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Interval oscillation criteria for a second order nonlinear differential equation
Cun-Hua Zhang 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5
This paper is concerned with the interval oscillation of the second order nonlinear ordinary differential equation (r (t) |y' (t)α−1 y'(t)' + p (t) |y' (t)α−1 y'(t)+q (t) f (y (t)) g (y'(t)) = 0. By constructing a generalized Riccati transformation and using the method of averaging techniques, we establish some interval oscillation criteria when f(y) is not differetiable but satisfies the condition<수식> μ0 > 0 for y ≠ 0. This paper is concerned with the interval oscillation of the second order nonlinear ordinary differential equation (r (t) |y' (t)α−1 y'(t)' + p (t) |y' (t)α−1 y'(t)+q (t) f (y (t)) g (y'(t)) = 0. By constructing a generalized Riccati transformation and using the method of averaging techniques, we establish some interval oscillation criteria when f(y) is not differetiable but satisfies the condition<수식> μ0 > 0 for y ≠ 0.
Taher S. Hassan,Qingkai Kong 대한수학회 2012 대한수학회지 Vol.49 No.5
We consider forced second order differential equation with p-Laplacian and nonlinearities given by a Riemann-Stieltjes integrals in the form of [수식] is strictly increasing such that [수식] with [수식] and [수식] R is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. These criteria are further extended to equations with deviating arguments. As special cases, our work generalizes, unies, and improves many existing results in the literature. We consider forced second order differential equation with p-Laplacian and nonlinearities given by a Riemann-Stieltjes integrals in the form of [수식] is strictly increasing such that [수식] with [수식] and [수식] R is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. These criteria are further extended to equations with deviating arguments. As special cases, our work generalizes, unies, and improves many existing results in the literature.
OSCILLATION THEOREMS FOR CERTAIN SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS
Sun, Yibing,Han, Zhenlai,Zhao, Ping,Sun, Ying The Korean Society for Computational and Applied M 2011 Journal of applied mathematics & informatics Vol.29 No.5
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.
OSCILLATION THEOREMS FOR CERTAIN SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS
Yibing Sun,Zhenlai Han,Ping Zhao,Ying Sun 한국전산응용수학회 2011 Journal of applied mathematics & informatics Vol.29 No.5
In this paper, we consider the oscillation of the following certain second order nonlinear differential equations (r(t)(x'(t))^α+q(t)x^β(t)=0,where α and β are ratios of positive odd integers. New oscillation theorems are established, which are based on a class of new functions Φ=Φ(t,s,l) defined in the sequel. Also, we establish some interval oscillation criteria for this equation.
INTERVAL OSCILLATION CRITERIA FOR A SECOND ORDER NONLINEAR DIFFERENTIAL EQUATION
Zhang, Cun-Hua The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.5
This paper is concerned with the interval oscillation of the second order nonlinear ordinary differential equation (r(t)|y'(t)|$^{{\alpha}-1}$ y'(t))'+p(t)|y'(t)|$^{{\alpha}-1}$ y'(t)+q(t)f(y(t))g(y'(t))=0. By constructing ageneralized Riccati transformation and using the method of averaging techniques, we establish some interval oscillation criteria when f(y) is not differetiable but satisfies the condition $\frac{f(y)}{|y|^{{\alpha}-1}y}$ ${\geq}{\mu}_0$ > 0 for $y{\neq}0$.
Hassan, Taher S.,Kong, Qingkai Korean Mathematical Society 2012 대한수학회지 Vol.49 No.5
We consider forced second order differential equation with $p$-Laplacian and nonlinearities given by a Riemann-Stieltjes integrals in the form of $$(p(t){\phi}_{\gamma}(x^{\prime}(t)))^{\prime}+q_0(t){\phi}_{\gamma}(x(t))+{\int}^b_0q(t,s){\phi}_{{\alpha}(s)}(x(t))d{\zeta}(s)=e(t)$$, where ${\phi}_{\alpha}(u):={\mid}u{\mid}^{\alpha}\;sgn\;u$, ${\gamma}$, $b{\in}(0,{\infty})$, ${\alpha}{\in}C[0,b)$ is strictly increasing such that $0{\leq}{\alpha}(0)<{\gamma}<{\alpha}(b-)$, $p$, $q_0$, $e{\in}C([t_0,{\infty}),{\mathbb{R}})$ with $p(t)>0$ on $[t_0,{\infty})$, $q{\in}C([0,{\infty}){\times}[0,b))$, and ${\zeta}:[0,b){\rightarrow}{\mathbb{R}}$ is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. These criteria are further extended to equations with deviating arguments. As special cases, our work generalizes, unifies, and improves many existing results in the literature.
Forced Oscillation Criteria for Nonlinear Hyperbolic Equations via Riccati Method
Shoukaku, Yutaka Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.1
In this paper, we consider the nonlinear hyperbolic equations with forcing term. Some suffcient conditions for the oscillation are derived by using integral averaging method and a generalized Riccati technique.