• Title/Summary/Keyword: nonlinear oscillation

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OSCILLATIONS OF CERTAIN NONLINEAR DELAY PARABOLIC BOUNDARY VALUE PROBLEMS

  • Zhang, Liqin;Fu, Xilin
    • Journal of applied mathematics & informatics
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    • v.8 no.1
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    • pp.137-149
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    • 2001
  • In this paper we consider some nonlinear parabolic partial differential equations with distributed deviating arguments and establish sufficient conditions for the oscillation of some boundary value problems.

OSCILLATION CRITERIA FOR SECOND-ORDER NONLINEAR DIFFERENCE EQUATIONS WITH 'SUMMATION SMALL' COEFFICIENT

  • KANG, GUOLIAN
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.2
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    • pp.245-256
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    • 2005
  • We consider the second-order nonlinear difference equation (1) $$\Delta(a_nh(x_{n+1}){\Delta}x_n)+p_{n+1}f(x_{n+1})=0,\;n{\geq}n_0$$ where ${a_n},\;{p_n}$ are sequences of integers with $a_n\;>\;0,\;\{P_n\}$ is a real sequence without any restriction on its sign. hand fare real-valued functions. We obtain some necessary conditions for (1) existing nonoscillatory solutions and sufficient conditions for (1) being oscillatory.

OSCILLATORY PROPERTY OF SOLUTIONS FOR A CLASS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH PERTURBATION

  • Zhang, Quanxin;Qiu, Fang;Gao, Li
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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    • pp.883-892
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    • 2010
  • This paper is concerned with oscillation property of solutions of a class of second order nonlinear differential equations with perturbation. Four new theorems of oscillation property are established. These results develop and generalize the known results. Among these theorems, two theorems in the front develop the results by Yan J(Proc Amer Math Soc, 1986, 98: 276-282), and the last two theorems in this paper are completely new for the second order linear differential equations.

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
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    • v.47 no.1
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    • pp.1-20
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    • 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.

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Oscillation Criteria for Certain Nonlinear Differential Equations with Damping

  • Zheng, Zhaowen;Zhu, Siming
    • Kyungpook Mathematical Journal
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    • v.46 no.2
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    • pp.219-229
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    • 2006
  • Using the integral average method, we establish some oscillation criteria for the nonlinear differential equation with damped term $$a(t)|{x}^{\prime}(t)|^{\sigma-1}{x}^{\prime}(t)^{\prime}+p(t)|{x}^{\prime}(t)|^{\sigma-1}{x}^{\prime}(t)+q(t)f(x(t))=0,\;{\sigma}>1$$, where the functions $a,\;p$ and $q$ are real-valued continuous functions defined on $[t_o,{\infty})$ with $a(t)>0,\;f(x){\in}C^1(\mathbb{R})$ and $\frac{f^{\prime}(u)}{|f^{({\sigma}-1)/{\sigma}}(u)|}{\geq}k>0$ for $u{\neq}0$.

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Oscillation Results for Second Order Nonlinear Differential Equation with Delay and Advanced Arguments

  • Thandapani, Ethiraju;Selvarangam, Srinivasan;Vijaya, Murugesan;Rama, Renu
    • Kyungpook Mathematical Journal
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    • v.56 no.1
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    • pp.137-146
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    • 2016
  • In this paper we study the oscillation criteria for the second order nonlinear differential equation with delay and advanced arguments of the form $$([x(t)+a(t)x(t-{\sigma}_1)+b(t)x(t+{\sigma}_2)]^{\alpha})^{{\prime}{\prime}}+q(t)x^{\beta}(t-{\tau}_1)+q(t)x^{\gamma}(t+{\tau}_2)=0,\;t{\geq}t_0$$ where ${\sigma}_1$, ${\sigma}_2$, ${\tau}_1$ and ${\tau}_2$ are nonnegative constants and ${\alpha}$, ${\beta}$ and ${\gamma}$ are the ratios of odd positive integers. Examples are provided to illustrate the main results.

OSCILLATION BEHAVIOR OF SOLUTIONS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS ON TIME SCALES

  • Han, Zhenlai;Li, Tongxing;Sun, Shurong;Zhang, Meng
    • Communications of the Korean Mathematical Society
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    • v.26 no.3
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    • pp.499-513
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    • 2011
  • By using the Riccati transformation technique, we study the oscillation and asymptotic behavior for the third-order nonlinear delay dynamic equations $(c(t)(p(t)x^{\Delta}(t))^{\Delta})^{\Delta}+q(t)f(x({\tau}(t)))=0$ on a time scale T, where c(t), p(t) and q(t) are real-valued positive rd-continuous functions defined on $\mathbb{T}$. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our oscillation results are essentially new. Some examples are considered to illustrate the main results.