On the uniqueness problem of differential equations in normed spaces^†

^†This work was supported by the U.S.Army Research Office, Durham, N.C.

Abstract

Some uniqueness theorems for the Cauchy problem

u? = f(t, u),u(0) = u₀

in an arbitrary normed space are presented. These theorems extend the results by Diaz and Weinacht from an inner product space to a normed space X and improve upon their main assumptions. One consequence of these theorems is interesting in connection with a result by Perron on non-uniqueness. By considering f defined on [O,T]×D, where D is a subset of X, our uniqueness theorems can be applied to initial-boundary-value problems. Examples of such applications to classes of diffusion equations and wave equations under some typical initial and boundary conditions are given

^†This work was supported by the U.S.Army Research Office, Durham, N.C.

^†This work was supported by the U.S.Army Research Office, Durham, N.C.

Notes

^†This work was supported by the U.S.Army Research Office, Durham, N.C.