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Asymptotic behavior of increasing positive solutions of second order quasilinear ordinary differential equations in the framework of regular variation
Advances in Difference Equations volume 2015, Article number: 273 (2015)
Abstract
The existence and asymptotic behavior at infinity of increasing positive solutions of second order quasilinear ordinary differential equations \((p(t)\varphi (x'(t)) )^{\prime}+q(t)\psi(x(t))=0 \) are studied in the framework of regular variation, where p and q are continuous functions regularly varying at infinity and φ, ψ are both continuous functions regularly varying at zero and regularly varying at infinity, respectively.
1 Introduction
It is of particular interest in the theory of qualitative analysis of differential equations to determine the exact asymptotic behavior at infinity of the solutions under the appropriate assumptions for the coefficients of an equation. This problem is extremely complex when the coefficients are general continuous functions. Thus, the recent research shows that the problem should be studied in the framework of regularly varying functions (also known as Karamata functions). This approach was initiated by Avakumović in 1947 (see [1]), and followed by Marić and Tomić (see [2–4]). It turns out that the problem is completely solvable in the case when the coefficients are regularly varying or generalized regularly varying functions. Namely, in this case, it is possible to completely determine the existence of the solutions, as well as their asymptotic behavior at infinity.
In this paper, we study the differential equation of the form
under the following assumptions:
-
(i)
\(\varphi :(0,\infty) \rightarrow(0,\infty)\) is an increasing continuous function which is regularly varying at zero of index \(\alpha>0\);
-
(ii)
\(\psi:(0,\infty) \rightarrow(0,\infty)\) is a continuous function which is regularly varying at infinity of index \(\beta\in (0,\alpha)\);
-
(iii)
\(p:[a,\infty)\rightarrow(0,\infty)\) is a continuous function which is regularly varying at infinity of index \(\eta\in(0,\alpha)\);
-
(iv)
\(q:[a,\infty)\rightarrow(0,\infty)\) is a continuous function which is regularly varying at infinity of index \(\sigma\in \mathbb{R}\).
By a solution of (E) we mean a function \(x(t):[T,\infty)\rightarrow \mathbb{R}\), \(T\geq a \) which is continuously differentiable together with \(p(t)\varphi (x'(t)) \) on \([T,\infty) \) and satisfies (E) at every point of \([T,\infty)\).
It is easily seen (see [5]) that if \(x(t)\) is an increasing positive solution of (E), then we have the following classification of increasing positive solutions of (E) into three types according to their asymptotic behavior at infinity:
where the function \(P(t)\) is defined as
and \(\varphi^{-1}(\cdot)\) denotes the inverse function of \(\varphi (\cdot)\).
Solutions of type (I), (II), (III) are often called, respectively, subdominant, intermediate, and dominant solutions.
It is well known (see [5, 6]) that the existence of subdominant and dominant solutions for (E) with continuous coefficients \(p(t)\), \(q(t)\), \(\varphi (s)\), and \(\psi(s)\) can be completely characterized by the convergence of the integrals
Theorem 1.1
Let \(p(t),q(t)\in C[a,\infty)\) and \(\varphi (s),\psi(s)\in C[0,\infty)\).
-
(a)
Equation (E) has an increasing positive solution of type (I) if and only if \(J<\infty\).
-
(b)
Equation (E) has an increasing positive solution of type (III) if and only if \(I<\infty\).
-
(c)
Equation (E) has an increasing positive solution of type (II) if \(J=\infty\) and \(I<\infty\).
For the existence of intermediate solutions for (E), necessary conditions can be obtained with relative ease. But the problem of establishing necessary and sufficient conditions turns out to be extremely difficult and thus has been an open problem for a long time.
In this paper we establish the necessary and sufficient conditions for the existence of intermediate solutions for (E) and precisely determine their behavior at infinity, using the theory of regularly varying functions. The present work was motivated by the recent progress in the asymptotic analysis of differential equations by means of regularly varying functions in the sense of Karamata, which was initiated by the monograph of Marić [7]. Also, the equation under consideration in this paper is a generalization of the equation
considered in [8], as well as of the equation
considered in [9, 10]. See also [11–16] for related results regarding second order equations and first order systems, and [17–21] for high-order differential equations and systems.
The main body of the paper is divided into six sections. The definition and basic properties of regularly varying functions are given in Section 2. The main results are stated in Section 3 and proved in Section 5. In Section 4 we collect some preparatory results which will help us to simplify the proof of our main theorems. Finally, some illustrative examples are presented in Section 6.
2 Regularly varying functions
In our analysis we shall extensively use the class of regularly varying functions introduced by Karamata in 1930 by the following.
Definition 2.1
A measurable function \(f:[a,\infty)\rightarrow(0,\infty)\), \(a>0\) is said to be regularly varying at infinity of index \(\rho\in \mathbb{R}\) if
A measurable function \(f : (0,a)\rightarrow(0,\infty)\) is said to be regularly varying at zero of index \(\rho\in{\Bbb{R}}\) if
The set of regularly varying functions of index ρ at infinity and at zero, are denoted, respectively, with \(\operatorname{RV}(\rho)\) and \(\operatorname{RV}_{0}(\rho)\). If, in particular \(\rho=0\), the function f is called slowly varying at infinity or at zero. With SV and \(\mathrm{SV}_{0}\) we denote, respectively, the set of slowly varying functions at infinity and at zero. By an only regularly or a slowly varying function, we mean regularity at infinity.
