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

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]∖{t1,…,tp},Δx(tk)=x(tk+)−x(tk)=Ik(x(tk)),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.