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SomeResultsonIntegrationofSubdi erentialsZiliWuandJaneJ.YeJuly21,1999DepartmentofMathematicsandStatistics,UniversityofVictoria,Victoria,B.C.,CanadaV8W3P4Keywordsandphrases.Nonsmoothanalysis,integrationofsubdi erentials,Clarkesubdi erential,Michel-Penotsubdi erential,lowerDinisubdi erential,Fr echetsubdi erential,m-subdi erentialand@ -subdi erential.1IntroductionIntegrationofsubdi erentialsisafundamentalprobleminnonsmoothanalysis.Theproblemiswhetherornottheconditionthatthesubdi erentialoffcontainsthesubdi erentialofgimpliesthatfandgdi eronlybyaconstant.Forthisproblem,perhapstheearlistresultisTheorem24.9inRockafellar[16],whichassertsthatiffandgaretwoclosedproperconvexfunctionssuchthat@cg(x) @cf(x)8x2Rn;where@cisthesubdi erentialinthesenseofconvexanalysis,thenfandgdi erbyaconstant.Someprogressonthestudyofintegrationofsubdi erentialsfornonconvexfunc-tionshasbeenmadeoverthelastdecadeorso.Rockafellar[17]provedthatiff;g:Rn!RarelocallyLipschitzandfisClarkeregular,andiftheClarkesubdif-ferential@g(x) @f(x)8x2Rn,theng(x)=f(x)+CforsomeconstantC.Thisresultwasgeneralizedtoupper-upperandupper-lowerregularfunctionsinanyBa-nachspacebyCorreaandThibault[11].SomeclassesofnonconvexfunctionswhichcanbedeterminedbytheirClarkesubdi erentials(i.e.,anytwofunctionsfromthesameclassdi eronlybyaconstantiftheirClarkesubdi erentialscoincide)have1alsobeenfound.Qi[15]showedthataprimalfunctionwhosedomainisconnectedisdeterminedbyitsClarkesubdi erentialuptoanadditiveconstant.Poliquin[14]foundaclassoffunctionscalledprimallowernicefunctionswhichcanbedeterminedbytheirClarkesubdi erentials(hencetheirproximalsubdi erentialssincetheyco-incideforthisclassoffunctions)uptoaconstant.Otherresultsonintegrationofsubdi erentialsincludethoseofBorwein[2],CorreaandJofre[10],ThibaultandZagrodny[18].AlthoughmostoftheseintegrationresultsconcerntheClarkesubdi erential,in-tegratingsubdi erentialsotherthanthatofClarkeisalsoimportant.ClarkeandRedhe er[7]provedthatifg:Rn!( 1;1]isalowersemicontinuousfunction,fisacontinuouslytwicedi erentiablefunctiononRnand@ g(x) @ f(x),where@ denotestheproximalsubdi erential,thenf(x)=g(x)+CforsomeconstantC.Theyalsoindicatedthatiffisdi erentiableand@Fg(x) @Ff(x),where@FistheFr echetsubdi erential,thenfandgdi eronlybyaconstant.Inthispaper,westudyprimarilytheintegrationof@ -subdi erentials,aclassofsubdi erentialswhichincludesthelowerDinisubdi erential,theFr echetsubdif-ferentialandthem-subdi erential.Weprovethatiflowersemicontinuousfunctionsf:X!R,g:X!( 1;1]aresuchthatg fislowersemicontinuousonanonemptyopenconvexsubsetUofX,andforeachxinU,@ f(x)[@ ( f)(x)isnonemptyandbounded,thengdi ersfromfbyaconstantonUifandonlyifforsomesubdi erential@0andanyxinU,@ g(x) @ f(x)forxwith@ f(x)6=;,@ ( f)(x) @ ( g)(x)forxwith@ f(x)=;and@0(g f)(x) @ (g f)(x),where@0hisasubdi erentialhavingthepropertythat@0h(x) f0gonUimpliesthathisconstantonU.Thisresultservestounifyandextendseveralresultsintheliterature.Inparticular,aconsequenceoftheaboveresultisageneralizationofRockafellar’sresultonintegrabilityofregularfunctions,i.e.,iffisalocallyLipschitzfunctionandgisanextended-valuedlowersemicontinuousfunctiononanAsplundspaceoraseparableBanachspaceandiffisMichel-Penotregular(weakerthanClarkeregular)and@ g(x) @ f(x),where@ denotesthelowerDinisubdi erential(weakerthan@g(x) @f(x))forallx,thenfandgdi erbyaconstant.Wealsoobtainsimilarresultsforaseparatelyregularfunction,i.e.,abivariatefunctiononaproductoftwoAsplundspaces(ortwoseparableBanachspaces)X YwhichisClarkeregularasafunctionofxandyseparately.OurmainresultalsoextendsClarkeandRedhe er’sresultbyallowingftobeanylocalLipschitzfunctionwhoseproximalsubdi erential2orFr echetsubdi erentialisnonemptyeverywhere.Weorganizethepaperasfollows.Inx2,wegiveconditionsunderwhichsome@ -subdi erentialsarenonemptyandsomeresultsonthecalculusofthesesubdi er-entials.Inx3,wediscussourintegrationresultsindetail.Throughoutthispaper,XisarealBanachspacewhoseopenunitballanddualspacearedenotedbyBandX respectively.HdenotesarealHilbertspace.2Calculusofsubdi erentialsWebrie yreviewsomewell-knownnotionsofsubdi erentials.Letf:X!RbeLipschitzofrankLnearx2X. TheClarkederivativeoffatxinthedirectionvisde nedbyf (x;v):=limsupy!xt!0+f(y+tv) f(y)t: TheMichel-Penotderivativeoffatxinthedirectionvisde nedbyf (x;v):=supylimsupt!0+f(x+ty+tv) f(x+ty)t: TheClarkesubdi erentialoffatxistheset@f(x):=f 2X :h ;vi f (x;v)8v2Xg: TheMichel-Penotsubdi erentialoffatxistheset@ f(x):=f 2X :h ;vi f (x;v)8v2Xg:SincetheClarkeandMichel-Penotderivativesarebothsublinearfunctionsofvandtheirvaluesf (x;v)andf (x;v)areboundedbyLkvk,theircorrespondingsubdif-ferentialsarenonempty.Letf:X!R[f1gbelowersemicontinuous(l.s.c.)atx2domf:=fx2X:f(x)1gandvbeinX. ThelowerDiniderivativeoffatxinthedirectionvisde nedbyf (x;v):=liminft!0+f(x+tv) f(x)t:3 ThelowerDinisubdi erentialoffatxistheset@ f(x):=f 2X :h ;vi f (x;v)8v2Xg: TheFr echetsubdi erentialoffatxistheset@Ff(x):=f 2X :f(y) f(x)+ (ky xk) h ;y xiforsome (t)withlimt!0+ (t)t=0; 0andanyyinx+ Bg:Tounifycertainnotionsofsubdi erentials,wede nethem-subdi erent