Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations

Accurate asymptotic formulas for regularly varying solutions of the second order half-linear differential equation (|x′|αsgn x′)′ + q(t)|x|sgn x = 0, will be established explicitly, depending on the rate of decay toward zero of the function Qc(t) = t ∫ ∞ t q(s)ds− c as t→ ∞, where c < αα(α + 1)−α−1.

In this paper we are concerned primarily with nontrivial solutions of (A) which exist in a neighborhood of infinity, that is, in an interval of the form [t 0 , ∞), t 0 ≥ a.Such a solution is said to be oscillatory if it has a sequence of zeros clustering at infinity, and nonoscillatory otherwise.
Although equation (A) with α = 1 is nonlinear, it has many qualitative properties in common with the linear differential equation x + q(t)x = 0. See Elbert [2] and Došly and Řehák [3].For example, all nontrivial solutions of (A) are either oscillatory, in which case (A) is called oscillatory, or else nonoscillatory, in which case (A) is called nonoscillatory.Also, it is shown that (A) is nonoscillatory if and only if the generalized Riccati differential equation has a solution defined in some neighborhood of infinity.
In what follows our attention will be focused on the case where (A) is nonoscillatory.Since if x(t) satisfies (A), so does −x(t), it is natural to restrict our consideration to (eventually) positive solutions of (A).
The systematic study of equations of the form (A) by means of regularly varying functions (in the sense of Karamata) was proposed by Jaroš, Kusano and Tanigawa [5], who proved the following theorem.

2)
Equation (A) possesses a pair of regularly varying solutions x i (t), i = 1, 2, such that if and only if Recently, Řehák [10] considering only special case of the equation (A) with nonpositive differentiable coefficient q(t) established a condition which guarantees that all eventually positive increasing solutions are regularly varying.
Theorem B. Let q be negative differentiable function and lim t→∞ q (t)|q(t)| α+1 α = C < 0 . (1.5) Then, all positive eventually increasing solutions x(t) of (A) are such that lim t→∞ x(t) = ∞ and belongs to RV −αρ , where ρ 1 is the positive real root of the equation Although the integral condition (1.4) is more general then (1.5), Theorem A guarantees the existence of at least one positive increasing RV-solution, while Theorem B says that all positive increasing solutions are regularly varying.
A natural question arises about the possibility of acquiring detailed information on the asymptotic behavior at infinity of the solutions whose existence is assured by the above two theorems.This problem has been partially examined in [7,11].Namely, in [7] the equation (A) has been considered in the framework of regular variation, but only the case c = 0 in (1.4) has been considered, providing some asymptotic formulas for normalized slowly varying solutions of (A), while in [11] considering only special case of the equation (A) with negative differentiable coefficient q(t), a condition is established which ensures that the equation (A) has exponentially increasing solutions and exponentially decreasing solutions, providing some asymptotic estimates for such solutions.
Therefore, the objective of this paper is to extend and improve results obtained in [7,11], by indicating assumptions that make it possible to determine the accurate asymptotic formulas for regularly varying solutions (1.3) of (A).This can be accomplished by elaborating the proof of Theorem A so as to gain insight into the interrelation between the asymptotic behavior of solutions of (A) and the rate of decay toward zero of the function as t → ∞.In Section 2 we present the elaborated proof of Theorem A, thereby adding useful information to the exponential representations for regularly varying solutions (1.3) of (A) constructed in the paper [5].Using the results of Section 2, we then specify in Section 3 some classes of equations of the form (A) having solutions (1.3) whose asymptotic behaviors are governed by the precise formulas.Examples illustrating the main results are provided in Section 4.
For the convenience of the reader the definition and some basic properties of regularly varying functions are summarized in the Appendix at the end of the paper.

Existence of regularly varying solutions
Let c be a constant satisfying (1.1) and let λ i , i = 1, 2, (λ 1 < λ 2 ) denote the real roots of the equation (1.2).It is clear that The purpose of this section is to prove variants of Theorem A ensuring the existence of regularly varying solutions , for equation (A), and utilize them for pointing out the cases where one can determine the asymptotic behavior of these solutions as t → ∞.
As in [5], the cases where c = 0 and c = 0 are examined separately.

