 
Summary: Electronic Journal of Differential Equations, Vol. 2010(2010), No. 63, pp. 19.
ISSN: 10726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
FUNCTIONAL EXPANSION  COMPRESSION FIXED POINT
THEOREM OF LEGGETTWILLIAMS 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
LeggettWilliams. Neither the entire lower nor the entire upper boundary is
required to be mapped inward or outward.
1. Introduction
The spirit of the original LeggettWilliams 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 LeggettWilliams 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 LeggettWilliams fixed point theorem
concerning those elements with x > b and (x) = a, and hence they avoided any
invariancelike conditions with respect to one boundary. The entire upper boundary
was mapped inward (LeggettWilliams had invariancelike conditions with respect
