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Multi-valued solutions of a functional equation. (English) Zbl 0577.39013

Let X be a non-empty set, Y a topological Hausdorff space, F a mapping of X into X, \(\Phi\) an unknown mapping of X into the class of all non-empty subsets of Y, and H a mapping of \(X\times Y\) into the class of all non- empty subsets of Y. The authors consider the functional inequality \(\Phi\) (x)\(\supset H(X,\Phi [f(x)])\), and also the corresponding equality, and discusses the conditions under which the solution \(\phi\) is upper semi- continuous and either multi-valued or single-valued. The case when the range of \(\Phi\) consists of compact sets or of continua, with H connected, are also discussed. For the particular case of \(Y=R\), the monotonicity of H in the second variables is assumed so as to lead to single-valued mapping and solutions.
Reviewer: A.Buche

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B72 Systems of functional equations and inequalities
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