Nonoscillation of higher order half-linear differential equations

We establish nonoscillation criteria for even order half-linear differential equations. The principal tool we use is the Wirtinger type inequality combined with various perturbation techniques. Our results extend nonoscillation criteria known for linear higher order differential equations.


Introduction
In this paper we deal with the even order half-linear differential equation where Φ(y) = |y| p−2 y, p > 1, is the odd power function, r j are continuous functions, j = 0, . . ., n, and r n (t) > 0 in the interval under consideration.The terminology half-linear equation was introduced by I. Bihari [3] and reflects the fact that the solution space of (1.1) is homogeneous, but not additive, i.e., it has just one half of the properties characterizing linearity.In the case n = 1, equation (1.1) reduces to the classical second order half-linear differential equation − r 1 (t)Φ(x ) + r 0 (t)Φ(x) = 0 (1.2) whose oscillation theory is relatively deeply developed, see [1,16] and e.g. the recent papers [11,13,17,19,24,27,28].The theory of (1.1) is much less developed and as far as we known only [16,Sec. 9.4] and the paper [25] deal with this problem.The reason is that we miss the so-called Reid's roundabout theorem in the higher order case, in particular, the Riccati technique is not available for Corresponding author.Email: dosly@math.muni.cz 2 O. Došlý and V. R ůžička (1.1), in contrast to (1.2).Actually, necessary and sufficient conditions for (non)oscillation of (1.1) with p = 2, i.e., in the linear case, follow from the fact that this equation can be written as a linear Hamiltonian system (for which the Reid's roundabout theorem is well known, [26, Chap.V., Theorem 6.3]) and this enables to present oscillation and spectral theory of (1.1) with p = 2 as it is exhibited e.g. in the book [22], see also [20] and the references given therein.
The energy functional associated with (1.1) considered on the interval [T, ∞) is (equation (1.1) is the Euler-Lagrange equation of (1.3)).If there exists a nontrivial solution ỹ of (1.1) with two zeros of multiplicity n in [T, ∞), i.e., for some T ≤ t 1 < t 2 , then we define the function and obviously y ∈ W n,p 0 [T, ∞) (the definition of this Sobolev space will be recalled later).Multiplying (1.1) by y and integrating by parts over [T, ∞) gives F n (y) = 0. Hence, if we show that F n (y) > 0 for all nontrivial functions y ∈ W n,p 0 [T, ∞), we eliminate the existence of a solution of (1.1) satisfying (1.4) The paper is organized as follows.In the next section we concentrate our attention on basic properties of the higher order half-linear Euler differential equation and on the so-called Wirtinger inequality which is the principal tool in our investigation.Section 3 is devoted to nonoscillation criteria for Euler type even order differential equation.Section 4 deals with nonoscillation criteria for general two-term 2nth order half-linear differential equations and in the last section we present some remarks and comments concerning possible further investigation.
The "classical" Euler second order half-linear differential equation is the equation as can be verified by a direct computation.
Concerning equation (2.1), similarly to the linear case, we look for a solution in the form x(t) = t λ .Consider first the two-term equation with α ∈ {p − 1, . . ., np − 1} and γ ∈ R. Substituting into (2.3)we find that λ must be a root of the algebraic equation G(λ) + γ = 0 with Next we show that the function G has a stationary point λ * = np−1−α p .We have the equality x , therefore, by a direct calculation we obtain that for λ = j, n − α−j p−1 , j = 0, . . ., n − 1, .
Because 1 Substituting the value λ * into G gives the value of the so-called critical constant in the 2nth order Euler half-linear differential equation (2.3).We denote The previous computation shows that the equation G(λ) − γ n,p,α = 0 has a double root λ * = np−1−α p .The terminology critical constant is used by analogy with the linear case where its value is a "borderline" between oscillation and nonoscillation of equation (2.3) with p = 2.In the halflinear case, we are able to prove only "one half" of conditions for an oscillation constant yet, namely that (2.3) is nonoscillatory for γ > −γ n,p,α .The proof of an "oscillation counterpart" resists our effort till now, nevertheless, it is a subject of the present investigation.More details about this problem are given in the last section.
Therefore, (2.3) with γ = −γ n,p,α has a solution x(t) = t λ * .Note that linearly independent solutions cannot be computed explicitly even in the case n = 1 and α = 0 (i.e., for second order equation (2.2) with γ = −γ p , because γ p = γ 1,p,0 ).Nevertheless, as shown in [18], any solution of (2.2) with γ = −γ p , which is linearly independent of p t is also an "approximate" solution of the equation log t is a solution of this equation.Now we recall the definition of the Sobolev space, consisting of functions with a compact support.We denote for where AC[T, ∞) is the set of absolutely continuous functions with the domain [T, ∞).
We finish this section with a half-linear version of the classical Wirtinger inequality, which we use in the next sections.Its proof in the formulation presented here can be found in [7]. (2.5)

