Geodesic
A survey and some new results on the existence of solutions of IPBVPs for first order functional differential equations
Applications of Mathematics, Tome 54 (2009) no. 6, pp. 527-549.

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This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation $$ \begin{cases} x'(t)=f(t,x(t),x(\alpha _1(t)),\cdots ,x(\alpha _n(t))) \text {for a.e.} \ t\in [0,T], \Delta x(t_k)=I_k(x(t_k)), \ k=1,\cdots ,m, x(0)=x(T). \end{cases} $$ We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin's continuation theorem. Examples are presented to illustrate the main results.
DOI : 10.1007/s10492-009-0032-6
Classification : 34B10, 34B15, 34K10, 34K45, 47N20
Keywords: periodic boundary value problem; impulsive differential equation; fixed-point theorem; growth condition 
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     title = {A survey and some new results on the existence of solutions of {IPBVPs} for first order functional differential equations},
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Liu, Yuji. A survey and some new results on the existence of solutions of IPBVPs for first order functional differential equations. Applications of Mathematics, Tome 54 (2009) no. 6, pp. 527-549. doi : 10.1007/s10492-009-0032-6. http://geodesic.mathdoc.fr/articles/10.1007/s10492-009-0032-6/

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