In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existe...In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.展开更多
This paper is concerned with the oscillation of second order linear functional equations of the form x(g(t)) = p(t)x(t) + Q(t)X(g(2)(t)), Where p, Q, g : [t(0), infinity) --> R+ = [0, infinity) are given real value...This paper is concerned with the oscillation of second order linear functional equations of the form x(g(t)) = p(t)x(t) + Q(t)X(g(2)(t)), Where p, Q, g : [t(0), infinity) --> R+ = [0, infinity) are given real valued functions such that g(t) not equivalent to t, lim(t-->infinity) g(t) = infinity. It is proved here that when 0 less than or equal to m := lim inf(t-->infinity) Q(t)P(g(t)) less than or equal to 1/4 all solutions of this equation oscillate if the condition lim(t-->infinity) sup Q(t)P(g(t)) > (1 + root1 -4m/2)(2) (*) is satisfied. It should be emphasized that the condition (*) can not be improved in some sense.展开更多
In this paper,some sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients of the formx′(t)+p(t)x(t-τ)=0are established,which improve and generalize some of t...In this paper,some sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients of the formx′(t)+p(t)x(t-τ)=0are established,which improve and generalize some of the known results in the literature.展开更多
This paper is concerned with the oscillatory (and nonoscillatory) behavior of solutions of second oder quasilinear difference equations of the type Some necessary and sufficient conditions are given for the equation t...This paper is concerned with the oscillatory (and nonoscillatory) behavior of solutions of second oder quasilinear difference equations of the type Some necessary and sufficient conditions are given for the equation to admit oscillatory and nonoscillatory solutions with special asymptotic properties. These results generalize and improve some known results.展开更多
In this paper, we study oscillation of solutions for a class of high order neutral delay difference equations with variable coefficients -τm [x（t） - c（t）x（t - τ）] = （-1）mp（t）x（t - σ）, t ≥ t0 〉 0. Some...In this paper, we study oscillation of solutions for a class of high order neutral delay difference equations with variable coefficients -τm [x（t） - c（t）x（t - τ）] = （-1）mp（t）x（t - σ）, t ≥ t0 〉 0. Some sufficient conditions are obtained for bounded oscillation of the solutions.展开更多
In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the s...In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the solutions.展开更多
The aim of this paper is to study the oscillation of second order neutral difference equations.Our results are based on the new comparison theorems,that reduce the problem of the oscillation of the second order equati...The aim of this paper is to study the oscillation of second order neutral difference equations.Our results are based on the new comparison theorems,that reduce the problem of the oscillation of the second order equation to that of the first order equation.The comparison principles obtained essentially simplify the examination of the equations.展开更多
基金supported by the National Natural Science Foundation of China(12071491,12001113)。
文摘In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.
文摘This paper is concerned with the oscillation of second order linear functional equations of the form x(g(t)) = p(t)x(t) + Q(t)X(g(2)(t)), Where p, Q, g : [t(0), infinity) --> R+ = [0, infinity) are given real valued functions such that g(t) not equivalent to t, lim(t-->infinity) g(t) = infinity. It is proved here that when 0 less than or equal to m := lim inf(t-->infinity) Q(t)P(g(t)) less than or equal to 1/4 all solutions of this equation oscillate if the condition lim(t-->infinity) sup Q(t)P(g(t)) > (1 + root1 -4m/2)(2) (*) is satisfied. It should be emphasized that the condition (*) can not be improved in some sense.
基金This project is supported by the NNSF of China (19831030).
文摘In this paper,some sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients of the formx′(t)+p(t)x(t-τ)=0are established,which improve and generalize some of the known results in the literature.
基金The Science Foundation (00C029) of Hunan Educational Committee.
文摘This paper is concerned with the oscillatory (and nonoscillatory) behavior of solutions of second oder quasilinear difference equations of the type Some necessary and sufficient conditions are given for the equation to admit oscillatory and nonoscillatory solutions with special asymptotic properties. These results generalize and improve some known results.
基金Supported by the National Natural Science Foundation of China (Grant No.10571050)the Science and Research Fund for Higher College of Hunan Province (Grant No.06C054)
文摘In this paper, we study oscillation of solutions for a class of high order neutral delay difference equations with variable coefficients -τm [x（t） - c（t）x（t - τ）] = （-1）mp（t）x（t - σ）, t ≥ t0 〉 0. Some sufficient conditions are obtained for bounded oscillation of the solutions.
基金This work are supported by the NNSF of China (No.10571050)the Science and Research Fund for Higher Colleges of Hunan (No.06C054).
文摘In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the solutions.
基金the NSF of Shanxi Province(No.2008011002-1)the Development Foundation of Higher Education Department of Shanxi Province(No.20111117)the Foundation of Datong University 2010-B-01
文摘The aim of this paper is to study the oscillation of second order neutral difference equations.Our results are based on the new comparison theorems,that reduce the problem of the oscillation of the second order equation to that of the first order equation.The comparison principles obtained essentially simplify the examination of the equations.
基金Supported by the Natural Science Foundation of Hunan Province(12JJ6006)Hunan Province Science and Technology Project(2012FJ3107)the National Natural Science Foundation of China(11071222)
