Existence of nonoscillatory solutions to neutral dynamic equations on time scales

Abstract

In this paper, we give an analogue of the Arzela-Ascoli theorem on time scales. Then, we establish the existence of nonoscillatory solutions to the neutral dynamic equation [x(t) + p(t)x(g(t))] + f (t, x(h(t))) = 0 on a time scale. To dwell upon the importance of our results, three interesting examples are also included.

Existence of nonoscillatory solutions to nonlinear third-order neutral dynamic equations on time scales

“…Some results for existence of oscillatory and nonoscillatory solutions to various classes of neutral functional dynamic equations were presented in [9,10,[13][14][15][17][18][19][20][21][22], and some open problems were given in Mathsen et al [18]. Zhu and Wang [22] studied the existence of nonoscillatory solutions to a class of first-order dynamic equations. Gao and Wang [10] considered the same problem of a second-order dynamic equation…”

Existence of nonoscillatory solutions to third-order neutral functional dynamic equations on time scales

“…In this paper, we further consider (1.2) on a time scale T satisfying sup T = ∞, where t ∈ [t 0 , ∞) T = [t 0 , ∞) ∩ T with t 0 ∈ T. The motivation originates from [10,12,20,23,24]. We shall establish the existence of nonoscillatory solutions to (1.2) by employing Krasnoselskii's fixed point theorem, and we will give three examples to show the versatility of the results.…”

“…Afterwards, some open problems were presented in the paper by Mathsen et al [19]. Zhu and Wang [24] discussed the existence of nonoscillatory solutions to a first-order nonlinear neutral dynamic equation…”

Hille and Nehari-Type Oscillation Criteria for Third-Order Emden–Fowler Neutral Delay Dynamic Equations

“…And for the oscillation and nonoscillation of the neutral delay dynamic equations, some excellent works have already been established, and we refer the reader to the various articles [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”