Oscillation of third order nonlinear functional dynamic equations on time scales

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.