Even Order Half-Linear Differential Equations with Regularly Varying Coefﬁcients

: We establish nonoscillation criterion for the even order half-linear differential equation ( − 1 ) n (cid:16) f n ( t ) Φ (cid:16) x ( n ) (cid:17)(cid:17) ( n ) + ∑ nl = 1 ( − 1 ) n − l β n − l (cid:16) f n − l ( t ) Φ (cid:16) x ( n − l ) (cid:17)(cid:17) ( n − l ) = 0, where β 0 , β 1 , . . . , β n − 1 are real numbers, n ∈ N , Φ ( s ) = | s | p − 1 sgn s for s ∈ R , p ∈ ( 1, ∞ ) and f n − l is a regularly varying (at inﬁnity) function of the index α − lp for l = 0, 1, . . . , n and α ∈ R . This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms f n ( t ) and f n − l ( t ) are replaced by the t α and t α − lp , respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.


Introduction
Consider the 2n-th order half-linear differential equation (−1) n f n (t)Φ x (n) (n) + n ∑ l=1 (−1) n−l β n−l f n−l (t)Φ x (n−l) (n−l) = 0, where β 0 , β 1 , . . . , β n−1 are real numbers, n ∈ N, Φ is the odd power function defined by the relation Φ(s) = |s| p−1 sgn s for s ∈ R, p ∈ (1, ∞) and for each l ∈ {0, 1, . . . , n} the function f n−l is defined, positive and continuous on [S, ∞), where S ∈ R. Moreover, we assume that f n−l is a regularly varying (at infinity) function of the index α − l p (the definition is given later) for l = 0, 1, . . . , n and α ∈ R. More briefly, we write f n−l ∈ RV (α − l p), where RV (ϑ) for ϑ ∈ R denotes the set of all regularly varying functions of the index ϑ. Denote SV := RV (0). Functions belonging to SV are called slowly varying functions and the function f n−l can be equivalently described for l = 0, 1, . . . , n as follows: there exists a function L n−l defined and continuous on [S, ∞) such that L n−l ∈ SV and f n−l (t) = t α−l p L n−l (t), t ∈ [S, ∞).
The function L n−l is called component of f n−l .
Equation (1) can be understood as a generalization of the 2n-th order Euler type half-linear differential equation studied in [1,2]. The two-term even order (Euler type and more general) half-linear differential equations are studied in [1,3,4] and in the book [5] (Section 9.4). The two-term 2n-th order Euler type linear differential equation with γ ∈ R is a special case of Equation (2) since Φ(s) = s for s ∈ R and p = 2. Equation (3) with α ∈ R M n,2 is nonoscillatory if and only if γ n,2,α + γ ≥ 0 (see [6] (p. 132) and for α = 0 see also [7] (pp. 97-98)), where Equation (2) with n = 1 and β 0 = γ is the second order Euler type half-linear differential equation which is nonoscillatory if and only if γ p,α + γ ≥ 0, see [5] (Theorem 1.4.4) for α ∈ R M 1,p and for the proof see [8]. For the case α ∈ M 1,p (that is α = p − 1) see Remark 2 in this article. Equation (4) with α = 0 and its various perturbations are also studied in [9][10][11][12][13][14]. This article is organized as follows. In Section 2, we define the concept of nonoscillation for (1), we formulate the variational principle for (1) and we recall basic concepts of the theory of regularly varying functions. The main results are given in Section 3. We conclude the article with several examples and comments in the last two sections.

Preliminaries
First, we define the concept of nonoscillation for Equation (1). Similarly as in the linear case, real points t 1 and t 2 are said to be conjugate relative to Equation (1), if t 1 = t 2 and there exists a nontrivial solution x of Equation (1), such that t 1 and t 2 are its zero points of multiplicity n, i.e., t 1 and t 2 satisfying Note that the concept of conjugate points does not need such strict assumptions on coefficients as they are given for Equation (1). Instead of β n−l f n−l (t) in (1) we can take r n−l (t) defined and continuous on the interval [S, ∞) for l = 1, 2, . . . , n; and instead of f n (t) we can take r n (t) defined, continuous and positive on the interval [S, ∞). (1) is said to be nonoscillatory if there exists T ∈ R such that no pair t 1 , t 2 of points from [T, ∞) conjugate relative to Equation (1) exists. In the opposite case, Equation (1) is said to be oscillatory.

