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Existence of positive solutions of a third order nonlinear differential equation with positive and negative terms
Advances in Difference Equations volume 2018, Article number: 87 (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.
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.
Let \(\mathbb{T}\) be a time scale such that \(t_{0}\in\mathbb{T}\). In this article, 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:
Throughout this paper we assume that
 (\({H}_{1}\)):

\(a(t), b(t), p(t), q(t), \tau(t), \sigma(t) \in C([t_{0},\infty))\) 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 \(f(u)\leq u\);
 (\({H}_{4}\)):

\(\tau(t)\leq t\), \(\lim_{t\rightarrow\infty}\tau (t)=\infty\), \(\lim_{t\rightarrow\infty}\sigma(t)=\infty\).
We consider the canonical case of (1.1), that is,
 (\({H}_{5}\)):

\(\int_{t_{0}}^{\infty}\frac{1}{a(s)}\,ds=\int _{t_{0}}^{\infty}\frac{1}{b(s)}\,ds=\infty\).
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 \(q(t)\equiv0\).
In what follows we shall assume that
 (\({H}_{6}\)):

\(\int_{t_{0}}^{\infty}\frac{1}{a(t)}\int_{t}^{\infty }\frac{1}{b(s)}\int_{s}^{\infty}q(u)\,du\,ds\,dt<\infty\).
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])
The differential equation
has property A if every solution of (2.1) for n even is oscillatory and for n odd either is oscillatory or satisfies the condition \(x^{(i)}(t)\rightarrow0\) as \(t\rightarrow +\infty\) (\(i=1,2,\ldots,n1\)).
Assume that (1.1) possesses an eventually positive solution \(x(t)\) on \([T_{x},\infty)\), \(T_{x}\geq t_{0}\). We introduce the auxiliary function \(z(t)\) associated with \(x(t)\) by
It follows from (\(H_{6}\)) and the boundedness of \(h(u)\) that the definition of function \(z(t)\) is correct and \(z(t)\) exists for all \(t\geq T_{x}\). It is useful to notice that \(z(t)>x(t)>0\), \(z'(t)< x'(t)\), and
Theorem 2.3
Assume that all the solutions of
are nonoscillatory. Then every positive solution \(z(t)\) of (1.1) satisfies either
or
Corollary 2.4
([1])
Assume that
then the set \(\mathcal{N}\) of all positive solutions of (1.1) has the following decomposition:
In this article, to ensure qualitative theory of the studied equation, some easily verifiable conditions must be established. To predigest our notation, we denote that
and
Definition 2.5
([1])
A solution \(x(t)\) of (1.1) defined on an interval \([t_{1},+\infty )\subset[t_{0},+\infty)\) is said to be a Kneser solution if
for \(t_{0}\leq t_{1}\) (\(i=0,1,\ldots,n1\)).
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
Let, for all \(t_{1}\) large enough,
Assume that
Then the positive solution class \(\mathcal{N}_{2}=\varnothing\).
Proof
Assume on the contrary that (1.1) possesses an eventually positive solution \(z(t)\in\mathcal{N}_{3}\). Using the fact that \(b(t) (a(t)z'(t) )'\) is decreasing, we have
In view of (3.3), we can see that \((\frac {a(t)z'(t)}{B(t)} )'\leq0\); consequently, \(\frac{a(t)z'(t)}{B(t)}\) is decreasing. Then
Setting the last estimate into (2.3), we see that \(y(t)=a(t)z'(t)\) is a positive solution of the differential inequality
What is more, \(\frac{y(t)}{B(t)}\) is decreasing and \(b(t)y'(t)>0\).
On the other hand, an integration of (3.4) from t to ∞ and then from \(t_{1}\) to t yields
Having replaced t by \(\tau(t)\) in the last inequality, we obtain
Employing (\(H_{3}\)) and the fact that \(y(t)\) is increasing and \(\frac {y(t)}{B(t)}\) is decreasing, we have
Therefore, setting \(u=\frac{y(\tau(t))}{B(\tau(t))}\), we get
Condition (3.1) guarantees that \(\frac{y(t)}{B(t)}\rightarrow0\) as \(t\rightarrow\infty\). Indeed, if we admit \(\frac {y(t)}{B(t)}\rightarrow\ell>0\), then \(\frac{y(t)}{B(t)}\geq\ell\) and setting the last inequality into (3.4), we obtain
An integration from \(t_{1}\) to ∞ yields
which contradicts condition (3.1). Now, we can take lim sup on both sides of (3.6), one gets a contradiction to (3.2). □
Obviously, we have the following easily verifiable criterion for some special cases of (1.1).
Corollary 3.2
Let, for all \(t_{1}\) large enough,
Then the positive solution class \(\mathcal{N}_{2}\) of third order trinomial differential equation
is empty.
Theorem 3.3
Let, for all \(t_{1}\) large enough,
Assume that
Then the positive solution class \(\mathcal{N}_{2}\) of (1.1) is empty.
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 \(z(t)\in\mathcal{N}_{2}\).
If \(z(t)\in\mathcal{N}_{2}\), then \(y(t)=a(t)z'(t)\) satisfies (3.5). We claim that condition (3.10) implies that \(y(t)\rightarrow\infty\) as \(t\rightarrow\infty\). Really, if not, then \(y(t)\rightarrow L\) as \(t\rightarrow\infty\). An integration of (3.4) from t to ∞ yields
