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EXISTENCE AND UNIQUENESS OF SOLUTIONS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS IN BANACH SPACES

LetX and Y be a pair of Banach spaces with Y being continuously embedded in X and let A be a linear operator whose graph is closed in Y ×X . In this paper, the operators A are characterized for which the zero solution is the unique solution of u′(ξ) = Au(ξ), u(0) = 0 on [0, ξ0] for some 0 < ξ0. Moreover, the initial values x ∈ X are characterized for which there are solutions of u′(ξ) = Au(ξ), u(0) = x on [0, ξ0] for some 0 < ξ0.