Abstract and Applied Analysis

Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

Jiang Zhu and Dongmei Liu

Full-text: Open access

Abstract

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 165429, 28 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412279730

Digital Object Identifier
doi:10.1155/2014/165429

Mathematical Reviews number (MathSciNet)
MR3212400

Zentralblatt MATH identifier
07021847

Citation

Zhu, Jiang; Liu, Dongmei. Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 165429, 28 pages. doi:10.1155/2014/165429. https://projecteuclid.org/euclid.aaa/1412279730


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References

  • S. S. Cheng, Partial Difference Equations, Taylor & Francis, London, UK, 2003.
  • H. J. Kuo and N. S. Trudinger, “On the discrete maximum principle for parabolic difference operators,” Modélisation Mathématique et Analyse Numérique, vol. 27, no. 6, pp. 719–737, 1993.
  • M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1967.
  • M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer, Berlin, Germany, 2004.
  • M. E. Mincsovics and T. L. Horváth, “On the differences of the discrete weak and strong maximum principles for elliptic operators,” in Large-Scale Scientific Computing, I. Lirkov, S. Margenov, and J. Waśniewski, Eds., pp. 614–621, Springer, Heidelberg, Germany, 2012.
  • P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, Switzerland, 2007.
  • P. G. Ciarlet, “Discrete maximum principle for finite-difference operators,” Aequationes Mathematicae, vol. 4, pp. 338–352, 1970.
  • G. Apostolakis and G. F. Dargush, “Variational methods in irreversible thermoelasticity: theoretical developments and minimum principles for the discrete form,” Acta Mechanica, vol. 224, no. 9, pp. 2065–2088, 2013.
  • F. Dong, “Maximum principle and applications of parabolic partial differential equations,” IERI Procedia, vol. 3, International Conference on Mechanical and Electronic Engineering, pp. 198–205, 2012.
  • I. Faragó, S. Korotov, and T. Szabó, “On continuous and discrete maximum principles for elliptic problems with the third boundary condition,” Applied Mathematics and Computation, vol. 219, no. 13, pp. 7215–7224, 2013.
  • J. Karátson and S. Korotov, “Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions,” Numerische Mathematik, vol. 99, no. 4, pp. 669–698, 2005.
  • J. Karátson and S. Korotov, “Discrete maximum principles for finite element solutions of some mixed nonlinear elliptic problems using quadratures,” Journal of Computational and Applied Mathematics, vol. 192, no. 1, pp. 75–88, 2006.
  • J. Karátson, S. Korotov, and M. Křížek, “On discrete maximum principles for nonlinear elliptic problems,” Mathematics and Computers in Simulation, vol. 76, no. 1–3, pp. 99–108, 2007.
  • A. Mareno, “Maximum principles and bounds for a class of fourth order nonlinear elliptic equations,” Journal of Mathematical Analysis and Applications, vol. 377, no. 2, pp. 495–500, 2011.
  • P. Pucci and J. Serrin, “The strong maximum principle revisited,” Journal of Differential Equations, vol. 196, no. 1, pp. 1–66, 2004.
  • S. Hilger, “Analysis on measure chains–-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
  • R. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, “Dynamic equations on time scales: a survey,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.
  • A. Slavík, “Dynamic equations on time scales and generalized ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 534–550, 2012.
  • B. Jackson, “Partial dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 186, no. 2, pp. 391–415, 2006.
  • C. D. Ahlbrandt and C. Morian, “Partial differential equations on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 35–55, 2002.
  • M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.
  • M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.
  • N. Martins and D. F. M. Torres, “Calculus of variations on time scales with nabla derivatives,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e763–e773, 2009.
  • P. Stehlík, “Maximum principles for elliptic dynamic equations,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1193–1201, 2010.
  • P. Stehlík and B. Thompson, “Maximum principles for second order dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 913–926, 2007. \endinput