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Asymptotic equivalence relations for rapidly varying solutions of sublinear differential equations of Emden–Fowler type
Advances in Continuous and Discrete Models volume 2022, Article number: 19 (2022)
Abstract
We discuss sublinear differential equations of the Emden–Fowler type \(x''=q(t) x^{\gamma }\) under the assumption that the coefficient q is a rapidly varying function. We show that all of their strongly decreasing and strongly increasing solutions are rapidly varying functions and are in the asymptotic equivalence relation with a precisely defined function determined by the coefficient q.
1 Introduction
This paper is concerned with positive solutions of differential equations of the Emden–Fowler type of the form
where \(\gamma \neq 1\) is a positive constant, and q is positive, continuous function on \([a,\infty )\).
Equation (E) is called sublinear or superlinear according to \(\gamma <1\) or \(\gamma >1\). We consider the sublinear case, i.e., when \(0<\gamma <1\).
Any positive solution x of (E), continuable at infinity and eventually different from zero, is either increasing or decreasing. A positive decreasing solution of (E) is said to be

strongly decreasing if \(\lim_{t\to \infty }x(t)=0\), \(\lim_{t\to \infty }x'(t)=0\),

asymptotically constant if \(\lim_{t\to \infty }x(t)=\mathrm{const}>0\), \(\lim_{t\to \infty }x'(t)=0\).
A positive increasing solution of (E) is said to be

asymptotically linear if \(\lim_{t\to \infty }x(t)=\infty \), \(\lim_{t\to \infty } \frac{x(t)}{t}=\mathrm{const}>0\),

strongly increasing if \(\lim_{t\to \infty }x(t)=\infty \), \(\lim_{t\to \infty } x'(t)= \infty \).
The existence of the above four types has been studied in [2, 25]. In the sublinear case, the existence of strongly increasing solutions is completely characterized, while for the existence of strongly decreasing solutions, only the sufficient condition is known, as it is stated in the following propositions.
Proposition 1.1
([25, Theorem 3.8])
Sublinear equation (E) has a strongly increasing solution if and only if
Proposition 1.2
([25, Theorem 3.2])
Sublinear equation (E) has a strongly decreasing solution if
The existence and asymptotic behavior of regularly varying solutions of nonlinear differential equations were extensively studied in [8, 9, 11, 13–16, 18–22, 24]. Unlike regularly varying solutions, rapidly varying solutions of linear and nonlinear equations are much less studied. The study of secondorder linear differential equation in the framework of rapid variation was initiated by Marić [23]. Halflinear differential equations in the framework of the Karamata theory and the de Haan theory were treated in [26–28]. Also, the existence of regularly and rapidly varying solutions of thirdorder nonlinear differential equations was studied in [17], while in [10, 12] the conditions for the existence and asymptotic representations of solutions are given assuming that the coefficient of the equation belongs to the subclass of rapidly varying functions. Although the results in [10, 12] can be applied to (E), the problem of determining the conditions for all solutions to be rapidly varying functions is not considered in these papers. Therefore, our goal in this paper is to prove that all strongly decreasing and strongly increasing solutions are rapidly varying functions under the assumption that the coefficient q is rapidly varying and to examine the properties of these solutions in more detail. In addition, the existence conditions and asymptotic representations of solutions are given in [10, 12] under the assumption that the coefficient of the equation belongs to the subclass of rapidly varying functions. The solutions considered in these papers also belong to the subclass of rapidly varying functions. Therefore, the results obtained in this paper improve the results in [10, 12], since we consider the equation with rapidly varying coefficient and its rapidly varying solutions.
This paper is organized as follows: In Sect. 2, we give the basic definitions and properties of the regularly and rapidly varying functions. We also present asymptotic equivalence relations in the class of rapidly varying functions of index ∞, which are defined in [1, 5], and some of their basic properties that are useful for our research. In addition, we introduce analogous relations in the class of rapidly varying functions of index −∞ and examine their properties. The main results are stated in Sect. 3. In Sect. 4, we prove some important lemmas that significantly shorten the proof of the main results. Section 5 contains the proofs of the main results. Some illustrative examples are presented in Sect. 6.
2 Preliminaries
In this section, first, we recall basic information on the Karamata theory of regularly varying functions and the de Haan theory of rapidly varying functions.
Definition 2.1
A measurable function \(f:[a,\infty )\rightarrow (0,\infty )\) is said to be regularly varying of index \(\rho \in \mathbb{R}\) if
The set of all regularly varying functions of index ρ is denoted by \(\operatorname{RV}(\rho ) \).
Definition 2.2
A measurable function \(f:[a,\infty )\rightarrow (0,\infty )\) is said to be rapidly varying of index ∞ if
A measurable function \(f : [a,\infty )\rightarrow (0,\infty )\) is said to be rapidly varying of index −∞ if
The set of rapidly varying functions of index ∞ (or −∞) is denoted by \(\operatorname{RPV}(\infty )\) (or \(\operatorname{RPV}(\infty )\)). For more information on regular and rapid variation, the reader is referred to the monograph by Bingaham, Goldie, and Teugels [1]. For more recent contribution of the theory of rapid variation, see [6, 7].
Example 2.1

