Contractive maps on normed linear spaces and their applications to nonlinear matrix equations |
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Authors: | Martine CB Reurings |
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Institution: | SWOV Institute for Road Safety Research, P.O. Box 1090, 2260 BB Leidschendam, The Netherlands |
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Abstract: | In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = In + A∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A. |
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Keywords: | Contractions on cones Fixed point theorem Nonlinear matrix equations |
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