The formulation of second-order boundary value problems on time scales

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.

Existence of Weak Solutions for Second-Order Boundary Value Problem of Impulsive Dynamic Equations on Time Scales

“…The study of boundary value problems for dynamic equations on time scales has recently received a lot of attention, see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] . At the same time, there have been significant developments in impulsive differential equations, see the monographs of Lakshmikantham et al 17 and Samoȋlenko and Perestyuk 18 .…”

L2 spaces and boundary value problems on time-scales

“…(b) In the previous literature, when defining the operator L it has been assumed that p ∈ C 1 rd (T κ ), q ∈ C 0 rd (T κ 2 ), and the domain of L is the subset of C 2 rd (T) satisfying the boundary conditions (5.2), while the codomain of L is C 0 rd (T κ 2 ), see for example [2,[4][5][6]8,11]. We observe that precise conditions on p and q are not always explicitly stated, but these conditions, or stronger ones, always seem to be implicitly assumed.…”

“…We note also that the formulation of the boundary conditions in (5.2) is somewhat different to the usual formulation. The justification for this is discussed in detail in [6]. In particular, if γ a = 0 then α a = 0, δ a = 0, and the boundary condition at a becomes u(σ (a)) = 0.…”

Existence Results for Solutions of Integral Boundary Value Problems on Time Scales

“…For the details of basic notions connected to time scales, we refer the readers to the books [2,3] and the papers [4,5], which are useful references for calculus on time scales. Hereafter, we use the notation [ , ] T to indicate the time scale interval [ , ]∩T.…”

“…For more details of boundary value problems involving integral boundary conditions, see, for instance, [6,[13][14][15][16][17][18][19] and references therein. Also this type of problems includes twopoint, three-point, and multipoint boundary value problems as special cases [4,5,7,20] and the references therein.…”