Some Hybrid Fixed Point Theorems Related to Optimization
نویسنده
چکیده
state conditions under which there exist solutions to x = f(x) for the singlevalued function f or to Y E F(y ) for the set-valued mapping F. The prototype of the former is the contraction mapping theorem, while that of the latter is the Kakutani fixed point theorem. These results, as well as generalizations, applications, and other references, are surveyed in [ 14 ]. A representative sample of subsequent work is given by [ 4, 5, 8, 9, 10, 13, 15] and the references therein. In this paper we first show in Section 2 that maximization with respect to a cone, which subsumes ordinary and Pareto optimization, is equvalent to a fixed point problem of determining y such that {y} = F(y). In some cases this equivalence actually provides a computational procedure. This fixed point problem can be considered a hybrid between the previously described standard fixed point problems for single-valued functions and set-valued mappings. We then establish in Section 3 conditions under which such a hybrid fixed point exists; an immediate corollary is an existence result for maximization with respect to a cone. In Section 4 a generalization of the contraction mapping theorem is presented.
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