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Existence of positive solutions of a third order nonlinear differential equation with positive and negative terms
 Demou Luo^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366201815203
© The Author(s) 2018
 Received: 28 November 2017
 Accepted: 7 February 2018
 Published: 12 March 2018
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.
Keywords
 Positive solutions
 Asymptotic properties
 Delay argument
 Positive and negative term
MSC
 34A34
 34K13
 34K30
 34L30
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 [2–14] and the references therein.
 (\({H}_{1}\)):

\(a(t), b(t), p(t), q(t), \tau(t), \sigma(t) \in C([t_{0},\infty))\) are positive;
 (\({H}_{2}\)):

\(f(u), h(u) \in C(\mathbb{R})\), \(uf(u)>0\), \(uh(u)>0\) for \(u\neq0\), g is bounded, f is nondecreasing;
 (\({H}_{3}\)):

\(f(uv)\geq f(uv)\geq f(u)f(v)\) for \(uv>0\), and \(f(u)\leq u\);
 (\({H}_{4}\)):

\(\tau(t)\leq t\), \(\lim_{t\rightarrow\infty}\tau (t)=\infty\), \(\lim_{t\rightarrow\infty}\sigma(t)=\infty\).
 (\({H}_{5}\)):

\(\int_{t_{0}}^{\infty}\frac{1}{a(s)}\,ds=\int _{t_{0}}^{\infty}\frac{1}{b(s)}\,ds=\infty\).
By a solution of Eq. (1.1), we can easily understand a function \(x(t)\) with derivatives \(a(t)x'(t)\), \(b(t) (a(t)x'(t) )'\) continuous on \([T_{x},\infty)\), \(T_{x}\geq t_{0}\), which satisfies Eq. (1.1) on \([T_{x},\infty)\). We consider only those solutions \(x(t)\) of (1.1) which satisfy \(\sup\{x(t):t\geq T\}>0\) for all \(T\geq T_{x}\).
The research of the higher order ordinary differential equations (ODE) (see [1–7]) essentially takes advantage of some recapitulation of Kiguradze’s lemma [1, 2]. In the lemma, from the fixed sign of the highest derivative, we can infer the form of possible nonoscillatory solutions. We cannot fix the sign of the fourth order quasiderivative for an ultimately positive solution because the positive and negative terms are included in (1.1). So the authors primarily investigate the properties of (1.1) in the partial case when either \(p(t)\equiv0\) or \(q(t)\equiv0\).
 (\({H}_{6}\)):

\(\int_{t_{0}}^{\infty}\frac{1}{a(t)}\int_{t}^{\infty }\frac{1}{b(s)}\int_{s}^{\infty}q(u)\,du\,ds\,dt<\infty\).
The organization of this paper is as follows. In Section 2, we introduce some definitions and lemmas and declare some preliminary material needed in later sections. We will state some facts about the differential equations with deviating argument as well as Kiguradze’s lemma of classification of positive solutions. For details on Kiguradze’s theorem, we refer the reader to [1]. In Section 3, we establish our main results for positive solutions by applying Kiguradze’s classification of positive solutions theorem. In Section 4, we present the asymptotic properties of solutions. In Section 5, we give two examples to illustrate our results. The results presented in this paper extend the main results in [15].
2 Preliminaries
A time scale is an arbitrary nonempty closed subset of real numbers. The research of dynamic equations on time scales is an incredibly new area, and the number of studies on this subject is rapidly growing. The theory of dynamic equations unifies the theories of differential equations and difference equations. We suppose that the reader is familiar with the basic concepts concerning the calculus on time scales for dynamic equations. Otherwise one can find most of the material needed to read this paper in Kiguradze and Chanturia’s books [1].
Definition 2.1
([1])
A solution of (1.1) is termed oscillatory if it has arbitrarily large zeros on \([T_{x},\infty)\), otherwise it is termed nonoscillatory. Eq. (1.1) is said to be oscillatory if all its solutions are oscillatory.
Definition 2.2
([1])
Theorem 2.3
Corollary 2.4
([1])
3 Existence of positive solutions
In this section we shall investigate the existence of positive solutions for Eq. (1.1). The main result is in the following theorem.
Theorem 3.1
Proof
Obviously, we have the following easily verifiable criterion for some special cases of (1.1).
Corollary 3.2
Theorem 3.3
Proof
Assume that \(x(t)\) is a positive solution of (1.1). Proceeding exactly as in the proof of Theorem 3.1, we verify that the associated function \(z(t)\) belongs to the situation of \(z(t)\in\mathcal{N}_{2}\).
Theorem 3.4
Proof
Theorem 3.5
Let (2.5) hold. Assume that all the conditions of Theorems 3.1 and 3.4 hold. Then Eq. (1.1) has no positive solutions.
4 Asymptotic properties
Theorem 4.1
Proof
It follows from (4.7), (4.13), and (4.17), that for any positive integer k the function \(x_{k}(t)\) is a solution of Eq. (4.1) on the interval \([t_{1},t_{1}+k]\). By Lemma 10.2 of [1], \(\{x_{k}(t)\}_{k=1}^{+\infty}\) contains a subsequence \(\{ x_{k_{l}}(t)\}_{l=1}^{+\infty}\) such that \(\{x_{k_{l}}(t)^{(i)}\} _{l=1}^{+\infty}\), \(i=0,1,2\), converge uniformly on every finite subinterval of \([t_{1},+\infty)\), and \(x(t)=\lim_{l\rightarrow+\infty}\{ x_{k_{l}}(t)\}\) for \(t\geq t_{1}\) is a solution of Eq. (4.1). In view of (4.11) and (4.13), \(x(t)\) satisfies conditions (4.4) and (4.6).
The proof of Theorem 4.1 is complete. □
5 Examples
Example 5.1
Our results are also applicable for the case when \(\tau(t)\equiv t\).
Example 5.2
Declarations
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant No. 61673121, in part by the Natural Science Foundation of Guangdong Province under Grant No. 2014A030313507, and in part by the Projects of Science and Technology of Guangzhou under Grant No. 201508010008. The authors are grateful to the referee for careful reading of the paper and for his or her useful comments which helped them to improve the paper.
Authors’ contributions
All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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