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Oscillation of third order nonlinear functional dynamic equations on time scales

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Abstract

It is the purpose of this paper to give oscillation criteria for the third order nonlinear functional dynamic equation

$$ \left( {a\left( t \right)\left[ {\left( {r\left( t \right)x^\Delta \left( t \right)} \right)^\Delta } \right]^\gamma } \right)^\Delta + f\left( {t,x\left( {g\left( t \right)} \right)} \right) = 0 $$

on a time scale \( \mathbb{T} \), where γ is the quotient of odd positive integers, a and r are positive rd-continuous functions on \( \mathbb{T} \), and the function g: \( \mathbb{T} \to \mathbb{T} \) satisfies limt→∞ g(t) = ∞ and f ∈ C \( \left( {\mathbb{T} \times \mathbb{R}, \mathbb{R}} \right) \). Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equations. Some examples are given to illustrate the main results.

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Author notes

  1. Taher S. Hassan

    Present address: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588-0130, USA

Authors and Affiliations

  1. Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588-0130, USA

    Lynn Erbe & Allan Peterson

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    Taher S. Hassan

Authors
  1. Lynn Erbe
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  2. Taher S. Hassan
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  3. Allan Peterson
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Correspondence to Lynn Erbe.

Additional information

This paper is dedicated to Bernd Aulbach.

Supported by the Egyptian Government while visiting the University of Nebraska-Lincoln.

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Erbe, L., Hassan, T.S. & Peterson, A. Oscillation of third order nonlinear functional dynamic equations on time scales. Differ Equ Dyn Syst 18, 199–227 (2010). https://doi.org/10.1007/s12591-010-0005-y

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