# American Institute of Mathematical Sciences

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

## Subdifferentials of convex functions on time scales

 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.
Citation: Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete and 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. [2] D. R. Anderson, Taylor polynomials for nabla dynamic equations on time scales, Panamer. Math. J., 12 (2002), 17-27. [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. [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. [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. [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. [7] M. Bohner and A. Peterson, "Dynamic Equations on Time Scales. An Introduction with Applications," Birkhäuser Boston, Boston, MA, 2001. [8] M. Bohner and A. Peterson (eds.), "Advances in Dynamic Equations on Time Scales," Birkhäuser Boston, Boston, MA, 2003. [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. [10] C. Dinu, Convex functions on time scales, An. Univ. Craiova Ser. Mat. Inform., 35 (2008), 87-96. [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. [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. [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. [14] D. Mozyrska and D. F. M. Torres, The natural logarithm on time scales, J. Dyn. Syst. Geom. Theor., 7 (2009), 41-48. [15] K. Murota, "Discrete Convex Analysis," SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. [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. [17] U. M. Özkan and H. Yildirim, Steffensen's integral inequality on time scales, J. Inequal. Appl., (2007), Art. ID 46524, 10 pp. [18] R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510. [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. [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. [21] Q. Sheng, Hybrid approximations via second order combined dynamic derivatives on time scales, Electron. J. Qual. Theory Differ. Equ., 17 (2007), 13 pp.

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. [2] D. R. Anderson, Taylor polynomials for nabla dynamic equations on time scales, Panamer. Math. J., 12 (2002), 17-27. [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. [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. [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. [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. [7] M. Bohner and A. Peterson, "Dynamic Equations on Time Scales. An Introduction with Applications," Birkhäuser Boston, Boston, MA, 2001. [8] M. Bohner and A. Peterson (eds.), "Advances in Dynamic Equations on Time Scales," Birkhäuser Boston, Boston, MA, 2003. [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. [10] C. Dinu, Convex functions on time scales, An. Univ. Craiova Ser. Mat. Inform., 35 (2008), 87-96. [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. [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. [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. [14] D. Mozyrska and D. F. M. Torres, The natural logarithm on time scales, J. Dyn. Syst. Geom. Theor., 7 (2009), 41-48. [15] K. Murota, "Discrete Convex Analysis," SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. [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. [17] U. M. Özkan and H. Yildirim, Steffensen's integral inequality on time scales, J. Inequal. Appl., (2007), Art. ID 46524, 10 pp. [18] R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510. [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. [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. [21] Q. Sheng, Hybrid approximations via second order combined dynamic derivatives on time scales, Electron. J. Qual. Theory Differ. Equ., 17 (2007), 13 pp.
 [1] Jan L. Cieśliński. Some implications of a new approach to exponential functions on time scales. Conference Publications, 2011, 2011 (Special) : 302-311. doi: 10.3934/proc.2011.2011.302 [2] Chao Wang, Ravi P Agarwal. Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 781-798. doi: 10.3934/dcdsb.2019267 [3] Le Thi Hoai An, Tran Duc Quynh, Kondo Hloindo Adjallah. A difference of convex functions algorithm for optimal scheduling and real-time assignment of preventive maintenance jobs on parallel processors. Journal of Industrial and Management Optimization, 2014, 10 (1) : 243-258. doi: 10.3934/jimo.2014.10.243 [4] Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223 [5] Leandro M. Del Pezzo, Nicolás Frevenza, Julio D. Rossi. Convex and quasiconvex functions in metric graphs. Networks and Heterogeneous Media, 2021, 16 (4) : 591-607. doi: 10.3934/nhm.2021019 [6] Christophe Cheverry, Thierry Paul. On some geometry of propagation in diffractive time scales. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 499-538. doi: 10.3934/dcds.2012.32.499 [7] Sung Kyu Choi, Namjip Koo. Stability of linear dynamic equations on time scales. Conference Publications, 2009, 2009 (Special) : 161-170. doi: 10.3934/proc.2009.2009.161 [8] Yunfei Peng, X. Xiang, W. Wei. Backward problems of nonlinear dynamical equations on time scales. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1553-1564. doi: 10.3934/dcdss.2011.4.1553 [9] Petr Hasil, Petr Zemánek. Critical second order operators on time scales. Conference Publications, 2011, 2011 (Special) : 653-659. doi: 10.3934/proc.2011.2011.653 [10] Massimiliano Berti, Philippe Bolle. Fast Arnold diffusion in systems with three time scales. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 795-811. doi: 10.3934/dcds.2002.8.795 [11] B. Kaymakcalan, R. Mert, A. Zafer. Asymptotic equivalence of dynamic systems on time scales. Conference Publications, 2007, 2007 (Special) : 558-567. doi: 10.3934/proc.2007.2007.558 [12] Akio Ito, Noriaki Yamazaki, Nobuyuki Kenmochi. Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces. Conference Publications, 1998, 1998 (Special) : 327-349. doi: 10.3934/proc.1998.1998.327 [13] Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial and Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004 [14] Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070 [15] Khalida Inayat Noor, Muhammad Aslam Noor. Higher order uniformly close-to-convex functions. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1277-1290. doi: 10.3934/dcdss.2015.8.1277 [16] Luigi Ambrosio, Camillo Brena. Stability of a class of action functionals depending on convex functions. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022055 [17] Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577 [18] Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control and Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002 [19] Paul Fife, Joseph Klewicki, Tie Wei. Time averaging in turbulence settings may reveal an infinite hierarchy of length scales. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 781-807. doi: 10.3934/dcds.2009.24.781 [20] Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145

2020 Impact Factor: 1.392