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

Yang-Cong Qiu

doi:10.1186/s13662-019-2451-3

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.

Highlights

1 Introduction In this paper, we consider the existence of nonoscillatory solutions to fourth-order nonlinear neutral dynamic equations of the form r1(t) r2(t) r3(t) x(t) + p(t)x g(t)
The majority of the scholars obtained the sufficient conditions to ensure that the solutions of the equations oscillate or tend to zero by using the generalized Riccati transformation and integral averaging technique
3 Sufficient conditions we firstly present some sufficient conditions for the existence of each type of eventually positive solutions to (1)

Summary

Introduction

Note that there exist only two cases for every eventually positive solution x to (2): limt→∞ x(t) = a > 0 or limt→∞ x(t) = 0. Lemma 2.2 Suppose that x is an eventually positive solution to (1) and z(t) a, λ = 0, 1, where λ = 1 only if (4) holds. Proof In view of (C2) and (C3), for any eventually positive solution x to (1), there always exist t1 ∈ [t0, ∞)T and p1 with |p0| < p1 < 1 such that x(t) > 0, x(g(t)) > 0, x(h(t)) > 0, and |p(t)| ≤ p1, t ∈ [t1, ∞)T.

Results

Conclusion