It follows from Definition 2.1 that any function \(f(t)\in \operatorname{RV}(\rho) \) can be written as
and so the class SV of slowly varying functions is of fundamental importance in the theory of regular variation. If, in particular, the function \(g(t)\rightarrow k>0\) as \(t\to\infty \), it is called a trivial slowly varying, denoted by \(g(t)\in \mbox{tr-}\mathrm{SV}\), and the function \(f(t) \) is called a trivial regularly varying of index ρ, denoted by \(f(t)\in \mbox{tr-RV}(\rho) \). Otherwise, the function \(g(t)\) is called a nontrivial slowly varying, denoted by \(g(t)\in \mbox{ntr-}\mathrm{SV}\), and the function \(f(t)\) is called a nontrivial regularly varying of index ρ, denoted by \(f(t)\in \mbox{ntr-RV}(\rho) \).
Since regularly variation of \(f(\cdot)\) at zero of index α means in fact regularly variation of \(f(1/t)\) at infinity of index −α, the properties of RV0 functions can be deduced from theory of RV functions.
For a comprehensive treatise on regular variation the reader is referred to Bingham et al. [22]. See also Seneta [23]. However, to help the reader we present here some elementary properties of regularly varying functions and a fundamental result, called Karamata’s integration theorem, which will be used throughout the paper.
Proposition 2.1
(Karamata’s integration theorem)
Let \(L(t)\in \mathrm{SV}\). Then:
-
(i)
If \(\alpha>-1 \),
$$\int_{a}^{t} s^{\alpha}L(s)\,ds\sim \frac{t^{\alpha+1} L(t)}{\alpha+1},\quad t\to\infty. $$ -
(ii)
If \(\alpha<-1 \),
$$\int_{t}^{\infty}s^{\alpha}L(s)\,ds\sim- \frac{t^{\alpha+1} L(t)}{\alpha+1},\quad t\to\infty. $$ -
(iii)
If \(\alpha=-1 \), the integral \(\int_{a}^{\infty}s^{-1}L(s)\,ds \) may or may not be convergent. The integral \(m_{1}(t)=\int_{a}^{t} s^{-1} L(s)\,ds \) is a new slowly varying function and \(L(t)/m_{1}(t)\rightarrow0\), \(t\to\infty \). In the case \(\int_{a}^{\infty}s^{-1}L(s)\,ds<\infty\), again \(m_{2}(t)=\int_{t}^{\infty}s^{-1} L(s)\,ds\in \mathrm{SV}\) and \(L(t)/m_{2}(t)\rightarrow0\), \(t\to\infty \).
The symbol ∼ denotes the asymptotic equivalence of two positive functions, i.e.,
We shall also use the following results:
Proposition 2.2
Let \(g_{1}(t)\in \operatorname{RV}(\sigma_{1})\), \(g_{2}(t)\in \operatorname{RV}(\sigma_{2})\), \(g_{3}(t)\in \operatorname{RV}_{0}(\sigma_{3})\). Then:
-
(i)
\((g_{1}(t))^{\alpha}\in \operatorname{RV}(\alpha\sigma_{1})\) for any \(\alpha\in {\Bbb{R}}\);
-
(ii)
\(g_{1}(t)+g_{2}(t)\in \operatorname{RV}(\sigma)\), \(\sigma=\max(\sigma_{1},\sigma_{2})\);
-
(iii)
\(g_{1}(t)g_{2}(t)\in \operatorname{RV}(\sigma_{1}+\sigma_{2})\);
-
(iv)
\(g_{1}(g_{2}(t))\in \operatorname{RV}(\sigma_{1}\sigma_{2})\), if \(g_{2}(t)\rightarrow\infty\), as \(t\to\infty \); \(g_{3}(g_{2}(t))\in \operatorname{RV}(\sigma _{3}\sigma _{2})\), if \(g_{2}(t)\rightarrow0\), as \(t\to\infty \);
-
(v)
for any \(\varepsilon> 0\) and \(L(t)\in \mathrm{SV}\) one has \(t^{\varepsilon}L(t) \rightarrow\infty\), \(t^{-\varepsilon }L(t)\rightarrow0\), as \(t\to\infty \).
Proposition 2.3
If \(f(t)\sim t^{\alpha}l(t)\) as \(t\to\infty \) with \(l(t)\in \mathrm{SV}\), then \(f(t)\) is a regularly varying function of index α i.e. \(f(t)=t^{\alpha}l^{*}(t)\), \(l^{*}(t)\in \mathrm{SV}\), where in general \(l^{*}(t)\neq l(t)\), but \(l^{*}(t)\sim l(t)\) as \(t\to\infty \).
Proposition 2.4
A positive measurable function \(l(t)\) belongs to SV if and only if for every \(\alpha>0\) there exist a non-decreasing function Ψ and a non-increasing function ψ with
Proposition 2.5
For the function \(f(t)\in \operatorname{RV}(\alpha)\), \(\alpha>0\), there exists \(g(t)\in \operatorname{RV}(1/\alpha)\) such that
Here g is an asymptotic inverse of f (and it is determined uniquely to within asymptotic equivalence).
