Smajdor, A.; Smajdor, W. Multi-valued solutions of a functional equation. (English) Zbl 0577.39013 Ann. Pol. Math. 41, 89-97 (1983). 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 Cited in 1 Document MSC: 39B52 Functional equations for functions with more general domains and/or ranges 39B72 Systems of functional equations and inequalities Keywords:multi-valued; compact; upper semi-continuous, Hausdorff space PDFBibTeX XMLCite \textit{A. Smajdor} and \textit{W. Smajdor}, Ann. Pol. Math. 41, 89--97 (1983; Zbl 0577.39013) Full Text: DOI