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

Abstract

Abstract In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation d dt x ( t ) = − a   ( t )   h   ( x   ( t ) ) + d dt Q   ( 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.