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Summary: Electronic Journal of Differential Equations, Vol. 2010(2010), No. 63, pp. 19.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
FUNCTIONAL EXPANSION - COMPRESSION FIXED POINT
THEOREM OF LEGGETT-WILLIAMS TYPE
DOUGLAS R. ANDERSON, RICHARD I. AVERY, JOHNNY HENDERSON
Abstract. This paper presents a fixed point theorem of compression and
expansion of functional type in the spirit of the original fixed point work of
Leggett-Williams. Neither the entire lower nor the entire upper boundary is
required to be mapped inward or outward.
1. Introduction
The spirit of the original Leggett-Williams fixed point theorem [10] is to take
a subset of the elements in the cone in which (x) = a and map these outward
in the sense that (Tx) a, where is a concave positive functional defined on
the cone. The subset that Leggett-Williams considered can be thought of as the
set of all elements of the cone in which x b and (x) = a. There were no
outward conditions on the operator T in the Leggett-Williams fixed point theorem
concerning those elements with x > b and (x) = a, and hence they avoided any
invariance-like conditions with respect to one boundary. The entire upper boundary
was mapped inward (Leggett-Williams had invariance-like conditions with respect
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