In this paper we show that the local monotonicity in the sense of Minty and Browder on some residual sets assure the global monotonicity and, according to an earlier result, the convexity of the inverse images. We pay some special attention to the residual sets arising as complements of some special first Baire category sets, namely the $$\sigma $$ -affine sets, the $$\sigma $$ -compact sets and the $$\sigma $$ -algebraic varieties. We achieve this goal gradually by showing, at first, that the continuous real valued functions of one real variable, which are locally nondecreasing on sets whose complements have no nonempty perfect subsets, are globally nondecreasing. The convexity of the inverse images combined with their discreteness, in the case of local injective operators, ensure the global injectivity. Note that the global monotonicity and the local injectivity of regular enough operators is guaranteed by the positive definiteness of the symmetric part of their Gâteaux differentials on the involved residual sets. We close this work with a short subsection on the global convexity which is obtained out of its local counterpart on some residual sets.
Positivity – Springer Journals
Published: Aug 25, 2012
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