Behavior of solutions of a third-order dynamic equation on time scales

M Tamer Şenel

doi:10.1186/1029-242x-2013-47

Abstract

In this paper, we will establish some sufficient conditions which guarantee that every solution of the third-order nonlinear dynamic equation oscillates or converges to zero on an arbitrary time scale .

Highlights

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D
Assume b > and we show that this leads to a contradiction
If Case (ii) holds, as before, limt→∞ x(t) exists and the proof is complete

Summary

Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. By a solution of ( ), we mean a nontrivial real-valued function satisfying equation ( ) for t ≥ a. Proof Let x be an eventually positive solution of ( ). It follows from this that either r (t)x (t) < on [t , ∞) or r (t)x (t) is eventually positive and the proof is complete.

Results

Conclusion