Integrating once more, we get
This is in contradiction to (3.10), and we conclude that \(y(t)\rightarrow\infty\) as \(t\rightarrow\infty\).
Thus, we can set that \(u=y(\tau(t))\), and we obtain
Now, we can take lim sup on both sides of (3.12), one gets a contradiction to (3.11). The proof is complete now. □
Theorem 3.4
Let, for all \(t_{1}\) large enough,
Then the positive solution class \(\mathcal{N}_{0}=\varnothing\).
Proof
Assume on the contrary that (1.1) possesses an eventually positive solution \(z(t)\in\mathcal{N}_{0}\). Let us denote that \(Z'(t)=z(t)\) and \(Z(t)>0\). Due to \(z(t)>0\) and \(z'(t)<0\), then \(\int _{t_{1}}^{t}z(s)\,ds\geq tz(t)t_{1}z(t)\) and \(Z(t_{1})t_{1}z(t)\geq 0\), we obtain
Using the fact that \(uf(u)>0\) and f is nondecreasing, we have \(f(x(\tau(t)))\geq\varepsilon>0\). An integration of (2.3) from t to ∞ yields
Integrating from t to ∞ and consequently from \(t_{1}\) to ∞, we obtain
We get a contradiction to (3.13), and the proof is complete. □
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.
Theorem 3.6
Let (2.5) hold. Assume that all the conditions of Theorems 3.3 and 3.4 hold. Then Eq. (1.1) has no positive solutions.
Theorem 3.7
Let (2.5) hold. Assume that all the conditions of Corollary 3.2 and Theorem 3.4 hold. Then Eq. (3.9) has no positive solutions.
4 Asymptotic properties
In this section some asymptotic properties of positive solutions of (1.1) are treated. Firstly, we need to transform (1.1). Let us denote that
and
Obviously, Eq. (1.1) will be transformed as follows:
Below we establish the existence criteria for Kneser solutions of (4.1) and study the asymptotics of these solutions.
Theorem 4.1
Let
for \(t\geq t_{1}\), \(m\in\{0,1\}\), \(r\in(0,+\infty)\) and
for \(t\geq t_{1}\), \(0\leq(1)^{m+i1}x_{i}\leq rt^{1i}\), \(i=1,2,3\). Then Eq. (4.1) has a continuum of Kneser solutions satisfying the conditions
for \(t\geq t_{1}\), \(i=0,1,2\).
Proof
Set
and choose \(\delta\in (0,\frac{1}{t_{1}+1} )\) so small that
To prove the theorem, it suffices to show that if \(r_{0}= (\frac {\delta}{t_{1}+\delta} )^{2}\frac{r}{12}\), then for any \(c_{0}\in [0,r_{0}]\) Eq. (4.1) together with the condition
Consider the functions
and \(\widetilde{\mathcal{F}}(t,x_{1},x_{2},x_{3})=\mathcal{F}(t,\chi _{1}(t,x_{1}),\chi_{2}(t,x_{2}),\chi_{3}(t,x_{3}))\). Obviously,
for \(t\geq t_{1}\), \(0\leq(1)^{m+i1}x_{i}\leq rt^{1i}\), \(i=1,2,3\). And
for \(t\geq t_{1}\), \((x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\).
On the other hand, (4.2) and (4.3) yield
for \(t\geq t_{1}\), \((1)^{m+i1}x_{i}\leq0\), \(i=1,2,3\). And
for \(t\geq t_{1}\), \((1)^{m+i1}x_{i}\geq0\), \(i=1,2,3\).
By Lemma 10.1 of [1], for any positive integer k, the differential equation
has a solution \(x_{k}(t)\) satisfying the boundary conditions
Assuming that \((1)^{m+2}x''_{k}(t_{1}+k)<0\), from (4.9) we obtain
for \(t_{1}\leq t\leq t_{1}+k\), \((1)^{m}x_{k}(t_{1})<0\).
But this is impossible because \((1)^{m}x_{k}(t_{1})=c_{0}\geq0\). So
According to (4.10), (4.11), and (4.12),
for \(t_{1}\leq t\leq t_{1}+k\), \(i=0,1,2,3\),
for \(t_{1}\leq t\leq t_{1}+k\), and
Applying (4.5), (4.7), and (4.15), we get
for \(t_{1}\leq t\leq t_{1}+\delta\), \(i=0,1,2\). On the other hand, (4.14) implies
for \(t_{1}+\delta\leq t\leq t_{1}+k\), \(i=0,1,2\). Therefore,
for \(t_{1}\leq t\leq t_{1}+k\), \(i=0,1,2\).
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
Consider the third order trinomial differential equation
with \(p>0\), \(q>0\), \(\lambda\in(0,1)\). Now \(f(x)=x\) is nondecreasing and \(h(x)=\arctan(x)\) is bounded. We can easily calculate that
Then conditions (3.1) and (3.7) hold true. Due to
condition (3.13) holds true. Therefore, the positive solution class \(\mathcal{N}_{0}=\varnothing\). Simple computation shows that
and (3.2), (3.10) take the form
and
Therefore, the positive solution class \(\mathcal{N}_{2}=\varnothing\). We can conclude that Eq. (5.1) has no positive solutions.
Our results are also applicable for the case when \(\tau(t)\equiv t\).
Example 5.2
Consider the third order differential equation
with \(p>0\), \(q>0\). We can easily calculate that
The following steps are the same as in Example 5.1, thus we omit them.
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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.
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Luo, D. Existence of positive solutions of a third order nonlinear differential equation with positive and negative terms. Adv Differ Equ 2018, 87 (2018). https://doi.org/10.1186/s1366201815203
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DOI: https://doi.org/10.1186/s1366201815203