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Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 254720, 18 pages
http://dx.doi.org/10.1155/2009/254720
Research Article

Implicit Difference Inequalities Corresponding to First-Order Partial Differential Functional Equations

Institute of Mathematics, University of Gdańsk, Wit Stwosz Street 57, 80-952 Gdańsk, Poland

Received 19 August 2008; Accepted 5 January 2009

Academic Editor: Donal O'Regan

Copyright © 2009 Z. Kamont and K. Kropielnicka. 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. Z. Kowalski, “A difference method for the non-linear partial differential equation of the first order,” Annales Polonici Mathematici, vol. 18, pp. 235–242, 1966. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Pliś, “On difference inequalities corresponding to partial differential inequalities of the first order,” Annales Polonici Mathematici, vol. 20, pp. 179–181, 1968. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. P. Brandi, Z. Kamont, and A. Salvadori, “Differential and differential-difference inequalities related to mixed problems for first order partial differential-functional equations,” Atti del Seminario Matematico e Fisico dell'Università di Modena, vol. 39, no. 1, pp. 255–276, 1991. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. K. Prządka, “Difference methods for non-linear partial differential-functional equations of the first order,” Mathematische Nachrichten, vol. 138, pp. 105–123, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  5. M. Malec, C. Mączka, and W. Voigt, “Weak difference-functional inequalities and their application to the difference analogue of non-linear parabolic differential-functional equations,” Beiträge zur Numerischen Mathematik, no. 11, pp. 69–79, 1983. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. Malec and M. Rosati, “Weak monotonicity for nonlinear systems of functional-finite difference inequalities of parabolic type,” Rendiconti di Matematica. Serie VII, vol. 3, no. 1, pp. 157–170, 1983. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Malec and M. Rosati, “A convergent scheme for nonlinear systems of differential functional equations of parabolic type,” Rendiconti di Matematica. Serie VII, vol. 3, no. 2, pp. 211–227, 1983. View at Google Scholar · View at MathSciNet
  8. M. Malec and A. Schiaffino, “Méthode aux différences finies pour une équation non-linéaire differentielle fonctionnelle du type parabolique avec une condition initiale de Cauchy,” Bollettino della Unione Matemática Italiana. Serie VII. B, vol. 1, no. 1, pp. 99–109, 1987. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. W. Czernous, “Generalized solutions of mixed problems for first-order partial functional differential equations,” Ukraïnskyj Matematychnyj Zhurnal, vol. 58, no. 6, pp. 803–828, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Z. Kamont, Hyperbolic Functional Differential Inequalities and Applications, vol. 486 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. View at Zentralblatt MATH · View at MathSciNet
  11. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, vol. 118 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1996. View at Zentralblatt MATH · View at MathSciNet
  12. K. M. Magomedov and A. S. Kholodov, Mesh Characteristics Numerical Methods, Nauka, Moscow, Russia, 1988. View at MathSciNet
  13. A. A. Samarskii, The Theory of Difference Schemes, vol. 240 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2001. View at Zentralblatt MATH · View at MathSciNet
  14. A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich, Difference Schemes with Operator Factors, vol. 546 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002. View at Zentralblatt MATH · View at MathSciNet
  15. N. N. Luzin, “On the method of approximate integration of academician S. A. Chaplygin,” Uspekhi Matematicheskikh Nauk, vol. 6, no. 6(46), pp. 3–27, 1951 (Russian). View at Google Scholar · View at MathSciNet
  16. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatulina, Introduction to the Theory of Functional Differential Equations, vol. 3 of Advanced Series in Mathematical Science and Engineering, World Federation, Atlanta, Ga, USA, 1995. View at Zentralblatt MATH · View at MathSciNet
  17. V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, vol. 55 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1969. View at Zentralblatt MATH · View at MathSciNet
  18. A. Kępczyńska, “Implicit difference methods for first order partial differential functional equations,” Nonlinear Oscillations, vol. 8, no. 2, pp. 198–213, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet