Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations

Abstract

This paper is concerned with the solvability of periodic boundary value problems for nonlinear impulsive functional differential equations (∗) { x ′ ( t ) + a ( t ) x ( t ) = f ( t , x ( t ) , x ( α 1 ( t ) ) , … , x ( α n ( t ) ) ) , t ∈ J ∖ { t 1 , … , t p } , x ( t k + ) − x ( t k ) = I k ( x ( t k ) ) , k = 1 , 2 , … , p , x ( 0 ) = x ( T ) . We obtain sufficient conditions for the existence of at least one solution of problem (∗) at resonance and nonresonance cases, respectively. Examples are presented to illustrate the main results.