Asymptotic Behavior of Positive Solutions of a Class of Systems of Second Order Nonlinear Differential Equations

The two-dimensional system of nonlinear differential equations (A) x′′ = p(t)y, y′′ = q(t)x , with positive exponents α and β satisfying αβ < 1 is analyzed in the framework of regular variation. Under the assumption that p(t) and q(t) are nearly regularly varying it is shown that system (A) may possess three types of positive solutions (x(t), y(t)) which are strongly monotone in the sense that (i) both components are strongly decreasing, (ii) both components are strongly increasing, and (iii) one of the components is strongly decreasing, while the other is strongly increasing. The solutions in question are sought in the three classes of nearly regularly varying functions of positive or negative indices. It is also shown that if we make a stronger assumption that p(t) and q(t) are regularly varying, then the solutions from the above three classes are fully regularly varying functions, too.


Introduction
This paper is concerned with positive solutions of the two-dimensional system of second order nonlinear differential equations (A) x ′′ = p(t)y α , y ′′ = q(t)x β , where (a) α and β are positive constants; (b) p(t) and q(t) are positive continuous functions on [t 0 , ∞), t 0 > 0. By a positive solution of (A) we mean a vector function (x(t), y(t)) on an interval of the form [T, ∞), T ≥ t 0 , with positive components x(t) and y(t) satisfying system (A) for t ≥ T .Our aim is to acquire as precise information as possible about the asymptotic behavior of positive solutions of (A).Let (x(t), y(t)) be a positive solution of (A) existing on [T, ∞).Then, we see from (A) that x ′′ (t) > 0 and y ′′ (t) > 0, so that x ′ (t) and y ′ (t) are increasing for t ≥ T and tend to finite or infinite limits as t → ∞.If x ′ (t) is eventually positive, then either lim t→∞ x ′ (t) = const > 0 or lim t→∞ x ′ (t) = ∞, in which case x(t) satisfies (I) lim t→∞ x(t) t = ∞ or (II) lim t→∞ x(t) t = const > 0 respectively, while if x ′ (t) is eventually negative, then lim t→∞ x ′ (t) = 0, in which case x(t) satisfies (III) lim t→∞ x(t) = const > 0 or (IV) lim t→∞ x(t) = 0, respectively.Naturally the same is true of the asymptotic behavior of the component y(t).
A function x(t) (or y(t)) is referred to as primitive if it is of type (II) or (III), and as non-primitive it it is of type (I) or (IV).
Positive solutions of (A) may exhibit a variety of asymptotic behavior at infinity depending on which of the four cases (I), (II), (III) and (IV) holds for each of their components.In view of the symmetry of x(t) and y(t) there are ten different types of asymptotic behavior beginning with type (I,I) and ending with type (IV,IV), of which the three types (II,II), (II,III) and (III,III) are special in the sense that the existence of solutions of these types for (A) can be completely characterized without difficulty.(iii) System (A) has solutions (x(t), y(t)) such that if and only if To prove each of these statements it suffices to construct a suitable system of integral equations from (A) and solve it routinely by means of the Schauder-Tychonoff fixed point theorem.The proof may be omitted.
Of the remaining types of solutions of (A) which seem to be difficult to deal with in the case of general positive continuous p(t) and q(t) we take up the extreme three types, i.e., (I,I), (IV,IV) and (I,IV) types of solutions and show that, if analyzed in the framework of regular variation, it is possible to indicate the situation in which system (A) EJQTDE, 2013 No. 23, p. 2 possesses solutions of these types having accurate order of growth or decay as t → ∞.More specifically, the exponents α and β in (A) are restricted to the case αβ < 1, the coefficients p(t) and q(t) in (A) are assumed to be nearly regularly varying, and the above-mentioned types of solutions are sought in the classes of nearly regularly varying solutions of suitable but definite indices, positive or negative.
The present work was motivated by the recent progress of the asymptotic analysis of positive solutions of nonlinear differential equations by means of regular variation which was triggered by the publication of Marić's book [8]; see, for example, the papers [3 -7].A prototype of existence results we are going to prove here is Theorem 8 from [6] concerning the fourth order sublinear differential equation of the Thomas-Fermi type (1.7) x (4) = q(t)|x| β sgn x, 0 < β < 1, (equivalent to a special case of (A) where p(t) ≡ 1 and α = 1) which states that if and ρ is given by for some constants T ≥ t 0 and a, A > 0. Our purpose here is to generalize the above result to nonlinear systems of the form (A) with "nearly regularly varying" coefficients p(t) and q(t) and to show that the solutions of (A) satisfying (1.8) are actually fully regularly varying if it is assumed that both p(t) and q(t) are regularly varying functions.

