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Set differential equations in Frechet spaces

Galanis G., N., Bhaskar T., G., Lakshmikantham V.

Journal of Applied Analysis

2008

Vol. 14, nr 1

103-113

It is known that a frachet space F can be realized as a projective limit of a sequence of Banach spaces Ei. The space Kc(F) of all compact, convex subsets of a Frechet space, F, is realized as a projective limit of the semilinear metric spaces Kc(Ei). Using the notion of Hukuhara derivative for maps with values in Kc(F), we prove the local and global existence theorems for an initial value problem associated with a set differential equation.

Nonlinear fundamental systems for linear differential equations in Frechet spaces

Herzog G., Lemmert R.

Demonstratio Mathematica

2000

Vol. 33, nr 2

313-318

Let E be a Frechet space. We prove that ex (E) = ex1 (E), that is that the IVP u' = Au + f, u(0) = uo is always solvable if the homogeneous problem u' = Au, u(0) = uo is always solvable (even if this solution is not unique). Moreover we prove that there is a continuous, in general nonlinear selection of solutions, which can be applied to prove an existence theorem for u = Au u+ g(',u), u(0) = uo.