Oscillation Criteria of Second-Order Dynamic Equations with Damping on Time Scales

Abstract

Using functions in some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order dynamic equations with damping on time scales of the form(r(t)(xΔ(t))γ)Δ+p(t)(xΔ(t)γ)+f(t,x(g(t)))=0. Two examples are included to show the significance of the results.

Oscillation Criteria of Third-Order Nonlinear Damped Dynamic Equations on Time Scales

“…When 0 < < 1, if ( ) = , the conclusions above are not applicable. Similarly, the assumption that ( ) ≥ in Şenel [12] and Qiu and Wang [10] should be changed to (C3) in this paper.…”

“…There has been much research achievement about the oscillation of dynamic equations on time scales in the last few years; see the papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the references therein.…”

“…However, it seemed that the obtained theorems and corollaries are incorrect. Qiu and Wang [10] corrected some mistakes in [12] and established correct oscillation criteria for (11) by employing functions in some function classes and the generalized Riccati transformation. Şenel [13] considered the third-order nonlinear dynamic equation…”

New Oscillation Results of Second-Order Damped Dynamic Equations withp-Laplacian on Time Scales

“…In 1988, the theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in order to unify continuous and discrete analysis; see also [2]. In recent years, there has been much research activity concerning the oscillation of solutions of dynamic equations on time scales; for example, see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein.…”

Oscillation results for nonlinear second-order damped dynamic equations

“…For oscillation of damped dynamic equations on time scales, we refer the reader to the papers [1,2,4,7,8,9,11,15,16,18,19] and the references cited therein. Erbe and Peterson [7,8] investigated a second-order nonlinear damped dynamic equation…”