Note that the same result holds for \(t\to0\) i.e. when \(f(t)\in \operatorname{RV}_{0}(\alpha)\), \(\alpha>0\).
Proposition 2.6
For the function \(f(t)\in \operatorname{RV}_{0}(\alpha)\), \(\alpha>0\), there exists \(g(t)\in \operatorname{RV}_{0}(1/\alpha)\) such that
Proof
Since \(f(t)\in \operatorname{RV}_{0}(\alpha)\), we have \(f(1/t)\in \operatorname{RV}(-\alpha)\) and \(1/f(1/t)\in \operatorname{RV}(\alpha)\). We can apply the Proposition 2.5 to the function \(\tilde{f}(t)=1/f(1/t)\). Then there exists \(\tilde{g}\in \operatorname{RV}(1/\alpha)\) such that
Then it is easy to show that the function \(g(t)=1/\tilde{g}(1/t)\in \operatorname{RV}_{0}(1/\alpha)\) is an asymptotic inverse of f. □
Next result is proved in [24] and we are going to use it very often in our proofs. It helps us with manipulation of the asymptotic relations.
Let \(H=\{x|x:[a,\infty)\rightarrow(0,\infty)\}\) and \(H_{1}=\{x\in H|x(t)\to\infty, t\to\infty \}\). If ≃ denotes the asymptotic similarity of two positive functions, i.e.,
and \(\rho_{1}\), \(\rho_{2}\) are arbitrary relations from the set \(\{\sim, \simeq \}\), then let \(\operatorname{Hom}((H_{1},\rho_{1});(H,\rho_{2}))\) be the set of all measurable functions \(F:[a,\infty)\rightarrow(0,\infty)\) such that
Proposition 2.7
Let \(F:[a,\infty)\rightarrow(0,\infty)\) be a measurable function. Then
To avoid repetitions we state here basic conditions imposed of the functions φ, ψ, p, q. In what follows we always assume
Using the notation (2.3), we can express \(\varphi (s)\), \(\psi(s)\), \(p(t)\), and \(q(t)\) as
By assumption (i), \(\varphi (s)\) is an increasing function, so \(\varphi (s)\) has the inverse function, denoted by \(\varphi^{-1}(s)\) and from (2.5) we conclude that
We also need the additional requirements for the slowly varying parts of φ and ψ:
It is easy to check that this is satisfied by e.g.
where \(\log_{k} t=\log\log_{k-1} t\), \(k=1,2,\ldots \) .
Remark 2.1
The condition (2.8) implies an useful property of the function \(\varphi^{-1}\). For \(u(t)\in \mathrm{SV}\cap C^{1}(\mathbb{R})\) and \(\lambda\in \mathbb{R}^{-}\), applying Proposition 2.2(iv), we have \(u(s^{\frac{1}{\lambda}})\in \mathrm{SV}_{0}\cap C^{1}(\mathbb{R})\). Using substitution \(t^{\lambda}=s\) (\(s\to0\) as \(t\to\infty \)) and (2.8) we obtain
from which it follows that
Similarly, the condition (2.9) implies an useful property of the function ψ:
3 Main results
This section is devoted to the study of the existence and asymptotic behavior of an intermediate regularly varying solutions of (E) with functions φ, ψ, p, q satisfying (2.4). We seek such solutions \(x(t)\) of (E) expressed in the form
Since \(\eta>0\), applying Proposition 2.2(v), we have \(\lim_{t\to\infty } p(t)=\infty\). Then, applying Proposition 2.2(iv), we get \(\varphi^{-1}(p(t)^{-1} )\in \operatorname{RV}_{0} (-\frac{\eta}{\alpha} ) \) so that the assumption \(\eta<\alpha\) ensures that we may apply Karamata’s integration theorem (Proposition 2.1) to the integral in (1.1). Using (2.6), (2.10), (2.7), and Proposition 2.1 we obtain
implying that \(P(t)\in \operatorname{RV}(1-\frac{\eta}{\alpha})\). Since \(\eta <\alpha\) by Proposition 2.2(v) we have \(\lim_{t\to\infty } P(t)=\infty\).
We emphasize that we exclude the case \(\eta=\alpha\) because of the computational difficulty and the fact that the integral
might be either convergent or divergent.
Since there are positive constants \(c_{1}\) and \(c_{2}\) such that \(c_{1}\leq x(t)\leq c_{2} P(t)\), for all large t, the regularity index ρ of \(x(t)\) must satisfy \(0\leq\rho\leq1-\frac{\eta}{\alpha}\). Therefore, the class of intermediate regularly varying solutions of (E) is divided into three types of subclasses:
To state our main results, we will need the function
which is clearly increasing on \((0,\infty)\). From (2.5), (3.3), and Proposition 2.1 we get
implying \(\Psi(y)\in \operatorname{RV}(\frac{\alpha-\beta}{\alpha})\) and \(\Psi ^{-1}(y)\in \operatorname{RV}(\frac{\alpha}{\alpha-\beta})\) with \(\frac{\alpha -\beta }{\alpha}>0\).
