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Abstract and Applied Analysis
Volume 2014, Article ID 268230, 6 pages
http://dx.doi.org/10.1155/2014/268230
Research Article

Ulam-Hyers Stability and Well-Posedness of Fixed Point Problems for α-λ-Contraction Mapping in Metric Spaces

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Rangsit Center, Pathum Thani 12121, Thailand

Received 13 January 2014; Revised 27 April 2014; Accepted 22 May 2014; Published 12 June 2014

Academic Editor: Adrian Petrusel

Copyright © 2014 Marwan Amin Kutbi and Wutiphol Sintunavarat. 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. S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, 1960. View at MathSciNet
  2. D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222–224, 1941. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M.-F. Bota, E. Karapınar, and O. Mleşniţe, “Ulam-Hyers stability results for fixed point problems via α-ψ-contractive mapping in (b)-metric space,” Abstract and Applied Analysis, vol. 2013, Article ID 825293, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  4. M. F. Bota-Boriceanu and A. Petruşel, “Ulam-Hyers stability for operatorial equations,” Analele Stiintifice ale Universitatii Al I Cuza din Iasi: Matematica, vol. 57, no. 1, pp. 65–74, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. V. L. Lazǎr, “Ulam-Hyers stability for partial differential inclusions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 21, pp. 1–19, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. I. A. Rus, “The theory of a metrical fixed point theorem: theoretical and applicative relevances,” Fixed Point Theory, vol. 9, no. 2, pp. 541–559, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. I. A. Rus, “Remarks on Ulam stability of the operatorial equations,” Fixed Point Theory, vol. 10, no. 2, pp. 305–320, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. F. A. Tişe and I. C. Tişe, “Ulam-Hyers-Rassias stability for set integral equations,” Fixed Point Theory, vol. 13, no. 2, pp. 659–668, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. Brzdȩk, J. Chudziak, and Z. Páles, “A fixed point approach to stability of functional equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 17, pp. 6728–6732, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. Brzdek and K. Ciepliski, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 18, pp. 6861–6867, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Brzdȩk and K. Ciepliński, “A fixed point theorem and the Hyers-Ulam stability in non-Archimedean spaces,” Journal of Mathematical Analysis and Applications, vol. 400, no. 1, pp. 68–75, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Cadariu, L. Gavruta, and P. Gavruta, “Fixed points and generalized Hyers-Ulam stability,” Abstract Applied Analysis, vol. 2012, 10 pages, 2012. View at Publisher · View at Google Scholar
  13. F. S. de Blasi and J. Myjak, “Sur la porosité de l'ensemble des contractions sans point fixe,” Comptes Rendus des Séances de l'Académie des Sciences I: Mathématique, vol. 308, no. 2, pp. 51–54, 1989. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. Reich and A. J. Zaslavski, “Well-posedness of fixed point problems,” Far East Journal of Mathematical Sciences (FJMS), Special volume, part III, pp. 393–401, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. B. K. Lahiri and P. Das, “Well-posedness and porosity of a certain class of operators,” Demonstratio Mathematica, vol. 38, no. 1, pp. 169–176, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. V. Popa, “Well-posedness of fixed point problem in orbitally complete metric spaces,” Studii şi Cercetări Ştiinţifice. Seria Matematică, no. 16, pp. 209–214, 2006, Proceedings of ICMI 45. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. V. Popa, “Well posedness of fixed point problem in compact metric spaces,” Bulletin Universităţii Petrol-Gaze din Ploieşti. Seria Matematică—Informatică—Fizică, vol. 60, no. 1, p. 14, 2008. View at Google Scholar
  18. B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for α-ψ-contractive type mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2154–2165, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. R. P. Agarwal, W. Sintunavarat, and P. Kumam, “PPF dependent fixed point theorems for an c-admissible non-self mapping in the Razumikhin class,” Fixed Point Theory and Applications, vol. 2013, article 280, 2013. View at Google Scholar
  20. M. A. Alghamdi and E. Karapinar, “G-beta-psi contractive type mappings and related fixed point theorems,” Journal of Inequalities and Applications, vol. 2013, article 70, 2013. View at Google Scholar
  21. N. Hussain, E. Karapınar, P. Salimi, and P. Vetro, “Fixed point results for Gm-Meir-Keeler contractive and G-(α,ψ)-Meir-Keeler contractive mappings,” Fixed Point Theory and Applications, vol. 2013, article 34, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  22. E. Karapnar and B. Samet, “Generalized (α-ψ) contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 793486, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. E. Karapinar and W. Sintunavarat, “The existence of an optimal approximate solution theorems for generalized α-proximal contraction non-self mappings and applications,” Fixed Point Theory and Applications, vol. 2013, article 323, 2013. View at Google Scholar
  24. M. A. Kutbi and W. Sintunavarat, “The existence of fixed point theorems via w-distance and α-admissible mappings and applications,” Abstract and Applied Analysis, vol. 2013, Article ID 165434, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  25. W. Sintunavarat, S. Plubtieng, and P. Katchang, “Fixed point result and applications on b-metric space endowed with an arbitrary binary relation,” Fixed Point Theory and Applications, vol. 2013, article 296, 2013. View at Google Scholar
  26. M. U. Ali, T. Kamran, W. Sintunavarat, and P. Katchang, “Mizoguchi-Takahashi's fixed point theorem with α,η functions,” Abstract and Applied Analysis, vol. 2013, Article ID 418798, 4 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet