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Interval criteria for oscillation of second-order functional differential equations

Authors
Journal
Computers & Mathematics with Applications
0898-1221
Publisher
Elsevier
Publication Date
Volume
50
Identifiers
DOI: 10.1016/j.camwa.2004.12.018
Keywords
  • Quasilinear Differential Equation
  • Oscillation
  • Interval Criteria
  • Generalized Riccati Technique
  • Integral Averaging Method
Disciplines
  • Mathematics

Abstract

Abstract By using averaging functions, new interval oscillation criteria are established for the second-order functional differential equation, ( r ( t ) | x ′ ( t ) | α − 1 x ′ ( t ) ) ′ + F ( t , x ( t ) , x ( τ ( t ) ) , x ′ ( t ) , x ′ ( τ ( t ) ) ) = 0 , t ≥ t 0 that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [ t 0, ∞], rather than on the whole half-line. Our results can be applied to three cases: ordinary, delay, and advance differential equations. In the case of half-linear functional differential equations, our criteria implies that the τ(t) ≤ t delay and Gt( t) ≥ t advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results.

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