(A, K)-Stability Of The Family Of Regular Presolving Operators

At the beginning of this century,Lizama proposed the concept of(a,k)-regularized resolvent operator family.It deals with a class of Volterra integial equations which are widely used in physics,engineering technology and biology,etc.It has important theoretical and practical significance.The stability of(a,k)-regularized resolvent operator family is investigated in this paper,including GGP type theorem,weak Lp type theorem,ABLV type theorem and ergodic theorem of(a,k)-regularized resolvent operator family.This paper is divided into four parts.In Chapter 1,we introduce the history and background of the(a,k)-regularized resolvent operator family,as well as the existing conclusions about the operator semigroup and resolvent operator family,including GGP type theorem,weak LT type theorem,ABLV type theorem and eigodicity.In Chapter 2,we give the definitions and related properties of(a,k)-regularized resolvent operator family,and other preparatory knowledge which need to be used in this paper.In Chapter 3,we study the stability theory of(a,k)-regularized resolvent operator family.This chapter is divided into three sections.In section 1,we mainly discuss GGP type theorem of(a,k)-regularized resolvent operator family.Sufficient conditions for the uniform stability to(a,k)-regularized resolvent operator family are given by the theory of Hilbert space and the theory of resolvent as well as the method of complex analysis,and some corollaries are derived.In section 2,we research weak Lr type theorem of(a,k)-regularized resolvent operator family.The new sufficient conditions for uniform stability to(a,k)-regularized resolvent operator family are given by empolying the GGP type theorem and the adjoint theory of operator family,and some corollaries are obtained.In section 3,we study ABLV type theorem of(a,k)-regularized resolvent operator family in Banach space.Sufficient conditions for strong stability to(a,k)-regularized resolvent operator family are obtained in Banach space by constructing a new operator-valued function,Cauchy theorem and Kiemann-Lebesgue lemma.In Chapter 4,we investigate the ergodicity of(a.k)-regularized resoiven,operator family.The Abel-ergodicity and Cesaro-ergodicity of unbounded(a,k)-regularized resolvent operator family are studied by direct sum decomposition and the theory of operators as well as the method of complex analysis,which generalize the results for operator semigroup and resolvent operator family.