Perturbations of Half-Linear Euler Differential Equation and Transformations of Modified Riccati Equation

and Applied Analysis 3 was considered. It was shown that if there exists k ∈ {1, . . . , n} such that βj − αj/4 1/4 for j 1, . . . , k − 1, and βk − αk / 1/4, then 1.9 is oscillatory if and only if βk − αk/4 > 1/4. If βj−αj 1/4 for all j 1, . . . , n, then 1.11 is nonoscillatory. This result was partially extended to half-linear equations in 3 . There, [( 1 α logt ) Φ ( x′ ) ]′ [ γp tp β tp logt ] Φ x 0 1.10 was investigated and it was shown that 1.10 is oscillatory if and only β−αγp > μp. For some related results see also 6 . In this paper we deal with perturbations of the Euler half-liner differential equation in full generality. We consider ⎡ ⎣ ⎛ ⎝1 n ∑ j 1 αj Logj t ⎞ ⎠Φ ( x′ ) ⎤ ⎦ ′ ⎡ ⎣ γp tp n ∑ j 1 βj tp Logj t ⎤ ⎦Φ x 0, 1.11 and we find an explicit formula for the relationship between constants αj , βj in 1.11 which implies non oscillation of this equation. In the last section of the paper we explain why perturbations are just in the above considered form. Our result is based on a new method which consists in transformations of the modified Riccati equations associated with 1.2 . The main result along this line is established in Section 3, while its application to the perturbed Euler equation is presented in Section 4. In the last section we present some remarks and comments concerning the results of our paper. In the next section we recall some essentials of the half-linear oscillation theory. 2. Preliminaries It is a well-known fact that many of the results of the linear oscillation theory can be directly extended to half-linear equation 1.2 , even if, in contrast to the case p 2 then 1.2 is a linear equation , the additivity of the solution space is lost and only homogeneity remains. In particular, the so-called Riccati technique, consisting in the relationship between 1.2 and its associated Riccati type equation related to 1.2 by the substitution w rΦ x′/x R w t : w′ c t ( p − 1r1−q t |w| 0, q : p p − 1 2.1 extends almost literally to 1.2 . More precisely, the following statement holds see 1, Theorem 2.2.1 . Proposition 2.1. Equation 1.2 is nonoscillatory if and only if there exists a differentiable function w such that R w t 0 for large t. The modified Riccati equation associated with 1.2 is introduced explicitly in 7 , but it can be found implicitly already in some earlier papers, for example, 8–10 . Suppose that 4 Abstract and Applied Analysis 1.2 is nonoscillatory i.e., every its nontrivial solution is eventually positive or negative and let h be a positive differentiable function. Consider the substitution v h t w −G t , G t : r t h t Φh′ t , 2.2 where w is a solution of 2.1 . Then v is a solution of the modified Riccati equation v′ c̃ t ( p − 1r1−q t h−q t H v,G t 0, 2.3 with H v,G : |v G| − qΦ−1 G v − |G|, 2.4 Φ−1 s |s|q−2s being the inverse function of Φ, and c̃ t h t [( r t Φ ( h′ t ))′ c t Φ h t ] . 2.5 Note that the function H v,G satisfies H v,G ≥ 0 for every v,G ∈ R and H v,G 0 Hv v,G if and only if v 0. Observe also that Riccati equation 2.1 is a special case of 2.3 with h t ≡ 1, that is, G t ≡ 0. In the investigation of perturbations of the half-linear Euler equation we will need the following criteria for non existence of a proper solution of 2.3 . Recall that a solution v of 2.3 is called proper if it exists on some interval T,∞ . Nonexistence of a proper solution of 2.3 is equivalent to oscillation of 1.2 since it eliminates via the transformation w h−p v G proper solutions of 2.1 . For more details concerning this method, as well as the proof of the next two propositions, we refer to 11 . For the sake of the later application, we will write 2.3 in the form v′ C t ( p − 1R−1 t H v,G t 0, 2.6 with continuous functions C, R, and R t > 0. Proposition 2.2. i If C t ≤ 0 for large t, then 2.6 possesses a (nonnegative) proper solution. In the remaining part of the proposition suppose that lim inf t→∞ |G t | > 0, C t ≥ 0, for large t. 2.7 Denote R t R−1 t |G t |q−2, 2.8 and suppose that ∫∞ R t dt ∞, ∫∞ C t dt < ∞. 2.9 Abstract and Applied Analysis 5 ii Ifand Applied Analysis 5 ii If lim sup t→∞ (∫ t R s ds )(∫∞ t C s ds ) < 1 2q , 2.10 then 2.6 has a proper solution. iii If lim inf t→∞ (∫ t R s ds )(∫∞ t C s ds ) > 1 2q , 2.11 then 2.6 possesses no proper solution. Proposition 2.3. Together with 2.6 consider the equation of the same form v′ D t ( p − 1R−1 t H v,G t 0, 2.12 with the function D satisfying D t ≥ C t for large t. If the (majorant) equation 2.12 has a proper solution, then 2.6 has a proper solution as well. Next, we recall basic properties of solutions of the “critical” half-linear Euler and Riemann-Weber differential equations as presented, for example, in 4 . Consider the halflinear Euler differential equation ( Φ ( x′ ))′ γ tp Φ x 0. 2.13 This equation is nonoscillatory if and only if γ ≤ γp p − 1 /p . In the critical case γ γp, 2.13 has the solution h t t p−1 , and every linearly independent solution is asymptotically equivalent up to a multiplicative factor to the function x t t p−1 /p logt. The Riemann-Weber half-linear differential equation ( Φ ( x′ ))′ [ γp tp μ tp logt ] Φ x 0 2.14 is nonoscillatory if and only if μ ≤ μp 1/2 p − 1 /p p−1. In the critical case μ μp, 2.14 has the so-called principal solution which is asymptotically equivalent up to a multiplicative factor to the function h t t p−1 /p logt, and every linearly independent solution is asymptotically equivalent to the function x t t p−1 /plogtlog log t , see 4 . Finally, we recall the transformation method of the investigation of 1.9 which we extend in a modified form to half-linear equations. The Sturm-Liouville differential equation ( r t x′ )′ c t x 0 2.15 6 Abstract and Applied Analysis is the special case p 2 in 1.2 . The transformation x f t y gives the identity suppressing the argument t f [( rx′ )′ cx ] ( rf2y′ )′ f [( rf ′ )′ cf ] y. 2.16 In particular, if f t / 0, then x is a solution of 2.15 if and only if y is a solution of the equation ( r t f2 t y′ )′ f t [( r t f ′ t )′ c t f t ] y 0. 2.17 Let us emphasize at this moment that we have in disposal no half-linear version of transformation identity 2.16 . Let us denote


