Abstract and Applied Analysis
Volume 2003 (2003), Issue 1, Pages 19-31

Relaxed submonotone mappings

Tzanko Donchev1 and Pando Georgiev2,3

1Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, 1 Christo Smirnenski blvd., Sofia 1046, Bulgaria
2Faculty of Mathematics and Informatics, Sofia University St. Kliment Ohridski, Department of Operational Research, Sofia 1126, Bulgaria
3Laboratory for Advanced Brain Signal Processing, Brain Science Institute, The Institute of Physical and Chemical Research (RIKEN), 2-1, Wako-shi, Saitama, Hirosawa 351-0198, Japan

Received 30 November 2001

Copyright © 2003 Tzanko Donchev and Pando Georgiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The notions of relaxed submonotone and relaxed monotone mappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hypotheses, the surjectivity of the relaxed monotone mappings.