Anti-periodic boundary value problems for first order impulsive functional differential equations

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1、AppliedMathematicsandComputation186(2007)45–53www.elsevier.com/locate/amcAnti-periodicboundaryvalueproblemsforfirstorderqimpulsivefunctionaldifferentialequationsWeiDing*,YepengXing,MaoanHanDepartmentofMathematics,ShanghaiNormalUniversity,Shanghai200234,ChinaAbstr

2、actThispaperconsidersexistenceofsolutionsforaclassoffirstorderfunctionaldifferentialequationwithanti-periodicboundaryvalueconditions.Weintroducenewconceptofloweranduppersolutionswhichisquitedifferentfromtheformerone.Andwepresentthatmonotoneiterativetechniquecouple

3、dwithloweranduppersolutionsisstillvalid.Ó2006PublishedbyElsevierInc.Keywords:Impulsivefunctionaldifferentialequation;Monotoneiterativetechnique;Lower(upper)solution;Anti-periodicboundaryvalueproblem1.IntroductionAsanimportanttooltostudypracticalproblemsofbiology

4、,engineering,andphysics,impulsivedifferentialequationshascausedmoreandmoreattention[1].Monotoneiterativemethodcoupledwithupperandlowersolutionsisausefulmethodtosolveperiodicboundaryvalueproblems(PBVP)forimpulsivedifferentialequa-tionsandtherehavemanypapersconcern

5、edwiththis,suchas[2–5]andthereferencestherein.However,inphys-icalfield,manyproblemscomedowntoanti-periodicboundaryvalueproblems.Byourknowledge,thepapersconcernedaboutthisisfew,[6–8].Inthispaper,weconsiderthefollowinganti-periodicboundaryvalueproblem:8y0ðtÞ¼gðt;y

6、ðtÞ;yðwðtÞÞÞ;t6¼t;t2J¼½0;T;>>k>>yð0Þ¼yðTÞ;>:yðtÞ¼yð0Þ;t2½r;0;2++whereg2C(J·R,R),w(t)2C(J,J),tr6w(t)6t,t2Jandtk0,0

7、yShanghaiOutstandingDisciplineLeaderProject(06XD14034),ShanghaiMunicipalEducationCommission(06ZZ16),andShanghaiLeadingAcademicDisciplineProject(T0401).*Correspondingauthor.E-mailaddresses:dingwei@shnu.edu.cn,dingwei@shnu.com(W.Ding).0096-3003/$-seefrontmatterÓ2

8、006PublishedbyElsevierInc.doi:10.1016/j.amc.2006.07.08746W.Dingetal./AppliedMathematicsandComputation186(2007)45–53Remark(i)Ifw(t)=t,thenEq.(1)isanordinarydifferentialequatio