Title: Ranges of operators and convex variational inequalities

Authors: Ana Isabel Garralda-Guillem; Manuel Ruiz Galán

Addresses: Department of Applied Mathematics, E.T.S. Ingeniería de Edificación, University of Granada, c/ Severo Ochoa s/n, 18071 Granada, Spain ' Department of Applied Mathematics, E.T.S. Ingeniería de Edificación, University of Granada, c/ Severo Ochoa s/n, 18071 Granada, Spain

Abstract: The main topic in this work is the analysis of the range of a continuous and linear operator between Banach spaces, approached here through non-linear techniques, more precisely, by means of the classical Fan minimax theorem. We characterise the elements in the range of such an operator as those satisfying a certain variational inequality and provide a numerical scheme of Galerkin type to determine approximately the preimage of an element that lies in that range. The passage from the theoretical setting to its numerical realisation is done by means of the use of bases in adequate spaces. In addition we deal with the extension of the previous results to the case of systems of variational inequalities.

Keywords: operator ranges; minimax inequalities; variational principles; Galerkin methods; bases; reflexivity; Banach spaces.

DOI: 10.1504/IJANS.2013.052753

International Journal of Applied Nonlinear Science, 2013 Vol.1 No.1, pp.52 - 66

Received: 22 Aug 2012
Accepted: 11 Sep 2012
Published online: 30 Jul 2014 *

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