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Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 42, pp. 111. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
 

Summary: Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 42, pp. 111.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OPERATOR TYPE EXPANSION-COMPRESSION 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
Leggett-Williams. We conclude with an application verifying the existence of
a positive solution to a second-order boundary-value problem.
1. Introduction
Mavridis [7] published the first extension to the work of Leggett-Williams [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

  

Source: Anderson, Douglas R. - Department of Mathematics and Computer Science, Concordia College

 

Collections: Mathematics