Nonoscillational Solutions Of A Higher Order Funtional Dynamic Equation On Time Scales

This paper mainly discusses the existence of nonoscillational solutions for a higher order funtional dynamic equation on time scales. The paper is organized as follows:In the first section we introduce the background, significance and current study status of dynamic equations on time scales, as well as the definitions and assumptions needed in the present paper. In the second section, by using the Schaefer type fixed point theorem and Bihari type inequalities on time scales, we prove the global existence of solutions for initial value problems to the equationIn the last section we give some criteria for the nonoscillation of all solutions of our equation, and the necessary and sufficient conditions for the existence of nonoscillational solutions.Our motivation stems from the work [32], and the results we obtained are more perfect in theory, and more comprehensive in practice.