The case where c = 0 in (1.2)
Let c = 0 in (1.2), so that its real roots are λ 1 = 0 and λ 2 = 1.Our task is to construct regularly varying solutions x i (t), i = 1, 2, of (A) such that x 1 ∈ SV = RV(0) and x 2 ∈ RV(1) under certain conditions on q(t) stronger than Our first result consists of the following two existence theorems indicative of how the asymptotic behavior of the SV-and RV(1)-solutions of (A) is affected by the decay property of Q(t) as t → ∞.Theorem 2.1.Suppose that there exists a continuous positive function φ(t) on [a, ∞) which decreases to 0 as t → ∞ and satisfies t α ∞ t q(s)ds ≤ φ(t) for all large t.Then, equation (A) possesses a slowly varying solution x 1 (t) which is expressed in the form for some T > a, with v Proof.We seek a solution x 1 (t) of (A) expressed in the form (2.2).For x 1 (t) to be a solution of (A), it is necessary that u(t) = (v 1 (t) + Q(t))/t α satisfies the Riccati-type equation (B) for t ≥ T. Further, if v 1 (t) tends to 0 as t → ∞, x 1 (t) would be slowly varying solution.An elementary computation shows that equation (B) for u(t) is transformed into the following differential equation for v 1 (t): the integrated version of which is With a help of fixed-point technique we show the existence of a solution of the integral equation (2.5).
Choose T ≥ a so that and then, using the decreasing nature of φ(t) and (2.6), we have ) we obtain This proves that F is a contraction mapping.It follows that F 1 has a unique fixed point v 1 (t) in V 1 , which clearly satisfies the integral equation (2.5), and hence the differential equation (2.4) on [T, ∞).From (2.7) it follows that v 1 (t) satisfies (2.3).Moreover, the function x 1 (t) defined by (2.2), with this v 1 (t), is a slowly varying solution of equation (A).This completes the proof of Theorem 2.1.Theorem 2.2.Suppose that there exists a continuous slowly varying function ψ(t) on [a, ∞) which decreases to 0 as t → ∞ and satisfies t α ∞ t q(s)ds ≤ ψ(t) for all large t.Then, equation (A) possesses a regulary varying solution x 2 (t) of index 1, which is expressed in the form for some T > a, with v 2 (t) (2.9) Proof.The desired solution x 2 ∈ RV(1) is sought in the form (2.8).From the requirement that u(t) = (1 + v 2 (t) + Q(t))/t α satisfy (B) we obtain the differential equation for v 2 (t) which is transformed as (2.10) It suffices to solve the special integrated version of (2.10) under the condition v 2 (t) → 0 as t → ∞, where (2.12) For this purpose we need detailed information about F(t, v).Let T 0 ≥ a be such that ψ(t) ≤ 1   4   for t ≥ T 0 , define D = {(t, v) : t ≥ T 0 , |v| ≤ 1 4 } and consider F(t, v) on the set D. It will be convenient to decompose F(t, v) as follows: where Using the mean value theorem, for some θ ∈ (0, 1) the following inequalities hold: , and we obtain that the following inequalities hold on D: where A is a positive constant such that We note that there exists a constant γ > 0 such that This follows from the relations which are implied by the Karamata integration theorem applied to slowly varying functions

The case where c = 0 in (1.2)
Let c be a nonzero number in the interval (−∞, E(α)) (cf.(1.1)).Then, the real roots and regardless of the sign of c.Our aim is to find regularly varying solutions under certain conditions on q(t) stronger than (1.4).Since λ 2 > 0, the asterisk sign may be The extreme case where Q c (t) ≡ 0 for all large t will be excluded from our consideration.Clearly, this case occurs only for the particular equation which, as easily checked, has exact two trivial RV-solutions The main results of this subsection are stated and proved as follows.
Theorem 2.3.Let c be a nonzero constant in (−∞, E(α)).Suppose that there exists a continuous positive function φ(t) on [a, ∞) which decreases to 0 as t → ∞ and satisfies t α ∞ t q(s)ds − c ≤ φ(t) for all large t.Then, equation (A) possesses a regularly varying solution x 1 ∈ RV(λ for some T > a, where v Using the notation we transform the above equation into where (2.26) We consider F 1 (t, v) on the set where T 1 > a is chosen so that ψ(t) ≤ min |λ 1 | 4 , 1 for t ≥ T 1 , and express it as where (2.28) By a similar procedure as in the proof of Theorem 2.2, using the mean value theorem the following inequalities are proved to hold in D 1 : where A 1 is a positive constant such that Let a constant l ∈ (0, 1) be given and let T > T 1 be large enough so that Define the set V 1 and the integral operator F 1 by and if v, w ∈ V 1 , then using (2.30) and (2.33), we see that This completes the proof of Theorem 2.3.Theorem 2.4.Let c be a nonzero constant in (−∞, E(α)).Suppose that there exists a continuous slowly varying function ψ(t) on [a, ∞) which tends to 0 as t → ∞ and satisfies t α ∞ t q(s)ds − c ≤ ψ(t) for all large t.Then, equation (A) possesses a regulary varying solution x 2 ∈ RV λ for some T > a, where v 2 (t) in the class of continuously differentiable functions tending to 0 as t → ∞.Exactly as in the proof of Theorem 2.3 this equation is transformed into where and (2.39) Here the variable of F 2 (t, v) is restricted to the domain where Noting that the constant µ 2 in (2.38) satisfies µ 2 > α because of (2.19), we form the following integrated version of (2.37) and solve it in the space C 0 [T, ∞) for some suitably chosen T > a.For this purpose use is made of the fact that there exists a constant γ > 0 such that This is an immediate consequence of the Karamata integration theorem applied to t µ 2 −α−1 f (t) for any f ∈ SV.
In order to solve the integral equation (2.40) it is convenient to use the decomposition of F 2 (t, v) corresponding precisely to (2.27) Let a constant l ∈ (0, 1) be given and choose T > T 2 so that Consider the integral operator and the set Using the estimates corresponding to (2.29)-(2.31) in combination with (2.41) and (2.43), we can show that if v ∈ V 2 , then 2 of (A).This completes the proof of Theorem 2.4.

