Acta Universitatis Sapientiae: Mathematica (Dec 2016)

Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

  • Mesmouli Mouataz Billah,
  • Ardjouni Abdelouaheb,
  • Djoudi Ahcene

DOI
https://doi.org/10.1515/ausm-2016-0017
Journal volume & issue
Vol. 8, no. 2
pp. 255 – 270

Abstract

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In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation ddtx(t)=−a (t) h (x (t))+ddtQ (t, x (t−τ (t)))+G (t, x(t), x (t−τ (t))).$${{\rm{d}} \over {{\rm{dt}}}}{\rm{x}}({\rm{t}}) = - {\rm{a}}\;({\rm{t}})\;{\rm{h}}\;({\rm{x}}\;({\rm{t}})) + {{\rm{d}} \over {{\rm{dt}}}}{\rm{Q}}\;({\rm{t}},\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))) + {\rm{G}}\;({\rm{t}},\;{\rm{x}}({\rm{t}}),\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))).$$ We invert this equation to construct a sum of a completely continuous map and a large contraction which is suitable for applying the modificatition of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G.

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