Euler equation
Following the linear terminology, we say that (1.1) is nonoscillatory if there exists T ∈ R such that no solution of this equation has two or more zeros of multiplicity n in [T, ∞).In the opposite case, i.e., when for every T ∈ R there exists a nontrivial solution of (1.1) with at least two zeros of multiplicity n in [T, ∞), then (1.1) is said to be oscillatory.
We start this section with a variational lemma which plays the fundamental role in our treatment, for its proof (whose outline we have already presented below (1.3)) see [16,Sec. 9.4].
The first statement of this section is a nonoscillation criterion which is essentially proved in [16,Theorem 9.4.5].This criterion is formulated in [16] for the equation but a small modification of the proof (via Wirtinger inequality) shows that it can be extended to a more general equation (2.3).
Note that the same statement (for α = 0) is proved via the weighted Hardy inequality in [25], we will mention this result later in our paper.Now we turn our attention to the "full term" 2nth order Euler differential equation.
with α ∈ {p − Proof.We apply the Wirtinger inequality to each term (except that one for k = n) in the energy functional We obtain for any y ∈ W n,p Then we have O. Došlý and V. R ůžička Remark 3.4.The reason why the case α ∈ {p − 1, . . ., np − 1} we needed to exclude from the previous considerations is the following.For α = p − 1 the Wirtinger inequality takes the form so, a logarithmic term appears.This more difficult case is treated in the next part of this section.
We start with an auxiliary statement.
Proof.The energy functional corresponding to (3.8) is The first term in the integral is estimated in Lemma 3.5.Concerning the terms under summation signs, for i = 0, . . ., j − 1 O. Došlý and V. R ůžička Substituting these computations into F n (y), we have .
Since the second term in the bracket tends to zero as T → ∞, we have F n (y; T, ∞) > 0 for T sufficiently large if (3.9) holds, which means that equation (3.8) is nonocillatory by Lemma 3.1.

General nonoscillation criteria
We start with two nonoscillation criteria from [25] (proved in [25] via the weighted Hardy inequality) which we later compare with our results.Both criteria are contained in the following theorem.

If one of the following conditions
holds, then the two-term differential equation is nonoscillatory.
} is the negative part of c, and for some j ∈ {1, . . ., n}.Then the equation is nonoscillatory.
Half-linear differential equations 9 Proof.Let T ∈ R be so large, that the limited expression in (4.4) is greater than where ε > 0 is sufficiently small.Then for any 0 ≡ y ∈ W n,p 0 [T, ∞) we have with z = y/t n−j (using the inequality ∫ b a f g ≤ ∫ b a | f | p 1/p ∫ b a |g| q 1/q between the fourth and fifth line and (3.3) (with β = α − (j − 1)p and x = z ) between the fifth and sixth line in the next computation) In the previous computation, we have used the equality |z(t)| p = p t T Φ(z(s))z (s) ds, which follows from the formula |z| p = pΦ(z)z and from the definition of z (z(T) = 0).We have also used the relation |Φ(z)| q = |z| p .Now, we apply Lemma 4.2 with m = n − j, i.e., n − m − 1 = j − 1, and we denote Then, using Wirtinger inequality (3.2) (in a slightly modified form), we get for y ∈ W n,p O. Došlý and V. R ůžička Summarizing the previous computations Now, according to the definition of the constant K we see that the energy functional corresponding to (4.5) is positive for large T and hence (4.5) is nonoscillatory.
Next we prove a statement which relates nonoscillatory behavior of a two-term 2nth order half-linear differential equation to nonoscillation of a certain second order half-linear equation.It also presents a simpler proof of the previous theorem with j = n.
Proof.Using the Wirtinger inequality (as in (3.2)) we can estimate the energy functional in (4.5) as follows The expression in brackets on the second line of the previous computation is the energy functional of (