it is very easy to check whether these conditions are satisfied or not parametric oscillation and nonoscillation regions are drawn to help understand the obtained results

oscillation criteria for second order nonlinear differential equations with nonlinear damping this paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping several new oscillation criteria are established under quite general assumptions which improve and extend the known results in the literature

interval oscillation of a general class of second order nonlinear differential equations with nonlinear damping abstract the paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping terms several new interval oscillation criteria are established for such a class of differential equations under quite general assumptions examples are also given to illustrate the results in particular it is shown that under some very mild conditions on k 1 k 2 and f the equation k 1 x x cos tk 2 x x x sin tf x 0 is oscillatory which extends the result obtained by kwong and wong j math anal appl 288 2003 15

interval oscillation of second order functional differential equations with oscillatory potentials abstract interval oscillation criteria are established for second order differential equations of the form e k t x t p t x g t q t x g t 1 x g t e t t 0 where 1 k p q e and g are continuous real valued functions k t 0 is nondecreasing g t is nondecreasing lim t g t we consider g t t and g t t separately illustrative examples are included

oscillation problems for hill s equation with periodic damping abstract this paper deals with the second order linear differential equation x a t x b t x 0 where a and b are periodic coefficients the main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory our nonoscillation theorem and oscillation theorem are proved by using the riccati technique in our theorem the composite function of an indefinite integral of b and a suitable multiple valued continuously differentiable function are focused and the composite function of them plays an important role the results obtained here include a result by kwong and wong 15 and a result by sugie and matsumura 26 an application to a equation of whittaker hill type is given to show the usefulness of our results finally simulations are also attached to illustrate that our oscillation criterion is sharp

averaging technique and nonlinear second order elliptic oscillation abstract by using the refinement of the standard integral averaging technique we obtain new oscillation criteria for certain nonlinear second order damped elliptic differential equations we do not impose any additional conditions on the damping terms except for the continuity in particular when n 1 our results extend those of rogovchenko and tuncay yu v rogovchenko f tuncay oscillation criteria for second order nonlinear differential equations with damping nonlinear analysis 69 2008 208 221 for second order nonlinear differential equations with damping to damped elliptic differential equations when n 2 our results improve the oscillation criteria of xu jia ma z xu b jia d ma oscillation theorems for elliptic equations with damping appl math comput 156 2004 93 106 for second order elliptic equations with damping four examples are provided to illustrate the relevance of the main results

integral averaging technique for the interval oscillation criteria of certain second order nonlinear differential equations abstract we present new interval oscillation criteria related to integral averaging technique for certain classes of second order nonlinear differential equations which are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line they generalize and improve some known results examples are also given to illustrate the importance of our results

on kamenev type oscillation theorems for second order differential equations with damping abstract oscillation criteria for the nonlinear second order ordinary differential equation with damping x p t x f x 0 are given the results are extensions of integral averaging technique of kamenev and include earlier results of yeh yan and chen

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation

retarded differential equations retarded differential equations rdes are differential equations having retarded arguments they arise in many realistic models of problems in science engineering and medicine where there is a time lag or after effect numerical techniques for such problems may be regarded as extensions of dense output methods for ordinary differential equations odes but scalar rdes are inherently infinite dimensional with a richer structure than their ode counterparts we give background material develop a theoretical foundation for the basic numerics and give some results not previously published