On the n-parameter abstract Cauchy problem

Abstract

Let H{(i = 1,2,..., n), be closed operators in a Banach space X. The generalised initial value problemof the abstract Cauchy problem is studied. We show that the uniqueness of solution u :• X of this n-abstract Cauchy problem is closely related to Co-n-parameter semigroups of bounded linear operators on X. Also as another application of Cb-n-parameter semigroups, we prove that many n-parameter initial value problems cannot have a unique solution for some initial values.

Resolvent positive operator and two-parameter abstract Cauchy problem

“…Consider a n-parameter semigroup of operators (X, IR n + , W ) and let H i , i = 1, 2, ..., n, be the infinitesimal generator of the component semigroup {u i (t)} t≥0 of W , i = 1, 2, ..., n. We shall think of (H 1 , H 2 , ..., H n ) as the infinitesimal generator of (X, IR n + , W ). N-parameter semigroups of operators introduced by Hille in 1944 and one can see some of their properties in [5] , [6] and [7]. If W is a C 0 -n-parameter semigroup of operators then by the Hille-Yosida theorem, H i , i = 1, 2, ..., n, is a closed and densely defined operator.…”