Second-order half-linear delay di erential equations: Oscillation tests

Di erential equations of second-order appear in physical applications such as uid dynamics, electromagnetism, acoustic vibrations and quantum mechanics. In this paper, necessary and su cient conditions are established of the solutions to second-order helf-linear delay di erential equations of the form ( r(w′)μ1 )′ (ι) + q(ι)w(κ(ι)) = 0 , under the assumption ∫∞ ι ( r(s) )−1/μ1ds = ∞. We consider two cases when μ1 > μ and μ1 < μ. Some examples are given to show e ectiveness and applicability of the main result, and state an open problem.

The interest in the study of functional dierential equations comes from their applications to engineering and natural sciences. Equations involving arguments that are delayed, advanced or a combination of both arise in models such as the lossless transmission lines in high speed computers that interconnect switching circuits. Moreover, delay dierential equations play an important role in modeling virtually every physical, technical, and biological process, from celestial motion, to bridge design, to interactions between neurons.
Below, we review some key results in the oscillation of second-order dierential equations which motivated our study.
In very recent paper [7], Bazighifan et al. have studied a second-order dierential equations with several delays and several sub-linear neutral coecients and established several sucient conditions for oscillation of solution of the considered equation. Brands [11] showed that for bounded delays, the solutions to are oscillatory if and only if the solutions to u (ι) + q(ι)u(ι) = 0 are oscillatory.
Recently, Chatzarakis et al. [12] have established sucient conditions for the oscillation and asymptotic behavior of all solutions of the second-order half-linear dierential equations of the form, In an another paper, Chatzarakis et al. [13] have considered (2) and established improved oscillation criteria for (2). Fisnarova and Marik [16] considered the half-linear dierential equation where Φ(ι) = |ι| p−2 ι, p ≥ 2. Karpuz and Santra [19] have established several sucient conditions for the oscillation and asymptotic behavior of the solutions to the equation for dierent ranges of the neutral coecient b. In [24,25], Pinelas et al. have considered the rst order nonlinear neutral delay dierential equations and obtained necessary and sucient conditions for the oscillation of solution of the considered equation for the various ranges of the neutral coecient.
By a solution to equation (1), we mean a function w ∈ C([T w , ∞), R), where T w ≥ ι 0 , such that rw ∈ C 1 ([T w , ∞), R), and w satises (1) on the interval [T w , ∞). A solution w of (1) is said to be proper if w is not identically zero eventually, i.e., sup{|w(ι)| : ι ≥ T } > 0 for all ι ≥ T w . We assume that (1) possesses such solutions. A solution of (1) is called oscillatory if it has arbitrarily large zeros on [T w , ∞); otherwise, it is said to be non-oscillatory. (1) itself is said to be oscillatory if all of its solutions are oscillatory.
By induction, it is easy to verify that for n > 1, Therefore, the point-wise limit of the sequence exists. Let Hence, (8) is a necessary condition. This completes the proof.

Conclusion
The aim of this work is to establish necessary and sucient conditions for the oscillation of solution to second-order half-linear dierential equation. The obtained oscillation theorems complement the well-known oscillation results present in the literature.
In this section, we state one remark and illustrate it by two examples.
We conclude the paper by two examples that show how Remark 3.1 can be applied.
Open problem This work, as well as [3,4,7,13,14,15,19,20,26,27,29], lead us to pose an open problem: Can we nd necessary and sucient conditions for the oscillation of the solutions to second-order neutral dierential equation with several delays and several half-linear neutral coecients for dierent ranges of the neutral coecients.