Regularly varying functions
For the reader's convenience we recall here the definition of regularly varying functions, basic terminologies and notations, and Karamata's integration theorem which will play a central role in establishing the main results of this paper.
or equivalently it is expressed in the form for some t 0 > 0 and some measurable functions c(t) and δ(t) such that The totality of regularly varying functions of index ρ is denoted by RV(ρ).We often use the symbol SV instead of RV(0) and call members of SV slowly varying functions.By definition any function f (t) ∈ RV(ρ) is written as f (t) = t ρ g(t) with g(t) ∈ SV.So, the class SV of slowly varying functions is of fundamental importance in theory of regular variation.Typical examples of slowly varying functions are: all functions tending to positive constants as t → ∞, N n=1 (log n t) αn , α n ∈ R, and exp N n=1 (log n t) βn , β n ∈ (0, 1), where log n t denotes the n-th iteration of the logarithm.It is known that the function Otherwise f (t) is called a nontrivial regularly varying function of index ρ.The symbol tr-RV(ρ) (or ntr-RV(ρ)) is used to denote the set of all trivial RV(ρ)-functions (or the set of all nontrivial RV(ρ)-functions).
The following proposition, known as Karamata's integration theorem, is particularly useful in handling slowly and regularly varying functions analytically and is extensively used throughout the paper.
Proposition 2.1.Let L(t) ∈ SV.Then, provided L(s)/s is integrable near the infinity in the latter case.
A measurable function f : (0, ∞) → (0, ∞) is called regularly bounded if for any λ 0 > 1 there exist positive constants m and M such that We now define the class of nearly regularly varying functions which is a useful subclass of RO including all regularly varying functions.To this end it is convenient to introduce the following Notation 2.1.Let f (t) and g(t) be two positive continuous functions defined in a neighborhood of infinity, say for t ≥ T .We use the notation f (t) ≍ g(t), t → ∞, to denote that there exist positive constants m and M such that , t → ∞, and if lim t→∞ g(t) = 0, then lim t→∞ f (t) = 0. Definition 2.2.If positive continuous function f (t) satisfies f (t) ≍ g(t), t → ∞, for some g(t) which is regularly varying of index ρ, then f (t) is called a nearly regularly varying function of index ρ.
Since 2 + sin t ≍ 2 + sin(log n t), t → ∞, for all n ≥ 2, the function 2 + sin t is nearly slowly varying, and the same is true of 2 + sin(log t).If g(t) ∈ RV(ρ), then the functions (2+sin t)g(t) and (2+sin(log t))g(t) are nearly regularly varying of index ρ, but not regularly varying of index ρ.
The reader is referred to Bingham et al [1] for the most complete exposition of theory of regular variation and its applications and to Marić [8] for the comprehensive survey of results up to 2000 on the asymptotic analysis of second order linear and nonlinear ordinary differential equations in the framework of regular variation.

Strongly decreasing solutions of (A)
We begin with the problem of existence of strongly decreasing solutions of system (A), by which we mean positive solutions of type (IV,IV), that is, those solutions (x(t), y(t)) such that Let (x(t), y(t)) be such a solution of (A) on [T, ∞), T > t 0 .Since x ′ (t) and y ′ (t) tend to 0 as t → ∞, integrating two equations in (A) twice from t to ∞, we have Our aim is to obtain strongly decreasing solutions of (A) as solutions of the system of integral equations (3.2) by means of fixed point techniques.For this purpose essential use is made of the fact that regularly varying solutions of the system of asymptotic relations EJQTDE, 2013 No. 23, p. 5 which is regarded as an approximation at infinity of (A), can be completely characterized provided αβ < 1 and p(t) and q(t) are regularly varying.Let x(t) = t ρ ξ(t), ξ(t) ∈ SV (resp.y(t) = t σ η(t), η(t) ∈ SV) satisfy (3.1).Then, it is clear that either ρ < 0, or ρ = 0 and ξ(t) → 0 as t → ∞ (resp.either σ < 0, or σ = 0 and η(t) → 0 as t → ∞).In this section we focus only on the case where ρ < 0 and σ < 0, leaving the remaining two possibilities to our later investigations.

