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 Advances in Pure Mathematics, 2012, 2, 211-215 http://dx.doi.org/10.4236/apm.2012.23030 Published Online May 2012 (http://www.SciRP.org/journal/apm) α-Times Integrated C-Semigroups Man Liu, Da-Qing Liao, Qian-Qian Zhu, Fu-Hong Wang Department of Basic Education, Xuzhou Air Force College, Xuzhou, China Email: liuman8866@163.com Received February 3, 2012; revised March 12, 2012; accepted March 20, 2012 ABSTRACT The α-times integrated C semigroups, α > 0, are introduced and analyzed. The Laplace inverse transformation for α-times integrated C semigroups is obtained, some known results are generalized. Keywords: α-Times Integrated C Semigroups; Laplace Inverse Transformation; Pseudo-Resolvent Identity 1. Introduction Integrated semigroups are more general than strongly continuous semigroups (i.e., 0 semigroups), cosine operator functions and exponentially bounded distribu- tion semigroups. Integrated exponentially bounded semi- groups were investigated in [1-15]. In this paper, we will introduce and analyze α-times integrated C semigroups, . In Theorem 2.6 we give a necessary and suffi- cient condition for an CCRR to be the pseudo-resol- vent of an α-times integrated C semigroups . At the same time we discuss the Laplace inverse transformation for α-times integrated C semigroups. The results obtained are generalizations of the corresponding results for inte- grated semigroups. StThroughout this paper, X is a Banach space, BXRA is the space of bounded linear operators from X into X, , , DAKA denote the domain, range, core of operator A respectively, . CBX2. Definitions and Properties of α-Times Integrated C Semigroups For 0, ,  denote the integral part and deci- mal part of α respectively. 1sx is well known Gamma function, and 0dsxe x 1s1, . ssFor :0,jR, we definite the function , and 1tjt 1j denotes 0-point Dirac meas- ure 0. For continuous function f, 1 , the definition of convolution product is as following  01tts fsjftd, 1,1sft0. At first we introduce the fractional differential and in- tegral of function. For arbitrary , -order differential of function u denotes 100nDu tt0. For arbitrary , -times cumulative integral of function u denotes 1IujutR. Definition 2.1. Let , a strongly continuous fam- ily St BX10t is called α-times integrated C- semigroups, if StCCSt, and (V)00S2 ; (V) 1101dd, ,0sttsStSsxts rSrCxrtsrSrCxr ts (2.1) If nn N0tSt , then  is called n-times integrated C semigroups. If nn N0tSt 0, and C = I, then is called n-times integrated semigroups. 0StxIf ,  (t) implies , then α- times integrated C semigroups is non-degen- erated. 00x0tSt RIf there exists M > 0, , such that tSt Me, 0t0tSt 0 is called exponentially bounded. , then Definition 2.2. Let , a strongly continuous fam- ily 0tSt BX is called α-times exponentially bounded integrated C semigroups generated by A, if 00S, and there exists , 0M0, such that ,A , tSt Me0t, , and for arbitrary Copyright © 2012 SciRes. APM M. LIU ET AL. 212 xX,dteStxt0t0, we have ,   10CRAx ACx . (2.2) Proposition 2.3. Let A be the generator of an α-times integrated C semigroups , St. Then 1) For all xDA and t, 0 x StAx ,Stx DA ASt (2.3)   0dtSsAxs dxs DA1tStx Cx (2.4) 2) , for all 0tSsxX0t, and and  0dt1tASsxsStxCxdteStxtRe (2.5) Proof. Letting , 0CRx , ,,d,CARuAxuAStxt Fix , then uA 00,dtCCtCeStRuAxt ReR for all Re, and xX,,0At. By the uniqueness theo- rem it follows that ,,CCRuASt StRuAu (2.6) This implies (2.3). Let xDReA, then for all ,  101001001100d1,,d(tCCtttttttttCxeCx tRAxRAAxeStxteSeStxtedeStxte       00)dddddd.tAxtSsAxsSsAxst Then (2.4) follows from the uniqueness theorem. In order to prove (2.5), let xX, and , 0Ret, then by (2.3), (2.4), (2.6) we have 0000d,,d,ttCtCtCCC SsxsRASsxRASsRASsRAStxd,d1sAxsxstCx,,RAAx  (2.7) Noting that C Hence, , and by (2.7), (2.5) follows. CRAxCx dSsxsDAR0tCorolla ry 2 .4. Let . Then Stx DA for all xX0t and . Then is right differenti- able in if . In that case 0tStSx x DA1d,0, .dtStxAStxCxtx Xt : Proposition 2.5. Let ADA X be closed linear operator, when uA, we have , 1) The pseudoresolvent identity  ,, ,,CC CCRACRAC RARA (2.8) 2)  1d,1!,dnnnnCCnRACnRA 1, 2,n    111,,,,,,,,CCCCCCCCRACAARACACARA (2.9) Proof. 