DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

FIXED-SIGN EIGENFUNCTIONS OF TWO-POINT RIGHT
FOCAL BOUNDARY VALUE PROBLEMS ON MEASURE CHAIN
K. L. BOEY AND PATRICIA J. Y. WONG
We consider the boundary value problem (−1) n−1 y Δn (t) = λ(−1) p+1 F(t, y(σ n−1 (t))), t ∈
[a,b] ∩ T, y Δi (a) = 0, 0 ≤ i ≤ p − 1, y Δi (σ(b)) = 0, p ≤ i ≤ n − 1, where λ>0, n ≥ 2, 1 ≤
p ≤ n − 1isfixed,andT is a measure chain. The values of λ are characterized so that this
problem has a fixed-sign solution. In addition, explicit intervals of λ are established.
Copyright © 2006 K. L. Boey and P. J. Y. Wong. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Throughout, for any c, d (>c), the interval [c,d]isdefinedas[c,d] ={t ∈ T | c ≤ t ≤ d}.
We also use the notation R[c,d] to denote the real interval {t ∈ R | c ≤ t ≤ d}.Analogous
notations for open and half-open intervals will also be used.
In this paper, we present results governing the existence of fixed-sign solutions to the
differential equation on measure chains of the form
(−1) n−1 y Δn (t) = λ(−1) p+1 F ( t, y ( σ n−1 (t) )) , t ∈ [a,b], (1.1)
subject to the two-point right focal boundary conditions:
y Δi (a) = 0, 0 ≤ i ≤ p − 1, y Δi (σ(b)) = 0, p ≤ i ≤ n − 1, (1.2)
where λ>0, p,n are fixed integers satisfying n ≥ 2, 1 ≤ p ≤ n − 1, a,b ∈ T with a