1.
It is easy to see that function \(f(t)=a^{t}\), \(a>1\) is a typical representative of the class \(\operatorname{RPV}(\infty )\), while the function \(f(t)=a^{t}\), \(a\in (0,1)\) is a typical representative of the class \(\operatorname{RPV}(\infty )\).

2.
The function \(f(t)=g(t) a^{h(t)}\), \(g\in \operatorname{RV}(\rho )\), \(\rho \in \mathbb{R}\), \(h\in \operatorname{RV}(m)\), \(m>0\) belongs to the \(\operatorname{RPV}(\infty )\), when \(a>1\) and \(\operatorname{RPV}(\infty )\), when \(a\in (0,1)\).
Next, we give some properties of rapidly varying functions.
Proposition 2.1

(1)
\(f\in \operatorname{RPV}(\infty )\) if and only if \(1/f\in \operatorname{RPV}(\infty )\).

(2)
If \(f,g\in \operatorname{RPV}(\infty )\) and \(h\in \operatorname{RV}(\rho )\), \(\rho \in \mathbb{R}\), then

(i)
\(f^{p}\in \operatorname{RPV}(\infty )\) for any \(p>0\).

(ii)
\(f\cdot h\in \operatorname{RPV}(\infty )\).

(iii)
\(f\cdot g\in \operatorname{RPV}(\infty )\).

(i)
Proof
(1) This part of the proposition is shown in [29] on time scales.
(2) Since \(\lim_{t\to \infty }\frac{f(\lambda t)}{f(t)}=\infty \), \(\lim_{t\to \infty }\frac{g(\lambda t)}{g(t)}=\infty \) and \(\lim_{t\to \infty }\frac{h(\lambda t)}{h(t)}=\lambda ^{\rho }\) for all \(\lambda >1\), we have

(i)
\(\lim_{t\to \infty }\frac{(f(\lambda t))^{p}}{(f(t))^{p}}= ( \lim_{t\to \infty }\frac{f(\lambda t)}{f(t)} )^{p}=\infty \), for any \(p>0\),

(ii)
\(\lim_{t\to \infty } \frac{f(\lambda t)\cdot h(\lambda t)}{f(t)\cdot h(t)}=\lambda ^{\rho } \cdot \lim_{t\to \infty }\frac{f(\lambda t)}{f(t)}=\infty \)

(iii)
\(\lim_{t\to \infty } \frac{f(\lambda t)\cdot g(\lambda t)}{f(t)\cdot g(t)}=\infty \).
□
Next, we consider some useful equivalence relations on the classes \(\operatorname{RPV}(\infty )\) and \(\operatorname{RPV}(\infty )\). The following relation is introduced in [1] and further considered in [3, 4].
Definition 2.3
Let f and g be positive functions in \([a,\infty ) \). These two functions are called mutually inversely asymptotic at ∞, denoted by \(f(t)\stackrel {\star }{\sim }g(t), t\to \infty \), if for every \(\lambda >1\), there exists \(t_{0}=t_{0}(\lambda ) \) such that
Definition of the next relation and its properties are given in [5].
Definition 2.4
Let f and g be positive functions in \([a,\infty ) \). These two functions are called mutually rapidly equivalent at ∞, denoted by \(f(t)\stackrel {r}{\sim }g(t), t\to \infty \), if
Proposition 2.2
Let f and g be positive functions in \([a,\infty )\). Then, the following assertions hold:

