SasymptoticallyperiodicSolutionsofRLFractionalDerivativeIntegralEquation.doc

SasymptoticallyperiodicSolutionsofRLFractionalDerivativeIntegralEquation WANGBing (SchoolofScience,Tianjin,UniversityofTechnologyandEducation,Tianjin300222,China) 【Abstract】TheaimofthispaperistostudytheS-asymptotically-periodicsolutionsofR-Lfractionalderivative-integralequation: 【Keywords】S-asymptotically-periodicmildsolutions;R-LFractionalDerivative-Integral;Equation 0Introduction Fractionalordercalculusisthetheoryofarbitraryorderdifferentialandintegral,itisunifiedwiththeintegerorderdifferentialandintegralcalculus,isageneralizationofthefractionalcalculus.Whentheproposedintegerorderdifferentialandintegralcalculus,fractionalordercalculusisalsotheinevitableisputforward.Thescoreisnotonlyarationalnumber,alsocanbeirrationalfraction,tosomeextent,thefractionalordercalculuscanbecalculatedintointegerorderdifferentialandintegralcalculus.Fractionalorderdifferentialequationswithdeepphysicalbackgroundandrichtheoreticalconnotation,referstothefractionalorderdifferentialequationwithfractionalorderderivativeorfractionalintegralequation.Thefractionalorderderivativeandfractionalintegralsinthephysical,biological,chemical,andotherdisciplineshasbeenwidelyused,suchasdynamicsystemswithchaoticbehavior,chaoticdynamicsystem,aplexmaterialorthedynamicsofporousmedia,suchasrandomwalkwithmemory1-5. 1Preliminaries Tostateourmainresult,weneedsomeelimmentarydeffinitions. Definition1.1(6)LetAbeaclosedandlinearoperator.Ifthereexist0/2，M0，Rsuchthatitsresolventexistoutsidethesector Ontheotherhand,itiseasytoderivethat 【Rreference】 D.Araya.C.Lizama,AlmostautomorphicmildsolutionstofractionaldifferentialequationsJ.NonlinearAnal.,xx,10.004. C.Cuevas,E,Hernandez,psedo-almostperiodicsolutionsforabstractpartialfunctionaldifferentialequationsJ.Appl.Math.Lett.,(xx)doi:10.1016. C.Cuevas,M.pinto,ExistenceanduniquenessofpseudoalmostperiodicsolutionsofsemilinearCauchyproblemswithnondensedomain,NonlinearAnal.,45(xx)73-83. Diagana.G.M.Almostautomorphicsolutionstosemilinearevolutionequations,J.Funct.differ.Equ.,13(2)(xx)195-206. T.diagana.G.M.,N.vanminh,AlmostautomorphicsolutionsofevolutionsequationsJProv.Amer.Math.Soc.,132(xx)3289-3298. ClaudioCuevas,JulioCesardesouza.S-asymptotically-periodicsolutionsofsemilinearfractionalintegral-differentialequationsJ.AppliedMathematics