Abstract
In this small book we consider variational problems in the context of the Hahn quantum calculus.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aldwoah KA (2009) Generalized time scales and associated difference equations. Ph.D. Thesis, University of Cairo, Cairo
Aldwoah KA, Hamza AE (2011) Difference time scales. Int J Math Stat 9(A11):106–125
Almeida R, Torres DFM (2009) Hölderian variational problems subject to integral constraints. J Math Anal Appl 359(2):674–681
Atici FM, McMahan CS (2009) A comparison in the theory of calculus of variations on time scales with an application to the Ramsey model. Nonlinear Dyn Syst Theory 9(1):1–10
Aulbach B, Hilger S (1990) A unified approach to continuous and discrete dynamics. In: Qualitative theory of differential equations (Szeged, 1988), Colloq Math Soc János Bolyai, 53, North-Holland, Amsterdam, pp 37–56
Bangerezako G (2004) Variational \(q\)-calculus. J Math Anal Appl 289(2):650–665
Bohner M (2004) Calculus of variations on time scales. Dyn Syst Appl 13(3–4):339–349
Bohner M, Peterson A (2001) Dynamic equations on time scales. An introduction with applications, Birkhäuser, Boston
Cresson J, Frederico GSF, Torres DFM (2009) Constants of motion for non-differentiable quantum variational problems. Topol Methods Nonlinear Anal 33(2):217–231
Cresson J, Malinowska AB, Torres DFM (2012) Time scale differential, integral, and variational embeddings of Lagrangian systems. Comput Math Appl 64(7):2294–2301
Fort T (1937) The calculus of variations applied to Nörlund’s sum. Bull Am Math Soc 43(12):885–887
Hahn W (1949) Über orthogonalpolynome, die \(q\)-differenzenlgleichungen genügen. Math Nachr 2:4–34
Hilger S (1990) Analysis on measure chains–a unified approach to continuous and discrete calculus. Results Math 18(1–2):18–56
Jackson FH (1910) On \(q\)-definite integrals. Quart J Pure Appl Math 41:193–203
Jackson FH (1951) Basic integration. Quart J Math Oxford Ser 2(2):1–16
Kac V, Cheung P (2002) Quantum calculus. Springer, New York
Malinowska AB, Torres DFM (2009) On the diamond-alpha Riemann integral and mean value theorems on time scales. Dyn Syst Appl 18(3–4):469–481
Malinowska AB, Torres DFM (2010) Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales. Appl Math Comput 217(3):1158–1162
Malinowska AB, Torres DFM (2011) Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete Cont Dyn Syst 29(2):577–593
Martins N, Torres DFM (2011a) L’Hôpital-type rules for monotonicity with application to quantum calculus. Int J Math Comput 10(M11):99–106.
Martins N, Torres DFM (2011b) Generalizing the variational theory on time scales to include the delta indefinite integral. Comput Math Appl 61(9):2424–2435
Martins N, Torres DFM (2012) Necessary optimality conditions for higher-order infinite horizon variational problems on time scales. J Optim Theory Appl 155(2):453–476
Mozyrska, D, Pawłuszewicz, E, Torres, DFM (2010) The Riemann-Stieltjes integral on time scales. Aust J Math Anal Appl 7(1): 14
Nörlund N (1924) Vorlesungen über Differencenrechnung. Springer Verlag, Berlin
Nottale L (1992) The theory of scale relativity. Internat J Mod Phys A 7(20):4899–4936
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 The Author(s)
About this chapter
Cite this chapter
Malinowska, A.B., Torres, D.F.M. (2014). Conclusion. In: Quantum Variational Calculus. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-02747-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-02747-0_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02746-3
Online ISBN: 978-3-319-02747-0
eBook Packages: EngineeringEngineering (R0)