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April  2011, 29(2): 577-593. doi: 10.3934/dcds.2011.29.577

Euler-Lagrange equations for composition functionals in calculus of variations on time scales

1. 

Faculty of Computer Science, Białystok University of Technology, 15-351 Białystok, Poland

2. 

Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received  May 2009 Revised  March 2010 Published  October 2010

In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H (\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.
Citation: Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
References:
[1]

R. P. Agarwal, M. Bohner and P. J. Y. Wong, Sturm-Liouville eigenvalue problems on time scales,, Appl. Math. Comput., 99 (1999), 153. doi: doi:10.1016/S0096-3003(98)00004-6. Google Scholar

[2]

C. D. Ahlbrandt and C. Morian, Partial differential equations on time scales,, J. Comput. Appl. Math., 141 (2002), 35. doi: doi:10.1016/S0377-0427(01)00434-4. Google Scholar

[3]

R. Almeida and D. F. M. Torres, Isoperimetric problems on time scales with nabla derivatives,, J. Vib. Control, 15 (2009), 951. doi: doi:10.1177/1077546309103268. Google Scholar

[4]

F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics,, Math. Comput. Modelling, 43 (2006), 718. doi: doi:10.1016/j.mcm.2005.08.014. Google Scholar

[5]

F. M. Atici and F. Uysal, A production-inventory model of HMMS on time scales,, Appl. Math. Lett., 21 (2008), 236. doi: doi:10.1016/j.aml.2007.03.013. Google Scholar

[6]

B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics,, in, 53 (1990), 37. Google Scholar

[7]

Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales,, J. Math. Anal. Appl., 342 (2008), 1220. doi: doi:10.1016/j.jmaa.2008.01.018. Google Scholar

[8]

M. Bohner, Calculus of variations on time scales,, Dynam. Systems Appl., 13 (2004), 339. Google Scholar

[9]

M. Bohner, R. A. C. Ferreira and D. F. M. Torres, Integral inequalities and their applications to the calculus of variations on time scales,, Math. Inequal. Appl., 13 (2010), 511. Google Scholar

[10]

M. Bohner and G. Sh. Guseinov, Double integral calculus of variations on time scales,, Comput. Math. Appl., 54 (2007), 45. doi: doi:10.1016/j.camwa.2006.10.032. Google Scholar

[11]

M. Bohner and A. Peterson, "Dynamic Equations on Time Scales,", Birkhäuser Boston, (2001). Google Scholar

[12]

M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales,", Birkhäuser Boston, (2003). Google Scholar

[13]

E. Castillo, A. Luceno and P. Pedregal, Composition functionals in calculus of variations. Application to products and quotients,, Math. Models Methods Appl. Sci., 18 (2008), 47. doi: doi:10.1142/S0218202508002607. Google Scholar

[14]

T. Ernst, The different tongues of $q$-calculus,, Proc. Est. Acad. Sci., 57 (2008), 81. doi: doi:10.3176/proc.2008.2.03. Google Scholar

[15]

R. A. C. Ferreira and D. F. M. Torres, Remarks on the calculus of variations on time scales,, Int. J. Ecol. Econ. Stat., 9 (2007), 65. Google Scholar

[16]

R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales,, in, (2008), 149. doi: doi:10.1007/978-3-540-69532-5_9. Google Scholar

[17]

R. A. C. Ferreira and D. F. M. Torres, Isoperimetric problems of the calculus of variations on time scales,, in, 514 (2010), 123. Google Scholar

[18]

R. Hilscher and V. Zeidan, Calculus of variations on time scales: Weak local piecewise $C_{rd}^1$ solutions with variable endpoints,, J. Math. Anal. Appl., 289 (2004), 143. doi: doi:10.1016/j.jmaa.2003.09.031. Google Scholar

[19]

V. Kac and P. Cheung, "Quantum Calculus,", Springer, (2002). Google Scholar

[20]

V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, "Dynamic Systems on Measure Chains,", Kluwer Acad. Publ., (1996). Google Scholar

[21]

A. B. Malinowska and D. F. M. Torres, Necessary and sufficient conditions for local Pareto optimality on time scales,, J. Math. Sci. (N. Y.), 161 (2009), 803. doi: doi:10.1007/s10958-009-9601-1. Google Scholar

[22]

A. B. Malinowska and D. F. M. Torres, Strong minimizers of the calculus of variations on time scales and the Weierstrass condition,, Proc. Est. Acad. Sci., 58 (2009), 205. doi: doi:10.3176/proc.2009.4.02. Google Scholar

[23]

A. B. Malinowska and D. F. M. Torres, The delta-nabla calculus of variations,, Fasc. Math., 44 (2010), 75. Google Scholar

[24]

A. B. Malinowska and D. F. M. Torres, Natural boundary conditions in the calculus of variations,, Math. Methods Appl. Sci., 33 (2010), 1712. Google Scholar

[25]

A. B. Malinowska and D. F. M. Torres, Leitmann's direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales,, Appl. Math. Comput., 217 (2010), 1158. doi: doi:10.1016/j.amc.2010.01.015. Google Scholar

[26]

N. Martins and D. F. M. Torres, Calculus of variations on time scales with nabla derivatives,, Nonlinear Anal., 71 (2009). doi: doi:10.1016/j.na.2008.11.035. Google Scholar

[27]

