Asymptotic behavior of positive solutions of odd order Emden-Fowler type differential equations in the framework of regular variation

The asymptotic behavior of solutions of the odd-order differential equation of Emden-Fowler type x(t) + q(t)|x(t)|sgn x(t) = 0, is studied in the framework of regular variation, under the assumptions that 0 < γ < 1 and q(t) : [a,∞) → (0,∞) is regularly varying function. It is shown that complete and accurate information can be acquired about the existence of all possible positive solutions and their asymptotic behavior at infinity.


Introduction
The objective of this paper is to make a detailed study of the existence and the asymptotic behavior of positive solutions of the nonlinear odd-order differential equation (A) x (2n+1) (t) + q(t)|x(t)| γ sgn x(t) = 0, where γ is a constant such that 0 < γ < 1 and q : [a, ∞) → (0, ∞) is a continuous function.Equation (A) is often referred to as sublinear differential equation in this case, while equation (A) for which γ > 1 is called superlinear differential equation.
A solution x(t) of (A) existing in an infinite interval of the form [T x , ∞) is said to be proper if sup{|x(t)| : t ≥ T } > 0 for any T ≥ T x .
A proper solution is called oscillatory if it has an infinite sequence of zeros clustering at infinity and nonoscillatory otherwise.Thus, a nonoscillatory solution is eventually positive or eventually negative.
Sublinear equation (A) may have both oscillatory and nonoscillatory solutions on [t 0 , ∞) for some t 0 > a.
Theorem A. Any proper solution x(t) of sublinear equation (A) is either oscillatory or satisfies (1.1) x (i) (t) ↓ 0 as t → ∞, i = 0, 1, . . ., 2n, if and only if Our main interest is in nonoscillatory solutions of equation (A).If x(t) satisfies (A), then so does −x(t), and so in studying nonoscillatory solutions of (A) it suffices to restrict our attention to its (eventually) positive solutions.Let P denote the set of eventually positive solutions of equation (A), while P k , 0 ≤ k ≤ 2n + 1 denote the set of all x ∈ P satisfying (1.3) By the well-known Kiguradze's lemma (see [5]) every positive solution x(t) ∈ P falls into one and only one class P k with k such that k ∈ {0, 2, . . ., 2n}.In other words, the set P has the decomposition (1.4) P = P 0 ∪ P 2 ∪ . . .∪ P 2n .
Since x (i) (t), i ∈ {0, 1, • • • , 2n}, are eventually monotone, they tend to finite or infinite limits as t → ∞, i.e. lim t→∞ If x ∈ P k , then the set of its asymptotic values {ω i : i = 0, 1, 2, . . ., 2n} falls into one of the following three cases: (1.5) for k ∈ {2, 4, . . ., 2n}, or into one of the following two cases: (1. 6) For simplicity of notation we introduce the symbols ∼ and ≺ to denote the asymptotic equivalence and the asymptotic dominance of two positive functions f (t) and g(t): and the classes of positive solutions: Sharp criteria for the existence of solutions belonging to P(I j ), j ∈ {0, 1, • • • , 2n} and P(II k ), k ∈ {2, 4, . . ., 2n} can be given explicitly (for the proof see Kiguradze, Chanturia [6, Theorem 16.9], Kusano, Naito [13] and Tanaka [14]). and Therefore, the following questions naturally arise: The recent development of the study of second order differential equations by means of regular variation (in the sense of Karamata) as demonstrated in the papers [3], [4], [7], [8], [11], [12] seem to suggest the possibility of investigating the higher-order problems in the framework of regularly varying functions, more specifically, by limiting ourselves to equation (A) with regularly varying coefficient q(t).The objective of this paper is to show that theory of regular variation can provide us with full information about the existence and asymptotic behavior of positive solutions of the odd order differential equation (A) with regularly varying coefficient q(t).
Recently, Evtukhov and Samoilenko in [2] studied the differential equation where The condition imposed on the function q(t) in main results of [2] means actually that it is either of regular or rapid variation.However, this fact is neither used nor mentioned by Evtukhov and Samoilenko, which makes their method of proofs different from ours and the statements on solutions somewhat weaker than ours.Some comments along this line is given in the last section of our paper.

