OSCILLATION AND NONOSCILLATION OF PERTURBED HIGHER ORDER EULER-TYPE DIFFERENTIAL EQUATIONS

Oscillatory properties of even order self-adjoint linear differential equations in the form n k=0 (−1) k ν k y (k) t 2n−2k−α (k) = (−1) m q m (t)y (m) , are real constants satisfying certain conditions, are investigated. In particular, the case when q m (t) = β t 2n−2m−α ln 2 t is studied.

As a consequence of the presented oscillation and nonoscillation criteria, we deal with the oscillation of (1) in case q m (t) = β t 2n−2m−α ln 2 t , m ∈ {0, 1}, β ∈ R. In particular, we show that, under the conditions (4) and (5) This paper can be regarded as a continuation of some recent papers where the two-term differential equation in the form has been investigated, see [4,5,6,8,10,11,12,13,14,16].Namely, we extend the results of [8] and also of [7] dealing with (1) in the case where n = 2 and α = 0.
Similarly, as in the above mentioned papers, we use the methods based on the factorization of disconjugate operators, variation techniques, and the relationship between self-adjoint equations and linear Hamiltonian systems.
The paper is organized as follows.In the next sections we recall necessary definitions and some preliminary results.Our main results, the oscillation and nonoscillation criteria for (1), are contained in Section 3 and Section 4. In the last section we formulate some technical results needed in the proofs.

Preliminaries
Here, we present some basic results which we will apply in the next section.We will need a statement concerning factorization of formally self-adjoint differential operators.Consider the equation ( 8) Lemma 1. ( [1]) Suppose that equation (8) possesses a system of positive solutions y 1 , . . ., y 2n such that Wronskians W (y 1 , . . ., y k ) = 0, k = 1, . . ., 2n, for large t.Then the operator L (given by the left-hand side of (8)) admits the factorization for large t where and Using the previous result we can factor the differential operator L ν .The proof of the following statement is almost the same as that of [ the first roots (ordered by their size) of the polynomial P (λ) given by (3).Now we recall basic oscillatory properties of self-adjoint differential equations (8).These properties can be investigated within the scope of the oscillation theory of linear Hamiltonian systems (LHS) (10) x where A, B, C are n × n matrices with B, C symmetric.Indeed, if y is a solution of (8) and we set ; then (x, u) solves (10) with A, B, C given by In this case we say that the solution (x, u) of ( 10) is generated by the solution y of (8).Moreover, if y 1 , . . ., y n are solutions of (8) and the columns of the matrix solution (X, U ) of ( 10) are generated by the solutions y 1 , . . ., y n , we say that the solution (X, U ) is generated by the solutions y 1 , . . ., y n .Recall that two different points t 1 , t 2 are said to be conjugate relative to system (10) if there exists a nontrivial solution (x, u) of this system such that x(t 1 ) = 0 = x(t 2 ).Consequently, by the above mentioned relationship between (8) and (10), these points are conjugate relative to (8) if there exists a nontrivial solution y of this equation such that y (i) (t 1 ) = 0 = y (i) (t 2 ), i = 0, 1, . . ., n − 1. System (10) (and hence also equation (8)) is said to be oscillatory if for every T ∈ R there exists a pair of points EJQTDE, 2005, No. 13, p. 4 ) which are conjugate relative to (10) (relative to (8)), in the opposite case (10) (or (8)) is said to be nonoscillatory.
We say that a conjoined basis (X, U ) of (10) (i.e., a matrix solution of this system with n×n matrices X, U satisfying X T (t)U (t) = U T (t)X(t) and rank (X T , U T ) T = n) is the principal solution of (10) if X(t) is nonsingular for large t and for any other conjoined basis ( X, Ū ) such that the (constant) matrix X T Ū − U T X is nonsingular, lim t→∞ X−1 (t)X(t) = 0 holds.The last limit equals zero if and only if (11) 17]).A principal solution of ( 10) is determined uniquely up to a right multiple by a constant nonsingular n × n matrix.If (X, U ) is the principal solution, any conjoined basis ( X, Ū ) such that the matrix X T Ū − U T X is nonsingular is said to be a nonprincipal solution of (10).Solutions y 1 , . . ., y n of ( 8) are said to form the principal (nonprincipal) system of solutions if the solution (X, U ) of the associated linear Hamiltonian system generated by y 1 , . . ., y n is a principal (nonprincipal) solution.
Using the relation between ( 8), (10) and the so-called Roundabout Theorem for linear Hamiltonian systems (see e.g.[17]), one can easily prove the following variational lemma.

