Touching multifunctions on a Hilbert space

TL;DR

This paper introduces the concept of touching multifunctions on a Hilbert space and demonstrates their application in establishing unique fixed points and analyzing projections in convex spaces.

Contribution

It presents a novel concept of touching multifunctions and applies it to fixed point theory and projections in convex analysis.

Findings

Certain multifunctions have unique fixed points.

Application to generalized cycles and gap vectors.

Insights into projections onto convex sets.

Abstract

We introduce the concept of the touching of two multifunctions on a real Hilbert space, and deduce that certain multifunctions on the space have a unique fixed point. These result are applied to the theory of genaralized cycles and generalized gap vectors for the composition of the projections onto a finite number of closed convex space in a real Hilbert space.