(a)
if f and g are measurable functions such that \(f(t)\stackrel {r}{\sim }g(t)\) for \(t\to \infty \), then f and g both belong to \(\operatorname{RPV}(\infty )\);

(b)
the relation \(\stackrel {r}{\sim }\) is an equivalence relation in the class \(\operatorname{RPV}(\infty )\).
Proposition 2.3
Let \(f\in \operatorname{RPV}(\infty )\) be a locally bounded function on \([a,\infty )\). Also, let \(1/f\) be a locally bounded function on \([a,\infty )\). Then the following assertions are true:

(a)
\(f(t) \stackrel {r}{\sim }\frac{1}{t}\int _{a}^{t} f(s)\,ds\), \(t\to \infty \);

(b)
\(f(t) \stackrel {r}{\sim }\frac{1}{t\int _{t}^{\infty }\frac{ds}{s^{2}f(s)}}\), \(t\to \infty \);

(c)
\(F\in \operatorname{RPV}(\infty ) \), where \(F(t)=\int _{a}^{t} f(s)\,ds\), \(t>a\);

(d)
\(\varphi \in \operatorname{RPV}(\infty ) \), where \(\varphi (t)=\frac{1}{\int _{t}^{\infty }\frac{ds}{f(s)}}\), \(t>a\).
Remark 2.1
It is easy to prove that if \(f(t)\stackrel {r}{\sim }g(t)\), \(t\to \infty \), then

(a)
\(f(t)^{p}\stackrel {r}{\sim }g(t)^{p}\), \(t\to \infty\) for all \(p>0\),

(b)
\(h(t)\cdot f(t)\stackrel {r}{\sim }h(t)\cdot g(t) \), \(t\to \infty\) for \(h\in \operatorname{RV}(\rho )\), \(\rho \in \mathbb{R}\) or \(h\in \operatorname{RPV}(\infty )\).
Here, we introduce two new relations on \(\operatorname{RPV}(\infty )\).
Definition 2.5
Let f and g be positive functions in \([a,\infty ) \). These two functions are called mutually inversely asymptotic at −∞, denoted by \(f(t)\underset {\star }{\sim }g(t), t\to \infty \), if for every \(\lambda >1\), there exists \(t_{0}=t_{0}(\lambda ) \) such that
Definition 2.6
Let f and g be positive functions in \([a,\infty ) \). These two functions are called mutually rapidly equivalent at −∞, denoted by \(f(t)\underset {r}{\sim }g(t), t\to \infty \), if
The next proposition establishes a connection between relations \(\stackrel {r}{\sim }\) and \(\underset {r}{\sim }\).
Proposition 2.4
Let f and g be positive functions in \([a,\infty )\). Then
Proof
The proposition directly follows from the equalities
□
The next proposition directly follows from Proposition 2.4, Proposition 2.2, and Proposition 2.1.
Proposition 2.5
Let f and g be positive functions in \([a,\infty )\). Then the following assertions hold:

(a)
if f and g are measurable functions such that \(f(t)\underset {r}{\sim }g(t)\) for \(t\to \infty \), then f and g both belong to \(\operatorname{RPV}(\infty )\);

