Open Access Research

Fixed points of nonexpansive potential operators in Hilbert spaces

Biagio Ricceri

Author Affiliations

Department of Mathematics, University of Catania, Viale A. Doria 6, Catania, 95125, Italy

Fixed Point Theory and Applications 2012, 2012:123 doi:10.1186/1687-1812-2012-123

Published: 24 July 2012

Abstract

In this paper, we show the impact of certain general results by the author on the topic described in the title. Here is a sample:

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/123/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/123/mathml/M1">View MathML</a> be a real Hilbert space and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/123/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/123/mathml/M2">View MathML</a> be a nonexpansive potential operator.

Then, the following alternative holds: either T has a fixed point, or, for each sphere S centered at 0, the restriction to S of the functional <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/123/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/123/mathml/M3">View MathML</a> has a unique global maximum towards which each maximizing sequence in S converges.

MSC: 47H09, 47H10, 47J30, 47N10, 49K40, 90C31.

Keywords:
nonexpansive operator; potential operator; fixed point; well-posedness