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Oscillation Theory for Second Order Dynamic Equations
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- Oscillation for Higher Order Dynamic Equations on Time Scales.
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Oscillation for Higher Order Dynamic Equations on Time Scales
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Oscillation Theory for Second Order Dynamic Equations - CRC Press Book
Description Table of Contents Reviews. Summary The qualitative theory of dynamic equations is a rapidly developing area of research. The theory of time scale, which has recently received a lot of attention, was introduced by Hilger's landmark paper [ 1 ], a rapidly expanding body of the literature that has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus, where a time scale is an nonempty closed subset of the real numbers, and the cases when this time scale is equal to the real numbers or to the integers represent the classical theories of differential or of difference equations.
Many other interesting time scales exist, and they give rise to many applications see [ 2 ]. In this work, knowledge and understanding of time scales and time scale notation are assumed; for an excellent introduction to the calculus on time scales, see Bohner and Peterson [ 2 , 4 ]. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to the papers [ 5 — 20 ]. Recently, Erbe et al. Hassan [ 24 ] studied the third-order dynamic equations and obtained some oscillation criteria, which improved and extended the results that have been established in [ 21 — 23 ].
In this section, we investigate the oscillation of 2. To do this, we need the following lemmas. Lemma 1 see [ 25 ]. Then, 1 implies for ; 2 implies for. Lemma 2 see [ 25 ]. Assume that 5 holds. If and for , then there exists an integer with even such that 1 for and ; 2 if , then there exists such that for and.
Let , and let be the set of integers. Then, Lemmas 1 and 2 are Lemma and Theorem of [ 26 ], respectively. Furthermore, suppose that If is an eventually positive solution of 2 , then there exists sufficiently large such that 1 for ; 2 either for and or. Pick so that on. It follows from 2 that By Lemma 2 , we see that there exists an integer with even such that for and , and is eventually monotone.
We claim that implies. If not, then and , and there exist and a constant such that on. Integrating 2 from into , we get that for Thus, Again, integrating the above inequality from into , we obtain that for It follows from 6 that , which is a contradiction to. The proof is completed.
Assume that is an eventually positive solution of 2 such that for and for and. Then, and there exist and a constant such that where. Since , it follows that is strictly decreasing on. Then, for , On the other hand, we have that for , Thus, there exist and a constant such that Again, Thus, there exists a constant such that Again, Thus, there exists a constant such that The rest of the proof is by induction. Lemma 6 see [ 2 ].
Let be continuously differentiable and suppose that is delta differentiable.