Integral asymptotic relations
Lemma 3.1.Let αβ < 1 and suppose that p(t) ∈ RV(λ) and q(t) ∈ RV(µ) are expressed as System (3.3) has regularly varying solutions of index (ρ, σ) with ρ < 0 and σ < 0 if and only if (λ, µ) satisfies the system of inequalities in which case ρ and σ are given by and the asymptotic behavior of any such solution (x(t), y(t)) is governed by the formula PROOF.(The "only if" part) Suppose that (3.3) has a regularly varying solution (x(t), y(t)) of negative index (ρ, σ) which is expressed in the form Since the functions p(t)y(t) α = t λ+ασ l(t)η(t) α and q(t)x(t) β = t µ+βρ m(t)ξ(t) β are integrable twice on [T, ∞) we see that λ + ασ ≤ −2 and µ + βρ ≤ −2, in which case we have by Karamata's integration theorem ((ii) of Proposition 2.1) and hence This excludes the possibility λ + ασ = −2.In fact, if this equality would hold, then (3.9) and the first relation of (3.3) would imply that which shows that the regularity index of x(t) is zero: ρ = 0, an impossibility.Thus, we must have λ+ασ < −2.In this case, applying (ii) of Proposition 2.1 to (3.9) and combining the result with the first relation in (3.3), we obtain the asymptotic equivalence , t → ∞.
The same argument applies to the second relation in (3.3) and leads to the conclusion that µ + βρ < −2 holds and y(t) satisfies the asymptotic relation , t → ∞.
(The "if" part) Suppose that (λ, µ) satisfies (3.5) and define (ρ, σ) by (3.6).We define (X(t), Y (t)) by (3.12) which can be rewritten as It suffices to prove that Using Karamata's integration theorem, we compute as follows: and hence Similarly we obtain This ensures the truth of (3.13).This completes the proof of Lemma 3.1.

