Inequalities among Eigenvalues of Second-Order Symmetric Equations on Time Scales
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Research Article Inequalities among Eigenvalues of Second-Order Symmetric Equations on Time Scales Chao Zhang and Shurong Sun School of Science, University of Jinan, Jinan, Shandong 250022, China Correspondence should be addressed to Chao Zhang, ss [email protected] Received 28 January 2010; Revised 2 May 2010; Accepted 5 May 2010 Academic Editor: A. Pankov Copyright q 2010 C. Zhang and S. Sun. 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. We consider coupled boundary value problems for second-order symmetric equations on time scales. Existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only unify the existing ones of coupled boundary value problems for second-order symmetric differential equations but also contain more complicated time scales.
1. Introduction In this paper we consider the following second-order symmetric equation: Δ − ptyΔ t qtyσ t λrtyσ t,
t ∈ ρ0, ρ1 ∩ T, and 0, 1 ∈ T
1.1
with the coupled boundary conditions:
y1 yΔ 1
y ρ0 e K
, yΔ ρ0 iθ
1.2
where T is a time scale; pΔ , q, and r are real and continuous functions in ρ0, ρ1 ∩T, p > 0 over ρ0, 1 ∩ T, r > 0 over ρ0, ρ1 ∩ T, and pρ0 p1 1; σt and ρt are the
2
Advances in Difference Equations
Δ is the delta derivative, and yσ t : yσt; forward and backward jump operators in T, y√ θ/ 0, −π < θ < π, is a constant parameter; i −1,
K
k11 k12 , k21 k22
kij ∈ R,
i, j 1, 2, with det K 1.
1.3
The boundary condition 1.2 contains the two special cases: the periodic and antiperiodic conditions. In fact, 1.2 is the periodic boundary condition in the case where θ 0 and K I, the identity matrix, and 1.2 is the antiperiodic condition in the case where θ π and K I. Equation 1.1 with 1.2 is called a coupled boundary value problem. Hence, according to 1, Theorem 3.1 , the periodic and antiperiodic boundary value problems have Nd 1 real eigenvalues and they satisfy the following inequality: D D D −∞ < λ0 I < λ0 −I ≤ λD 0 ≤ λ1 −I < λ1 I ≤ λ1 ≤ λ2 I < λ2 −I ≤ λ2 ≤ λ3 −I ≤ λ3 ≤ · · · , 1.4
where Nd : |0, 1 ∩ T| − defμρ0 − 1, λD n denote the nth Dirichlet eigenvalues. Denote the number of point of a set S ⊂ R by |S| and introduce the following notation for α ∈ R:
def α
⎧ ⎨0,
if α / 0,
⎩1,
if α 0.
1.5
Furthermore, if Nd < ∞, then λ0 I < λ0 −I ≤ λ1 −I < λ1 I ≤ λ2 I < λ2 −I ≤ λ3 −I < λ3 I ≤ · · · ≤ λNd −1 −I < λNd −1 I ≤ λNd I < λNd −I,
if Nd 1 is odd,
λ0 I < λ0 −I ≤ λ1 −I < λ1 I ≤ λ2 I < λ2 −I ≤ λ3 −I < λ3 I ≤ · · · ≤ λNd −1 I < λNd −1 −I ≤ λNd −I < λNd I,
1.6
if Nd 1 is even.
In 2 , Eastham et al. considered the second-order differential equation:
− ptx t q
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