Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

We provide a generalized version of the nonlinear small gain theorem for the case of more than two coupled input-to-state stable systems. For this result the interconnection gains are described in a nonlinear gain matrix, and the small gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than 1. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated lower-dimensional discrete time dynamical system.

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An ISS small gain theorem for general networks

This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately)from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. We remark that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume.

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Continuous Selections of Multivalued Mappings

Best Proximity Pair Theorems for Multifunctions with Open Fibres

Foundations of the KKM Theory on Generalized Convex Spaces

Contractibility and generalized convexity

Maximal elements and fixed points for binary relations on topological ordered spaces

Nash points, Ky Fan inequality and equilibria of abstract economies in Max-Plus and B-convexity

A B-convex production model for evaluating performance of firms

We show that if n sets in a topological space are given so that all the sets are closed or all are open, and for each k ≤ n every k of the sets have a (k − 2)-connected union, then the n sets have a point in common. As a consequence, we obtain the following starshaped version of Helly's theorem: If every n + 1 or fewer members of a finite family of closed sets in Rn have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.

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Charles Horvath

Papers

Contractibility and generalized convexity

Maximal elements and fixed points for binary relations on topological ordered spaces

Nash points, Ky Fan inequality and equilibria of abstract economies in Max-Plus and B-convexity

A B-convex production model for evaluating performance of firms

Intersection of sets with -connected unions