PHILOS' INEQUALITY ON TIME SCALES AND ITS APPLICATION IN THE OSCILLATION THEORY

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KARPUZ B.

MATHEMATICAL INEQUALITIES & APPLICATIONS, vol.21, no.4, pp.1029-1046, 2018 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 21 Issue: 4
  • Publication Date: 2018
  • Doi Number: 10.7153/mia-2018-21-70
  • Journal Name: MATHEMATICAL INEQUALITIES & APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1029-1046
  • Keywords: Asymptotic behaviour, dynamic equations, higher-order, oscillation, time scales, NEUTRAL DIFFERENTIAL-EQUATIONS, DELAY DYNAMIC EQUATIONS, SUFFICIENT CONDITIONS, BEHAVIOR
  • Dokuz Eylül University Affiliated: Yes

Abstract

In [Bull. Acad. Polon. Sci. Ser. Sci. Math. 29 (1981), no. 7-8, 367-370], Philos proved the following result: Let f : [t(0), infinity)(R )-> R it be an n-times differentiable function such that f((n))(t)<= 0 (not equivalent to 0) and f(t) > 0 for all t <= t(0). If f is unbounded, then f(t) >= lambda t(n-1)/(n-1)! f((n-1))(t) for all sufficiently large t, where lambda is an element of (0, 1)(R). In this work, we first present time scales unification of this result. Then, by using it, we provide sufficient conditions for oscillation and asymptotic behaviour of solutions to higher-order neutral dynamic equations.