The Application Of A Class Linear Semigroup In The Abstract Cauchy Problem

In this article， the application of a class linear operator semigroups (local-timeintegrated C-semigroups and local m-time integrated C-semigroups) in the abstractCauchy problem on Banach Spaces is be explored．On the base of definition andproperties of these linear operator semigroups，the conditions of the existence anduniqueness about Cauchy problem solution will be illustrated．The paper is divided intothree parts．In the first part， the definition of local-time integrated C-semigroups isintroduced． Combined with the properties of-time repeated integral and convolutionformula， some theorem of local-time integrated C-semigroups are also obtained．In the second part， definition between local-time integrated C-semigroups isintroduced．With the definition of classical solution， the existence and uniquenessabout Cauchy problem solution is illustrated．In the third part，with the concept and properties of local m-time integratedC-semigroups，their relations with C-wellposedness of an abstract Cauchy problemon a finite interval are investigated．The result show that a closed linear operator A (sub)generates local m-time integrated C-semigroups if and only if the associatedACP m1is C-wellposedness．...