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In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type ${(r (t) {(z^{'} (t))}^{γ})}^{'} + \sum_{i = 1}^{m} q_{i} (t) x^{α_{i}} (σ_{i} (t)) = 0, t \geq t_{0},$ where $z (t) = x (t) + p (t) x (τ (t))$ . Under the assumption $\int^{\infty} {(r (η))}^{- 1 / γ} d η = \infty$ , we consider two cases when $γ > α_{i}$ and $γ < α_{i}$ . Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

Necessary and sufficient conditions for the oscillation a third-order differential equation.

Das, Pitambar, Pati, Jitendra Kumar (2010)

Electronic Journal of Differential Equations (EJDE) [electronic only]

New oscillation criteria for first order nonlinear delay differential equations

Xianhua Tang, Jianhua Shen (2000)

Colloquium Mathematicae

New oscillation criteria are obtained for all solutions of a class of first order nonlinear delay differential equations. Our results extend and improve the results recently obtained by Li and Kuang [7]. Some examples are given to demonstrate the advantage of our results over those in [7].

New results on the nonoscillation of solutions of some nonlinear differential equations of third order.

Tunç, Ercan (2009)

Journal of Inequalities and Applications [electronic only]

Nichteuklidischer Nullstellenabstand der Lösungen von w'' + p(z)w = 0.

H. Herold (1990)

Mathematische Annalen

Nichtoszillatorische lineare Differentialgleichungen 2. Ordnung

Jan Mařík, Miloš Ráb (1963)

Czechoslovak Mathematical Journal

Nichtoszillatorische und oszillatorische Differentialgleichungen dritter Ordnung

Milan Gera (1971)

Časopis pro pěstování matematiky

Niektoré vlastnosti diferenciálnej rovnice $y^{''} + f (x) | y | = 0$

Jozef Rovder (1970)

Matematický časopis

Niektoré vlastnosti diferenciálnej rovnice $y^{''} + f (x) | y | = 0$

Jozef Rovder (1969)

Matematický časopis

Nonlinear eigenvalue problems with the parameter near resonance

Jean Mawhin, Klaus Schmitt (1990)

Annales Polonici Mathematici

Nonnegative linearization of orthogonal polynomials

Ryszard Szwarc (1996)

Colloquium Mathematicae

Nonnegativity of functionals corresponding to the second order half-linear differential equation

Robert Mařík (1999)

Archivum Mathematicum

In this paper we study extremal properties of functional associated with the half–linear second order differential equation E $_{p}$ . Necessary and sufficient condition for nonnegativity of this functional is given in two special cases: the first case is when both points are regular and the second is the case, when one end point is singular. The obtained results extend the theory of quadratic functionals.

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