Monotone Relations in Hadamard Spaces

In this paper, the notion of W-property for subsets of X x X-lozenge is introduced and investigated, where X is an Hadamard space and X-lozenge is its linear dual space. It is shown that an Hadamard space X is flat if and only if X x X-lozenge has W-property. Moreover, the notion of monotone relation from an Hadamard space to its linear dual space is introduced. A characterization result for monotone relations with W-property (and hence in flat Hadamard spaces) is given. Finally, a type of Debrunner-Flor Lemma concerning extension of monotone relations in Hadamard spaces is proved.