• Title/Summary/Keyword: Second order nonlinear differential equations

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INTERVAL OSCILLATION THEOREMS FOR SECOND-ORDER DIFFERENTIAL EQUATIONS

  • Bin, Zheng
    • Journal of applied mathematics & informatics
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    • v.27 no.3_4
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    • pp.581-589
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    • 2009
  • In this paper, we are concerned with a class of nonlinear second-order differential equations with a nonlinear damping term and forcing term: $$(r(t)k_1(x(t),x'(t)))'+p(t)k_2(x(t),x'(t))x'(t)+q(t)f(x(t))=0$$. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity. And, as a consequence, our results apply to wider classes of nonlinear differential equations. Some illustrative examples are considered.

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Oscillation of Second Order Nonlinear Elliptic Differential Equations

  • Xu, Zhiting
    • Kyungpook Mathematical Journal
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    • v.46 no.1
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    • pp.65-77
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    • 2006
  • By using general means, some oscillation criteria for second order nonlinear elliptic differential equation with damping $$\sum_{i,j=1}^{N}D_i[a_{ij}(x)D_iy]+\sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0$$ are obtained. These criteria are of a high degree of generality and extend the oscillation theorems for second order linear ordinary differential equations due to Kamenev, Philos and Wong.

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ON THE STABILITY AND INSTABILITY OF A CLASS OF NONLINEAR NONAUTONOMOUS ORDINARY DIFFERENTIAI, EQUATIONS

  • Sen, M.DeLa
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.2
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    • pp.243-251
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    • 2003
  • This note Presents sufficient conditions for Lyapunov's stability and instability of a class of nonlinear nonautonomous second-order ordinary differential equations. Such a class includes as particular cases a remarkably large number of differential equations with specific physical applications. Two successive nonlinear transformations are applied to the original differential equation in order to convert it into a more convenient form for stability analysis purposes. The obtained stability / instability conditions depend closely on the parameterization of the original differential equation.

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 THEOREMS FOR CERTAIN SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS

  • Sun, Yibing;Han, Zhenlai;Zhao, Ping;Sun, Ying
    • Journal of applied mathematics & informatics
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    • v.29 no.5_6
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    • pp.1557-1569
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    • 2011
  • In this paper, we consider the oscillation of the following certain second order nonlinear differential equations $(r(t)(x^{\prime}(t))^{\alpha})^{\prime}+q(t)x^{\beta}(t)=0$>, where ${\alpha}$ and ${\beta}$ are ratios of positive odd integers. New oscillation theorems are established, which are based on a class of new functions ${\Phi}={\Phi}(t,s,l)$ defined in the sequel. Also, we establish some interval oscillation criteria for this equation.

A NEW METHOD FOR SOLVING NONLINEAR SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS

  • Gachpazan. M.;Kerayechian, A.;Kamyad, A.V.
    • Journal of applied mathematics & informatics
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    • v.7 no.2
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    • pp.453-465
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    • 2000
  • In this paper, a new method for finding the approximate solution of a second order nonlinear partial differential equation is introduced. In this method the problem is transformed to an equivalent optimization problem. them , by considering it as a distributed parameter control system the theory of measure is used for obtaining the approximate solution of the original problem.

OSCILLATION AND NONOSCILLATION THEOREMS FOR NONLINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER

  • Kim, Rak-Joong;Kim, Dong-Il
    • Journal of the Korean Mathematical Society
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    • v.44 no.6
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    • pp.1453-1467
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    • 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.

BOUNDED OSCILLATION FOR SECOND-ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS

  • Song, Xia;Zhang, Quanxin
    • Journal of applied mathematics & informatics
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    • v.32 no.3_4
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    • pp.447-454
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    • 2014
  • Two necessary and sufficient conditions for the oscillation of the bounded solutions of the second-order nonlinear delay differential equation $$(a(t)x^{\prime}(t))^{\prime}+q(t)f(x[{\tau}(t)])=0$$ are obtained by constructing the sequence of functions and using inequality technique.