 
Summary: Electronic Journal of Qualitative Theory of Differential Equations
2010, No. 5, 16; http://www.math.uszeged.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 LeggettWilliams 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 nontrivial example.
1. Introduction
The recent topological proof and extension of the LeggettWilliams 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 LeggettWilliams 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