Theorem 3.1
Suppose that (2.4), (2.8), and (2.9) hold. Equation (E) possesses intermediate solutions \(x(t)\in \mbox{ntr-}\mathrm{SV}\) if and only if
in which case any such solution \(x(t)\) has the asymptotic behavior \(x(t)\sim X_{1}(t)\), \(t\to\infty\), where
Theorem 3.2
Suppose that (2.4), (2.8), and (2.9) hold. Equation (E) possesses intermediate solutions \(x(t)\in \operatorname{RV}(\rho)\) with \(\rho\in(0,1-\frac{\eta}{\alpha})\) if and only if
in which case ρ is given by
and any such solution \(x(t)\) has the asymptotic behavior \(x(t)\sim X_{2}(t)\), \(t\to\infty\), where
Theorem 3.3
Suppose that (2.4), (2.8), and (2.9) hold. Equation (E) possesses intermediate solutions \(x(t)\in \mbox{ntr-RV}(1-\frac {\eta}{\alpha})\) if and only if
in which case any such solution \(x(t)\) has the asymptotic behavior \(x(t)\sim X_{3}(t)\), \(t\to\infty\), where
4 Preparatory results
Let \(x(t)\) be an intermediate solution of (E) defined on \([t_{0},\infty )\). Since \(\lim_{t\to\infty }p(t)\varphi (x'(t))=\lim_{t\to\infty }x'(t)=0\), \(\lim_{t\to\infty }x(t)=\infty \), integrating of (E) first on \((t_{0},\infty)\) and then on \([t_{0},t]\) gives
It follows therefore that \(x(t)\) satisfies the integral asymptotic relation
for any \(b\geq a\), which is regarded as an ‘approximation’ of (4.1) at infinity. A common way of determining the desired intermediate solution of (E) would be by solving the integral equation (4.1) with the help of a fixed point technique. For this purpose the Schauder-Tychonoff fixed point theorem should be applied to the integral operator
acting on some closed convex subsets \(\mathcal{X}\) of \(C[t_{0},\infty)\), which should be chosen in such a way that \({\mathcal{F}}\) is a continuous self-map on \(\mathcal{X}\) and send it into a relatively compact subset of \(C[t_{0},\infty)\). That such choices of \(\mathcal{X}\) are feasible is guaranteed by the existence of three types of regularly varying functions that determine exactly the asymptotic behavior of all possible solutions of (4.2).
The purpose of this section is to collect preparatory results which will help us to simplify the proof of both ‘if’ and ‘only if’ parts of our main theorems. We begin by proving three results verifying that regularly varying functions \(X_{i}(t)\), \(i=1,2,3\) defined, respectively by (3.6), (3.9), and (3.11) satisfy the integral asymptotic relation (4.2).
Lemma 4.1
Suppose that (3.5) holds. The function \(X_{1}(t) \) given by (3.6) satisfies the asymptotic relation (4.2).
Proof
Let (3.5) hold. Since \(\eta<\alpha\), from (3.5) we have \(\sigma<-1\), so we can apply Proposition 2.1 to the integral
Using the above relation, (2.6), (2.10), and (2.7) we get
Since \(\sigma=\eta-\alpha-1\) we can rewrite (4.3) in the form
Application of Proposition 2.1(iii) to (4.4) gives
From (3.6) and (4.5), by Proposition 2.2(iv), we find that \(X_{1}(t)\in \mbox{ntr-}\mathrm{SV}\) and \(\psi(X_{1}(t))\in \mbox{ntr-}\mathrm{SV}\). We integrate \(q(t) \psi(X_{1}(t))\) on \([t,\infty)\). Applying Proposition 2.1 (which is possible since \(\sigma<-1\)) and using (2.6) we obtain
as \(t\to\infty \), from which it readily follows that
From the above relation, using Proposition 2.7, (2.10), and (2.7) we conclude
In view of (4.4), integrating (4.6) from \(t_{0}\) to t, we get
On the other hand, we rewrite (3.6) as
Since
differentiation of (4.8) gives
Integrating (4.9) on \([t_{0},t]\) and combining with (4.7) we obtain
This completes the proof of Lemma 4.1. □
Lemma 4.2
Suppose that (3.7) holds and let ρ be defined by (3.8). The function \(X_{2}(t) \) given by (3.9) satisfies the asymptotic relation (4.2).
Proof
Let (3.7) hold. Using (2.6) and (2.7) we rewrite (3.9) in the form
from which using (3.4) follows
Since \(\frac{\sigma+\alpha+1-\eta}{\alpha}>0\), by Proposition 2.2(v), we conclude that the function on the right-hand side of (4.10) tends to ∞ as \(t\to\infty \). From (4.10) using the previous conclusion and \(\Psi^{-1}\in \operatorname{RV}(\frac{\alpha}{\alpha-\beta})\) with application of Proposition 2.2(iv), we obtain \(X_{2}(t)\in \operatorname{RV}(\rho)\), with ρ given by (3.8). Thus, \(X_{2}(t)\) is expressed as \(X_{2}(t)=t^{\rho}l_{2}(t)\), \(l_{2}(t)\in \mathrm{SV}\). Then, using (4.11), we get
Since \(\sigma+\beta+1<\frac{\beta}{\alpha}\eta\), we have \(\alpha (\rho -1)+\eta<0\), implying that we can apply Proposition 2.1 on the last integral in (4.12) and then multiplying the result with \(p(t)^{-1}\) we obtain
from which, applying Proposition 2.7, it readily follows as \(t\to\infty \) that
where we use (2.7) and (2.10) in the two last steps. Integration on the above relation from \(t_{0}\) to t with application of Proposition 2.1 (which is possible since \(\rho>0\)) then yields
This completes the proof of Lemma 4.2. □
Lemma 4.3
Suppose that (3.10) holds. The function \(X_{3}(t) \) given by (3.11) satisfies the asymptotic relation (4.2).
