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April  2011, 29(2): 671-691. doi: 10.3934/dcds.2011.29.671

Subdifferentials of convex functions on time scales

Małgorzata Wyrwas 1, , Dorota Mozyrska 2, and  Ewa Girejko 2,

1. 

Faculty of Computer Science, Białystok University of Technology, ul. Wiejska 45a, 15-351 Bia lystok, Poland
2. 

Faculty of Computer Science, Białystok University of Technology, ul. Wiejska 45a, 15-351 Białystok, Poland, Poland

Received  August 2009 Revised  March 2010 Published  October 2010

The paper studies the notion of subdifferentials of functions defined on a time scale. The subdifferential of a given function $f$ is defined as the set of certain extended functions. Since the convexity of the given function guarantees its subdifferentiability, properties of convex functions on time scales are presented. We show that the convexity of a function is the necessary and sufficient condition for its subdifferentiability. The relations between the delta, nabla, diamond-$\alpha$ derivatives and subdifferentials of convex functions are given.
Keywords: convex functions, subdifferentials., Time scales.
Mathematics Subject Classification: Primary: 26D15, 39A13; Secondary: 52A4.
Citation: Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671
References:
[1]

R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35 (1999), 3-22.  Google Scholar

[2]

D. R. Anderson, Taylor polynomials for nabla dynamic equations on time scales, Panamer. Math. J., 12 (2002), 17-27.  Google Scholar

[3]

J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984.  Google Scholar

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B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in "Qualitative Theory of Differential Equations" (eds. B. Sz-Nagy and L. Hatvani), Colloq. Math. Soc. János Bolyai, 53, North Holland, Amsterdam, (1990), 37-56.  Google Scholar

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B. Aulbach and S. Hilger, Linear dynamic process with inhomogeneous time scale, in "Nonlinear Dynamics and Quantum Dynamical Systems" (eds. G. Leonov, V. Reitmann and W. Timmermann), Math. Res., 59, Akademie-Verlag, Berlin, (1990), 9-20.  Google Scholar

[6]

N. S. Barnett, P. Cerone and S. S. Dragomir, Majorisation inequalities for Stieltjes integrals, Appl. Math. Lett., 22 (2009), 416-421. doi: doi:10.1016/j.aml.2008.06.009.  Google Scholar

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M. Bohner and A. Peterson, "Dynamic Equations on Time Scales. An Introduction with Applications," Birkhäuser Boston, Boston, MA, 2001.  Google Scholar

[8]

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

[9]

F. H. Clarke, "Optimization and Nonsmooth Analysis," 2nd edition, Classics in Applied Mathematics, 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.  Google Scholar

[10]

C. Dinu, Convex functions on time scales, An. Univ. Craiova Ser. Mat. Inform., 35 (2008), 87-96.  Google Scholar

[11]

J.-B. Hiriart-Urruty and C. Lemaréchal, "Convex Analysis and Minimization Algorithms. I. Fundamentals," Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 305, Springer-Verlag, Berlin, 1993.  Google Scholar

[12]

A. B. Malinowska and D. F. M. Torres, On the diamond-alpha Riemann integral and mean value theorems on time scales, Dynam. Systems Appl., 18 (2009), 469-481.  Google Scholar

[13]

A. B. Malinowska and D. F. M. Torres, Necessary and sufficient conditions for local Pareto optimality on time scales, Journal of Mathematical Sciences, 161 (2009), 803-810, arXiv:0801.2123v1. Google Scholar

[14]

D. Mozyrska and D. F. M. Torres, The natural logarithm on time scales, J. Dyn. Syst. Geom. Theor., 7 (2009), 41-48.  Google Scholar

[15]

K. Murota, "Discrete Convex Analysis," SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.  Google Scholar

[16]

C. Niculescu and L. E. Persson, "Convex Functions and Their Applications. A Contemporary Approach," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23, Springer, New York, 2006.  Google Scholar

[17]

U. M. Özkan and H. Yildirim, Steffensen's integral inequality on time scales, J. Inequal. Appl., (2007), Art. ID 46524, 10 pp.  Google Scholar

[18]

R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510.  Google Scholar

[19]