Asymptotic behavior of regularly varying solutions
It is natural to ask whether one can accurately determine the asymptotic behavior at infinity of the regularly varying solutions of equation (A) whose existence was established in the above four theorems.An answer to this question is provided in this section by way of the exponential representations for the solutions which, in some cases, make it possible to reveal the effect of the functions Q(t) or Q c (t) upon the behavior of the solutions under study.We begin by indicating the situation in which the asymptotic behavior of the SV-and RV(1)-solutions of (A) described in Theorems 2.1 and 2.2 can be determined precisely.
Throughout the text "t ≥ T" means that t is sufficiently large, so that T need not to be the same at each occurrence.Theorem 3.1.Let φ(t) be a positive continuous function on [a, ∞) which decreases to 0 as t → ∞ and satisfies ∞ a φ(t) Suppose that the function Q(t) defined by (2.1) is eventually of one-signed and satisfies Then, equation (A) possesses a nontrivial slowly varying solution x 1 (t) such that for some constant c > 0. Proof where Q = sgn Q, using (2.3) we see that Then, equation (A) possesses a nontrivial regularly varying solution x 2 (t) of index 1 such that for some constant c > 0.
Proof.Because of (3.6) there is a constant κ ≥ 1 such that |Q(t)| ≤ κψ(t) for all large t, and so applying Theorem 2.2 (with ψ(t) replaced by κψ(t)), we see that (A) has an RV(1)solution x 2 (t) expressed in the form (2.8), where v 2 (t) satisfies the decay condition (2.9) and the integral equation (2.11), with F(t, v) being given by (2.12).Suppose that Q(t) defined by (2.1) is one-signed on [T, ∞) for some T > a.
For more information about the decay of v 2 (t) we are going to use the decomposition (2.13) of F(t, v) and the estimates for G(t, v), H(t, v) and k(t) obtained in (2.14), which we may assume holding on [T, ∞).First, note that (2.9) and (2.14) implies while denoting by Q = sgn Q and rewriting (3.6) as Using (3.8) and (3.9) in (2.11) and taking into account the relation which follows from the Karamata integration theorem, we obtain This, combined with On the other hand it is clear that v 2 (t) + Q(t) = O(ψ(t) 2 ) as t → ∞.Bringing the above observations together, we find (3.10) We now combine (2.8) with (3.10) to obtain for t ≥ T Therefore, implying from (3.11) the desired asymptotic formula (3.7) for x(t).
Our next task is to establish the accurate asymptotic formulas for the regularly varying solutions of (A) constructed in Theorems 2.3 and 2.4.The non-zero constant c satisfying (1.1), the function Q c (t) defined by (1.6), the real roots λ i , i = 1, 2, of (1.2) satisfying (2.19) and the constants µ i , i = 1, 2, given by (2.23) and (2.38) will be used below.Theorem 3.3.Let φ(t) be a positive continuously differentiable function on [a, ∞) which decreases to 0 as t → ∞, has the property that t|φ (t)| is decreasing and satisfies Suppose that the function Q c (t) defined by (1.6) is eventually one-signed and satisfies Then, equation (A) possesses a nontrivial regularly varying varying solution x 1 (t) of index λ for some constant c > 0.
Proof.Suppose that the function Q c (t) defined by (1.6) is one-signed on [T, ∞), for some T > a, so that we may rewrite (3.13) as where Q c = sgn Q c .Since (3.13) implies the existence of a constant κ ≥ 1 such that |Q c (t)| ≤ κφ(t) for all large t, by Theorem 2.3 (with φ(t) replaced by κφ(t)) there exists an RV λ 1 α * 1solution x 1 (t) of (A) which is expressed as (2.20), where v 1 (t) is a solution of the integral equation (2.26) satisfying (2.21) with F 1 (t, v) defined by (2.25).As in the proof of Theorem 2.3 we express F 1 (t, v) as in (2.27) and utilize estimates presented in (2.29), which without lost of generality is assumed to be valid on [T, ∞).By combining (2.29) with (3.15) we obtain Thus, Using (3.16) and (3.18) in (2.26) we obtain from which, via integration by parts, it follows that where Combining (3.15) and (3.19) we obtain which due to (3.17) gives Therefore, the representation formula (2.20) for x 1 (t) becomes Since O(φ(t) 2 /t) is integrable on [T, ∞) by (3.12) as well as O(J(t)/t) because the desired asymptotic formula (3.14) for follows from (3.20).This completes the proof of Theorem 3.3.Theorem 3.4.Let φ(t) be a positive continuously differentiable slowly varying function on [a, ∞) which decreases to 0 as t → ∞, has the property that t|ψ (t)| is slowly varying and satisfies Suppose that the function Q c (t) defined by (1.6) is eventually one-signed and satisfies Then, equation (A) possesses a nontrivial regularly varying solution x 2 (t) of index λ for some constant c > 0.
Proof.Suppose that the function Q c (t) Moreover, as regards where Q c = sgn Q c .Using (3.24) and (3.25) in (2.40), we obtain where Our final step is to show that (3.26) is crucial in determining the asymptotic behavior of x 2 (t) given by (3.23).Employing (3.26), we find that which, substituted for (2.35), shows that x 2 (t) is expressed as which is a consequence of the Karamata integration theorem.Therefore, 2 ).The proof of Theorem 3.4 has thus been completed.