Basic properties of regularly varying functions
The class of regularly varying functions was introduced in 1930 by J. Karamata by the following: EJQTDE, 2012 No. 45, p. 4 We denote by RV(ρ) the set of all regularly varying functions of index ρ.If in particular ρ = 0, we often use SV instead of RV(0) and refer to members of SV as slowly varying functions.It is clear that an RV(ρ)-function f (t) is expressed as f (t) = t ρ L(t) with L(t) ∈ SV, and so the class SV of slowly varying functions is of fundamental importance in the theory of regular variation.
Otherwise f (t) is called a nontrivial regularly varying function of index ρ.The symbol tr-RV(ρ) (or ntr-RV(ρ)) denotes the set of all trivial RV(ρ)-functions (or the set of all nontrivial RV(ρ)functions).
Typical examples of slowly varying functions are all functions tending to positive constants as t → ∞, (log n t) βn , β n ∈ (0, 1), where log n t denotes the n-th iteration of the logarithm.
The following result concerns operations which preserve slow variation.A slowly varying function may grow to infinity or decay to 0 as t → ∞.But its order of growth or decay is severely limited as is shown in the following Proposition 2.2 If L(t) ∈ SV, then for any ε > 0, The following result, termed Karamata's integration theorem, will play a central role in establishing our main results in Sections 3. The reader is referred to Bingham, Goldie and Teugels [1] for the most complete exposition of theory of regular variation and its applications and to Marić [16] 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.

