Fixed Points for Non-Invariant Maps
by
Daiga Grundmane
University of Latvia

Let Y be a metric space with distance d, let X be a nonempty subset of Y, and let f be a mapping from space X to space Y.

For the case when we actually have the equality X = Y there are known many nice results which show that function f : X --> X has a fixed point, if f and X satisfy some required conditions.

Here we consider more general case, when X subset or equal Y. In [3] we have shown that a convenient condition which guarantees the existence of a fixed point for f is an existence of a special subset O subset or equal X which we call a reflector.

Definition 1. Let Y be a metric space with distance d, \emptyset =/= X subset or equal Y, and let f : X --> Y. We call subset O of a metric space X a reflector if the following two conditions are satisfied:

1. f(O) subset or equal X, and
2. forall x in X, if f(x) not in X, then there exists r_x in O such that d(x, f(x)) = d(x, r_x) + d(r_x, f(x)).

Theorem 1 Let Y be a metric space with distance d, \emptyset =/= X subset or equal Y, and let f : X --> Y be strongly nonexpansive mapping, such that there exists a reflector O subset or equal X. Then there exists the unique fixed point of f.

However, the requirement that f must be strongly nonexpansive appears to be too strong, and it seems that actually some weaker conditions on f could be sufficient to guarantee existence of fixed points. Our next theorem shows that this requirement of strong nonexpansiveness can be substituted by requirement of existence of mappings A : X --> PY and \phi: PY --> R which satisfy some natural properties. We expect that such mappings A and \phi could be used to generalize also several other known fixed point theorems.

Theorem 2 Let Y be a metric space with distance d, \emptyset =/= X subset or equal Y, and let f : X --> Y be a mapping, such that there exist a reflector O subset or equal X and mappings A : X --> PY, \phi: PY --> R which for all x in X satisfy the following properties:

1. \phi(A(f(x))) <= q\phi(A(x)) for some q in (0, 1), and
2. d(x, f(x)) <= \phi(A(x)).

1 Kirk W. A. An abstract fixed point theorem for nonexpansive mappings Proc. Amer. Math. Soc. 42 1981 40 - 642
2 Liepi ns A. A cradle-song for a little tiger on fixed points Topological spaces and their mappings, Riga, 1983 61 - 69
3 Zabarovska D. and Liepi ns A. Mirror and existence of fixed points for non-invariant mappings Acta Universitatis Latviensis 1989 88 - 92

Date received: July 22, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaj-48.