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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 582161, 8 pages
http://dx.doi.org/10.1155/2013/582161
Research Article

Measure Functional Differential Equations in the Space of Functions of Bounded Variation

1Departamento de Matemática, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista (UNESP), Campus de Rio Claro, CP 178, 13506-900 Rio Claro, SP, Brazil
2Institute of Mathematics, Academy of Sciences of the Czech Republic, Branch in Brno, 22 Žižkova Street, 616 62 Brno, Czech Republic

Received 21 May 2013; Accepted 8 August 2013

Academic Editor: Rodrigo Lopez Pouso

Copyright © 2013 S. M. Afonso and A. Rontó. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Kurzweil, “Generalized ordinary differential equations and continuous dependence on a parameter,” Czechoslovak Mathematical Journal, vol. 7, no. 82, pp. 418–449, 1957. View at Zentralblatt MATH
  2. S. Schwabik, Generalized Ordinary Differential Equations, vol. 5 of Series in Real Analysis, World Scientific, River Edge, NJ, USA, 1992.
  3. S. Schwabik, M. Tvrdý, and O. Vejvoda, Differential and Integral Equations, D. Reidel Publishing, Dordrecht, The Netherlands, 1979, Boundary value problems and adjoints.
  4. M. Tvrdý, “Differential and integral equations in the space of regulated functions,” Memoirs on Differential equations and Mathematical Physics, vol. 25, pp. 1–104, 2002. View at Zentralblatt MATH
  5. M. Tvrdý, Stieltjesův integrál. Kurzweilova teorie, Palacky University, Olomouc, Czech Republic, 2012, Czech.
  6. J. Kurzweil, Generalized Ordinary Differential Equations, vol. 11 of Series in Real Analysis, World Scientific, Hackensack, NJ, USA, 2012, Not absolutely continuous solutions.
  7. M. Tvrdý, “Regulated functions and the Perron-Stieltjes integral,” Časopis pro Pěstování Matematiky, vol. 114, pp. 187–209, 1989. View at Zentralblatt MATH
  8. P. C. Das and R. R. Sharma, “Existence and stability of measure differential equations,” Czechoslovak Mathematical Journal, vol. 22, no. 97, pp. 145–158, 1972. View at Zentralblatt MATH
  9. S. G. Pandit and S. G. Deo, Differential Systems Involving Impulses, vol. 954 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1982.
  10. R. R. Sharma, “An abstract measure differential equation,” Proceedings of the American Mathematical Society, vol. 32, pp. 503–510, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995.
  12. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäauser, Boston, Mass, USA, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. M. Federson, J. G. Mesquita, and A. Slavík, “Measure functional differential equations and functional dynamic equations on time scales,” Journal of Differential Equations, vol. 252, no. 6, pp. 3816–3847, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. M. Federson, J. G. Mesquita, and A. Slavík, “Basic results for functional differential and dynamic equations involving impulses,” Mathematische Nachrichten, vol. 286, pp. 181–204, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. N. Azbelev, V. Maksimov, and L. Rakhmatullina, Introduction to the Theory of Linear Functional-Differential Equations, vol. 3 of Advanced Series in Mathematical Science and Engineering, World Federation, Atlanta, Ga, USA, 1995. View at Zentralblatt MATH
  16. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations: Methods and Applications, vol. 3 of Contemporary Mathematics and Its Applications, Hindawi, Cairo, Egypt, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. E. I. Bravyi, Solvability of Boundary Value Problems for Linear Functional Differential Equations, R&C Dynamics, Izhevsk, Russia, 2011, (Russian). View at Zentralblatt MATH
  18. R. Hakl, A. Lomtatidze, and J. Šremr, Some Boundary Value Problems for First Order Scalar Functional Differential Equations, Masaryk University, Brno, Czech Republic, 2002.
  19. I. Kiguradze and B. Půža, Boundary Value Problems for Systems of Linear Functional Differential Equations, Masaryk University, Brno, Czech Republic, 2003.
  20. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. N. N. Krasovskii, Stability of Motion. Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay, Translated by J. L. Brenner, Stanford University Press, Stanford, Calif, USA, 1963.
  22. M. H. Chang and P. T. Liu, “Maximum principles for control systems described by measure functional differential equations,” Journal of Optimization Theory and Applications, vol. 15, no. 5, pp. 517–531, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993.
  24. N. Dilna and A. Rontó, “General conditions guaranteeing the solvability of the Cauchy problem for functional differential equations,” Mathematica Bohemica, vol. 133, pp. 435–445, 2008.
  25. N. Dilna and A. Rontó, “Unique solvability of a non-linear non-local boundary-value problem for systems of non-linear functional differential equations,” Mathematica Slovaca, vol. 60, no. 3, pp. 327–338, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. A. Rontó, V. Pylypenko, and N. Dilna, “On the unique solvability of a non-local boundary value problem for linear functional differential equations,” Mathematical Modelling and Analysis, vol. 13, no. 2, pp. 241–250, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  27. R. Hakl, A. Lomtatidze, and B. Půža, “On a boundary value problem for first-order scalar functional differential equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 53, no. 3-4, pp. 391–405, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  28. A. Lomtatidze and H. Štěpánková, “On sign constant and monotone solutions of second order linear functional differential equations,” Memoirs on Differential equations and Mathematical Physics, vol. 35, pp. 65–90, 2005. View at Zentralblatt MATH
  29. J. Šremr, “On the Cauchy type problem for systems of functional differential equations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 67, no. 12, pp. 3240–3260, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  30. M. Tvrdý, “Linear integral equations in the space of regulated functions,” Mathematica Bohemica, vol. 123, pp. 177–212, 1998. View at Zentralblatt MATH
  31. N. Z. Dilna and A. M. Rontó, “General conditions for the unique solvability of initial-value problem for nonlinear functional differential equations,” Ukrainian Mathematical Journal, vol. 60, no. 2, pp. 191–198, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  32. M. A. Krasnoselskii, E. A. Lifshits, Y. V. Pokornyi, and V. Y. Stetsenko, “Positively invertible linear operators and the solvability of non-linear equations,” Doklady Akademii Nauk Tadzhikskoj SSR, vol. 17, pp. 12–14, 1974.
  33. M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, New York, NY, USA, 1984.
  34. M. A. Krasnoselskii, Positive Solutions of Operator Equations, Translated from the Russian by R. E. Flaherty, P. Noordhoff, Groningen, The Netherlands, 1964.