General quantum variational calculus
AbstractWe develop a new variational calculus based in the general quantum difference operator recently introduced by Hamza et al. In particular, we obtain optimality conditions for generalized variational problems where the Lagrangian may depend on the endpoints conditions and a real parameter, for the basic and isoperimetric problems, with and without fixed boundary conditions. Our results provide a generalization to previous results obtained for the $q$- and Hahn-calculus.
K. A. Aldwoah, Generalized time scales and associated difference equations, Ph.D. thesis, Cairo University, 2009.
K. A. Aldwoah, A. B. Malinowska and D. F. M. Torres, The power quantum calculus and variational problems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 19, 2012, no 1-2, 93–116
R. Almeida and D. F. M. Torres, H¨olderian variational problems subject to integral constraints, Journal of Mathematical Analysis and Applications 359, 2009, Issue 2, 2009, 674–681.
R. Almeida and D. F. M. Torres, Nondifferentiable variational principles in terms of a quantum operator, Math. Methods Appl. Sci. 34, 2011, no. 18, 2231–2241.
R. A´ lvarez-Nordase, On characterizations of classical polinomials, J. Comput. Appl. Math. 196, 2016, 320-337.
G. Bangerezako, Variational q-calculus, J. Math. Anal. Appl. 289, 2004, 650-665.
A. M. C. Brito da Cruz and N. Martins, The q-symmetric variational calculus, Comput. Math. Appl., Volume 64, Issue 7, 2012, 2241–2250.
A. M. C. Brito da Cruz, N. Martins and D. F. M. Torres, Higher-order Hahn’s quantum variational calculus, Nonlinear Anal. 75, 2012, no. 3, 1147–1157.
J. L. Cardoso and J. Petronilho, Variations around Jackson’s quantum operator, Methods and Applications of Analysis 22 (4), 2015, 343–358.
J. Cresson, G. S. F. Frederico and D. F. M. Torres, Constants of motion for non-differentiable quantum variational problems, Topol. Methods Nonlinear Anal. 33, 2009, no. 2, 217–231.
N. Faried, E. M. Shehata and R. M. El Zafarani, On homogeneous second order linear general quantum difference equations, J. Inequal. Appl. 2017: 198.
M. Foupouagnini, Laguerre-Hahn Orthogonal Polynomials with respect to the Hahn Operator: Fourth-order Difference Equation for the rth Associated and the Laguerre-Freud Equations for the Recurrence Coefficients, Ph.D thesis, Universite Nationale du Benin, Benin, 1998.
W. Hahn, Uber Orthogonalpolynome, die q-Differenzengleichungen genu¨gen, Math. Nachr. 2, 1949, 4–34.
A. E. Hamza, Abdel-Shakoor M. Sarhan, Enas M. Shehata and Khaled A. Aldwoah, A general quantum difference calculus, Advances in Difference Equations, 2015:182.
K. A. Hoffman, Stability results for constrained calculus of variations problems: an analysis of the twisted elastic loop, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 461, issue 2057, 2005, pp 1357–1381.
F. H. Jackson, q-Difference equations, Amer. J. Math. 32, 1910, no. 4, 305–314.
V. Kac and P. Cheung, Quantum calculus, Universitext, Springer, New York, 2002.
R. Koekoek, P. A. Lesky and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer, Berlin, 2010.
A. B. Malinowska and N. Martins, Generalized transversality conditions for the Hahn quantum variational calculus, Optimization, 2011, Vol. 62 , Iss. 3., 323–344.
A. B. Malinowska and D. F. M. Torres, The Hahn quantum variational calculus, J. Optim. Theory Appl. 147, 2010, no. 3, 419–442.
A. B. Malinowska and D. F. M. Torres, Quantum variational calculus, Springer Briefs in Electrical and Computer Engineering: Control, Automation and Robotics, Springer, New York, 2014.
N. Martins and D. F. M. Torres, Higher-order infinite horizon variational problems in discrete quantum calculus, Comput. Math. Appl., Volume 64, Issue 7, 2012, 2166–2175.
J. Petronilho, Generic formulas for the values at the singular points of some special monic classical Hq;!-ortogonal polynomials, J. Comput. Appl. Math. 205, 2007, 314–324.
A. Zinober and S. Sufahani, A non-standard optimal control problem arising in an economics application, Pesqui. Oper., Volume 33, no.1, Rio de Janeiro, 2013.
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