The Degenerate Resolvent Operator Families And Its Applications

In this thesis we mainly study degenerate -times integrated operater families and its applications to abstract Cauchy problems,and we study the mean ergodicty theorems and the convergence rates of ergodic limits and approximation for K-regularized resolvent families. The paper is divided into four parts.In first part we give out the definition of multivalued linear operators and their basic properties and investigate degenerate a-times integrated semigroups and prove its some basic properties. We also give out the notion of mild degenerate a-times integrated existence family, and prove that the wellposedness of the a-times abstract Cauchy problems is equivalent to mild degenerate a-times integrated existence famliy generated by operator A where A satisfies with some conditions and degenerate a-times integrated semigroups mild generated by A. At last,we conclude the generation theorem of degenerate a-times integrated semigroups.And we prove that degenerate a-times integrated semigroups mild generated by A is equivalent to generated by A .In two part we give out the definition of multivalued linear operators and their basic properties and examine degenerate a-times integrated C-semigroups and its some basic properties. We also write out the notion of mild degenerate a-times integrated C-existence family. Similarly, We prove that the C-wellposedness of the a-times abstract Cauchy problems is equivalent to mild degenerate a-times integrated C-existence famliy mild generated by operator A where A satisfies with some conditions and degenerate a-times integrated semigroups mild generated by A. Finally, we obtain the generation theorem of degenerate a-times integrated C-semigroups.And we prove that degenerate a-times integrated C-semigroups mild generated by A is equivalent to generated by A .In three part we study the ergodicty for K-regularized resolvent operator families including the mean ergodicty,Abel-ergodicity and Cesaro-ergodicity.We prove the mean ergodic theorems of K-regularized resolventoperator families.And we give out the definition of Abel-ergodicity and Cesaro-ergodicity for K-regularized resolvent operator families.Moreover,we give the relationship between the two kinds of ergodicity and their basic properties.In the final part we concern with the convergence rates of ergodic limits and approximation for K- regularized resolvent families for a linear Volterra integral equation.We give the ergodicty for K -regularized resolvent operator families at 0 and we also prove their basic properties by means of K-functional and relative completion.Finally, we obtain some results of the convergence rates of ergodic limits and approximation for K-regularized resolvent families.