Positive solutions of periodic boundary value problems for nonlinear first-order impulsive differential equations

Abstract

Motivated by Liu [Y. Liu, Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations, J. Math. Anal. Appl. 327 (2007) 435–452], Li et al. [X. Li, X. Lin, D. Jiang, X. Zhang, Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects, Nonlinear Anal. 62 (2005) 683–701] and Liu and Ge [Y. Liu, W. Ge, Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients, Nonlinear Anal. 57 (2004) 363–399], this paper is concerned with the periodic boundary value problem (PBVP for short) for the nonlinear impulsive differential equation { x ′ ( t ) + a ( t ) x ( t ) = f ( t , x ( t ) ) , a.e. t ∈ [ 0 , T ] ∖ { t 1 , … , t p } , Δ x ( t k ) = x ( t k + ) − x ( t k ) = I k ( x ( t k ) ) , k = 1 , … , p , x ( T ) = x ( 0 ) . We obtain new sufficient conditions for the existence of at least one, two or three positive solutions of the above PBVP, respectively. Examples are presented to illustrate the main results.