Uniqueness for algebraically regular solutions to pathological differential equations |
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Authors: | John M Bownds |
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Institution: | Department of Mathematics, University of Arizona, Tucson, Arizona 85721 USA |
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Abstract: | A uniqueness theorem is proved for algebraically regular solutions to the unbounded initial value problem P′ = AP, P(0) = diag(1, 1, 1,…) in the real Banach algebra of infinite matrices with standard norm. It is not assumed that ∈ , but it is required that A have an inverse in , a property which is seen to be implied quite naturally by certain divergent or pathological systems. The conditions for the theorem are motivated by a particular system, previously considered by Hille and Feller, which arises from a divergent, purebirth, time dependent stochastic process, although no restriction requiring the solution matrix to be either stochastic or substochastic is necessary.The theorem may be easily generalized to any Banach algebra with identity. |
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