Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T00:51:01.637Z Has data issue: false hasContentIssue false
  • English
  • Français

ON MODULATED TOPOLOGICAL VECTOR SPACES AND APPLICATIONS

Part of: Normed linear spaces and Banach spaces; Banach lattices Nonlinear operators and their properties

Published online by Cambridge University Press:  10 July 2019

WOJCIECH M. KOZLOWSKI Open the ORCID record for WOJCIECH M. KOZLOWSKI [Opens in a new window]
WOJCIECH M. KOZLOWSKI*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a notion of modulated topological vector spaces, that generalises, among others, Banach and modular function spaces. As applications, we prove some results which extend Kirk’s and Browder’s fixed point theorems. The theory of modulated topological vector spaces provides a very minimalist framework, where powerful fixed point theorems are valid under a bare minimum of assumptions.

Keywords

topological vector spacesBanach spacesmodular spacesmodular function spacesfixed pointsnonexpansive mappingsnormal structure

MSC classification

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Secondary: 46B20: Geometry and structure of normed linear spaces 47H10: Fixed-point theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 2 , April 2020 , pp. 325 - 332
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alspach, D., ‘A fixed point free nonexpansive mapping’, Proc. Amer. Math. Soc. 82 (1981), 423–424.Google Scholar
Bin Dehaish, B. and Khamsi, M. A., ‘On monotone mappings in modular function spaces’, J. Nonlinear Sci. Appl. 9 (2016), 5219–5228.Google Scholar
Garkavi, A. L., ‘On the optimal net and best cross-section of a set in a normed space’ (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87–106.Google Scholar
Gillespie, A. and Williams, B., ‘Fixed point theorem for nonexpansive mappings on Banach spaces’, Appl. Anal. 9 (1979), 121–124.Google Scholar
Goebel, K. and Kirk, W. A., Topics in Metric Fixed Point Theory (Cambridge University Press, Cambridge, 1990).Google Scholar
Hudzik, H., ‘The problems of separability, duality, reflexivity and comparison for generalized Orlicz–Sobolev spaces W Mk(𝛺)’, Comment. Math. 21 (1979), 315–324.Google Scholar
Khamsi, M. A. and Kirk, W. A., An Introduction to Metric Spaces and Fixed Point Theory (John Wiley, New York, 2001).Google Scholar
Khamsi, M. A. and Kozlowski, W. M., ‘On asymptotic pointwise contractions in modular function spaces’, Nonlinear Anal. 73 (2010), 2957–2967.Google Scholar
Khamsi, M. A. and Kozlowski, W. M., ‘On asymptotic pointwise nonexpansive mappings in modular function spaces’, J. Math. Anal. Appl. 380(2) (2011), 697–708.Google Scholar
Khamsi, M. A. and Kozlowski, W. M., Fixed Point Theory in Modular Function Spaces (Springer, Cham–Heidelberg–New York–Dordrecht–London, 2015).Google Scholar
Khamsi, M. A., Kozlowski, W. M. and Reich, S., ‘Fixed point theory in modular function spaces’, Nonlinear Anal. 14 (1990), 935–953.Google Scholar
Kirk, W. A., ‘A fixed point theorem for mappings that do not increase distances’, Amer. Math. Monthly 72 (1965), 1004–1006.Google Scholar
Kirk, W. A., ‘Nonexpansive mappings in metric and Banach spaces’, Rend. Sem. Mat. Milano LI (1981), 133–144.Google Scholar
Kozlowski, W. M., ‘Notes on modular function spaces I’, Comment. Math. 28 (1988), 91–104.Google Scholar
Kozlowski, W. M., ‘Notes on modular function spaces II’, Comment. Math. 28 (1988), 105–120.Google Scholar
Kozlowski, W. M., Modular Function Spaces, Series of Monographs and Textbooks in Pure and Applied Mathematics, 122 (Dekker, New York–Basel, 1988).Google Scholar
Kozlowski, W. M., ‘Advancements in fixed point theory in modular function’, Arab. J. Math. 1(4) (2012), 477–494.Google Scholar
Kozlowski, W. M., ‘An introduction to fixed point theory in modular function spaces’, in: Topics in Fixed Point Theory (eds. Almezel, S., Ansari, Q. H. and Khamsi, M. A.) (Springer, New York–Heidelberg–Dordrecht–London, 2014).Google Scholar
Musielak, J., Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, 1034 (Springer, Berlin, 1983).Google Scholar
Nakano, H., Modulared Semi-ordered Linear Spaces (Maruzen, Tokyo, 1950).Google Scholar
Sims, B., ‘A class of spaces with weak normal structure’, Bull. Aust. Math. Soc. 50 (1994), 523–528.Google Scholar
Turett, B., ‘Fenchel–Orlicz spaces’, Dissertationes Math. 181 (1980), 1–60.Google Scholar