New Hille Type and Ohriska Type Criteria for Nonlinear Third-Order Dynamic Equations.

The objective of this paper is to derive new Hille type and Ohriska type criteria for third-order nonlinear dynamic functional equations in the form of a 2 (ζ) φ α 2 a 1 ζ φ α 1 x Δ (ζ) Δ Δ + q (ζ) φ α x (g (ζ)) = 0 , on a time scale T , where Δ is the forward operator on T ,   α 1 ,   α 2 ,   α > 0 , and g ,   q ,   a i , i   =   1 ,   2 , are positive r d -continuous functions on T , and φ θ (u) : = u θ − 1 u . Our results in this paper are new and substantial for dynamic equations of the third order on arbitrary time scales. An example is included to illustrate the results. [ABSTRACT FROM AUTHOR]