Strongly decreasing solutions of (A)
We now consider system (A) with nearly regularly varying p(t) and q(t) and show that strongly decreasing solutions of (A) can be found in the class of nearly regularly varying functions of negative indices.Theorem 3.1.Let αβ < 1 and let p(t) and q(t) be nearly regularly varying of index λ and µ, respectively, such that Suppose that λ and µ satisfy (3.5) and define ρ and σ by (3.6).Then, system (A) possesses nearly regularly varying solutions (x(t), y(t)) of index (ρ, σ) with the property that where ∆(τ ) = τ (τ − 1) for τ < 0.
PROOF.Let p λ (t) and q µ (t) denote the functions By hypothesis there exist positive constants k, l, K and L such that Define the function (X λ (t), Y µ (t)) by the formula (3.12) with p(t) and q(t) replaced by p λ (t) and q µ (t), respectively.Since by Lemma 3.1 (X λ (t), Y µ (t)) satisfies the asymptotic relation (3.3), i.e. (3.13), there exists T > t 0 such that for t ≥ T It is elementary to see that such a choice of (a, b), (A, B) is really possible.For example, one may choose as follows: We define X to be the subset of where F and G denote the integral operators It can be checked that Φ fulfils the hypotheses of the Schauder-Tychonoff fixed point theorem.
(iii) Φ is continuous.Let {(x n (t), y n (t))} be a sequence in X converging to (x(t), y(t)) ∈ X uniformly on any compact subinterval of [T, ∞).We need to prove that Φ(x n (t), y n (t)) → Φ(x(t), y(t)), that is, uniformly on compact subintervals of [T, ∞).But this follows immediately from the Lebesgue dominated convergence theorem applied to the integrals in the following inequalities holding for t ≥ T Therefore, the Schauder-Tychonoff fixed point theorem ensures the existence of a function (x(t), y(t)) ∈ X such that Φ(x(t), y(t)) = (x(t), y(t)), t ≥ T , that is, for t ≥ T .It follows that (x(t), y(t)) gives a strongly decreasing solution of system (A).The membership (x(t), y(t)) ∈ X implies that (x(t), y(t)) is a nearly regularly varying of negative index (ρ, σ).This completes the proof of Theorem 3.1.
As for the solutions constructed in Theorem 3.1, their regularity can be characterized completely under the stronger assumption that p(t) and q(t) are regularly varying functions.
The generalized L'Hospital's rule given in the following lemma (see [2]) plays a crucial role in the proof of this theorem.Then, Theorem 3.2.Suppose that p(t) and q(t) are regularly varying of indices λ and µ, respectively.System (A) possesses regularly varying solutions (x(t), y(t)) such that x(t) ∈ RV(ρ), y(t) ∈ RV(σ), ρ < 0, σ < 0 if and only if (3.5) holds, in which case ρ and σ are given by (3.6) and the asymptotic behavior of any such solution (x(t), y(t)) is governed by the formulas EJQTDE, 2013 No. 23, p. 10 (3.23) PROOF OF THEOREM 3.2.(The "only if" part) It follows from Lemma 3.1.(The "if" part) Suppose that (3.5) holds and define the negative constants ρ and σ by (3.6).By Theorem 3.1 system (A) has a nearly regularly varying solution (x(t), y(t)) on [T, ∞) such that for some positive constants T, a, A, b and B, where It is clear that (x(t), y(t)) satisfies Let U(t) and V (t) denote the functions defined by Note that U(t) and V (t) satisfy the asymptotic relations From (3.24) and (3.26) we see that 0 < k ≤ K < ∞ and 0 < l ≤ L < ∞.Applying the generalized L'Hospital rule twice, we obtain where (3.28) has been used in the final step of each of the above computations.Since αβ < 1, the inequalities k ≥ l α and l ≥ k β thus obtained imply Similarly, we obtain K ≤ L α and L ≤ K β , from which it follows that From (3.30) and (3.31) we conclude that k = K = 1 and l = L = 1, that is, which combined with (3.28) shows that This completes the proof.

If in particular
then this system has an exact regularly varying solution ((t log t) −1 , t −2 log t).
EJQTDE, 2013 No. 23, p. 12 4 Strongly increasing solutions of (A) We turn our attention to strongly increasing solutions of system (A), by which we mean positive solutions of type (I,I), that is, those solutions (x(t), y(t)) such that (4.1) lim Let (x(t), y(t)) be one such solution of (A) on [T, ∞).Note that x ′ (t) and y ′ (t) tend to infinity as t → ∞.Integrating (A) twice on [T, t] gives (4.2) for t ≥ T , where x 0 = x(T ), x 1 = x ′ (T ), y 0 = y(T ) and y 1 = y ′ (T ).The wanted strongly increasing solutions of (A) are obtained by solving the system of integral equations (4.2) with the help of the Schauder-Tychonoff fixed point theorem.For this purpose an essential role is played by some of the basic properties of regularly varying solutions satisfying (4.1) and the system of integral asymptotic relations in which αβ < 1 and p(t) and q(t) are regularly varying.Let (x(t), y(t)) be a regularly varying solution of (4.3) which is expressed as Then, it is easy to see that (4.1) holds for (x(t), y(t)) if ρ > 1, or if ρ = 1 and ξ(t) → ∞ as t → ∞ on the one hand, and if σ > 1, or if σ = 1 and η(t) → ∞ as t → ∞ on the other.In this section we consider only the case where ρ > 1 and σ > 1, leaving the other possibilities to a later occasion because of computational difficulty.

Integral asymptotic relations
This subsection is concerned with the asymptotic system (4.3) which is regarded as an approximation of the system of integral equations (4.2).As a result of the analysis of (4.3) by means of regular variation full knowledge can be acquired of its regularly varying solutions satisfying (4.1) as the following lemma shows.Lemma 4.1.Let αβ < 1 and suppose that p(t) ∈ RV(λ) and q(t) ∈ RV(µ) are expressed in the form (4.4).System (4.3) has regularly varying solutions of index (ρ, σ) with ρ > 1 and σ > 1 if and only if (λ, µ) satisfies the system of inequalities in which case ρ and σ are given by (3.6), and the asymptotic behavior of any such solution (x(t), y(t)) is governed by the formula , t → ∞.
This shows that x(t) and y(t) are regularly varying of indices λ+ασ+2 > 1 and µ+βρ+2 > 1, respectively, and so we must have from which it readily follows that ρ and σ are given by (3.6).Since ρ > 1 and σ > 1, (3.6) determines the range of (λ, µ) to be the subset of R 2 defined by the inequalities Noting that (4.9) can be rewritten as we easily conclude that x(t) and y(t) satisfy EJQTDE, 2013 No. 23, p. 14 from which the asymptotic formula (4.6) immediately follows.

