The purpose of this paper is to investigate the asymptotic behavior of solutions of the forced nonlinear delay differential equations with impulses x’(t)+sum from i=1 to n(p_i(t)f(x(t-т_i))=h(t)). t≠t_k, x(t_k^+)...The purpose of this paper is to investigate the asymptotic behavior of solutions of the forced nonlinear delay differential equations with impulses x’(t)+sum from i=1 to n(p_i(t)f(x(t-т_i))=h(t)). t≠t_k, x(t_k^+)-x(t_k)=b_kx(t_k). Our results. which hold for linear and nonlinear equations, forced and unforced equations, impulsive and nonimpulsive equations. improve and generalize the known results recently obtained in [8].展开更多
Sufficient conditions are obtained respectively for tending to zero of the nonoscillatory solutions of equation with impulses and for tending to infinite of every nonoscillatory solution. Conditions are also obtained ...Sufficient conditions are obtained respectively for tending to zero of the nonoscillatory solutions of equation with impulses and for tending to infinite of every nonoscillatory solution. Conditions are also obtained for all solutions to be oscillatory when the equation is of oscillating coefficients.展开更多
Sufficient conditions are established for the asymptotic behavior of solutions of nonlinear delay differential equationswhere equipped with the sup norm for some r > 0. A new result is established, some known resul...Sufficient conditions are established for the asymptotic behavior of solutions of nonlinear delay differential equationswhere equipped with the sup norm for some r > 0. A new result is established, some known results are improved.展开更多
The boundedness of the every solution and the asymptotic behavior of all solutions of the nonlinear neutral delay differential equation [x（t） - P（t）x（t - r）]＇ ＋ Q1（t）f（x（t -σ1）） - Q2（t）f（x（t - σ2）...The boundedness of the every solution and the asymptotic behavior of all solutions of the nonlinear neutral delay differential equation [x（t） - P（t）x（t - r）]＇ ＋ Q1（t）f（x（t -σ1）） - Q2（t）f（x（t - σ2）） = 0, t 〉 t0 are investigated, where τ,σ1, σ2 ∈ （0, ∞）, P∈C（[t0, ∞）, R）, and Q1, Q2 ∈C（[t0,∞）,R）, f ∈ C（R, R）. The sufficient conditions obtained improve the existing results in the literatures.展开更多
A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive...A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.展开更多
研究了一类具有脉冲的二阶非线性时滞微分方程(r(t)x′(t))′-p(t)x′(t)+sum from i=1 to n qi(t)x(t-σ_i+f(t)=0,t≠t_k,x(t_k^+)-x(t_k)=a_kx(t_k),x′(t_k^+)-x′(t_k)=b_kx′(t_k),k∈Z^+的解的渐近性,并得到了一系列相关的充分条件.
基金This work is supported by the NNSF of China and the NSF of Hunan Province
文摘The purpose of this paper is to investigate the asymptotic behavior of solutions of the forced nonlinear delay differential equations with impulses x’(t)+sum from i=1 to n(p_i(t)f(x(t-т_i))=h(t)). t≠t_k, x(t_k^+)-x(t_k)=b_kx(t_k). Our results. which hold for linear and nonlinear equations, forced and unforced equations, impulsive and nonimpulsive equations. improve and generalize the known results recently obtained in [8].
文摘Sufficient conditions are obtained respectively for tending to zero of the nonoscillatory solutions of equation with impulses and for tending to infinite of every nonoscillatory solution. Conditions are also obtained for all solutions to be oscillatory when the equation is of oscillating coefficients.
文摘Sufficient conditions are established for the asymptotic behavior of solutions of nonlinear delay differential equationswhere equipped with the sup norm for some r > 0. A new result is established, some known results are improved.
基金Supported by the NNSF of China (No.10571050)the Key Project of Chinese Ministry of Education.
文摘The boundedness of the every solution and the asymptotic behavior of all solutions of the nonlinear neutral delay differential equation [x（t） - P（t）x（t - r）]＇ ＋ Q1（t）f（x（t -σ1）） - Q2（t）f（x（t - σ2）） = 0, t 〉 t0 are investigated, where τ,σ1, σ2 ∈ （0, ∞）, P∈C（[t0, ∞）, R）, and Q1, Q2 ∈C（[t0,∞）,R）, f ∈ C（R, R）. The sufficient conditions obtained improve the existing results in the literatures.
文摘A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.
文摘研究了一类具有脉冲的二阶非线性时滞微分方程(r(t)x′(t))′-p(t)x′(t)+sum from i=1 to n qi(t)x(t-σ_i+f(t)=0,t≠t_k,x(t_k^+)-x(t_k)=a_kx(t_k),x′(t_k^+)-x′(t_k)=b_kx′(t_k),k∈Z^+的解的渐近性,并得到了一系列相关的充分条件.
