Table of Contents
Journal of Operators
Volume 2013, Article ID 408791, 6 pages
http://dx.doi.org/10.1155/2013/408791
Research Article

Fibre Contraction Principle with respect to an Iterative Algorithm

Department of Mathematics, Babeş-Bolyai University, 1 M. Kogălniceanu, 400048 Cluj-Napoca, Romania

Received 6 March 2013; Accepted 2 September 2013

Academic Editor: Lingju Kong

Copyright © 2013 Marcel-Adrian Şerban. 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. I. A. Rus, “Picard operators and applications,” Scientiae Mathematicae Japonicae, vol. 58, no. 1, pp. 191–219, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. von Trotha, “Contractivity in certain D-R spaces,” Mathematische Nachrichten, vol. 101, pp. 207–213, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. von Trotha, “Structure properties of D-R spaces,” Polska Akademia Nauk, vol. 184, 72 pages, 1981. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Fréchet, Les Espaces Abstraits, Gauthier-Villars, Paris, France, 1928.
  5. L. M. Blumenthal, Theory and Applications of Distance Geometry, Clarendon Press, Oxford, UK, 1953. View at MathSciNet
  6. I. A. Rus, “Fiber Picard operators theorem and applications,” Universitatis Babeş-Bolyai, vol. 44, no. 3, pp. 89–97, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. W. Hirsch and C. C. Pugh, “Stable manifolds and hyperbolic sets,” in Global Analysis, pp. 133–163, American Mathematical Society, Providence, RI, USA, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Sotomayor, “Smooth dependence of solutions of differential equations on initial data: a simple proof,” Boletim da Sociedade Brasileira de Matemática, vol. 4, no. 1, pp. 55–59, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. Tămăşan, “Differentiability with respect to lag for nonlinear pantograph equations,” Pure Mathematics and Applications, vol. 9, no. 1-2, pp. 215–220, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. I. A. Rus, “A fiber generalized contraction theorem and applications,” Mathematica, vol. 41(64), no. 1, pp. 85–90, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M.-A. Şerban, “Fiber φ-contractions,” Universitatis Babeş-Bolyai, vol. 44, no. 3, pp. 99–108, 1999. View at Google Scholar · View at MathSciNet
  12. M.-A. Şerban, The Fixed Point Theory for the Operators on Cartesian Product, Cluj University Press, Cluj-Napoca, Romania, 2002.
  13. M.-A. Şerban, “Fiber contraction theorem in generalized metric spaces,” Automation Computers Applied Mathematics, vol. 16, no. 1-2, pp. 9–14, 2007. View at Google Scholar · View at MathSciNet
  14. S. Andrász, “Fibre φ−contraction on generalized metric spaces and applications,” Mathematica, vol. 45(68), no. 1, pp. 3–8, 2003. View at Google Scholar
  15. C. Bacoţiu, “Fiber Picard operators,” Seminar on Fixed Point Theory Cluj-Napoca, vol. 1, pp. 5–8, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Petruşel, I. A. Rus, and M. A. Şerban, “Fixed points for operators on generalized metric spaces,” Cubo, vol. 10, no. 4, pp. 45–66, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. I. A. Rus, A. Petruşel, and M. A. Şerban, “Fibre Picard operators on gauge spaces and applications,” Zeitschrift für Analysis und ihre Anwendungen, vol. 27, no. 4, pp. 407–423, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. Chiş-Novac, R. Precup, and I. A. Rus, “Data dependence of fixed points for non-self generalized contractions,” Fixed Point Theory, vol. 10, no. 1, pp. 73–87, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. V. Ilea and D. Otrocol, “On a D. V. Ionescu problem for functional-differential equations,” Fixed Point Theory, vol. 10, no. 1, pp. 125–140, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. Dobriţoiu, “An integral equation with modified argument,” Universitatis Babeş-Bolyai, vol. 49, no. 3, pp. 27–33, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Dobriţoiu, “Properties of the solution of an integral equation with modified argument,” Carpathian Journal of Mathematics, vol. 23, no. 1-2, pp. 77–80, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. Dobriţoiu, W. W. Kecs, and A. Toma, “An application of the fiber generalized contractions theorem,” WSEAS Transactions on Mathematics, vol. 5, no. 12, pp. 1330–1335, 2006. View at Google Scholar · View at MathSciNet
  23. I. M. Olaru, “An integral equation via weakly Picard operators,” Fixed Point Theory, vol. 11, no. 1, pp. 97–106, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. L. Barreira and C. Valls, “Parameter dependence of stable manifolds for difference equations,” Nonlinearity, vol. 23, no. 2, pp. 341–367, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. I. A. Rus, “An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations,” Fixed Point Theory, vol. 13, no. 1, pp. 179–192, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. V. Berinde, Iterative Approximation of Fixed Points, vol. 1912 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2nd edition, 2007. View at MathSciNet