Asymptotic equivalence relations for rapidly varying solutions of sublinear differential equations of Emden–Fowler type

We discuss sublinear differential equations of the Emden–Fowler type x″=q(t)xγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x''=q(t) x^{\gamma }$\end{document} 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.


Introduction
This paper is concerned with positive solutions of differential equations of the Emden-Fowler type of the form where γ = 1 is a positive constant, and q is positive, continuous function on [a, ∞). Equation (E) is called sublinear or superlinear according to γ < 1 or γ > 1. We consider the sublinear case, i.e., when 0 < γ < 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→∞ x(t) = 0, lim t→∞ x (t) = 0, • asymptotically constant if lim t→∞ x(t) = const > 0, lim t→∞ x (t) = 0. A positive increasing solution of (E) is said to be • asymptotically linear if lim t→∞ x(t) = ∞, lim t→∞ x(t) t = const > 0, • strongly increasing if lim t→∞ x(t) = ∞, lim t→∞ x (t) = ∞. 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. 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 second-order linear differential equation in the framework of rapid variation was initiated by Marić [23]. Half-linear differential equations in the framework of the Karamata theory and the de Haan theory were treated in [26][27][28]. Also, the existence of regularly and rapidly varying solutions of third-order 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

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.
The set of all regularly varying functions of index ρ is denoted by RV(ρ).
The set of rapidly varying functions of index ∞ (or -∞) is denoted by RPV(∞) (or RPV(-∞)). 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].
1. It is easy to see that function f (t) = a t , a > 1 is a typical representative of the class RPV(∞), while the function f (t) = a t , a ∈ (0, 1) is a typical representative of the class RPV(-∞).
Next, we give some properties of rapidly varying functions.
Proof (1) This part of the proposition is shown in [29] on time scales.
Next, we consider some useful equivalence relations on the classes RPV(∞) and RPV(-∞). 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, ∞). These two functions are called mutually inversely asymptotic at ∞, denoted by Definition of the next relation and its properties are given in [5].

Definition 2.4
Let f and g be positive functions in [a, ∞). These two functions are called mutually rapidly equivalent at ∞, denoted by f (t) ( Here, we introduce two new relations on RPV(-∞).

Definition 2.5
Let f and g be positive functions in [a, ∞). These two functions are called mutually inversely asymptotic at -∞, denoted by Definition 2.6 Let f and g be positive functions in [a, ∞). These two functions are called mutually rapidly equivalent at -∞, denoted by The next proposition establishes a connection between relations r ∼ and ∼ r .

Proposition 2.4 Let f and g be positive functions in [a, ∞).
Then , t → ∞.
Proof The proposition directly follows from the equalities The next proposition directly follows from Proposition 2.4, Proposition 2.2, and Proposition 2.1. Remark 2.2 Proposition 2.3(b) will be easier to use if we rewrite it in a different form. Denote g(t) = 1 t 2 f (t) . Hence, due to Remark 2.1, we have by using the Proposition 2.4. Since f ∈ RPV(∞), from Proposition 2.1, we conclude that g ∈ RPV(-∞). Also, since 1/f is a locally bounded function on [a, ∞), so is g.
Therefore, we have the following proposition.

Main results
Theorem 3.1 Suppose that q ∈ RPV(∞) 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

1)
where the function X is given by

Auxiliary lemmas
Let us denote First, we show that functions X, X 1 , and X 2 are in the relation r ∼ under the assumption that q is a rapidly varying function of index ∞.

Lemma 4.1 Suppose that q ∈ RPV(∞).
Then where the functions X and X 1 are given by

Lemma 4.2 Suppose that q ∈ RPV(∞).
Then    Denote by Next, we show that functions X, Y 1 , and Y 2 are in the relation ∼ r under the assumption that q is a rapidly varying function of index -∞.

Lemma 4.3 Suppose that q ∈ RPV(-∞). Then
where the functions X and Y 1 are given by (3.2) and (

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, ∞), T ≥ 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 t T q(s) ds → ∞ as t → ∞, 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 right-hand side of the inequality (5.1) is proved.
Next, we prove the left-hand side of the inequality (5.1). Set w(t) = x(t)x (t) and An application of Young's inequality gives Since, γ μν = νμ = -κ, we get After dividing (5.5) by w(t) 1-κ and integrating the obtained inequality on [T, t], we get that there is k 1 > 0 such that Integrating (5.6) from T to t, we find k 2 > 0 and T * ≥ T sufficiently large such that From (5.7), we obtain that there exists m > 0 such that the left-hand side of the inequality (5.1) is satisfied. Next, we prove that x is a rapidly varying function of index ∞. Fix arbitrary λ > 1. Indeed, from (5.1) for sufficiently large t, we have From (5.8) and (5.9), we obtain m M for sufficiently large t. By Lemma 4.1 and Lemma 4.2, we have X 1 (t) r ∼ X 2 (t), t → ∞, which means lim t→∞ X 2 (λt) Since λ was arbitrary, combining (5.10) and (5.11) gives us lim t→∞ It remains to prove that the solution x satisfies the asymptotic relation (3.1). Fix arbitrary λ > 1. Let m and M be positive numbers, satisfying (5.1) for t ≥ T 1 ≥ T. By Lemma 4.1 and Lemma 4.2, we have (4.3) and (4.5), so there exists T 2 = T 2 (λ) ≥ T 1 such that Therefore, from (5.1), we conclude that implying x(t) ∼ X(t), t → ∞. 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, ∞), T ≥ 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) → 0, t → ∞, and x is decreasing, integrating (E) from t to ∞, we get Dividing (5.14) by x(t) γ and then integrating from t to ∞, since x(t) → 0, t → ∞, we have implying that there exists M > 0 such that the right-hand side of the inequality (5.13) is satisfied. Next, we prove the left-hand 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) κ-1 and integrating from t to ∞, we find k 1 > 0 such that Since x(t) → 0, t → ∞, integrating (5.15) from t to ∞ yields that there is k 2 > 0 such that

Examples
Now, we present two examples that illustrate main results stated by Theorem 3.1 and Theorem 3.2.