Strongly increasing solutions of (A)
It is shown that if p(t) and q(t) are nearly regularly varying, then strongly increasing solutions of system (A) can be found in the class of nearly regularly varying functions of indices greater than 1.
PROOF.Put p λ (t) = t λ l(t) and q µ (t) = t µ m(t).There exist positive constants k, K, l and L such that Define the vector function (X λ (t), Y µ (t)) by (4.10) with p(t) and q(t) replaced by p λ (t) and q µ (t), respectively.Since (X λ (t), Y µ (t)) satisfies (4.12), there exists T 0 > t 0 such that (4.14) We may assume that X λ (t) and Y µ (t) are non-decreasing for t ≥ T 0 because any regularly varying function of positive index is asymptotic to a monotone non-decreasing function (cf [1, Theorem 1.5.3]).Noting that Let us choose (a, b), (A, B) ∈ R 2 so that a < A, b < B and the following inequalities hold: It is easy to check that such choice of (a, b) and (A, B) is possible by taking, if necessary, k and l sufficiently small and K and L sufficiently large.Let X denote the closed convex subset of C[T 0 , ∞)×C[T 0 , ∞) consisting of the vector functions (x(t), y(t)) such that and consider the mapping Φ : for t ≥ T 0 , and x 0 and y 0 are positive constants satisfying We prove that Φ is a continuous self-map of X and sends X into a relatively compact subset of C[T 0 , ∞)×C[T 0 , ∞).
(iii) Φ is continuous.Let (x n (t), y n (t)) be a sequence in X converging to (x(t), y(t)) ∈ X as n → ∞ uniformly on any compact subinterval of [T 0 , ∞).Noting that and applying the Lebesgue dominated convergence theorem to the right-hand sides of the above inequalities, it follows that uniformly on compact subintervals of [T 0 , ∞).This implies the continuity of Φ.
Therefore, the Schauder-Tychonoff fixed point theorem guarantees the existence of an element (x(t), y(t)) ∈ X such that (x(t), y(t)) = Φ(x(t), y(t)), t ≥ T 0 , that is, No. 23, p. 17 for t ≥ T 0 , which is a special case of the system of integral equations (4.2).Thus, (x(t), y(t)) is a solution of the system of differential equations (A) on [T 0 , ∞).Since (x(t), y(t)) is a member of X , it is a nearly regularly varying solution of index (ρ, σ) with ρ > 1 and σ > 1, which clearly provides a strongly increasing solution of (A).This completes the proof of Theorem 4.1.
As the next theorem demonstrates, the full regularity of the strongly increasing solutions obtained in Theorem 4.1 can be proved with the help of the generalized L'Hospital rule (Lemma 3.2), if we make a stronger assumption that the coefficients p(t) and q(t) are regularly varying functions.Thus, the existence of regularly varying solutions with indices greater than one is characterized completely in this particular case.The proof is similar to that of Theorem 3.2 and we omit it.Theorem 4.2.Suppose that p(t) and q(t) are regularly varying of indices λ and µ, respectively.System (A) possesses regularly varying solutions (x(t), y(t)) such that if and only if (4.5) holds, in which case ρ and σ are given by (3.6) and the asymptotic behavior of any such solution is governed by the formulas where ∆(τ ) = τ (τ − 1) for τ > 1.
Here λ = −3α and µ = 1 − 2β, and hence which implies that condition (4.5) holds.It is easy to see that (3.6) defines ρ = 2 and σ = 3, so that ∆(σ) = 2 and ∆(σ) = 6.On the other hand, since l(t) = 2 exp (α + 1) √ log t and m(t) = 6 exp −(β + 1) √ log t , we obtain Combining the above calculations, we conclude from Theorem 4.2 that the system (A) under consideration possesses a strongly increasing solution (x(t), y(t)) such that As is easily checked, if 5 Mixed strongly monotone solutions of (A) The purpose of the final section is to indicate the situation in which the system (A) possesses a positive solution (x(t), y(t)) such that (5.1) lim Such a solution is referred to as a mixed strongly monotone solution of (A).As in the preceding sections it is assumed that αβ < 1 and that p(t) and q(t) are nearly regularly varying functions.Mixed strongly monotone solutions of (A) are sought as solutions of the system of integral equations of the form (5.2) for some constants T > t 0 and y 0 > 0, belonging to the class of nearly regularly varying vector functions of indices ρ < 0 and σ > 1.The construction of such solutions of (A) is based on the accurate asymptotic behavior of regularly varying solutions of the system of asymptotic relations with regularly varying coefficients p(t) and q(t).

