 
Summary: Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 42, pp. 111.
ISSN: 10726691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OPERATOR TYPE EXPANSIONCOMPRESSION FIXED POINT
THEOREM
DOUGLAS R. ANDERSON, RICHARD I. AVERY, JOHNNY HENDERSON, XUEYAN LIU
Abstract. This article presents an alternative to the compression and expan
sion fixed point theorems of functional type by using operators and functions
to replace the functionals and constants that are used in functional compres
sion and expansion fixed point theorems. Only portions of the boundaries are
required to be mapped outward or inward in the spirit of the original work of
LeggettWilliams. We conclude with an application verifying the existence of
a positive solution to a secondorder boundaryvalue problem.
1. Introduction
Mavridis [7] published the first extension to the work of LeggettWilliams [6] that
replaced the arguments involving functionals with arguments involving operators.
An invariance condition was a key component of the arguments in that paper, that
is, T(KA,B(u, v) Kc) KA,B(u, v) (condition (i) of Theorem 2.8, the main result
therein). A similar approach was taken in the topological generalizations of fixed
point theorems presented by Kwong [5] which required boundaries to be mapped
