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Kamke's Uniqueness Theorem
Author(s) -
Ramankutty P.
Publication year - 1980
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-22.1.110
Subject(s) - mathematics , uniqueness , uniqueness theorem for poisson's equation , scalar (mathematics) , picard–lindelöf theorem , pure mathematics , generalization , product (mathematics) , mathematical analysis , fixed point theorem , ordinary differential equation , combinatorics , mathematical physics , differential equation , geometry
A generalization of Kamke's uniqueness theorem in ordinary differential equations is obtained for the limit Cauchy problem, viz x′ ( t ) = f ( t , x ( t )), x { t ) → x 0 as t t 0 , where f and x take values in an arbitrary normed linear space X and the initial point ( t 0 , x 0 ) is permitted to be on the boundary of the domain of f . Kamke's hypothesis that ∥ f ( t , x )− f ( t , y )∥ ⩽ ø(| t − t o |, ∥ x −, y ∥) is replaced by a weaker dissipative‐type hypothesis formulated in terms of the duality map of X and a semi‐inner product derived from it. Even in the scalar case in which X = R , the generalization obtained is still an extension of Kamke's theorem and some of its later analogues.