1) ACCRA RARACRARA    It follows that  ,, ,,CC CCRACRAC RARA1n 2) We apply the mathematical induction when , by (2.8)  2d,,dCCRACRAnk , (2.9) is complete. i.e., we suppose  1d,1!,dkkkkCCkRAC kRA  then   111121ddd,,dddd1! ,d11!,kkkkCCkkkkCkkCRACRACCkRA CkRA  1nk i.e., it follows . The proof is complete. Theorem 2.6. Let St be a stongly continuous op-erator function, and tSt Me0t 0dtCReStxt, , , letting Re. Then ReCR satisfies the pseudoresolvent CCRRCCRCRC (2.10) if and only if St satisfies (2V). hcondition. Proof. One can easily prove te necessary prove it is sufficiet. Let us thatnLetting Re, Reu, and u. Then the re-solvent equation implies Copyright © 2012 SciRes. APM M. LIU ET AL. 213  CC CCR CRCuuCCCR RuRCuRCuRCuuuu   (2.11)   dtStSst 00CC tuRR eeu (2.12)  0000000000dd1dd1dd1dd1ddCCuttCtut usut usut ustus ttttusRCuRCuueStCteuueeSsCsteeSsCstueeSsCstueeSsCstueeSstCstu11ddRuCtu      Noting that 1dvu0uvev  Then    1dddddddddvCvststCvstSrCvst    (2.13) Moreover, 00 01000100CCus vtrturtsturtRCuRCuueeSstrsee Stsree    11dddddddddddduvut usurttsrtturttrut rtCeeeSsC vvsteeSsCrts rstrt seeS sCrsttrsee SsCrst       (2.14) and 00 0001001000 CRuuu 100 01000100 0dddddd() dddCvut ustrut rttut rRCuueeeSsC vvsttrse eSsCrsttrsee SsCrst   Using (2.14) and (2.15), we obtain  (2.15)  100 0dddCtut rRCuuutrsee SsCrst (2.16) Assertion (2V) follows from (2.13) and (2.16uniqueness of the Laplace transformation. 3. Laplace Inverse Transformation for α-Times Integrated C-Semigroups Lemma 3.1. [16] Let0) and the , :,X , FF is0dt Laplace-type expression: Fett , 00, and ththt Mhe , ,0th, then  1d,2πititeFiTheorem 3.2. Let 0 , then the following condi- tioalns are equivent: 1) A generates an α-times exponentially bounded inte- grated semigroups St ; 0t2) There exists 0, such that ,Afor all u, and , A generates an uA C exponeunded semigroups ntially boTt , and t0 1T t. Proo If A generates an α-times exponentially bounded integrated semigroups 0tSt , then  dSt u Ajt df. 1) 01dttAWrxr WtxCx, oup gBy ([17], Proposition 3.7(a)), 0tWt  is an uAC semigrenerated by A is the exten- tion of A, By ([17],A Proposition 3.11), A. mbnheorem 3.4]2) Coig [18] with [17, T, we can prove  01tdtASrxr StxCx, an d the space of op- Copyright © 2012 SciRes. APM M. LIU ET AL. 214 erator is exchangeable, by Proposition 2.3, This ends the proof . Theorem 3.3. Let A be closed linear operator on X, A, A, an α-times exponentially bounded integrated C semigroups 0t wititesimal geneSt h infinrator A, and ,tMe St0, , then for xDA,  0,CAx1ddtitRSsxse2π1diitiIe R,2πCiAxi (3.1) Proof. Let  0dttSss,  1() ,ACxFxDA by Lemma 3.1   1dtAC00000ddddttttxFeSex SsseSsxsttxt    So F satisfies Lemma 3.1,   ,1d,2πtitCiRAxs ei0dSsx On the other hand, by Theorem 3.2 A generates uAC exponentially bounroups ded semig0tTt . So for xDA, we have  0111d1d2π1d,2πd2πtitiitiTsxseAuACxieARuACxiRCx,itiuAeAi  AIT t.  It follows that St uWhence   0001,d2π1d,.2πtitiitCiSsAI TRuAuAIeACxiIe RAxiddxsuAITsxsdusxstt   And the integral on the right converges uniformly on any bounded intervals. Corollary 3.4. The conditions are same as Theorem 3.3, then for xX, ,1d2π1,d2πitCiitCiRAxStx eiIe RAxi (3.2) rem 3.3 Proof. by Theo0,1dd2πitCiRAxSsxseit (3.3) Then  0,1dd2πtitCiRAAxASsxs ei . By (2.5) and noting that 11d12πititei   Therefore  1,1d1d2π,1d2π,1d,1d2πiitiiCtiiCtiRAAxeCxiRAAxeiRAxAARAxei12πCtiStxCx ei2πiCtieit ining Theorem 3.3 we can prove the next pary 3.5. The conditions are same as The 3. Combrt. Corolla o rem3, then for xX, 202,1dd2π1d,2πtitCiitCiRAxSsxseiIe RAxi (3.4) Proof. by Theorem 3.3  0,1dd2πtitCiRAxSs sei (3.5) integrating (3.5) from 0 to t, i.e.,  002,1ddd2π,1d12πttitCiitCiRAxtsSsxsesiRAxei Noting that ,1d02πiCiRAxi  Copyright © 2012 SciRes. APM M. LIU ET AL. Copyright © 2012 SciRes. APM 215Consequently,  2d2πtCiSsx ei0,1dtiRAxts s. The next part is easy to prove. REFERENCES [1] of Linear Operators and Applica-tial Equations,” Springer-VerlagNew York, 1983. doi:10.1007/978-1-4612-55 A. Pazy, “Semigroups tions to Partial Differen , 61-1 lized Opera- [2] Q. Zhao, G. T. Zhu and D. X. Feng, “Generaness of Singular Distrib-ce in China, Series A, Vol40, No. 5, 2010, pp. 477-495. [3] T.-J. Xiao and J. 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