Let H{(i = 1,2,..., n), be closed operators in a Banach space X. The generalised initial value problem Q£ufa,t2,...,ti,...,tn) = Haifa,...,tn), <i£(0,Ti] i = l , 2 , . . . , n | u(0) = x, x e n D{Hi), of the abstract Cauchy problem is studied. We show that the uniqueness of solution u : [0,Ti] x [0,T2] x • • • x [0,Tn] -• 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.

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