Examples and concluding remarks
We now present some examples illustrating our main results and showing that our results extend and improve results obtained in [7,11].
on [1, ∞), where θ ∈ (0, 1 2 ), A and B are constants.Put An easy calculation shows that , for all large t ≥ 1, and so integrating the above from t to ∞ and multiplying with t α , we see that implying that Q(t) is eventually positive.Moreover, because of θ ∈ (0, 1 2 ), it holds that ∞ φ(s) Thus all the hypotheses of Theorem 3.1 are fulfilled for equation (E 1 ), and so there exists a nontrivial SV-solution x 1 (t) of (E 1 ) having the precise asymptotic behavior for some c > 0. Note that if in particular A = −θ and B = 1 − θ, (E 1 ) has an exact SV-solution x(t) = exp((log t) θ ).
It should be noticed that if α ≤ 1, then (4.1) implies 2 is satisfied with ψ(t) = φ(t).Thus, Theorem 3.2 is applicable to (E 1 ) and ensures the existence of its nontrivial RV(1)-solution x 2 (t) with the precise asymptotic behavior for some constant c 2 > 0.
Example 4.2.Consider the half-linear equation on [1, ∞), where A ≥ 0, B ≥ 0 and γ are constants.Putting ψ(t) = 1/ log t, it is shown that for all large t, from which we see that and Q(t) is eventually negative.Since ψ(t) satisfies applying Theorem 3.2 to equation (E 2 ), we conclude that (E 2 ) possesses a nontrivial RV(1)solution x 2 (t) with the precise asymptotic behavior Note that if in particular A = 1, B = 0 and γ = α − 1, then (E 2 ) has an exact RV(1)-solution for some constant c 1 > 0.
Example 4.3.Consider the half-linear equation on [1, ∞), where A is a constant.As is easily checked, q 3 (t) satisfies for all large t, which implies that the hypotheses of Theorems 3.3 and 3.4 are fulfilled with the choice while Q c (t) is eventually negative.We need to find the two real roots of the equation λ for some constant c 1 > 0. If in particular A = 1, then (E 3 ) has an exact RV(-1)-solution x(t) = 1/t log t.
often said to be in the border case.Since in this case the equation |λ| 1+ 1 α − λ + E(α) = 0 has the only one real root ( α α+1 ) α , the regularity index of regularly varying solutions of (A), if exists, must be equal to α α+1 .In [5] a sufficient condition is presented for equation (A) in the border case possesses a trivial regularly varying solution of index α α+1 .It would be of interest to answer the question: Is it possible to find conditions under which equation (A) in the border case possesses nontrivial RV( α α+1 )-solutions and to determine their precise asymptotic behavior as t → ∞?
(2) In the paper [6] an attempt is made to generalize the results for (A) obtained in [5] to the half-linear differential equations of the form (p(t)|x | α sgn x ) + q(t)|x| α sgn x = 0, (4.4) where α > 0 is a constant and p(t) > 0, q(t) are continuous functions on [a, ∞).Naturally the qualitative properties of positive solutions of (4.4) depend heavily on the coefficient p(t).
In order to precisely describe the effect of the function p(t) upon the behavior of positive solutions of (4.4) the authors of [6] used the class of generalized Karamata functions, introduced in [4], as the framework for the asymptotic analysis of (4.4), and demonstrate how to build in the new framework the existence theory of generalized Karamata solutions for (4.4) which extends the results on regularly varying solutions of (A) developed in [5].It is expected that one can possibly indicate a class of equations of the form (4.4) possessing generalized Karamata solutions whose asymptotic behaviors at infinity are determined accurately and explicitly.The set of all regularly varying functions of index ρ is denoted by RV(ρ).The symbol SV is often used to denote RV(0), in which case members of SV are called slowly varying functions.Since any function f (t) ∈ RV(ρ) is expressed as f (t) = t ρ g(t) with g(t) ∈ SV, the class SV of slowly varying functions is of fundamental importance in the theory of regular variation.Here in defining m(t) it is assumed that L(t)/t is integrable on a neighborhood of infinity.

