Perturbed generalized half-linear Riemann – Weber equation – further oscillation results

We establish new oscillation and nonoscillation criteria for the perturbed generalized Riemann–Weber half-linear equation with critical coefficients (Φ(x′))′ + ( γp tp + n ∑ j=1 μp tLogj t + c̃(t) ) Φ(x) = 0 in terms of the expression 1 logn+1 t ∫ t c̃(s)sp−1 Log ns log 2 n+1 s ds. The obtained criteria complement results of [O. Došlý, Electron. J. Qual. Theory Differ. Equ., Proc. 10’th Coll. Qualitative Theory of Diff. Equ. 2016, No. 10, 1–14].


Introduction
Consider the half-linear differential equation of the form where r, c are continuous functions, r(t) > 0 and t ∈ [T, ∞) for some T ∈ R. The terminology half-linear comes from the fact that the solution space of (1.1) is homogenous, but generally not additive for p = 2.In the special case p = 2 this equation reduces to the linear Sturm-Liouville differential equation (r(t)x ) + c(t)x = 0. (1.2) In this paper we deal with oscillatory properties of equations of the form (1.1).It is well known that the classical linear Sturmian theory of (1.2) can be naturally extended also to (1.1), see [8].
Corresponding author.Email: fisnarov@mendelu.cz In particular, (1.1) is called oscillatory if all of its solutions are oscillatory, i.e., it has infinitely many zeros tending to infinity.In the opposite case all solutions of (1.1) are nonoscillatory, i.e., they are eventually positive or negative and (1.1) is said to be nonoscillatory.Let us emphasize that oscillatory and nonoscillatory solutions of (1.1) cannot coexist.
If we suppose that (1.1) is nonoscillatory, one can study the influence of the perturbation c on the oscillatory behavior of the equation of the form (r(t)(Φ(x )) + (c(t) + c(t))Φ(x) = 0. (1. 3) The concrete (non)oscillation criteria measure the positiveness of the function c (generally of arbitrary sign).If c is "sufficiently positive" then the perturbed equation (1.3) becomes oscillatory, if c is negative or "not too much positive", then (1.3) remains nonoscillatory.This approach is sometimes referred to as the perturbation principle and leads, e.g., to the Hille-Nehari type (non)oscillation criteria for (1.3) which compare limits inferior and superior of certain integral expressions with concrete constants.These integral expressions are usually either of the form where h is a solution of (1.1) (or a function which is asymptotically close to a solution of (1.1)) and R = rh 2 |h | p−2 .Criteria of this type can be found in [1-3, 5-7, 9, 10, 13], see also the references therein.Note that the divergence or convergence of the integral ∞ R −1 (t) dt is closely connected with the so called principality of the solution h of (1.1), see [4,8] for details.
Let us summarize the known results concerning the above mentioned criteria which apply to perturbations of the Euler and Rieman-Weber type equations.Denote An example of a nonoscillatory equation of the form (1.1) is the half-linear Euler type equation with the critical coefficient γ p (called also the oscillation constant) and the second one (linearly independent of h 1 ) is asymptotically equivalent to h 2 (t) = t p−1 p log 2 p t, see [11].Note that the criticality of γ p in (1.6) means that if we replace γ p in (1.6) by another constant γ, then (1.6) is oscillatory for γ > γ p and nonoscillatory for γ < γ p .It was shown in [7] that the perturbed Euler type equation where E(t) = log t ∞ t c(s)s p−1 ds.Došlý and Řezníčková [9] proved the same couple of nonoscillation and oscillation criteria with E(t) =  [12].The (non)oscillation criteria for the perturbed equation were formulated in terms of which complies with (1.4) taking h(t) = h 1 (t).The relevant nonoscillation criterion for (1.9) was proved in [2] and oscillatory criterion in [10].The case which corresponds to (1.5) and to the second function h 2 remained open.
Recently, the criteria from [2,10] were generalized in [3] to perturbations of the following generalized Riemann-Weber half-linear equation with critical coefficients where n ∈ N and Elbert and Schneider in [12] derived the asymptotic formulas for the two linearly independent nonoscillatory solutions of (1.10).These solutions are asymptotically equivalent to the functions Došlý in [3] studied the equation and proved the following statement.
(ii) Suppose that there exists a constant γ > 2γ p p(p−2) 3(p−1) 2 such that c(t)t p log 3 t ≥ γ for large t and Then (1.12) is oscillatory.
Observe that the integral expression from Theorem A relates to (1.4) with h(t) = h 1 (t) from (1.11).If n = 1, then (1.12) reduces to (1.9) and the criteria from Theorem A reduce to that obtained in [2,10].
The aim of this paper is to complement Theorem A (and also the corresponding results of [2,10] in case n = 1).We utilize the second function h 2 from (1.11) and find a related couple of criteria for equation (1.12) formulated in terms of the expression n+1 s ds which corresponds to (1.5).

Auxiliary statements
In this section we present the known statements which will be used in the proofs of our main results in the next section.Denote and recall that q = p p−1 is the so called conjugate number of p.The following statement comes from [13].

Theorem B.
Let h be a function such that h(t) > 0 and h (t) = 0, both for large t.Suppose that the following conditions hold: for some α > 0, then (1.1) is nonoscillatory.
The following theorem was proved in [6].
Theorem C. Let h be a positive continuously differentiable function satisfying the following conditions: Then (1.1) is oscillatory.
In the following lemma we summarize some technical facts which are either evident or were derived in [3].
Lemma 2.1.For n ≥ 2 and large t we have Moreover, for h(t) = t p−1 p Log 1 p n t and the operator defined in (1.12) we have as t → ∞. (2.7)

Main results
Our main result concerning nonoscillation of (1.12) reads as follows.
Proof.We prove the statement with the use of the function n+1 t in Theorem B. By a direct differentiation (and using Lemma 2.1) we have By a direct differentiation (and using Lemma 2.1 again) we obtain Observe that the expression in the square brackets can be rearranged as follows: Denote by A(t) the expression in the curly brackets.By a direct computation with using the fact that Next, denote Using the power expansion we obtain Next observe that if at least one of the indices i, j, k is greater than one, then Hence we can write B(t) in the form as t → ∞.From (3.3) and (3.4), we obtain as t → ∞.Summarizing the above computations, we have as t → ∞.Consequently, for the operator L RW defined in (1.12) we have as t → ∞.In order to check conditions (2.2), express R(t) and G(t) from (2.1): and and hence Finally, we show that conditions (2.3) and (2.4) hold.To this end, let ε ∈ (0, All assumptions of Theorem B are true, which finishes the proof.
To obtain the oscillatory counterpart of Theorem 3.

Remark 3 . 4 .
If α = 1 2 in Theorem 3.1, then 2µ p (−α + √ 2α) = µ p , 2µ p (−α − √ 2α) = −3µ pand the constants from (3.1) and (3.2) in Theorem 3.1 reduce to the constants in Theorem A, part (i).The generalization for α = 1 2 is due to Theorem B. Note also that the constants in the nonoscillatory part of Theorem A could be generalized in the same way by utilizing[13, Theorem 3.2]  in the proof of Theorem A.