Padmavathi, S.

Pinelas, Sandra

Thandapani, Ethiraju

Dynamic Systems and Applications

English

Repositório Comum

Dynamic Systems and Applications 23 (2014) 63-74OSCILLATION RESULTS FOR ODD-ORDER NONLINEARNEUTRAL DIFFERENTIAL EQUATIONS OF MIXED TYPEE. THANDAPANIa, S. PADMAVATHIb, AND S. PINELASca,bRamanujan Institute for Advanced Study in Mathematics, University ofMadras, Chennai 600 005, IndiaE-mail: ethandapani@yahoo.co.incAcademia Militar, Departamento de Cieˆncias Exactas e Naturais, Av. CondeCastro Guimara˜es, 2720-113 Amadora, PortugalE-mail: sandra.pinelas@gmail.comABSTRACT. In this paper the authors establish some new comparison theorems and Philos-typecriteria for oscillation of solutions to the odd order neutral mixed type differential equation(x(t) + ax(t− τ1) + bx(t+ τ2))(n) + p(t)xα(t− σ1) + q(t)xβ(t+ σ2) = 0, t ≥ t0,where n ≥ 3 is an odd integer, α ≥ 1 and β ≥ 1, are ratio of odd positive integers. Examples areprovided to illustrate the main results.AMS (MOS) Subject Classification. 34C15.1. PRELIMINARIESThis paper is concerned with the oscillation and asymptotic behavior of solutionsof odd order nonlinear neutral mixed type differential equation of the form(1.1) (x(t) + ax(t− τ1) + bx(t+ τ2))(n) + p(t)xα(t− σ1) + q(t)xβ(t+ σ2) = 0, t ≥ t0,where n ≥ 3 is an odd integer, α ≥ 1 and β ≥ 1 are the ratios of odd positive integers,p(t) and q(t) are continuous and positive functions for all t ≥ t0, and a, b, τ1, τ2, σ1, σ2are non-negative constants. We set z(t) = x(t)+ax(t−τ1)+ bx(t+ τ2). By a solutionof equation (1.1), we mean a function x(t) ∈ C([Tx,∞),R), Tx ≥ t0, which has theproperty z(t) ∈ Cn([Tx,∞),R) and satisfies equation (1.1) on [Tx,∞). We consideronly those solutions x(t) of equation (1.1) which satisfy sup {|x(t)| : t ≥ T} > 0 forall T ≥ Tx. We assume that equation (1.1) possesses such a solution. A solutionof equation (1.1) is called oscillatory if it has infinitely large zeros in [Tx,∞) andotherwise, it is said to be nonoscillatory. Equation (1.1) is said to be almost oscillatoryif all its solutions are either oscillatory or convergent to zero asymptotically.Received April 29, 2013 1056-2176 $15.00 c©Dynamic Publishers, Inc.

Oscillation Results For Odd-order Nonlinear Neutral Differential Equations of Mixed Type

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Oscillation Results For Odd-order Nonlinear Neutral Differential Equations of Mixed Type

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