Abstract
Throughout this work, new criteria for the asymptotic behavior and oscillation of a class of odd-order delay differential equations with distributed deviating arguments are established. Our method is essentially based on establishing sharper estimates for positive solutions of the studied equation, using an iterative technique. Moreover, the iterative technique allows us to test the oscillation, even when the related results fail to apply. By establishing new comparison theorems that compare the nth-order equations with one or a couple of first-order delay differential equations, we obtain new conditions for oscillation of all solutions of the studied equation. To show the importance of our results, we provide two examples.
Highlights
We study the asymptotic and oscillatory behavior of odd-order delay differential equations with distributed deviating arguments of the form α 0 Z
Oscillatory behavioral nature of solutions of various classes of neutral and delay differential equations is of great interest, and often encountered in applied problems in natural sciences, technology, and engineering, see [1,2]
Using the principles of comparison, we obtained new oscillation criteria that can be used to test for oscillations, even when the previously known criteria fail to apply
Summary
We study the asymptotic and oscillatory behavior of odd-order delay differential equations with distributed deviating arguments of the form α 0 Z r ( ξ ) x ( n −1) ( ξ ). Oscillatory behavioral nature of solutions of various classes of neutral and delay differential equations is of great interest, and often encountered in applied problems in natural sciences, technology, and engineering, see [1,2]. It has been noticed the rising interest of many researchers and papers in studying the qualitative properties of different classes of linear and non-linear differential equations, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Li and Rogovchenko [27] investigated asymptotic behavior of solutions to an odd-order delay differential equation α. All functional inequalities and properties including increasing, decreasing, positive, etc. are assumed to hold eventually, that is, they are satisfied for all ξ large enough
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