Nonoscillation of all solutions of a higher order nonlinear delay dynamic equation on time scales

Abstract

The authors give sufficient conditions for all solutions of a higher order nonlinear delay dynamic equation on time scales to be nonoscillatory. The results cover both the superlinear and sublinear cases and, in the differential equations case, resolve a question that has been open for 30 years. In order to prove their results, the authors first prove a new Bihari type inequality for functions on time scales.

Estimates of some applicable inequalities on time scales

“…In 1988 Stefan Hilger [10] in his Ph.D thesis introduced the calculus on time scales which unifies the continuous and discrete analysis. As a response to the diverse need of the applications recently in last decade many authors have studied the properties of solutions of dynamic equations on time scales [1,2,3,4,7,8,9,11,12,13,14,15,16,17,18]. Motivated by the above results in this paper we find inequalities with explicit estimates which can found to be important tool in the study of dynamical systems on time scales.…”

Nonoscillatory solutions for first-order neutral dynamic equations with continuously distributed delay on time scales

“…Throughout this paper, we assume that T = [t 0 , ∞) T := {t ∈ T : t ≥ t 0 }. In recent years, there has been much research activity concerning the nonoscillation of solutions of various equations on time scales, and we refer the reader to [1][2][3][4]. Mathsen, Wang and Wu [5] established some sufficient conditions for the existence of positive solutions of the delay equation…”

Structure of Non-Oscillatory Solutions for Second Order Dynamic Equations on Time Scales

“…Oscillation of the Equation (1) has been studied by Doslý and Hilger [6], Grace, Agarwal, Bohner and O'Regan [7], Zhou, Ahmad and Alsaedi [20]. A non-oscillatory of Equation (1) is also considered by Graef and Hill [21], Erbe, Baoguo and Peterson [22]. For more details, we refer the reader to see the references cited therein.…”