J. W. Rogers and Q. Sheng, Notes on the diamond-$\alpha$ dynamic derivative on time scales, J. Math. Anal. Appl., 326 (2007), 228-241. doi: doi:10.1016/j.jmaa.2006.03.004.  Google Scholar

[20]

Q. Sheng, M. Fadag, J. Henderson and J. M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl., 7 (2006), 395-413. doi: doi:10.1016/j.nonrwa.2005.03.008.  Google Scholar

[21]

Q. Sheng, Hybrid approximations via second order combined dynamic derivatives on time scales, Electron. J. Qual. Theory Differ. Equ., 17 (2007), 13 pp.  Google Scholar

show all references

References:
[1]

R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35 (1999), 3-22.  Google Scholar

[2]

D. R. Anderson, Taylor polynomials for nabla dynamic equations on time scales, Panamer. Math. J., 12 (2002), 17-27.  Google Scholar

[3]

J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984.  Google Scholar

[4]

B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in "Qualitative Theory of Differential Equations" (eds. B. Sz-Nagy and L. Hatvani), Colloq. Math. Soc. János Bolyai, 53, North Holland, Amsterdam, (1990), 37-56.  Google Scholar

[5]

B. Aulbach and S. Hilger, Linear dynamic process with inhomogeneous time scale, in "Nonlinear Dynamics and Quantum Dynamical Systems" (eds. G. Leonov, V. Reitmann and W. Timmermann), Math. Res., 59, Akademie-Verlag, Berlin, (1990), 9-20.  Google Scholar

[6]

N. S. Barnett, P. Cerone and S. S. Dragomir, Majorisation inequalities for Stieltjes integrals, Appl. Math. Lett., 22 (2009), 416-421. doi: doi:10.1016/j.aml.2008.06.009.  Google Scholar

[7]

M. Bohner and A. Peterson, "Dynamic Equations on Time Scales. An Introduction with Applications," Birkhäuser Boston, Boston, MA, 2001.  Google Scholar

[8]

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

[9]

F. H. Clarke, "Optimization and Nonsmooth Analysis," 2nd edition, Classics in Applied Mathematics, 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.  Google Scholar

[10]

C. Dinu, Convex functions on time scales, An. Univ. Craiova Ser. Mat. Inform., 35 (2008), 87-96.  Google Scholar

[11]

J.-B. Hiriart-Urruty and C. Lemaréchal, "Convex Analysis and Minimization Algorithms. I. Fundamentals," Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 305, Springer-Verlag, Berlin, 1993.  Google Scholar

[12]

A. B. Malinowska and D. F. M. Torres, On the diamond-alpha Riemann integral and mean value theorems on time scales, Dynam. Systems Appl., 18 (2009), 469-481.  Google Scholar

[13]

A. B. Malinowska and D. F. M. Torres, Necessary and sufficient conditions for local Pareto optimality on time scales, Journal of Mathematical Sciences, 161 (2009), 803-810, arXiv:0801.2123v1. Google Scholar

[14]

D. Mozyrska and D. F. M. Torres, The natural logarithm on time scales, J. Dyn. Syst. Geom. Theor., 7 (2009), 41-48.  Google Scholar

[15]

K. Murota, "Discrete Convex Analysis," SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.  Google Scholar

[16]

C. Niculescu and L. E. Persson, "Convex Functions and Their Applications. A Contemporary Approach," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23, Springer, New York, 2006.  Google Scholar

[17]

U. M. Özkan and H. Yildirim, Steffensen's integral inequality on time scales, J. Inequal. Appl., (2007), Art. ID 46524, 10 pp.  Google Scholar

[18]

R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510.  Google Scholar

[19]

J. W. Rogers and Q. Sheng, Notes on the diamond-$\alpha$ dynamic derivative on time scales, J. Math. Anal. Appl., 326 (2007), 228-241. doi: doi:10.1016/j.jmaa.2006.03.004.  Google Scholar

[20]

Q. Sheng, M. Fadag, J. Henderson and J. M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl., 7 (2006), 395-413. doi: doi:10.1016/j.nonrwa.2005.03.008.  Google Scholar

[21]

Q. Sheng, Hybrid approximations via second order combined dynamic derivatives on time scales, Electron. J. Qual. Theory Differ. Equ., 17 (2007), 13 pp.  Google Scholar

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