Bi-continuous N-times Integrated C Semigroups And The Abstract Cauchy Problem

In this article, the Bi-continuous n-times integrated C semigroups and the abstractCauchy problem in Banach space is studied. Using the concepts and properties of theBi-continuous n-times integrated C semigroups, the existence of solution of someabstract Cauchy problems is discussed, when the coefficient is generator of theBi-continuous n-times integrated C semigroups.The paper is divided into four parts．In the first chapter, the concepts and properties of the Bi-continuous n-timesintegrated C semigroups is introduced.In the second chapter, the Bi-continuous n-times integrated C semigroups and thefirst order abstract Cauchy problem is discussed. The existence of the first orderhomogeneous and inhomogeneous abstract Cauchy problem’s solution is proposed.In the third chapter, the Bi-continuous n-times integrated C semigroups and the firstorder abstract Cauchy problem’s solution is stated. The definition of the abstract Cauchyproblem’s strong solution is given. A necessary and sufficient condition of strongsolution’s existence of a class of first order inhomogeneous abstract Cauchy problem isdiscussed. What’s more, the definition of the abstract Cauchy problem’s weak solution isgiven. A sufficient condition of weak solution’s existence of a class of first orderinhomogeneous abstract Cauchy problem is discussed.In the fourth chapter, the Bi-continuous n-times integrated C semigroups and higherorder abstract Cauchy problem is focused on. The sufficient and necessary condition of M generated Bi-continuous n-times integrated C semigroups is presented, when M iscoefficient matrix of a kind of higher order abstract Cauchy problem.