Appendix: Regularly varying functions
For the almost complete exposition of theory of regular variation and its applications the reader is referred to the treatise of Bingham et al. [1].See also Seneta [12].A comprehensive survey of results up to the year 2000 on the asymptotic analysis of second order ordinary differential equations by means of regular variation can be found in the monograph of Marić [9].

Example 4 . 1 .
Consider the half-linear differential equation
.25) By(2.19) and (2.23), we see that µ 1 < α, so that it is natural to integrate (2.24) on [t, ∞) to obtain the integral equation .36) Proof.Note that the function x 2 (t) defined by (2.35) is a regularly varying solution of inof (A) if v 2 (t) tends to 0 as t → ∞ and has the property that the function u(t) = (λ 2 + v 2 (t) + Q c (t))/t α satisfies the equation (B) for all large t.The existence of such a v 2 (t) is equivalent to the solvability of the differential equation 2 .42) where G 2 , H 2 and k 2 stand, respectively, for G 1 , H 1 and k 1 in (2.28) with λ 1 replaced with λ 2 .Naturally, as regards G 2 , H 2 and k 2 exactly the same type of estimates as (2.29)-(2.31)hold true in D 2 provided λ 1 in (2.32) is replaced by λ 2 .
This confirms that F 2 is a contraction on V 2 , and consequently F 2 has a fixed point v 2 (t) ∈ V 2 which solves the integral equation (2.40).The property (2.36) of v 2 (t) follows from (2.44).The function x 2 (t) defined by (2.35) with this v 2 (t) (t, v) defined by(2.39).As in the proof of Theorem 2.4 we express F 2 (t, v) as in(2.42)whereG 2 , H 2 and k 2 stand, respectively, for G 1 , H 1 and k 1 in (2.28) with λ 1 replaced with λ 2 .As regards G 2 and H 2 exactly the same type of estimates as (2.29) hold for all large t provided λ 1 is replaced by λ 2 , which together with (2.36) gives defined by(1.6) is one-signed on [T, ∞) for some T > a.By (3.22) there is a constant κ ≥ 1 such that |Q c (t)| ≤ κψ(t) for all large t.Consequently, Theorem 2.4 ensures that (A) has a solution x ∈ RV(λ Definition 4.6.If f ∈ RV(ρ) has the property that lim t→∞ f (t) t ρ = const > 0, then it is called a trivial regularly varying function of index ρ and is denoted by f ∈ tr-RV(ρ).Otherwise f (t) is called a nontrivial regularly varying solution of index ρ and is denoted by f ∈ ntr-RV(ρ).One of the most important properties of regularly varying functions is the following representation theorem.Proposition 4.7.f(t)∈ RV(ρ) if and only if f (t) is represented in the form f(t) = c(t) expThe following result called Karamata's integration theorem is of highest importance in handling slowly and regularly varying functions analytically.Proposition 4.12.Let L(t) ∈ SV.Then, (i) if α > −1,