Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 165, pp. 1-8.
Title: An optimal existence theorem for positive solutions
of a four-point boundary value problem
Authors: Man Kam Kwong (Hong Kong Polytechnic Univ., China)
James S. W. Wong (Univ. of Hong Kong, China)
Abstract:
We are interested in the existence of positive solutions to a
four-point boundary value problem of the differential equation
$ y''(t) + a(t)f(y(t))=0 $ on $ [0,1] $. The value of $y$ at
$0$ and $1$ are each a multiple of $y(t)$ at an interior point.
Many known existence criteria are based on the limiting values
of $ f(u)/u $ as $u$ approaches $0$ and infinity.
In this article we obtain an optimal criterion (thereby improving
all existing results of kind mentioned above) by comparing these
limiting values to the smallest eigenvalue of the corresponding
four-point problem of the associated linear equation.
In the simpler case of three-point boundary value problems, the
same result has been established in an earlier paper by the first
author using the shooting method.
The method of proof is based upon a variant of Krasnoselskii's fixed
point theorem on cones, the classical Krein-Rutman theorem,
and the Gelfand formula relating the spectral radius of a linear
operator to its norm.
Submitted February 12, 2009. Published December 22, 2009.
Math Subject Classifications: 34B10, 34B15, 34B18.
Key Words: Four-point boundary value problem; second-order ODE;
Krasnoselskii fixed point theorem; mappings on cones.