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The formulation of second-order boundary value problems on time scales
Advances in Difference Equations volume 2006, Article number: 031430 (2006)
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.
Agarwal RP, O'Regan D: Nonlinear boundary value problems on time scales. Nonlinear Analysis 2001,44(4):527–535. 10.1016/S0362-546X(99)00290-4
Amster P, Rogers C, Tisdell CC: Existence of solutions to boundary value problems for dynamic systems on time scales. Journal of Mathematical Analysis and Applications 2005,308(2):565–577. 10.1016/j.jmaa.2004.11.039
Atici FM, Guseinov GSh: On Green's functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics 2002,141(1–2):75–99. 10.1016/S0377-0427(01)00437-X
Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2001:x+358.
Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2003:xii+348.
Chyan CJ, Davis JM, Henderson J, Yin WKC: Eigenvalue comparisons for differential equations on a measure chain. Electronic Journal of Differential Equations 1998,1998(35):1–7.
Chyan CJ, Henderson J: Eigenvalue problems for nonlinear differential equations on a measure chain. Journal of Mathematical Analysis and Applications 2000,245(2):547–559. 10.1006/jmaa.2000.6781
Davidson FA, Rynne BP: Global bifurcation on time scales. Journal of Mathematical Analysis and Applications 2002,267(1):345–360. 10.1006/jmaa.2001.7780
Erbe L, Peterson A: Eigenvalue conditions and positive solutions. Journal of Difference Equations and Applications 2000,6(2):165–191. 10.1080/10236190008808220
Erbe L, Peterson A: Positive solutions for a nonlinear differential equation on a measure chain. Mathematical and Computer Modelling 2000,32(5–6):571–585. 10.1016/S0895-7177(00)00154-0
Erbe L, Peterson A, Mathsen R: Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain. Journal of Computational and Applied Mathematics 2000,113(1–2):365–380. 10.1016/S0377-0427(99)00267-8
Henderson J, Peterson A, Tisdell CC: On the existence and uniqueness of solutions to boundary value problems on time scales. Advances in Difference Equations 2004,2004(2):93–109. 10.1155/S1687183904308071
Henderson J, Tisdell CC: Topological transversality and boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004,289(1):110–125. 10.1016/j.jmaa.2003.08.030
Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.
Hong C-H, Yeh C-C: Positive solutions for eigenvalue problems on a measure chain. Nonlinear Analysis 2002,51(3):499–507. 10.1016/S0362-546X(01)00842-2
Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic, Dordrecht; 1996:x+285.
Sun H-R, Li W-T: Positive solutions for nonlinear three-point boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004,299(2):508–524. 10.1016/j.jmaa.2004.03.079
Topal SG: Second-order periodic boundary value problems on time scales. Computers & Mathematics with Applications 2004,48(3–4):637–648. 10.1016/j.camwa.2002.04.005
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Davidson, F.A., Rynne, B.P. The formulation of second-order boundary value problems on time scales. Adv Differ Equ 2006, 031430 (2006). https://doi.org/10.1155/ADE/2006/31430
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Functional Equation