• Title, Summary, Keyword: nonoscillation

Search Result 12, Processing Time 0.041 seconds

Oscillation and Nonoscillation of Nonlinear Neutral Delay Differential Equations with Several Positive and Negative Coefficients

  • Elabbasy, Elmetwally M.;Hassan, Taher S.;Saker, Samir H.
    • Kyungpook Mathematical Journal
    • /
    • v.47 no.1
    • /
    • pp.1-20
    • /
    • 2007
  • In this paper, oscillation and nonoscillation criteria are established for nonlinear neutral delay differential equations with several positive and negative coefficients $$[x(t)-R(t)x(t-r)]^{\prime}+\sum_{i=1}^{m}Pi(t)H_i(x(t-{\tau}_i))-\sum_{j=1}^{n}Q_j(t)H_j(x(t-{\sigma}_j))=0$$. Our criteria improve and extend many results known in the literature. In addition we prove that under appropriate hypotheses, if every solution of the associated linear equation with constant coefficients, $$y^{\prime}(t)+\sum_{i=1}^{m}(p_i-\sum_{k{\in}J_i}qk)y(t-{\tau}_i)=0$$, oscillates, then every solution of the nonlinear equation also oscillates.

  • PDF

OSCILLATION AND NONOSCILLATION CRITERIA FOR DIFFERENTIAL EQUATIONS OF SECOND ORDER

  • Kim, RakJoong
    • Korean Journal of Mathematics
    • /
    • v.19 no.4
    • /
    • pp.391-402
    • /
    • 2011
  • We give necessary and sufficient conditions such that the homogeneous differential equations of the type: $$(r(t)x^{\prime}(t))^{\prime}+q(t)x^{\prime}(t)+p(t)x(t)=0$$ are nonoscillatory where $r(t)$ > 0 for $t{\in}I=[{\alpha},{\infty})$, ${\alpha}$ > 0. Under the suitable conditions we show that the above equation is nonoscillatory if and only if for ${\gamma}$ > 0, $$(r(t)x^{\prime}(t))^{\prime}+q(t)x^{\prime}(t)+p(t)x(t-{\gamma})=0$$ is nonoscillatory. We obtain several comparison theorems.

OSCILLATION AND NONOSCILLATION THEOREMS FOR NONLINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER

  • Kim, Rak-Joong;Kim, Dong-Il
    • Journal of the Korean Mathematical Society
    • /
    • v.44 no.6
    • /
    • pp.1453-1467
    • /
    • 2007
  • By means of a Riccati transform some oscillation or nonoscillation criteria are established for nonlinear differential equations of second order $$(E_1)\;[p(t)|x#(t)|^{\alpha}sgn\;x#(t)]#+q(t)|x(\tau(t)|^{\alpha}sgn\;x(\tau(t))=0$$. $$(E_2),\;(E_3)\;and\;(E_4)\;where\;0<{\alpha}$$ and $${\tau}(t){\leq}t,\;{\tau}#(t)>0,\;{\tau}(t){\rightarrow}{\infty}\;as\;t{\rightarrow}{\infty}$$. In this paper we improve some previous results.

OSCILLATION OF NEUTRAL DIFFERENCE EQUATIONS

  • Koo, Nam Jip
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.12 no.1
    • /
    • pp.125-131
    • /
    • 1999
  • We obtain some sufficient conditions for oscillation of the neutral difference equation with positive and negative coefficients $${\Delta}(x_n-cx_{n-m})+px_{n-k}-qx_{n-l}=0$$, where ${\Delta}$ denotes the forward difference operator, m, k, l, are nonnegative integers, and $c{\in}[0,1),p,q{\in}\mathbb{R}^+$.

  • PDF

WEIGHTED HARDY INEQUALITIES WITH SHARP CONSTANTS

  • Kalybay, Aigerim;Oinarov, Ryskul
    • Journal of the Korean Mathematical Society
    • /
    • v.57 no.3
    • /
    • pp.603-616
    • /
    • 2020
  • In the paper, we establish the validity of the weighted discrete and integral Hardy inequalities with periodic weights and find the best possible constants in these inequalities. In addition, by applying the established discrete Hardy inequality to a certain second-order difference equation, we discuss some oscillation and nonoscillation results.

OSCILLATION CRITERIA OF DIFFERENTIAL EQUATIONS OF SECOND ORDER

  • Kim, Rae Joong
    • Korean Journal of Mathematics
    • /
    • v.19 no.3
    • /
    • pp.309-319
    • /
    • 2011
  • We give sufficient conditions that the homogeneous differential equations : for $t{\geq}t_0$(> 0), $$x^{{\prime}{\prime}}(t)+q(t)x^{\prime}(t)+p(t)x(t)=0,\\x^{{\prime}{\prime}}(t)+q(t)x^{\prime}(t)+F(t,x({\phi}(t)))=0$$, are oscillatory where $0{\leq}{\phi}(t)$, 0 < ${\phi}^{\prime}(t)$, $\lim_{t\to{\infty}}{\phi}(t)={\infty}$. and $F(t,u){\cdot}sgn$ $u{\leq}p(t)|u|$. We obtain comparison theorems.