×

Double integral calculus of variations on time scales. (English) Zbl 1131.49019

Summary: We consider a version of the double integral calculus of variations on time scales, which includes as special cases the classical two-variable calculus of variations and the discrete two-variable calculus of variations. Necessary and sufficient conditions for a local extremum are established, among them an analogue of the Euler-Lagrange equation.

MSC:

49K10 Optimality conditions for free problems in two or more independent variables
39A10 Additive difference equations
26B99 Functions of several variables
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akhiezer, N. I., The Calculus of Variations (1962), Blaisdell Publishing Company: Blaisdell Publishing Company New York · Zbl 0119.05604
[2] Gelfand, I. M.; Fomin, S. V., Calculus of Variations (1963), Prentice Hall, Inc.: Prentice Hall, Inc. Englewood Cliffs · Zbl 0127.05402
[3] Ahlbrandt, C. D.; Peterson, A. C., (Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations. Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Texts in the Mathematical Sciences, vol. 16 (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Boston) · Zbl 0860.39001
[4] G.Sh. Guseinov, Discrete calculus of variations, in: K. Tas, D. Baleanu, D. Krupka, O. Krupka (Eds.), Global Analysis and Applied Mathematics, vol. 729, New York, 2004, pp. 170-176. American Institute of Physics Conference Proceedings; G.Sh. Guseinov, Discrete calculus of variations, in: K. Tas, D. Baleanu, D. Krupka, O. Krupka (Eds.), Global Analysis and Applied Mathematics, vol. 729, New York, 2004, pp. 170-176. American Institute of Physics Conference Proceedings · Zbl 1113.65313
[5] Hilscher, R.; Zeidan, V., Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: Survey, J. Difference Equ. Appl., 11, 857-875 (2005) · Zbl 1098.49025
[6] Kelley, W. G.; Peterson, A. C., Difference Equations: An Introduction with Applications (2001), Academic Press: Academic Press San Diego · Zbl 0970.39001
[7] Ahlbrandt, C. D.; Harmsen, B., Discrete versions of continuous isoperimetric problems, J. Difference Equ. Appl., 3, 449-462 (1998) · Zbl 0908.49031
[8] Aulbach, B.; Hilger, S., A unified approach to continuous and discrete dynamics, (Qualitative Theory of Differential Equations (Szeged, 1988). Qualitative Theory of Differential Equations (Szeged, 1988), Colloq. Math. Soc. János Bolyai, vol. 53 (1990), North-Holland: North-Holland Amsterdam), 37-56 · Zbl 0713.34050
[9] Hilger, S., Analysis on measure chains — a unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990) · Zbl 0722.39001
[10] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[11] Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston · Zbl 1025.34001
[12] Agarwal, R. P.; Bohner, M., Quadratic functionals for second order matrix equations on time scales, Nonlinear Anal., 33, 7, 675-692 (1998) · Zbl 0938.49001
[13] Bohner, M., Calculus of variations on time scales, Dynam. Systems Appl., 13, 339-349 (2004) · Zbl 1069.39019
[14] Hilscher, R.; Zeidan, V., Calculus of variations on time scales: Weak local piecewise \(C_{rd}^1\) solutions with variable endpoints, J. Math. Anal. Appl., 289, 143-166 (2004) · Zbl 1043.49004
[15] Ahlbrandt, C. D.; Morian, C.; Agarwal, R. P.; Bohner, M.; O’Regan, D., Partial differential equations on time scales, Dynamic Equations on Time Scales. Dynamic Equations on Time Scales, J. Comput. Appl. Math., 141, 1-2, 35-55 (2002), (special issue) · Zbl 1134.35314
[16] Bohner, M.; Guseinov, G. Sh., Partial differentiation on time scales, Dynam. Systems Appl., 13, 351-379 (2004) · Zbl 1090.26004
[17] Bohner, M.; Guseinov, G. Sh., Multiple integration on time scales, Dynam. Systems Appl., 14, 3-4, 579-606 (2005) · Zbl 1095.26006
[18] Bohner, M.; Guseinov, G. Sh., Line integrals and Green’s formula on time scales, J. Math. Anal. Appl., 326, 2, 1124-1141 (2007) · Zbl 1118.26009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.