Existence of positive solutions of a third order nonlinear differential equation with positive and negative terms

Demou Luo

doi:10.1186/s13662-018-1520-3

Abstract

In this article we investigate the existence of positive solutions for a third order nonlinear differential equation with positive and negative terms. The main tool employed here is Kiguradze’s lemma of classification of positive solutions. The asymptotic properties of solutions are also discussed. Two examples are also given to illustrate our result.

Highlights

1 Introduction In 1993, Kiguradze and Chanturia [1] introduced the theory of asymptotic properties of solutions of nonautonomous ordinary differential equations as a method of continuum calculi
Since Kiguradze’s groundbreaking work, there has been a significant growth in the theory of nonautonomous differential equations with deviating argument covering a variety of different problems; see [2,3,4,5,6,7,8,9,10,11,12,13,14] and the references therein
We are interested in the analysis of qualitative theory of positive solutions of third order nonlinear differential equations

Summary

Introduction

In 1993, Kiguradze and Chanturia [1] introduced the theory of asymptotic properties of solutions of nonautonomous ordinary differential equations as a method of continuum calculi. We are interested in the analysis of qualitative theory of positive solutions of third order nonlinear differential equations. Motivated by the papers [1, 15] and the references therein, we consider the following dynamic equation: b(t) a(t)x (t) + p(t)f x τ (t) – q(t)g x σ (t) = 0, t0 ≤ t. We consider only those solutions x(t) of (1.1) which satisfy sup{|x(t)| : t ≥ T} > 0 for all

Results

Conclusion