Coincidence Continuation Theory for Multivalued Maps with Selections in a Given Class

Donal O’Regan

doi:10.3390/axioms9020037

Abstract

This paper considers the topological transversality theorem for general multivalued maps which have selections in a given class of maps.

Highlights

To motivate this study first fix a map Φ.Many coincidence problems between a map F and Φ (i.e., finding a point x withF ( x ) ∩ Φ( x ) 6= ∅) arise naturally in applications
This paper considers the topological transversality theorem for general multivalued maps which have selections in a given class of maps
For a complicated map F the idea here is to try to relate it to a simpler and solvable coincidence problem between a map G and Φ (i.e., we assume we have a point y with G (y) ∩ Φ(y) 6= ∅) where the map G is homotopic to F and from this we hope to deduce that there is a coincidence point between F and Φ (i.e., we hope to deduce that there is a point x with F ( x ) ∩ Φ( x ) 6= ∅). To achieve this we consider general classes of maps and we present the notion of homotopy for this class of maps which are coincidence free on the boundary of the set considered

Summary

Introduction

To motivate this study first fix a map Φ (an important case is when Φ is the identity).Many coincidence problems between a map F and Φ (i.e., finding a (coincidence) point x withF ( x ) ∩ Φ( x ) 6= ∅) arise naturally in applications. In this paper let E be a completely regular topological space and U an open subset of E. Let E be a completely regular (respectively, normal) topological space and let Ψ, Λ ∈ D∂U (U, E).

Results

Conclusion