Existence of nonoscillatory solutions to nonlinear higher-order neutral dynamic equations

Yang-Cong Qiu,Tongxing Li,Irena Jadlovská,Kuo-Shou Chiu

doi:10.1186/s13662-020-02928-y

Abstract

We investigate the existence of different types of nonoscillatory solutions to a class of higher-order nonlinear neutral dynamic equations on a time scale. Two examples are provided to show the significance of the conclusions.

Highlights

The time scale theory has been introduced and developed rapidly since 1988; see, for instance, [1,2,3,4, 7, 8]
Afterwards, many scholars were concerned with the oscillation of dynamic equations on time scales and they obtained abundant achievements
Since 2007, numerous researchers have investigated the existence of nonoscillatory solutions to several classes of nonlinear neutral dynamic equations x(t) + p(t)x g(t) + f t, x h(t) = 0, (1)

Summary

Introduction

The time scale theory has been introduced and developed rapidly since 1988; see, for instance, [1,2,3,4, 7, 8]. Lemma 2.1 Suppose that x is an eventually positive solution to (4) and limt→∞ z(t)/Rλ(t) = a for λ = 0, 1. Theorem 2.2 If x is an eventually positive solution to (4), one of the following four cases holds: (A1) x ∈ A(0, 0); (A2) x ∈ A(b, 0); (A3) x ∈ A(∞, b); (A4) lim supt→∞ x(t) = ∞ and limt→∞ x(t)/R(t) = 0.

Results

Conclusion