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Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 5, 1-6; http://www.math.u-szeged.hu/ejqtde/
 

Summary: Electronic Journal of Qualitative Theory of Differential Equations
2010, No. 5, 1-6; http://www.math.u-szeged.hu/ejqtde/
EXISTENCE OF A POSITIVE SOLUTION TO A RIGHT FOCAL
BOUNDARY VALUE PROBLEM
RICHARD I. AVERY, JOHNNY HENDERSON AND DOUGLAS R. ANDERSON
Abstract. In this paper we apply the recent extension of the Leggett-Williams Fixed Point
Theorem which requires neither of the functional boundaries to be invariant to the second order
right focal boundary value problem. We demonstrate a technique that can be used to deal with
a singularity and provide a non-trivial example.
1. Introduction
The recent topological proof and extension of the Leggett-Williams fixed point theorem [3]
does not require either of the functional boundaries to be invariant with respect to a functional
wedge and its proof uses topological methods instead of axiomatic index theory. Functional
fixed point theorems (including [2, 4, 5, 6, 8]) can be traced back to Leggett and Williams [7]
when they presented criteria which guaranteed the existence of a fixed point for a completely
continuous map that did not require the operator to be invariant with regard to the concave
functional boundary of a functional wedge. Avery, Henderson, and O'Regan [1], in a dual of
the Leggett-Williams fixed point theorem, gave conditions which guaranteed the existence of a
fixed point for a completely continuous map that did not require the operator to be invariant
relative to the concave functional boundary of a functional wedge. We will demonstrate a

  

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

 

Collections: Mathematics