Abstract
AbstractOur aim is twofold: first, we want to introduce a partial quasiordering in cone uniform spaces with generalized pseudodistances for giving the general maximality principle in these spaces. Second, we want to show how this maximality principle can be used to obtain new and general results of Ekeland and Caristi types without lower semicontinuity assumptions, which was not done in the previous publications on this subject.
Highlights
The famous Banach contraction principle 1, fundamental in fixed point theory, has been extended in many different directions
The several generalizations of the variational principle of Ekeland type, for lower semicontinuous maps and fixed point and endpoint theorem of Caristi type for dissipative single-valued and set-valued dynamic systems with lower semicontinuous entropies in metric and uniform spaces are given, and various techniques and methods of
Fixed Point Theory and Applications investigations notably based on maximality principle are presented
Summary
The famous Banach contraction principle 1 , fundamental in fixed point theory, has been extended in many different directions. For many applications of these distances, see the papers 30–48 where, among other things, in metric and uniform spaces with generalized distances 30–34 , the new fixed point theorems of Caristi’ type for dissipative maps with lower semicontinuous entropies and variational principles of Ekeland type for lower semicontinuous maps are given. In cone uniform spaces with the families of generalized pseudodistances, the general variational principle of Ekeland type for not necessarily lower semicontinuous maps and a fixed point and endpoint theorem of Caristi type for dissipative set-valued dynamic systems with not necessarily lower semicontinuous entropies are established see Section 4. The definitions, the results, the ideas and the methods presented here are new for set-valued and single-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces and even in uniform, locally convex, and metric spaces
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