(b)
the relation \(\underset {r}{\sim }\) is an equivalence relation in the class \(\operatorname{RPV}(\infty )\).
Remark 2.2
Proposition 2.3(b) will be easier to use if we rewrite it in a different form. Denote \(g(t)=\frac{1}{t^{2} f(t)}\). Hence, due to Remark 2.1, we have
yielding
by using the Proposition 2.4. Since \(f\in \operatorname{RPV}(\infty )\), from Proposition 2.1, we conclude that \(g\in \operatorname{RPV}(\infty )\). Also, since \(1/f\) is a locally bounded function on \([a,\infty )\), so is g.
Therefore, we have the following proposition.
Proposition 2.6
Let \(g\in \operatorname{RPV}(\infty )\) be a locally bounded function on \([a,\infty )\). Then
3 Main results
Theorem 3.1
Suppose that \(q\in \operatorname{RPV}(\infty )\) satisfies the condition (1.1). Every strongly increasing solution of (E) is rapidly varying of index ∞. Moreover, any such solution x satisfies the asymptotic relation
where the function X is given by
Theorem 3.2
Suppose that \(q\in \operatorname{RPV}(\infty )\) satisfies the condition (1.2). Every strongly decreasing solution of (E) is rapidly varying of index −∞. Moreover, any such solution x satisfies the asymptotic relation
4 Auxiliary lemmas
Let us denote
First, we show that functions X, \(X_{1}\), and \(X_{2}\) are in the relation \(\stackrel {r}{\sim }\) under the assumption that q is a rapidly varying function of index ∞.
Lemma 4.1
Suppose that \(q\in \operatorname{RPV}(\infty )\). Then
where the functions X and \(X_{1}\) are given by (3.2) and (4.1), respectively.
Proof
Using Proposition 2.3(a), we have
Multiplying (4.4) by \(t^{2}\), in the view of Remark 2.1, we get
implying
since \(\frac{1}{1\gamma }>0\). This completes the proof of Lemma 4.1. □
Lemma 4.2
Suppose that \(q\in \operatorname{RPV}(\infty )\). Then
where the functions X and \(X_{2}\) are given by (3.2) and (4.2), respectively.
Proof
Applying Proposition 2.3(a), we conclude that
Since \(\frac{3+\gamma }{1\gamma }>0\), due to Remark 2.1, we get
On the other hand, another application of Proposition 2.3(a) gives us
By combining (4.6) and (4.7), we have
implying
This completes the proof of Lemma 4.2. □
Denote by
Next, we show that functions X, \(Y_{1}\), and \(Y_{2}\) are in the relation \(\underset {r}{\sim }\) under the assumption that q is a rapidly varying function of index −∞.
Lemma 4.3
Suppose that \(q\in \operatorname{RPV}(\infty )\). Then
where the functions X and \(Y_{1}\) are given by (3.2) and (4.8), respectively.
Proof
Using Proposition 2.6, we get
On the other hand, another application of Proposition 2.6 gives us
From (4.11) and (4.12), we conclude
implying (4.10). □
Lemma 4.4
Suppose that \(q\in \operatorname{RPV}(\infty )\). Then
where the functions X and \(Y_{2}\) are given by (3.2) and (4.9), respectively.
Proof
Applying Proposition 2.6, we conclude that
implying
since \(\frac{3+\gamma }{1\gamma }>0\). On the other hand, another use of Proposition 2.6 gives us
By combining (4.14) and (4.15), we get
yielding (4.13). □
5 Proofs of main results
Proof of Theorem 3.1
Since q satisfies the condition (1.1), we obtain that the equation (E) has a strongly increasing solution.
Let x be arbitrary strongly increasing solution of (E) defined on \([T,\infty ), T\geq a\). First, we show that there exist positive constants m, M such that
where \(X_{1}\) and \(X_{2}\) are given by (4.1) and (4.2), respectively. Integrating \(x'\) on \([T,t]\), we get
because \(x'\) is increasing. Hence, we find \(K_{1} > 0\) such that
Since x is increasing, integration of (E) from T to t gives
implying, due to the fact \(\int _{T}^{t} q(s)\,ds \rightarrow \infty \) as \(t\to \infty \), that we find \(K_{2}>0\) such that
By combining (5.2) and (5.3), we have
Thus, there exists \(M>0\) such that
The righthand side of the inequality (5.1) is proved.
Next, we prove the lefthand side of the inequality (5.1). Set \(w(t)=x(t)x'(t)\) and
An application of Young’s inequality gives
Since, \(\gamma \mu \nu =\nu \mu =\kappa \), we get
After dividing (5.5) by \(w(t)^{1\kappa }\) and integrating the obtained inequality on \([T,t]\), we get that there is \(k_{1}>0\) such that
or
Integrating (5.6) from T to t, we find \(k_{2}>0\) and \(T^{*}\geq T\) sufficiently large such that
