分数阶脉冲微分方程初值问题解的存在性.pdf

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1、CHINESEJOURNALApr.2012Vol.29No.2OFENGINEERINGMATHEMATICSArticleID:1005-3085(2012)02-0278-13ForcedOscillationforSecondOrderEllipticEquationswithMixedNonlinearities¤YANGGe-feng1,XUZhi-ting2(1-SchoolofCiscoInformatics,GuangdongUniversityofForeignStudies,Guangzhou51000

2、6;2-SchoolofMathematicalSciences,SouthChinaNormalUniversity,Guangzhou510631)Abstract:Inthispaper,theoscillationcriteriaareestablishedforaforcedsecondordermixednon-linearitiesellipticpartialdi®erentialequationwithdamping.Incontrasttotheexistingresultsintheliterature

3、,ourresultsincludeandimprovesomerecentoscillationcriteriaforsecondorderforcedordinarydi®erentialequationsandcertainsecondordersuperlinearellipticdi®erentialequations.Keywords:oscillation;forced;mixednonlinearities;ellipticequation;dampingClassi¯cation:AMS(2000)35B0

4、5;35J15;35J60CLCnumber:O175.25Documentcode:A1IntroductionConsidertheforcedsecondordermixednonlinearitiesellipticpartialdi®erentialequationwithdamping¡¢•®divA(x)jryj®¡1ry+b(x);jryj®¡1ry+C(x;y)=e(x);x2•(r);(1)0withXmC(x;y)=c(x)jyj®¡1y+c(x)jyj®i¡1y;ii=1where®>¢¢¢>®>®>

5、®>¢¢¢>®>0;x=(x;x;¢¢¢;x)2RN;1kk+1m12NhXNi1¡¢22jxj=xi;r=@=@x1;@=@x2;¢¢¢;@=@xN;i=1theoperatorh;idenotestheusualscalarproductinRN,and•(r)=fx2RN:jxj¸rgfor00somer0¸0.Throughoutthispaperweassumethat(A1)A(x)=(aij(x))N£Nisarealsymmetricpositivede¯nitematrixfunctionwith1+¹ai

6、j2Cloc(•(r0);R)foralli;j=1;2;¢¢¢;N;¹2(0;1);¹(A2)b(x)=(b1(x);b2(x);¢¢¢;bN(x))withbi2Cloc(•(r0);R)foralli=1;2;¢¢¢;N;Received:24Nov2010.Biography:YangGefeng(Bornin1974),Male,Master,Lecturer.Accepted:20June2011.Research¯eld:di®erentialequations.¤Foundationitem:TheDocto

7、ralProgramofHigherEducationofChina(20094407110001).NO.2YangGefeng,XuZhiting:ForcedOscillationforSecondOrderEllipticEquations279¹(A3)c;ci;e2Cloc(•(r0);R);i=1;2;¢¢¢;m.Afunctiony2C1+¹(•(r);R)withthepropertyajryj®¡1@y=@x2C1+¹(•(r);R)forloc0ijiloc0alli;jissaidtobeasolut

8、ionof(1)in•(r0),ify(x)satis¯es(1)forallx2•(r0).Forexistenceofsolutionsof(1),wereferthereadertothemonograph[1].Wehererestrictourattentiononlytothe