Proof
Let (3.10) hold. Since \(\sigma=\frac{\beta}{\alpha}\eta-\beta-1\), using (2.5), (2.6), and (3.2), by Proposition 2.2 we get \(q(t) \psi(P(t))\in \operatorname{RV}(-1)\) so that \(\int_{t}^{\infty}q(s) \psi(P(s))\,ds\in \mathrm{SV}\) by Proposition 2.1(iii). In view of (3.2) and (3.11), we conclude that \(X_{3}(t)\in \mbox{ntr-RV}(1-\frac{\eta}{\alpha})\). Using (2.11) and (3.2) we have
This, combined with (3.11), gives the following expression for \(X_{3}(t)\):
Next, we integrate \(q(t)\psi(X_{3}(t))\) on \([t,\infty)\). Since \(X_{3}(t)=t^{1-\frac{\eta}{\alpha}}l_{3}(t)\), \(l_{3}(t)\in \mathrm{SV}\), due to (2.11), we obtain
Changing (4.14) in the last integral in (4.15), by a simple calculation we have
where we use (4.13) in the last step. Since \(\int_{t}^{\infty}q(s)\psi (X_{3}(s))\,ds\in \mathrm{SV}\), (2.6), (2.7), and (2.10) give
as \(t\to\infty \). Integrating (4.17) from \(t_{0}\) to t, we conclude via Proposition 2.1 that
as \(t\to\infty \). This, combined with (3.2) and (4.16), shows that \(X_{3}(t)\) satisfies the asymptotic relation (4.2). This completes the proof of Lemma 4.3. □
After the construction of intermediate solutions with the help of the Schauder-Tychonoff fixed point theorem, to finish the proof of the ‘if’ part of our main results we prove the regularity of those solutions using the generalized L’Hospital rule (see [25]).
Lemma 4.4
Let \(f,g\in C^{1}[T,\infty)\). Let
Then
If we replace (4.18) with condition
then the same conclusion holds.
5 Proof of main results
Proof of the ‘only if’ part of Theorems 3.1, 3.2, 3.3
Suppose that (E) has an intermediate solution \(x(t)\in \operatorname{RV}(\rho)\) with \(\rho\in[0,1-\frac{\eta}{\alpha}]\) defined on \([t_{0},\infty)\). Since \(\lim_{t\to\infty } p(t)\varphi (x'(t))=0\), integration of (E) on \((t,\infty)\) using (2.5), (2.6), and (3.1) gives
implying the convergence of the last integral in (5.1) i.e. implying that \(\sigma+\beta\rho\leq-1\). We distinguish the two cases:
Assume that (a) holds. Multiplying (5.1) with \(p(t)^{-1}\) we get
Clearly, \(\xi(t)\in \mathrm{SV}\) and \(\lim_{t\to\infty }\xi(t)=0\). From (5.2), using (2.6) and (2.10) we have
Integrating (5.3) from \(t_{0}\) to t and using (2.7) we get
From (5.4) we find via Karamata’s integration theorem that
Using (3.2) we rewrite (5.5) in the form
Assume that (b) holds. Applying Proposition 2.1 to the last integral in (5.1) we have
Multiplying (5.7) with \(p(t)^{-1}\) and using (2.6) we get
Using Proposition 2.7, (2.10), and (2.7) we have
as \(t\to\infty \). Integration of (5.9) on \([t_{0},t]\) leads to
as \(t\to\infty \). Since the above integral tends to infinity as \(t\to\infty \) (note that \(x(t)\to\infty\), \(t\to\infty \)), we consider the following two cases separately.
Assume that (b.1) holds. Applying Proposition 2.1 to the integral in (5.10), we get
Assume that (b.2) holds. Then (5.10) shows that \(x(t)\in \mathrm{SV}\), that is, \(\rho=0\), and hence \(\sigma=\eta-\alpha-1\). Since \(\sigma +\beta \rho+1=\eta-\alpha\), (5.10) reduced to
Let us now suppose that \(x(t)\) is an intermediate solution of (E) belonging to ntr-SV. From the above observation this is possible only when the case (b.2) holds, in which case \(\rho=0\), \(\sigma=\eta -\alpha -1\), and \(x(t)=l_{x}(t)\) must satisfy the asymptotic behavior (5.12). Denote the right-hand side of (5.12) by \(\mu(t)\). Then \(\mu (t)\to \infty\), \(t\to\infty \) and it satisfies
where we use (2.5) in the last step. Since (5.12) is equivalent to \(x(t)\sim\mu(t)\), \(t\to\infty \), from the above using (4.4) we obtain
An integration of the last relation over \([t_{0},t]\) gives
or
Thus, it has been shown that \(x(t)\sim X_{1}(t)\), \(t\to\infty \), where \(X_{1}(t)\) is given by (3.6). Notice that the verification of (3.5) is included in the above discussions. This proves the ‘only if’ part of Theorem 3.1.
