Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations
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Next, we recall the well-known Sturm half-linear also called the Sturm—Picone comparison theorem.
If Eq. See, e. Now, we formulate the condition which is equivalent to the non-oscillation of Eq. The upcoming lemma describes the connection between the behaviour of solutions of Eq. We apply Theorem 2. From the positivity of s see also [ 1 ] , it is seen that 16 cannot be valid for all large t , i. Hence, there exists a non-negative solution w of Eq. The last lemma contains the opposite implication to the one in Lemma 2. The statement of the lemma follows directly from the half-linear version of the Reid roundabout theorem see [ 3 , Theorem 1. In this section, we formulate and prove our results.
Note that s can be non-positive. The only difference between Eq.
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The reason for this modification is purely technical it leads to more transparent calculations below and it does not mean any restriction consider 6. We will consider the associated adapted generalized Riccati equation in the form of Eq. For given a and function s , let us consider M s from Lemma 2.
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Using 12 in Lemma 2. Hence, 27 is valid.
We use the mean value theorem of the integral calculus to get this estimation. Immediately, from 34 , we obtain see 11 in Lemma 2.
An improved approach for studying oscillation of second-order neutral delay differential equations
It is seen that 35 gives Part I. By contradiction, let us suppose that Eq. From Lemma 2. Hence, we can approach to Next, 46 is valid. To prove 47 , we use the form of Eq. The proof of part I is complete. Part II. Let t 0 be a sufficiently large number.
Based on Lemma 2. We have see Eq. In addition, as in the first part of the proof see 39 , we have. Hence, from 60 and 61 , we obtain cf. To obtain this inequality, it suffices to show see Evidently, 69 is valid, because t 0 is sufficiently large. We show that 66 is valid. We have see 25 , Now we prove For a sufficiently large number t 0 , inequality 67 follows from It remains to prove We have see 58 , 63 , and Using 59 consider 73 , one can see that 74 gives 68 for large t 0.
Finally, applying 66 , 67 , 68 , and 69 in 65 , we have. Considering the fact that a can be arbitrarily large see Definition 2. The proof is complete. Now we describe the connection of Theorem 3. We repeat that our basic motivation comes from papers [ 1 , 7 , 8 , 9 ]. To the best of our knowledge , the strongest known results about non - perturbed conditionally oscillatory half - linear differential equations are proved just in those articles.
Note that , in [ 8 ], the considered type of equations differs from Eq. In [ 7 ] , the general form of Eq. Hence , it is entirely different from the method used in this paper which enables us to cover new types of equations.
The coefficients of equations considered in [ 7 ] has to be restricted in a certain sense. This restriction is removed in the presented results. Theorem 3. It is known see any of papers [ 10 , 11 , 12 , 13 , 14 , 15 ] that this case is not generally solvable. More precisely , for. Keywords: Riemann-Weber half-linear equation; Riccati technique; power comparison theorem; perturbation principle; principal solution.
Summary: We consider the half-linear second order differential equation which is viewed as a perturbation of the so-called Riemann-Weber half-linear differential equation. We present a comparison theorem with respect to the power of the half-linearity in the equation under consideration. Our research is motivated by the recent results published by J. Sugie, N. Yamaoka, Acta Math.
Similar articles:. We make the standing hypothesis that Eq.
As is customary, a solution of Eq. The equation itself is called oscillatory if all its solutions are oscillatory. Remark 1.
https://tsakdeaddispbega.ga All the functional inequalities considered in the sequel are assumed to hold eventually, that is, they are satisfied for all t large enough. His long and detailed memoir [ 1 ] was one of the first contributions in Liouville's newly founded journal and initiated a whole new research into the qualitative analysis of differential equations. Heretofore, the theory of differential equations was primarily about finding solutions of a given equation and so was very limited.
Contrarily, the main idea of Sturm was to obtain geometric properties of solutions such as sign changes, zeros, boundaries, and oscillation directly from the differential equation, without benefit of solutions themselves. Henceforth, the oscillation theory for ordinary differential equations has undergone a significant development. This is caused by the fact that differential equations, especially those with deviating argument, are deemed to be adequate in modeling of the countless processes in all areas of science.
For a summary of the most significant efforts and recent findings in the oscillation theory of FDEs and vast bibliography therein, we refer the reader to the excellent monographs [ 2 — 6 ]. The study of qualitative properties of solutions of such equations has, besides its theoretical interest, significant practical importance. We refer the reader to the monograph [ 7 ] for further applications in science and technology. So far, most of the results obtained in the literature has centered around the special undamped form of Eq.
For instance, in one of the pioneering works on the subject, Grammatikopoulos et al. Later on, Grace and Lalli [ 9 ] extended the results from [ 8 ] to the more general equation. In Ref. Afterward, his approach has been further developed by several authors, see, e. However, it appears that very little is known regarding the oscillation of Eq. Motivated by the results of Ref.