Asymptotic Properties Of Fractional Presolvable Families And Functional Calculus Of Generator

This thesis focuses on the following three aspects:First,we investigate the asymptotic stability of fractional resolvent families on Banach spaces and ordered Banach spaces.We show that an α-times resolvent family S_α（t）with generator A is uniformly Abel-stable if and only if 0 ∈ρ（A）;and if in addition S_α（t）is analytic and bounded,then S_α（t）is uniformly stable with ‖S_α（t）‖=O(t^-α)（t→∞）.For a bounded positive α-times resolvent family on an ordered Banach space,we show that it cannot be uniformly stable if α∈（1,2）;when α∈（0,1）,0 ∈ρ（A）implies the same decay rate t^-α.Several results on strong stability are also given by using contour integrals,Tauberian theorems and subordination principles.Second,we investigate the Bernstein functional calculus for the generators of fractional resolvent families on Banach spaces and ordered Banach lattices.If the generator of a bounded α-times fractional resolvent family on a Hilbert space is a normal operator,then for every completely Bernstein function f,-f（-A）defined by Bernstein functional calculus,extend functional calculus and spectral measure integral are coincide and also generates an α-times fractional resolvent family.A subordination principle is established for the Bernstein functional calculus of the generator.And we prove that if A is an ω-accretive operator,then for every Bernstein function f,f（A）also be an ω-accretive operator.We also show that if the generator A is resolvent positive and f is a complete Bernstein function,then-f（-A）is resolvent positive.If A generates a C0-semigroup and f is complete Bernstein function,we show that the C0-semigrolp generated by-f（-A）preserves some asymptotic properties of the semigroup generated by A.Third,we consider the decay estimate and integral representation of solution to abstract Cauchy problem with distributed order Caputo derivative:（?）where X is a Banach space and A is an invertible sectorial operator on X.We also consider the special cases that X is a Hilbert space and A is an invertible self-adjoint operator.We first investigate analyticity and decay estimates of the solution to（0.0.2）with operator A=λ>0,then by using these results we obtain the integral representation of the solution to the abstract problem.We also present some results on maximal regularity,strict regularity and Holder continuity of fractional resolvent families and distributed resolvent families.