Lemma 3. ([15]
) Equation ( 8) is nonoscillatory if and only if there exists T ∈ R such that We will also need the following Wirtinger-type inequality.
We finish this section with one general oscillation criterion based on the concept of principal solutions.The proof of this statement can be found in [2].Let us consider the equation ( 12) where where (X, U ) is the solution of the linear Hamiltonian system associated with (8) generated by y 1 , . . ., y n .
3. Oscillation and nonoscillation criteria for (1) in case m = 0 In this section, we deal with (1) in the case m = 0, i.e., with the equation We start with a nonoscillation criterion for (13).
Proof.Let T ∈ R be such that the statement of Lemma 3 holds for (14) and let y ∈ W n,2 (T, ∞) be any function with compact support in (T, ∞).Using Lemma 5, Wirtinger's inequality (Lemma 4), which we apply (n − 1)-times, and Lemma 6 from the last section, we obtain Hence, we have according to Lemma 3, since ( 14) is nonoscillatory (take u = y/t 2n−1−α 2 ) and consequently, nonoscillation of (13) follows from this Lemma as well.
Proof.Let T ∈ R be arbitrary, T < t 0 < t 1 < t 2 < t 3 (these values will be specified later).We show that for t 2 , t 3 sufficiently large, there exists a function 0 ≡ y ∈ W n,2 (T, ∞) with compact support in (T, ∞) and such that and then, nonoscillation of (1) will be a consequence of Lemma 3. We construct the function y as follows: where and g is the solution of (2) satisfying the boundary conditions ( 16) Denote By a direct computation and using Lemma 7, we have for k = 1, . . ., n, as t → ∞, where Consequently, Since as t → ∞, k = 0, . . ., n (take A 0 = B 0 = 0), we obtain Concerning the interval [t 2 , t 3 ], since q 0 (t) ≥ 0 for large t, we have Next we use the relationship between equation (2) and corresponding LHS (10).Since g is a solution of (2), we have EJQTDE, 2005, No. 13, p. 9 and Then, using conditions (16), Further, let (X, U ) be the principal solution of the LHS associated with (2).Then ( X, Ũ ) defined by X(t) = X(t) is also a conjoined basis of this LHS, and according to ( 16), if we let h = (h, h , . . ., h (n−1) ) T , we obtain x(t) = X(t) EJQTDE, 2005, No. 13, p. 10 (see e.g.[1]) and hence Using the fact that the principal solution (X, U ) is generated by , where α 1 , . . ., α n−1 , α 0 = 2n−1−α 2 are the first roots (ordered by size) of ( 3), by a direct computation (similarly to that in [8, Theorem 3.2]), we get where L 2 is a real constant and If we summarize all the above computations, we obtain It follows from Lemma 9 that Kn,α = Kn,α and νn,α = νn,α , and according to (15), it is possible to choose t 2 > t 1 so large that νn,α and that the sum of all the terms o( 1) is less than 1.Moreover, since (X, U ) is the principal solution, we can choose All together means that Proof.If β > νn,α , then and ( 22) is oscillatory according to Theorem 2.

Oscillation and nonoscillation criteria for (1) in case m = 1
In this section we turn our attention to the equation (23) L ν (y) = −(q 1 (t)y ) .
Proof.Let T ∈ R be such that the statement of Lemma 3 holds for (24) and let y ∈ W n,2 (T, ∞) with compact support in (T, ∞) be arbitrary.We show that the quadratic functional associated with ( 23) is positive for any nontrivial y ∈ W n,2 (T, ∞) with compact support in (T, ∞) by using the same argument as in the proof of EJQTDE, 2005, No. 13, p. 12 Theorem 1 and the transformation of this functional by the substitution y = t 2n−1−α 2 u (applied to the term The following oscillation criterion is based on Proposition 1 and we prove it similarly to [10, Theorem 4.1]. Theorem 4. Let q 1 (t) ≥ 0 for large t, (4), (5) hold and Then (23) is oscillatory.
Proof.To prove the first equality, it suffices to show that This follows from the definition of A k by (17) and from formula (27) of Lemma 8, where we take ᾱ = α + 2k − 2n.
Proof.We have x [1] , where x [n] , u [n] denote the n-th column of X, U respectively and x [1] , ũ [1] denote the first column of X, Ũ respectively, i.e., ln t.By a direct computation and using the fact that L is a constant matrix (so that we don't need to take into account the terms EJQTDE, 2005, No. 13, p. 19 j + 1)(λ − 2n + j + α) and 2n − 2 distinct real roots.This is equivalent to the following two conditions. 1 − 2j − α) 2 = 0.