Intermediate regularly varying solutions of (A)
We first study intermediate regularly varying solutions of equation (A) with regularly varying coefficient q(t).Thus, in what follows the function q(t) is assumed to be regularly varying of index σ expressed as and k is assumed to be even integer such that 2 ≤ k ≤ 2n.Let x(t) ∈ P(II k ) be a regularly varying solution of (A).In view of (1.3) there exists positive constants c 1 , c 2 and T > a such that Therefore, the class of regularly varying solutions of type P(II k ), if non-empty, is divided into three types of subclasses composed of regularly varying solutions belonging respectively to It will be shown that the class of regularly varying solutions of type P(II k ) of equation (A) coincides with only one of the three subsets in (3.2), depending on the regularity index of q(t), and that all members belonging to that subset has one and the same asymptotic behavior at infinity.Moreover, since x(t) ∈ P(II k ) ⊂ P k , in view of (1.5), we may integrate (A) (2n + 1 − k)−times from t to ∞ and then k−times from t 0 to t, to get 3) we obtain the integral asymptotic relation which can be concerned as an "approximation" of (3.3) at infinity.Let us interpret the conditions (1.8) and (1.9) in the language of regular variation.Since Similarly, This observation, with the statement of Theorem 1.2, suggests us to carry out the study of intermediate solutions belonging to the class P(II k ) by distinguishing the cases: Actually, we verify that the above conditions, respectively, are necessary and sufficient for the existence of three types of regularly varying solutions of (A) listed in (3.2) with precise asymptotic behavior at infinity and that the regularity index ρ of such solution is uniquely determined by γ, n and the regularity index σ of q(t).
For the proof of our main results we make use of the following lemma -general L'Hospital's rule (see [15]): = 0 and g ′ (t) < 0 for all large t . Then EJQTDE, 2012 No. 45, p. 7 We first show two preparatory results.
Proof of the "only if" part of Theorem 3.2: Suppose that x(t) ∈ RV(ρ) for some ρ ∈ (k − 1, k) is the solution of (A).Then, clearly only the statement (ii) of Lemma 3.2 could hold.EJQTDE, 2012 No. 45, p. 12 Thus, we must have ρ = σ + ργ + 2n + 1, which implies that ρ is given by (3.17).This combined with ρ ∈ (k − 1, k) determines the range of σ to be Using (3.17) we have P k Q k = L(ρ, k) , and (3.9) can be rewritten as so that asymptotic formula for x(t) is be given by (3.29).
Proof of the "if" part of Theorems 3.1, 3.2 and 3.3: Suppose that either (3.5) or (3.6) or (3.7) holds for q(t) ∈ RV(σ) and let ρ, L(ρ, k) be defined by (3.17), (3.20).We perform simultaneous proof of all three theorems, so to simplify notation we introduce the function Φ holds; By Lemma 3.3 the function Φ k (t) satisfies (3.4).Thus, there exists T 1 > a such that EJQTDE, 2012 No. 45, p. 13 Let such a T 1 be fixed.From (3.4) we have so that there exists T 2 > T 1 such that Let T 2 > T 1 be a fixed constant such that (3.36) hold, m ∈ (0, 1) be a fixed positive constant such that and choose a constant M > 1 such that ) .
We define the set X to be the set of continuous functions It is clear that X is a closed convex subset of the locally convex space C[T 1 , ∞) equipped with the topology of uniform convergence on compact subintervals of [T 1 , ∞).We now define the integral operator and let it act on the set X defined above.It can be shown that F a self-map on X and sends X continuously on a relatively compact subset of C[T 1 , ∞).
(b) F(X ) is relatively compact: The inclusion F(X ) ⊂ X implies that F(X ) is locally uniformly bounded on [T 1 , ∞).For all x(t) ∈ X , we have from (3.40) Thus, it follows that F(X ) is locally equicontinuous on [T 1 , ∞).The relative compactness of the set X then follows from the Arzela-Ascoli lemma.
(c) F is a continuous on X : Let {x n (t)} be a sequence in X converging to x(t) ∈ X as n → ∞ on any compact subinterval of [T 1 , ∞).We need to verify that Fx n (t) → Fx(t) uniformly on compact subintervals of [T 1 , ∞).From inequality , by the application of the Lebesgue dominated convergence theorem, we conclude that Fx n (t) → Fx(t) uniformly on any compact subinterval of [T 1 , ∞) as n → ∞, which proves the continuity of F. Thus all the hypotheses of the Schauder-Tychonoff fixed point thereom are fulfilled for F, and so there exists an element x(t) ∈ X such that x(t) = Fx(t), t ≥ T 1 , which satisfies the integral equation Differentiating the above 2n + 1 times we conclude that x(t) is a solution of (A) on [T 2 , ∞) satisfying Therefore, Since the function Φ k (t) satisfies (3.4), denoting Since γ < 1, from above we conclude that Similary, we can see that From (3.45) and (3.46) we obtain that l = L = 1, which means that This section is devoted to the study of regularly varying solutions belonging to P(II 0 ), that is those solutions which decay to 0 as t → ∞.It is assumed that coefficient q(t) is regularly varying of index σ and expressed as in (3.1).Let x(t) ∈ P(II 0 ) be a regularly varying solution of (A) expressed as x(t) = t ρ ξ(t), ξ(t) ∈ SV.In view of (1.3) there exists positive constants c and T > a such that x(t) ≤ c, t ≥ T , so that the regularity index ρ of x(t) clearly satisfies ρ ≤ 0, while if ρ = 0 slowly varying part ξ(t) must satisfy ξ(t) → 0 as t → ∞.We will prove two theorems stated below which show that the totality of decaying regularly varying solutions of (A) always consists of only one class: (4.1) ntr-RV(0) or RV(ρ) for some ρ < 0.

(i) If ( 1 . 2 )
holds, does (A) really possess decaying positive solutions?If so, what can be said about the exact asymptotic decay of such solutions?(ii) Is it possible to determine the accurate asymptotic behavior at infinity of intermediate solutions of equation (A)?