Integral asymptotic relations
Lemma 5.1.Let αβ < 1 and suppose that p(t) and q(t) are regularly varying functions of indices λ and µ, respectively.System of relations (5.3) has regularly varying solutions of index (ρ, σ) with ρ < 0 and σ > 1 if and only if in which case (ρ, σ) is given by (3.6) and the asymptotic behavior of any such solution (x(t), y(t)) is governed by the formula SKETCH OF PROOF.Since the right asymptotic relation (resp.the left asymptotic relation) in (5.3) can be analyzed exactly as in Lemma 3.1 (resp.Lemma 4.1), we need only to give a brief sketch of the proof, in which use is made of the expressions (3.4) and (3.8) for p(t), q(t), x(t) and y(t).

Mixed strongly monotone solutions of (A)
It is shown that system (A) with nearly regularly varying coefficients may have strongly monotone solutions of the mixed type which are nearly regularly varying vector functions of indices ρ < 0 and σ > 1.
It is clear that X which is a closed convex subset of C[T, ∞)×C[T, ∞) and it can be verified in a routine manner that Φ is a continuous self-map of X and sends X into a relatively compact subset of C[T, ∞)×C[T, ∞).Consequently, Φ has a fixed point (x(t), y(t)) ∈ X which satisfies the integral equation This implies that (x(t), y(t)) is a positive solution of system (A) belonging to the class of nearly regularly varying functions of index (ρ, σ).Since ρ < 0 and σ > 1, this solution EJQTDE, 2013 No. 23, p. 21 provides a mixed strongly monotone solution of (A).This sketches the proof of Theorem 5.1.
As in the preceding sections, under a stronger assumption that both coefficients in (A) are regularly varying functions, the following theorem characterizing the existence of fully regularly varying mixed strongly monotone solutions can be proved.The proof is similar to that of Theorem 3.2 and we omit it.Theorem 5.2.Suppose that p(t) and q(t) are regularly varying of indices λ and µ, respectively.System (A) possesses regularly varying solutions (x(t), y(t)) such that x(t) ∈ RV(ρ), y(t) ∈ RV(σ), ρ < 0, σ > 1, if and only if (5.4) holds, in which case ρ and σ are given by (3.6) and the asymptotic behavior of any such solution is governed by the formulas (5.13) x(t) ∼ t 2(α+1) p(t)q(t) α ∆(ρ)∆(σ) α 1 1−αβ , y(t) ∼ t 2(β+1) p(t) β q(t) ∆(ρ) β ∆(σ) from Theorem 5.2 it follows that our system (A) possesses mixed strongly monotone solutions belonging to class of regularly varying functions of index (−1, 2) and that the asymptotic behavior of any such solution (x(t), y(t)) is governed by the formula x(t) ∼ t −1 (log t), y(t) ∼ t 2 (log t) −1 , t → ∞.
The totality of regularly bounded functions is denoted by RO.It is clear that RV(ρ) ⊂ RO for any ρ ∈ R. Any function which is bounded both from above and from below by positive constants is regularly bounded.For example, 2 + sin t and 2 + sin(log t) are regularly bounded.Note that 2 + sin t and 2 + sin(log t) are not slowly varying, whereas 2 + sin(log n t), n ≥ 2, are slowly varying.
for all large t.