A Kneser theorem for ordinary differential equations in Banach spaces
Abstract
We show that the set of solutions of the initial-value problem
u(τ)=a, u'(t) = g(t,u(t)) + k(t,u(t)), τ≤t≤T,
in a Banach space is compact and connected, whenever g and k are bounded and continuous functions such that g is one-sided Lipschitz and k is Lipschitz with respect to the Kuratowski measure of noncompactness. The existence of solutions is already known from Sabina Schmidt [10].
Keywords
ordinary differential equations in Banach spaces; theorem of Sabina Schmidt; theorem of Hellmuth Kneser
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Institut für Analysis, Karlsruher Institut für Technologie, Germany Germany
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