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
It’s your single place to instantly
discover and read the research
that matters to you.
Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.
All for just $49/month
Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly
Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.
Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.
Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.
All the latest content is available, no embargo periods.
“Hi guys, I cannot tell you how much I love this resource. Incredible. I really believe you've hit the nail on the head with this site in regards to solving the research-purchase issue.”Daniel C.
“Whoa! It’s like Spotify but for academic articles.”@Phil_Robichaud
“I must say, @deepdyve is a fabulous solution to the independent researcher's problem of #access to #information.”@deepthiw
“My last article couldn't be possible without the platform @deepdyve that makes journal papers cheaper.”@JoseServera