Introduction
The half-linear Euler differential equation with the so-called oscillation constant γ p : p − 1 /p p−1 plays an important role in the oscillation theory of the half-linear differential equation Our investigation is mainly motivated by the papers 3-5 . In 4 , perturbations of 1.1 of the form, were investigated. Here, the notation log k t log k−1 log t , log 1 t log t 1.8 is used. It was shown that the crucial role in 1.7 plays the constant μ p : 1 In particular, if n 1 in 1.7 , that is, this equation reduces to the so-called Riemann-Weber half-linear differential equation, then this equation is oscillatory if β 1 > μ p and nonoscillatory in the opposite case. In general, if β j μ p for j 1, . . . , n − 1, then 1.7 is oscillatory if and only if β n > μ p . In 5 , the perturbations of the linear Euler differential equation were investigated and a perturbation was also allowed in the term-involving derivative. More precisely, the differential equation was investigated and it was shown that 1.10 is oscillatory if and only β − αγ p > μ p . For some related results see also 6 .
In this paper we deal with perturbations of the Euler half-liner differential equation in full generality. We consider and we find an explicit formula for the relationship between constants α j , β j in 1.11 which implies non oscillation of this equation. In the last section of the paper we explain why perturbations are just in the above considered form. Our result is based on a new method which consists in transformations of the modified Riccati equations associated with 1.2 . The main result along this line is established in Section 3, while its application to the perturbed Euler equation is presented in Section 4. In the last section we present some remarks and comments concerning the results of our paper. In the next section we recall some essentials of the half-linear oscillation theory.

