Fixed Point Theory for 1-Set Contractions: a Survey

Abstract

In this report, we first review some classical results concerning the fixed point theory for an important class of mappings for which the Banach contraction principle fails, namely nonexpansive mappings. Both metric and topological fixed point theory will be surveyed. We will also discuss some known results regarding the extension to nonlinear contractions and to˛-contractive mappings with respect to some measure of noncompactness˛. The second part of this survey paper will be devoted to some recent progress and development of the fixed point theory of 1-set contractions that have been achieved during the last couple of years. The theory for different boundary conditions and when the corresponding space is endowed with the weak topology are also discussed. Finally, some applications to equations of Krasnosels'kȋi type and to the solvability of nonlinear integral equations of Volterra type are presented.