Hille-Nehari type oscillation and nonoscillation criteria for linear and half-linear differential equations

. Di ﬀ erential equations attract considerable attention in many applications. In particular, it was found out that half-linear di ﬀ erential equations behave in many aspects very similar to that in linear case. The aim of this contribution is to investigate oscillatory properties of the second-order half-linear di ﬀ erential equation and to give oscillation and nonoscillation criteria for this type of equation. It is also considered the linear Sturm-Liouville equation which is the special case of the half-linear equation. Main ideas used in the proof of these criteria are given and Hille-Nehari type oscillation and nonoscillation criteria for the Sturm-Liouville equation are formulated. In the next part, Hille-Nehari type criteria for the half-linear di ﬀ erential equation are presented. Methods used in this investigation are based on the Riccati technique and the quadratic functional, that are very useful instruments in proving oscillation / nonoscillation both for linear and half-linear equation. Conclude that there are given further criteria which guarantee either oscillation or nonoscillation of linear and half-linear equation, respectively. These criteria can be used in the next research in improving some conditions given in theorems of this paper.


Introduction
In this paper we investigate oscillatory properties of the half-linear second-order differential equation of the form where Φ(x) := |x| p−2 x, p > 1, t ∈ I := [T, ∞) and r, c are real-valued continuous functions and r(t) > 0. Oscillation theory of (1) attracted considerable attention in the past years and it was shown that solutions of (1) behave in many aspects like those of the linear Sturm-Liouville differential equation which is the special case p = 2 of (1). The aim of this paper is to present some results of the investigation oscillatory properties of equation (1) in comparison with that one of (2).
Note that the term half-linear equations is motivated by the fact that the solution space of (1) has just one half of the properties which characterize linearity, namely homogeneity, but not additivity.
The paper is organized as follows. In Section 2 we present basic concepts and properties of solutions of (1) and (2). Section 3 is devoted to the investigation of properties of solutions of (1) and (2), in particular, we present oscillation criteria for (1) and (2). Section 4 gives some nonoscillation criteria for (1) and (2). * Corresponding author:reznickova@utb.cz

Preliminary results
In this section we define basic concepts concerning the half-linear differential equation (1). Definition 1 Two points t 1 , t 2 ∈ R are said to be conjugate relative to (1) if there exists a nontrivial solution x of this equation such that x(t 1 ) = x(t 2 ) = 0. (1) is said to be disconjugate on a closed interval [a, b] if this interval contains no pair of points conjugate relative to (1) (i.e., every nontrivial solution has at most one zero in I). In the opposite case, (1) is said to be conjugate on I (i.e., there exists a nontrivial solution with at least two zeros in I).

Definition 2 Equation
Note that by a zero of a solution x we mean such a t 0 that x(t 0 ) = 0. The following property of zeros of linearly independent solutions is one of the most characteristic properties which justifies the definition of oscillation/nonoscillation of the equation.It is known as the Sturmian separation theorem and reads as follows (see [5]).
Theorem 2 Let t 1 < t 2 be two consecutive zeros of a nontrivial solution x of (1). Then any other solution of this equation which is not proportional to x has exactly one zero on (t 1 , t 2 ). Along with (1) consider another equation of the same form where the functions R and C satisfy the same assumptions as r and c in (1). The next theorem is known as the Sturmian comparison theorem and reads as follows (see [5]).
Theorem 3 Let t 1 < t 2 be two consecutive zeros of a nontrivial solution x of (1) and suppose that . Then any solution of (3) has a zero in (t 1 , t 2 ) or it is a multiple of the solution x.
In the opposite case, (1) is said to be oscillatory, i.e., if every nontrivial solution has infinitely many zeros tending to ∞.
The previous definition says that one solution of (1) is oscillatory if and only if any other solution of (1) is oscillatory. Oscillation of a nontrivial solution of (1) means the existence of zeros of this solution tending to ∞.
One of the most important tools in the investigation of the qualitative properties of solutions of (1) is the Riccati technique and variational principle.
Riccati substitution. Let x be a solution of (2) such that x(t) 0 in an interval I. Then w(t) = r(t)x x is a solution of the associated Riccati differential equation Equation (2) is nonoscillatory if and only if there exists a solution w of (5) defined for large t, and this is equivalent to the positivity of the quadratic functional over the class of continuously differentiable functions y such that y(T ) = 0 and y(t) ≡ 0 on [T 1 , ∞) for some T 1 > T . Now we introduce the half-linear version of the Riccati type equation associated with equation (1). Let x be a solution of (1) such that x(t) 0 in an interval I. Then is a solution of the Riccati type differential equation of the form w + c(t) + (p − 1) r 1−q (t)|w| q = 0, where q is the conjugate number of p, i.e., 1 p + 1 q = 1. Let us recall that the Riccati equation and the p−degree functional play the same role in the oscillation theory of (1) as (5) and (6) in the linear oscillation theory.

Oscillation criteria for Sturm-Liouville equation
Hille-Nehari type oscillation criteria are criteria formulated in terms of the asymptotic behavior of the functions depending on the convergence/divergence of the integrals appearing in these formulas. Note that if both integrals (2) is oscillatory by the Leighton-Wintner oscillation criterion, see [10]. For the case (9), the Hille-Nehari criterion reads as follows (see, e.g. [1, Chap. 2]).

Theorem 4 Suppose that
and the integral ∞ c(t) dt is convergent. Equation (2) is oscillatory provided one of the following conditions holds: PROOF. (i) We prove this statement using the variational principle, i.e., we find, for every T ∈ R, a function y ∈ W 1,2 (T, ∞) with a compact support in (T, ∞) such that the functional F (y; T, ∞) ≤ 0. To this end, let T ∈ R be arbitrary, T < t 0 < t 1 < t 2 < t 3 , and let Now, by (12), there exists ε > 0 such that the last term in the brackets is less than −1 − ε for t 1 and t 2 sufficiently large, and (11) implies that the middle term is less than ε if t 3 is sufficiently large. Hence F (y; T, ∞) ≤ 0 for t 1 , t 2 , t 3 chosen in this way.
(ii) This part of the proof is based on the Riccati technique. Suppose, by contradiction, that (2) is nonoscillatory and w is a solution of the associated Riccati equation (5). Then, according to [6,Chap. XI], the solution w can be expressed in the form Multiplying the last equation by t r −1 (s) ds, we have Suppose first that lim inf t→∞ t r −1 (s) ds w(t) = λ exists finite. Then, using (13), there exists ε > 0 such that for large t, and hence, letting t → ∞ in the last inequality, denote Then m is nondecreasing and using (13) there exists ε > 0 such that Since m is nondecreasing, we have for s > t m(s) thus, which is a contradiction with (14).

Oscillation criteria for half-linear equation
Now, we turn out attention to Hille-Nehari type oscillation criteria for the half-linear differential equation (1). A direct modification of the proof of Theorem 4 shows that the criteria given in that theorem can be extended to (1) as follows, see [5, Sec 3.1.1].
Theorem 5 Suppose that ∞ r 1−q (t) dt = ∞ and the integral ∞ c(t) dt is convergent. Equation (1) is oscillatory provided one of the following conditions holds: Note that in the modification of the part (ii) of the proof of Theorem 5, one needs to use the fact that

Nonoscillation criteria
In this part we present Hille-Nehari type nonoscillation criteria for equation (1). We also present a brief survey of linear version of Hille-Nehari criteria.
PROOF. It is well known from the linear Sturmian theory that equation (2) is nonoscillatory provided there exists a differentiable function u which satisfies the Riccati-type inequality