Next, suppose that \(x(t)\) is an intermediate solution of (E) belonging to \(\operatorname{RV}(\rho)\), \(\rho\in(0,1-\frac{\eta}{\alpha})\). This is possible only when (b.1) holds, in which case \(x(t)\) must satisfy the asymptotic relation (5.11). Therefore,
which justifies (3.8). An elementary calculation shows that
which determines the range (3.7) of σ. Since \(\sigma+\beta\rho+1-\eta+\alpha=\alpha\rho\) and \(-(\sigma +\beta\rho +1)=\alpha(1-\rho)-\eta\), (5.11) reduced to
where we use (2.5), (2.6), (2.7), and (3.1) in the last step. From (5.13) using (3.4) we get
Thus, we conclude that \(x(t)\) enjoys the asymptotic formula \(x(t)\sim X_{2}(t)\), \(t\to\infty \), where \(X_{2}(t)\) is given by (3.9). This proves the ‘only if’ part of Theorem 3.2.
Finally, suppose that \(x(t)\) is an intermediate solution of (E) belonging to \(\mbox{ntr-RV}(1-\frac{\eta}{\alpha})\). Then the case (a) is the only possibility for \(x(t)\), which means that \(\rho=1-\frac{\eta }{\alpha}\), \(\sigma=\frac{\beta}{\alpha}\eta-\beta-1\), and (5.6) is satisfied by \(x(t)\). Differentiation of \(\xi(t)\), defined in (5.2), using (2.5), (2.6), and (3.1), leads to
Noting that \(x(t)\sim P(t)\xi(t)^{\frac{1}{\alpha}}\), \(t\to\infty \) and using (2.11), one can transform the above relation into
So, we get the differential asymptotic relation for \(\xi(t)\):
Due to fact that \(\alpha-\beta>0\) and \(\xi(t)\to0\) as \(t\to\infty \), the left-hand side of (5.14) can be integrated over \((t,\infty)\), assuring the integrability of \(q(t) \psi(P(t))\) on \((t,\infty)\), which implies the convergence of the integral in (3.10). Integration of (5.14) on \((t,\infty)\) yields
Combining (5.15) with (5.6) gives us \(x(t)\sim X_{3}(t)\), \(t\to\infty\), where \(X_{3}(t)\) is given by (3.11). This completes the ‘only if’ part of the proof of Theorem 3.3. □
Proof of the ‘if’ part of Theorems 3.1, 3.2, 3.3
Suppose that (3.5), (3.7) or (3.10) holds. From Lemmas 4.1, 4.2, and 4.3 it is well known that each \(X_{i}(t)\), \(i=1,2,3\), defined by (3.6), (3.9), and (3.11), satisfies the asymptotic relation (4.2) for any \(b\geq a\). We perform the simultaneous proof for \(X_{i}(t)\), \(i=1,2,3\) so the subscript \(i=1,2,3\) will be deleted in the rest of proof. By (4.2) there exists \(T_{0}>a \) such that
Let such a \(T_{0}\) be fixed. We may assume that \(X(t)\) is increasing on \([T_{0},\infty)\). Since (4.2) is satisfied with \(b=T_{0}\), there exists \(T>T_{0}\) such that
Applying Proposition 2.4 to the function \(\psi(s)\in \operatorname{RV}(\beta)\), \(\beta>0\) we see that there exists a constant \(A>1\) such that
Now we choose positive constants m and M such that
In addition, since \(X(t)\rightarrow\infty\) as \(t\to\infty \), from (2.1), for \(\lambda>0\) we have
Also, since \(Q(t)=1/p(t)\int_{t}^{\infty}q(s)\psi(X(s))\,ds\rightarrow0\) as \(t\to\infty \), from (2.2), for \(\lambda>0\) we have
Define the integral operator \(\mathcal{F}\) by
where \(x_{0}\) is constant such that
and let it act on the set
It is clear that \(\mathcal{X}\) is a closed convex subset of the locally convex space \(C[T_{0},\infty)\) equipped with the topology of uniform convergence on compact subintervals of \([T_{0},\infty)\).
Let \(x(t)\in\mathcal{X}\). Using first (5.18) and (5.24) and then (5.20) and (5.23) we get
from which, using (5.21), (5.16), and (5.19), it follows that
On the other hand, using (5.23) we have
and using (5.24), (5.18), and (5.20) we obtain
From the above using (5.21), (5.17), and (5.19) we conclude
This shows that \(\mathcal{F} x(t)\in\mathcal{X}\), that is, \(\mathcal{F}\) maps \(\mathcal{X}\) into itself.
Furthermore it can be verified (similarly to the proof of Theorem 3 in [5]) that \(\mathcal{F}\) is a continuous mapping and that \(\mathcal{F}(\mathcal{X})\) is relatively compact in \(C[T_{0},\infty)\).
