Abstract
We reconsider the basic formulation of second-order, two-point, Sturm-Liouville-type boundary value problems on time scales. Although this topic has received extensive attention in recent years, we present some simple examples which show that there are certain difficulties with the formulation of the problem as usually used in the literature. These difficulties can be avoided by some additional conditions on the structure of the time scale, but we show that these conditions are unnecessary, since in fact, a simple, amended formulation of the problem avoids the difficulties.
Highlights
In the time scale literature, there has been considerable interest in Sturm-Liouville boundary value problems of the form− puΔ Δ(t) + q(t)uσ (t) = f λ, t, uσ (t), [a, b] ∩ Tκ2, (1.1)for suitable functions p, q, and f and real parameter λ, together with boundary conditions, which are generally taken to have the form αu(a) + βuΔ(a) = 0, γu σ2(b) + δuΔ σ(b) = 0, (1.2)for some points a, b ∈ T, with a < b and (|α| + |β|)(|γ| + |δ|) > 0 there are other formulations of the boundary conditions
In this paper, we show that for general time scales, there are certain difficulties with this basic formulation. These difficulties can be avoided by some additional conditions on the structure of the time scale, but we show that these conditions are unnecessary, since a simple, amended formulation of the problem avoids the difficulties
The difficulties just mentioned concern the basic formulation of the linear operator formed from the left-hand side of (1.1), together with the boundary conditions (1.2), so on we consider the formulation of this linear operator and ignore the right-hand side of (1.1)
Summary
In the time scale literature, there has been considerable interest in Sturm-Liouville boundary value problems of the form. For suitable functions p, q, and f and real parameter λ, together with boundary conditions, which are generally taken to have the form αu(a) + βuΔ(a) = 0, γu σ2(b) + δuΔ σ(b) = 0,. For some points a, b ∈ T, with a < b and (|α| + |β|)(|γ| + |δ|) > 0 (see, e.g., [1, 2, 6, 7, 9, 11, 15] and in the case of systems [12]) there are other formulations of the boundary conditions (see Remark 3.1). The formulation (1.1)-(1.2) covers both linear eigenvalue problems and nonlinear problems.
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