Electron. J. Diff. Equ., Vol. 2014 (2014), No. 173, pp. 1-13.

Oscillation criteria for odd-order nonlinear differential equations with advanced and delayed arguments

Ethiraju Thandapani, Sankarappan Padmavathy, Sandra Pinelas

Abstract:
This article presents oscillation criteria for n-th order nonlinear neutral mixed type differential equations of the form
$$\displaylines{ \big((x(t)+ax(t-\tau_1)-bx(t+\tau_2))^{\alpha}\big)^{(n)} =q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2), \cr \big((x(t)-ax(t-\tau_1)+bx(t+\tau_2))^{\alpha}\big)^{(n)} =q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2), \cr \big((x(t)+ax(t-\tau_1)+bx(t+\tau_2))^{\alpha}\big)^{(n)} =q(t)x^{\beta}(t-\sigma_1)+p(t)x^{\gamma}(t+\sigma_2) }$$
where n is an odd positive integer, a and b are nonnegative constants, $\tau_1,\tau_2,\sigma_1$ and $\sigma_2$ are positive real constants, $q(t),p(t)\in C([t_0,\infty),(0,\infty))$ and $\alpha,\beta$ and $\gamma$ are ratios of odd positive integers with $\beta,\gamma\geq 1$. Some examples are provided to illustrate the main results.

Submitted March 14, 2014. Published August 14, 2014.
Math Subject Classifications: 34C15.
Key Words: Oscillation; odd order; neutral differential equation; mixed type.

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Ethiraju Thandapani
Ramanujan Institute for Advanced Study in Mathematics
University of Madras, Chennai 600 005, India
email: ethandapani@yahoo.co.in
Sankarappan Padmavathy
Ramanujan Institute for Advanced Study in Mathematics
University of Madras, Chennai 600 005, India
Sandra Pinelas
Academia Militar
Departamento de Ciências Exactas e Naturais
Av. Conde Castro Guimarães, 2720-113 Amadora, Portugal
email: sandra.pinelas@gmail.com

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