Thus, all the hypotheses of the Schauder-Tychonoff fixed point theorem are fulfilled and so there exists a fixed point \(x(t)\in\mathcal{X} \) of \(\mathcal{F}\), which satisfies integral equation
Differentiating the above twice shows that \(x(t) \) is a solution of (E) on \([T_{0},\infty)\). It is clear from (5.24) that \(x(t) \) is an intermediate solution of (E).
Therefore, the existence of three types of intermediate solutions of (E) has been established. The proof of our main results will be completed with the verification that the intermediate solutions of (E) constructed above are actually regularly varying functions.
We define the function
and put
Since \(x(t)\in\mathcal{X}\), it is clear that \(0< l\leq L<\infty\). By Lemmas 4.1, 4.2, and 4.3 we have
Using Lemma 4.4 and (2.5) we see that
Since \(m\leq\frac{x(t)}{X(t)}\leq M\), \(t\geq T_{0}\), using the uniform convergence theorem ([22], Theorem 1.2.1) we conclude
From (5.26), using (5.27) and (5.25), we get
Similarly, we conclude that
We denote \(\hat{x}(t)=p(t)^{-1}\int_{t}^{\infty}q(s)\psi(x(s))\,ds \) and \(\hat{X}(t)=p(t)^{-1}\int_{t}^{\infty}q(s)\psi(X(s))\,ds \). Using Lemma 4.4 and (2.7) we obtain
From (5.28) and (5.29) we see that \(\frac{\hat {x}(t)}{\hat {X}(t)}\) is bounded. So, we can apply the uniform convergence again, identically to (5.27), to get
In view of (5.28) and (5.30) we have \(l \geq l^{\frac {\beta }{\alpha}}\), implying that \(l \geq1\) because \(\alpha> \beta\). If we argue similarly by taking the superior limits instead of the inferior limits, we are led to the inequality \(L \leq L^{\frac{\beta}{\alpha}}\), which implies that \(L \leq1\). Thus we conclude that \(l = L = 1\), i.e. \(\lim_{t \to\infty} {x(t)}/{J(t)} = 1 \). This combined with (5.25) shows that \(x(t) \sim X(t)\), \(t \to\infty\), which shows that \(x(t)\) is a regularly varying function whose regularity index ρ is 0, \(\frac{\sigma+\alpha+1-\eta}{\alpha-\beta}\), or \(1-\frac {\eta }{\alpha}\) according to whether \(\sigma= \eta-\alpha-1\), \(\eta-\alpha-1 < \sigma<\frac{\beta }{\alpha }\eta-\beta-1\), or \(\sigma= \frac{\beta}{\alpha}\eta-\beta-1\). □
6 Examples
Example 6.1
Consider the equation
where \(p(t)=t^{\frac{\alpha}{2}}(\log t)^{\alpha}\in \operatorname{RV}(\frac {\alpha }{2})\), \(\varphi (s)=s^{\alpha}\in \operatorname{RV}_{0}(\alpha)\), and \(\psi(s)=s^{\beta}\log s\in \operatorname{RV}(\beta)\), \(\alpha>\beta>0\). We have \(\eta=\frac{\alpha }{2}\in (0,\alpha)\), \(P(t)\sim2\sqrt{t} (\log t)^{-1} \), and the functions \(\varphi^{-1}(s)\) and \(\psi(s)\) satisfy the additional requirements (2.8) and (2.9), respectively.
(i) Suppose that
where \(r(t)\) is continuous function on \([a,\infty)\) such that \(\lim_{t\to\infty }r(t)=1\). Then \(q(t)\in \operatorname{RV}(-1-\frac{\alpha}{2})\), so that \(\sigma=\eta -\alpha-1\) and we see that
implying that (3.5) holds. Therefore, by Theorem 3.1 there exist nontrivial slowly varying solutions of (E), and any such solution \(x(t)\) has asymptotic behavior
In view of (3.4) we have
implying that \(x(t)\sim\sqrt{\log t}\), \(t\to\infty \). If in (6.1) instead of ∼ one has =, and in particular \(r(t)=1-\frac{1}{\log t}\), then (E) possesses an exact increasing nontrivial SV-solution \(x(t)= \sqrt{\log t}\) on \([e,\infty)\).
(ii) Suppose that
where \(r(t)\) is continuous function on \([a,\infty)\) such that \(\lim_{t\to\infty }r(t)=1\). It is clear that \(q(t)\) is regularly varying function of index
and that \(\rho=\frac{\sigma+\alpha+1-\eta}{\alpha-\beta }=\frac{1}{3}\). By Theorem 3.2 there exist regularly varying solutions of index ρ of (E) and any such solution \(x(t)\) has asymptotic behavior
In view of (3.4) we have
implying that
Observe that in (6.2) instead of ∼ one has = and
then \(x(t)=\sqrt[3]{t}(\log t)^{-1}\) on \([e^{6},\infty)\) is an exact increasing solution.