Definition 1. Equation
Recall the definition of the Sobolev space. Denote where n ∈ N, T ∈ R and p ∈ (1, ∞). The symbol AC[T, ∞) indicates the set of all absolutely continuous functions of the form f : [T, ∞) → R and the symbol L p (T, ∞) indicates the space of (Lebesgue) measurable functions (equivalence classes of functions) such that . If we say that y is nontrivial, we mean that the function y is not identically zero on the interval [T, ∞).
The relation between Equation (1) and the energy functional F n , for which Equation (1) is its Euler-Lagrange equation, is formulated in the following lemma and is called the variational principle. (1) is nonoscillatory if there exists T ∈ R such that for every nontrivial function y ∈ W n,p

Lemma 1 ([5]). Equation
If Equation (1) is of the second order, condition (5) is even equivalent to nonoscillation of (1). Consider the general second order half-linear differential equation where r and c are continuous functions defined on a neighborhood of infinity and r is positive. (6) is nonoscillatory if and only if there exists T ∈ R such that
, the symbol q denotes the conjugate number of p, i.e., the number q is such that The auxiliary statement below is proved in [5] (Theorem 2.1.2). Proposition 1. Denote γ p := γ p,0 . The following statements hold.
Note that the assumptions of part (a) of Proposition 1 can be weakened, see [5] (Theorem 2.2.9). The function c − is replaced by c and instead of The last part of this section is devoted to the theory of regular varying functions. A comprehensive study of regular variation can be found in [15], where the proofs of the presented statements can be found.  (0)). The logarithm defined on [S, ∞) is also an element of SV (where SV = RV (0)) if S > 1. Let µ 1 , µ 2 , . . . , µ k be real numbers, k ∈ N and ln i+1 t := ln ln i t for i ∈ N, where ln 1 t := ln t. Then, the function defined by the relation is slowly varying for sufficiently large S. Examples of regularly varying functions of index ϑ have the form t ϑ L(t), where L is a slowly varying function; see Lemma 3.
Let f and g be real-valued functions, which are positive in a neighborhood of infinity. The functions f and g are said to be asymptotically equivalent if lim t→∞ f (t) g(t) = 1; we write f (t) ∼ g(t) as t → ∞.  (d) Let f ∈ RV (ϑ 1 ) and g ∈ RV (ϑ 2 ). Then f g ∈ RV (ϑ 1 + ϑ 2 ).
The following statement allows us to include equations with regularly varying coefficients in our investigation.
Proposition 2 (Karamata's theorem [15]). Let S be a real number and L be a slowly varying function defined on [S, ∞). The following statements hold.
Note that the case ϑ = −1 is not included in any part of Karamata's theorem since the integral ∞ S s −1 L(s) ds may or may not converge.

Equations with Regularly Varying Coefficients
It is worthy to note that the methods presented in this section have been previously used in [16], where we are dealing with the discrete case. As far as we know, our result in this section is new even in the linear case (p = 2). We use the following notation, which greatly simplifies the formulation of the main result. Recall Now we formulate the main theorem. It is an extension of the result in [1] (Theorem 3.3) obtained for Equation (2). The result in [1] is also extended in [2]. The extension from [2] generalizes the conditions on the coefficients of Equation (2). In this paper, moreover, a more general Equation (1) is considered.
The difference in the approach of this article and our previous articles [1,2] is that we do not utilize the so-called Wirtinger inequality, see [5] (Lemma 2.1.1). Consequently, we can consider more general coefficients, but we lose some potentially critical states of the constants β 0 , β 1 , . . . , β n−1 (especially, the case γ [n] (k) = 0 for k from an arbitrary subset of {1, 2, . . . , n − 1} and γ [n] (k) > 0 for k from the complement of this subset with respect to {1, 2, . . . , n}). Oscillation properties in the case γ [n] (n) = 0 are completely unknown to us.
Consider a special case of Equation (1), namely the second order half-linear differential equation where γ ∈ R and functions f and g are such that for some α ∈ R and some slowly varying functions K and L.
We start with the auxiliary nonoscillation criterion for Equation (8). In its proof, both parts of Propositions 1 and 2 (Karamata's theorem) are used. Proof. From Lemma 1 it follows that Equation (8) is nonoscillatory for γ ≥ 0.
Indeed, α(1 − q) > −1 if and only if α < p − 1 and K 1−q ∈ SV by part (c) of Lemma 3. Hence, Then c − ≡ c for γ < 0 and by part (a) of Proposition 2 we get Therefore, ∞ S γt α−p L(t) dt > −∞ holds. Further, the left-hand side of inequality (7) admits the form and The proof of the case α < p − 1 (with γ < 0) is analogous to the one of the case α(1 − q) > −1 and it uses part (b) of Proposition 1.

Remark 1.
The oscillation complement of Lemma 4 holds too. Indeed, instead of the parts (a) and (b) of Proposition 1, we use their oscillation complements (see [5] (Theorem 2.3.2 (ii)) in case of α < p − 1 and [5] (Theorem 3.1.4) in case of α > p − 1). Nevertheless, in this paper we only need the nonoscillation criterion shown in Lemma 4, and therefore we do not prove the oscillation complement explicitly.

Remark 2.
Due to the note below Proposition 2, we cannot decide on the convergence of integrals in Equation (8), then Equation (8) is the second order Euler type half-linear differential equation and it is nonoscillatory if and only if γ p,p−1 + γ ≥ 0 (γ p,p−1 = 0). Indeed, the "if" part immediately follows from Lemma 2 and the "only if" part follows from the half-linear version of the Leighton-Wintner oscillation criterion (see [5] (Theorem 1.2.9)).
The variational principle formulated in Lemma 2 allows obtaining a certain inequality from the knowledge of nonoscillation of an equation. This way, we obtain the inequalities, as shown in the following lemma.