From (5.7), we obtain that there exists \(m>0\) such that the lefthand side of the inequality (5.1) is satisfied.
Next, we prove that x is a rapidly varying function of index ∞. Fix arbitrary \(\lambda >1\). Indeed, from (5.1) for sufficiently large t, we have
and
From (5.8) and (5.9), we obtain
for sufficiently large t. By Lemma 4.1 and Lemma 4.2, we have \(X_{1}(t)\stackrel {r}{\sim }X_{2}(t)\), \(t\to \infty \), which means
Since λ was arbitrary, combining (5.10) and (5.11) gives us \(\lim_{t\to \infty }\frac{x(\lambda t)}{x(t)}=\infty \) for all \(\lambda >1 \), that is, \(x\in \operatorname{RPV}(\infty )\).
It remains to prove that the solution x satisfies the asymptotic relation (3.1). Fix arbitrary \(\lambda >1\). Let m and M be positive numbers, satisfying (5.1) for \(t\geq T_{1}\geq T\). By Lemma 4.1 and Lemma 4.2, we have (4.3) and (4.5), so there exists \(T_{2}=T_{2}(\lambda )\geq T_{1} \) such that
Therefore, from (5.1), we conclude that
implying \(x(t)\stackrel {\star }{\sim }X(t)\), \(t\to \infty \). This completes the proof of Theorem 3.1. □
Proof of Theorem 3.2
Assumption (1.2) ensures the existence of strongly decreasing solution of (E). Assume that x is the arbitrary strongly decreasing solution of (E) defined on \([T,\infty ), T\geq a\). First, we show that there exist positive constants m and M such that
where \(Y_{1}\) and \(Y_{2}\) are given by (4.8) and (4.9), respectively. Since \(x'(t)\to 0\), \(t\to \infty \), and x is decreasing, integrating (E) from t to ∞, we get
Dividing (5.14) by \(x(t)^{\gamma }\) and then integrating from t to ∞, since \(x(t)\to 0\), \(t\to \infty \), we have
implying that there exists \(M>0\) such that the righthand side of the inequality (5.13) is satisfied.
Next, we prove the lefthand side of the inequality (5.13). Setting \(w(t)=x(t)x'(t)\) and ν, μ, κ as in (5.4), application of Young’s inequality gives
afterwards multiplying by \(w(t)^{\kappa 1}\) and integrating from t to ∞, we find \(k_{1}>0\) such that
or
Since \(x(t)\to 0\), \(t\to \infty \), integrating (5.15) from t to ∞ yields that there is \(k_{2}>0\) such that
From (5.16), we obtain that there exists \(m>0\) such that the lefthand side of the inequality (5.13) is satisfied.
That \(x\in \operatorname{RPV}(\infty )\) and satisfies the asymptotic relation (3.3) can be proved in the same way as in the proof of Theorem 3.1, using Lemma 4.3 and Lemma 4.4. □
6 Examples
Now, we present two examples that illustrate main results stated by Theorem 3.1 and Theorem 3.2.
Example 6.1
Consider the equation
where \(q_{1}(t)=e^{t+(1\gamma )e^{t}} (1+e^{t})\). Since
by Theorem 3.1 follows that every strongly increasing solution of (6.1) is rapidly varying of index ∞, and any such solution x satisfies the asymptotic relation
where \(Q_{1}(t)=(t^{2} q_{1}(t))^{\frac{1}{1\gamma }}\). It is easy to check that \(x_{1}(t)=e^{e^{t}}\) is such a solution of (6.1), since \(x_{1}\in \operatorname{RPV}(\infty )\) and
implying that \(x_{1}\) satisfies the asymptotic relation (6.2).
Example 6.2
Consider the equation
where \(q_{2}(t)=k^{2} e^{k(\gamma 1)t}\), \(k>0\). Since
by Theorem 3.2 follows that every strongly decreasing solution of (6.3) is rapidly varying of index −∞, and any such solution x satisfies the asymptotic relation
where \(Q_{2}(t)=(t^{2} q_{2}(t))^{\frac{1}{1\gamma }}\). It is easy to check that \(x_{2}(t)=e^{k t}\), \(k>0\) is such a solution of (6.3), since \(x_{2}\in \operatorname{RPV}(\infty )\) and
implying that \(x_{2}\) satisfies the asymptotic relation (6.4).
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Corresponding author Jelena Milošević is not currently in receipt of any research funding. Author Jelena Manojlović acknowledges financial support through the Ministry of Education and Science of Republic of Serbia, agreement no. 4510368/202214/200124.
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Manojlović, J., Milošević, J. Asymptotic equivalence relations for rapidly varying solutions of sublinear differential equations of Emden–Fowler type. Adv Cont Discr Mod 2022, 19 (2022). https://doi.org/10.1186/s1366202203693w
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DOI: https://doi.org/10.1186/s1366202203693w