Asymptotic behaviors of higher order nonlinear dynamic equations on time scales

Abstract

In this paper, we study asymptotic behaviour of solutions of the following higher order nonlinear dynamic equationson an arbitrary time scale T with sup T = ∞, where n is a positive integer and δ = 1 or −1. We obtain some sufficient conditions for the equivalence of the oscillation of the above equations.

Oscillation criteria for higher‐order nonlinear dynamic equations with Laplacians and a deviating argument on time scales

“…Recently, using Riccati substitution, Hassan and Kong obtained asymptotics and oscillation criteria for the n th‐order half‐linear dynamic equation with deviating argument where α [1, n − 1]: = α 1 ⋯ α n − 1 ; and Grace and Hassan further studied the asymptotics and oscillation for the higher‐order nonlinear dynamic equation with Laplacians and deviating argument However, the establishment of the results in requires the restriction on the time scale that g ∗ ∘ σ = σ ∘ g ∗ with , which is hardly satisfied, see conclusion 1 in for such a special case. For more results on dynamic equations, we refer the reader to the papers .…”

Oscillation Criteria for Solutions to Nonlinear Dynamic Equations of Higher Order

“…In particular these papers present oscillatory criteria and asymptotic behavior for first, second and third order dynamic equations on time scales and some interesting results were obtained for special cases of (1.1); see [10,14,16,17,28].…”

Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

“…In recent years, more and more people have been interested in studying the oscillatory behavior of higher order dynamic equations on time scales, see [1][2][3][4][5][6][7][8][9][10][11][12][13] and references therein. For an introduction to time scale calculus and dynamic equations, we refer the reader to the landmark paper of Hilger [14] and the seminal book by Bohner and Peterson [15] for a comprehensive treatment of the subject.…”

“…Assume that ( 1 )-( 6 ) and(13) hold for all 1 ∈ [ 0 , ∞) T . If there exist a function ( ) ∈ 1 ([ 0 , ∞) T , (0, ∞))and a function ( ) ∈[ , ] such that ( ) > 0 on[ , ] T 1 and(11) and(12) hold,…”