(iii) Suppose that
where \(r(t)\) is continuous function on \([a,\infty)\) such that \(\lim_{t\to\infty }r(t)=1\). Here, \(q(t)\in \operatorname{RV}(-1-\frac{\beta}{2})\). Therefore, \(\sigma=\frac {\beta }{\alpha}\eta-\beta-1\) and
from which it follows that
implying that (3.10) holds. Therefore, by Theorem 3.3 there exist nontrivial regularly varying solutions of index \(1-\frac {\eta}{\alpha}=\frac{1}{2}\) of (E) and any such solution \(x(t)\) has asymptotic behavior
If in (6.3) instead of ∼ one has = and in particular
then (E) possesses an exact increasing solution \(x(t)= \sqrt{t}(\log t)^{-2}\) on \([e^{8},\infty)\).
References
Avakumović, V: Sur l’équation différentielle de Thomas-Fermi. Acad. Serbe Sci., Publ. Inst. Math. 1, 101-113 (1947) (in French)
Marić, V, Tomić, M: Asymptotic properties of solutions of the equation \(y''=f(x)\phi(y)\). Math. Z. 149(3), 261-266 (1976)
Marić, V, Tomić, M: Regular variation and asymptotic properties of solutions of nonlinear differential equations. Publ. Inst. Math. (Belgr.) 21(35), 119-129 (1977)
Marić, V, Tomić, M: Asymptotics of solutions of a generalized Thomas-Fermi equation. J. Differ. Equ. 35(1), 36-44 (1980)
Elbert, A, Kusano, T: Oscillation and non-oscillation theorems for a class of second order quasilinear differential equations. Acta Math. Hung. 56(3-4), 325-336 (1990)
Kusano, T, Ogata, A, Usami, H: On the oscillation of solutions of second order quasilinear ordinary differential equations. Hiroshima Math. J. 23, 645-667 (1993)
Marić, V: Regular Variation and Differential Equations. Lecture Notes in Mathematics, vol. 1726. Springer, Berlin (2000)
Kusano, T, Manojlović, J: Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type. Comput. Math. Appl. 62, 551-565 (2011)
Jaroš, J, Kusano, T, Manojlović, J: Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation. Cent. Eur. J. Math. 11(12), 2215-2233 (2013)
Kusano, T, Manojlović, J, Milošević, J: Intermediate solutions of second order quasilinear differential equations in the framework of regular variation. Appl. Math. Comput. 219, 8178-8191 (2013)
Kusano, T, Manojlović, J, Marić, V: Increasing solutions of Thomas-Fermi type differential equations - the sublinear case. Bull. - Acad. Serbe Sci. Arts, Cl. Sci. Math. Nat., Sci. Math. CXLIII(36), 21-36 (2011)
Manojlović, J, Marić, V: An asymptotic analysis of positive solutions of Thomas-Fermi type differential equations - sublinear case. Mem. Differ. Equ. Math. Phys. 57, 75-94 (2012)
Kusano, T, Marić, V, Tanigawa, T: An asymptotic analysis of positive solutions of generalized Thomas-Fermi differential equations - the sub-half-linear case. Nonlinear Anal. 75, 2474-2485 (2012)
Jaroš, J, Kusano, T: Existence and precise asymptotic behavior of strongly monotone solutions of systems of nonlinear differential equations. Differ. Equ. Appl. 5, 185-204 (2013)
Jaroš, J, Kusano, T: Slowly varying solutions of a class of first order systems of nonlinear differential equations. Acta Math. Univ. Comen. 82, 265-284 (2013)
Řehák, P: On decreasing solutions of second order nearly linear differential equations. Bound. Value Probl. 2014, 62 (2014)
Jaroš, J, Kusano, T: On strongly monotone solutions of a class of cyclic systems of nonlinear differential equations. J. Math. Anal. Appl. 417, 996-1017 (2014)
Jaroš, J, Kusano, T: Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities. Opusc. Math. 35(1), 47-69 (2015)
Kusano, T, Manojlović, J: Precise asymptotic behavior of intermediate solutions of even order nonlinear differential equations in the framework of regular variation. Mosc. Math. J. 13(4), 649-666 (2013)
Matucci, S, Řehák, P: Asymptotics of decreasing solutions of coupled p-Laplacian systems in the framework of regular variation. Ann. Mat. Pura Appl. 193(3), 837-858 (2014)
Řehák, P: Asymptotic behavior of increasing solutions to a system of n nonlinear differential equations. Nonlinear Anal. 77, 45-58 (2013)
Bingham, NH, Goldie, CM, Teugels, JL: Regular Variation. Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1987)
Seneta, E: Regularly Varying Functions. Lecture Notes in Mathematics, vol. 508. Springer, Berlin (1976)
Djurčić, D: \(\mathcal{O}\)-Regularly varying functions and some asymptotic relations. Publ. Inst. Math. (Belgr.) 61(75), 44-52 (1997)
Haupt, O, Aumann, G: Differential- und Integralrechnung. de Gruyter, Berlin (1938)
Acknowledgements
The author wishes to thank to Professor Jelena Manojlović for her valuable suggestions and comments, which resulted in great improvement of this paper. The author thanks both referees for their careful reading of the manuscript and helpful comments.
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Milošević, J. Asymptotic behavior of increasing positive solutions of second order quasilinear ordinary differential equations in the framework of regular variation. Adv Differ Equ 2015, 273 (2015). https://doi.org/10.1186/s13662-015-0599-z
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DOI: https://doi.org/10.1186/s13662-015-0599-z