B. van Brunt, "The Calculus of Variations,", Universitext, (2004). Google Scholar

show all references

References:
[1]

R. P. Agarwal, M. Bohner and P. J. Y. Wong, Sturm-Liouville eigenvalue problems on time scales,, Appl. Math. Comput., 99 (1999), 153. doi: doi:10.1016/S0096-3003(98)00004-6. Google Scholar

[2]

C. D. Ahlbrandt and C. Morian, Partial differential equations on time scales,, J. Comput. Appl. Math., 141 (2002), 35. doi: doi:10.1016/S0377-0427(01)00434-4. Google Scholar

[3]

R. Almeida and D. F. M. Torres, Isoperimetric problems on time scales with nabla derivatives,, J. Vib. Control, 15 (2009), 951. doi: doi:10.1177/1077546309103268. Google Scholar

[4]

F. M. Atici, D. C. Biles and A. Lebedinsky, An application of time scales to economics,, Math. Comput. Modelling, 43 (2006), 718. doi: doi:10.1016/j.mcm.2005.08.014. Google Scholar

[5]

F. M. Atici and F. Uysal, A production-inventory model of HMMS on time scales,, Appl. Math. Lett., 21 (2008), 236. doi: doi:10.1016/j.aml.2007.03.013. Google Scholar

[6]

B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics,, in, 53 (1990), 37. Google Scholar

[7]

Z. Bartosiewicz and D. F. M. Torres, Noether's theorem on time scales,, J. Math. Anal. Appl., 342 (2008), 1220. doi: doi:10.1016/j.jmaa.2008.01.018. Google Scholar

[8]

M. Bohner, Calculus of variations on time scales,, Dynam. Systems Appl., 13 (2004), 339. Google Scholar

[9]

M. Bohner, R. A. C. Ferreira and D. F. M. Torres, Integral inequalities and their applications to the calculus of variations on time scales,, Math. Inequal. Appl., 13 (2010), 511. Google Scholar

[10]

M. Bohner and G. Sh. Guseinov, Double integral calculus of variations on time scales,, Comput. Math. Appl., 54 (2007), 45. doi: doi:10.1016/j.camwa.2006.10.032. Google Scholar

[11]

M. Bohner and A. Peterson, "Dynamic Equations on Time Scales,", Birkhäuser Boston, (2001). Google Scholar

[12]

M. Bohner and A. Peterson, "Advances in Dynamic Equations on Time Scales,", Birkhäuser Boston, (2003). Google Scholar

[13]

E. Castillo, A. Luceno and P. Pedregal, Composition functionals in calculus of variations. Application to products and quotients,, Math. Models Methods Appl. Sci., 18 (2008), 47. doi: doi:10.1142/S0218202508002607. Google Scholar

[14]

T. Ernst, The different tongues of $q$-calculus,, Proc. Est. Acad. Sci., 57 (2008), 81. doi: doi:10.3176/proc.2008.2.03. Google Scholar

[15]

R. A. C. Ferreira and D. F. M. Torres, Remarks on the calculus of variations on time scales,, Int. J. Ecol. Econ. Stat., 9 (2007), 65. Google Scholar

[16]

R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales,, in, (2008), 149. doi: doi:10.1007/978-3-540-69532-5_9. Google Scholar

[17]

R. A. C. Ferreira and D. F. M. Torres, Isoperimetric problems of the calculus of variations on time scales,, in, 514 (2010), 123. Google Scholar

[18]

R. Hilscher and V. Zeidan, Calculus of variations on time scales: Weak local piecewise $C_{rd}^1$ solutions with variable endpoints,, J. Math. Anal. Appl., 289 (2004), 143. doi: doi:10.1016/j.jmaa.2003.09.031. Google Scholar

[19]

V. Kac and P. Cheung, "Quantum Calculus,", Springer, (2002). Google Scholar

[20]

V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, "Dynamic Systems on Measure Chains,", Kluwer Acad. Publ., (1996). Google Scholar

[21]

A. B. Malinowska and D. F. M. Torres, Necessary and sufficient conditions for local Pareto optimality on time scales,, J. Math. Sci. (N. Y.), 161 (2009), 803. doi: doi:10.1007/s10958-009-9601-1. Google Scholar

[22]

A. B. Malinowska and D. F. M. Torres, Strong minimizers of the calculus of variations on time scales and the Weierstrass condition,, Proc. Est. Acad. Sci., 58 (2009), 205. doi: doi:10.3176/proc.2009.4.02. Google Scholar

[23]

A. B. Malinowska and D. F. M. Torres, The delta-nabla calculus of variations,, Fasc. Math., 44 (2010), 75. Google Scholar

[24]

A. B. Malinowska and D. F. M. Torres, Natural boundary conditions in the calculus of variations,, Math. Methods Appl. Sci., 33 (2010), 1712. Google Scholar

[25]

A. B. Malinowska and D. F. M. Torres, Leitmann's direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales,, Appl. Math. Comput., 217 (2010), 1158. doi: doi:10.1016/j.amc.2010.01.015. Google Scholar

[26]

N. Martins and D. F. M. Torres, Calculus of variations on time scales with nabla derivatives,, Nonlinear Anal., 71 (2009). doi: doi:10.1016/j.na.2008.11.035. Google Scholar

[27]

B. van Brunt, "The Calculus of Variations,", Universitext, (2004). Google Scholar

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