Two Types Of Two-parameter Semigroups And N-th Order Abstract Cauchyproblem

It has been found that operator semigroups is a solution to solve the correspond-ding abstract cauchy problems. In Banach space, according to the one-parameter C-s-emigroups, one-parameter sine semigroups and two-parameterC0-semigroups, thispaper has extended the one-parameterC0-semigroups to the two-parameter C-semigr-oups and the one-parameter sine semigroups to the two-parameter sine semigroups.What’s more, it has given the application of two-parameter C-semigroups and two-par-ameter sine semigroups in the aspect of higher order abstract cauchy problems.Firstly, this paper contains the definitions and properties of one-parameter C-sem-igroups, one-parameter cosine semigroups, one-parameter sine semigroups, sparsesets and dense sets. Moreover, this chapter has given the definition of a more generaltwo-parameter C-semigroups and proof infinitesimal generator of two-parameterC0-semigroups.Secondly, two parameter C-semigroups’ total differential, partial differential, ind-ex of bounded have been studied.Thirdly, a equivalent description of the one-parameter sine semigroups has beenfound by using generalization from the one-parameter sine semigroups to the two-pa-rameter sine semigroups. Then the relationship between one-parameter sine semigrou-ps to the two-parameter sine semigroups has been found.The fourth chapter also hasgiven the promotion from the theory of one-parameter sine semigroups in laplace toone of two-parameter sine semigroups.Finally, this paper also contains the application of two-parameter C-semigroupsand two-parameter sine semigroups in the aspect of higher order abstract cauchyproblems. In the meantime, the two-parameter C-semigroups’ stability has been studi-ed.