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On the positive solutions of a higher order functional differential equation with a discontinuity

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  • John R. Graef
  • Paul W. Spikes
  • Myron K. Grammatikopoulos

Abstract

The n -th order nonlinear functional differential equation [ r ( t ) x ( n − υ ) ( t ) ] ( υ ) = f ( t , x ( g ( t ) ) ) is considered; necessary and sufficient conditions are given for this equation to have: (i) a positive bounded solution x ( t ) → B > 0 as t → ∞ ; and (ii) all positive bounded solutions converging to 0 as t → ∞ . Other results on the asymptotic behavior of solutions are also given. The conditions imposed are such that the equation with a discontinuity [ r ( t ) x ( n − υ ) ( t ) ] ( υ ) = q ( t ) x − λ ,       λ > 0 is included as a special case.

Suggested Citation

  • John R. Graef & Paul W. Spikes & Myron K. Grammatikopoulos, 1982. "On the positive solutions of a higher order functional differential equation with a discontinuity," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 5, pages 1-11, January.
    • Handle: RePEc:hin:jijmms:769678
    DOI: 10.1155/S0161171282000234
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