Abstract
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.
Highlights
Let f be a function defined on T. f is said to be delta differentiable at t ∈ Tk provided there exists a constant a such that for any ε > 0, there is a neighborhood U of t i.e., U t − δ, t δ ∩ T for some δ > 0 with f σ t − f s − a σ t − s ≤ ε |σ t − s|, ∀s ∈ U
In order to study the kind of separated boundary value problem for 1.1, we extend the above oscillation theorem to the more general equation 1.1 with
The boundary condition 1.2 in the cases of k11 ≤ 0, k12 > 0 or k11 < 0, k12 ≥ 0 and θ / 0, −π < θ < π, can be written as condition 1.2, where θ is replaced by π θ for θ ∈ −π, 0 and −π θ for θ ∈ 0, π, and K is replaced by −K
Summary
In this paper we consider the following second-order symmetric equation:. 1.2 where T is a time scale; pΔ, q, and r are real and continuous functions in ρ 0 , ρ 1 ∩T, p > 0 over ρ 0 , 1 ∩ T, r > 0 over ρ 0 , ρ 1 ∩ T, and p ρ 0 p 1 1; σ t and ρ t are the Advances in Difference Equations forward and backward jump operators in T, y√Δ is the delta derivative, and yσ t : y σ t ; θ / 0, −π < θ < π, is a constant parameter; i −1, K k11 k21. L1 a, b , R the space of real valued Lebesgue integrable functions on a, b They obtained the following results: the coupled boundary value problem 1.7 with 1.8 has an infinite but countable number of only real eigenvalues which can be ordered to form a nondecreasing sequence:.
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