Nonoscillatory solutions to fourth-order neutral dynamic equations on time scales

Abstract

In this paper, we present some sufficient conditions and necessary conditions for the existence of nonoscillatory solutions to a class of fourth-order nonlinear neutral dynamic equations on time scales by employing Banach spaces and Krasnoselskii's fixed point theorem. Two examples are given to illustrate the applications of the results.

Existence of nonoscillatory solutions tending to zero of fourth-order nonlinear neutral dynamic equations on time scales

“…The scientists have provided some sufficient conditions which guarantee that the equations have nonoscillatory solutions with certain characteristics. We refer the reader to [1][2][3][4][5][6] for details of the theory of time scale, and [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] with the references cited therein for the achievements on the existence of nonoscillatory solutions of nonlinear neutral dynamic equations on time scales.…”

“…was studied in [15,19,21], and the higher-order case was considered in [17,18,20]. To have a deeper understanding of the asymptotic behavior of nonoscillatory solutions of these equations, Qiu [16] studied (1) with some conditions. In their works, different groups of eventually positive solutions of the equations are summarized.…”

“…According to Krasnoselskii's fixed point theorem, some new results are presented. However, considering the cases such as g(t) = t -2, g(t) = t/3, and g(t) = t + cos t for t ∈ [t 0 , ∞) T , the conclusions in [24] are not applicable when g(t) ≥ t is not fulfilled eventually, especially for [7,8,[15][16][17][18][19][20][21][22]. Afterwards, Qiu et al [25] studied (2) under g(t) ≤ t for t ∈ [t 0 , ∞) T and partially solved the problem.…”