Preliminaries
It is a well-known fact that many of the results of the linear oscillation theory can be directly extended to half-linear equation 1.2 , even if, in contrast to the case p 2 then 1.2 is a linear equation , the additivity of the solution space is lost and only homogeneity remains. In particular, the so-called Riccati technique, consisting in the relationship between 1.2 and its associated Riccati type equation related to 1.2 by the substitution w rΦ x /x The modified Riccati equation associated with 1.2 is introduced explicitly in 7 , but it can be found implicitly already in some earlier papers, for example, 8-10 . Suppose that 1.2 is nonoscillatory i.e., every its nontrivial solution is eventually positive or negative and let h be a positive differentiable function. Consider the substitution In the investigation of perturbations of the half-linear Euler equation we will need the following criteria for non existence of a proper solution of 2.3 . Recall that a solution v of 2.3 is called proper if it exists on some interval T, ∞ . Nonexistence of a proper solution of 2.3 is equivalent to oscillation of 1.2 since it eliminates via the transformation w h −p v G proper solutions of 2.1 . For more details concerning this method, as well as the proof of the next two propositions, we refer to 11 .
For the sake of the later application, we will write 2.3 in the form with continuous functions C, R, and R t > 0.
In the remaining part of the proposition suppose that and suppose that Abstract and Applied Analysis 12 has a proper solution, then 2.6 has a proper solution as well.
Next, we recall basic properties of solutions of the "critical" half-linear Euler and Riemann-Weber differential equations as presented, for example, in 4 . Consider the halflinear Euler differential equation This equation is nonoscillatory if and only if γ ≤ γ p p − 1 /p p . In the critical case γ γ p , 2.13 has the solution h t t p−1 /p , and every linearly independent solution is asymptotically equivalent up to a multiplicative factor to the function x t t p−1 /p log 2/p t. The Riemann-Weber half-linear differential equation In the critical case μ μ p , 2.14 has the so-called principal solution which is asymptotically equivalent up to a multiplicative factor to the function h t t p−1 /p log 1/p t, and every linearly independent solution is asymptotically equivalent to the function x t t p−1 /p log 1/p tlog 2/p log t , see 4 . Finally, we recall the transformation method of the investigation of 1.9 which we extend in a modified form to half-linear equations. The Sturm-Liouville differential equation

2.22
If β 1 − α 1 /4 1/4, we can repeat the previous transformations and we obtain r e 2 t y β 2 − α 2 /4 t 2 · · · β n − α n /4 t 2 Log 2 n−2 t y 0, 2.23 here e 2 t : e e t . Now it should be clear how one can obtain the result of 5 concerning oscillation of 1.9 . We repeat the transformation of dependent variable y → √ ty followed by the change of independent variable t → e t as long as the condition β j − α j /4 1/4 is satisfied.
As we have emphasized above, we have no half-linear version of the linear transformation identity 2.16 . Consequently, the above procedure cannot be applied directly to 1

Transformation of Modified Riccati Equation
As a starting point of this section we consider the modified Riccati equation in the form where the function H is given by 2.4 , the functions R, C are supposed to be continuous and R t > 0. In this equation, we call the function C the absolute term since this term does not contain the unknown function v . We consider the transformation with a positive differentiable function f and with a function U which we determine as follows. We have again suppressing the argument t, this argument we will suppress also now and then in the next parts of the paper the following: 3.3 Next we determine the function U in such a way that the differential equation for z is again an equation of the form 3.
The terms on the fourth line of the previous computation we will take as the first two terms in the function of the same form as H in 3.1 . Differentiating 3.4 with respect to z, substituting z 0, and setting the obtained expression equal to zero, we obtain Consequently, we obtain the transformed modified Riccati equation

Perturbations of Euler Differential Equation
Now we apply the results of the previous section to the perturbed Euler half-linear differential equation The Riccati equation associated with 4.1 is In order to better understand the next transformation procedure, we recommend the reader to compare it with the linear transformation idea described at the end of the previous section. The transformation with c 1 /t given by 3.8 , that is, c 1 t /t C t with f t t p−1 /p , R r q−1 , and G 0. This means that

4.7
Hence, by a direct computation we obtain where B j β j − α j γ p . 4.9 In 4.6 , with the above given c 1 t /t, we change the independent variable t → e t and the resulting equation is with As the next step, we consider the modified Riccati equation where now , Ω 1 t : Ω 1 e t r e t Γ p r 1 t Γ p .

4.13
We apply the transformation v 2 tv 1 − U 2 , the quantity U 2 is again determined in such a way that we obtain a modified Riccati equation containing H type function for v 2 . Hence, using the results from formula 3.8 , with f t t 1/p , G t Ω 1 t r 1 t Γ p , and 4.14 and using the binomial expansion

4.16
Hence, the absolute term in the resulting modified Riccati equation is We apply the transformation v 3 tv 2 − U 3 to 4.22 . We obtain