Abstract and Applied Analysis

Ulam-Hyers Stability Results for Fixed Point Problems via α - ψ -Contractive Mapping in ( b )-Metric Space

Monica-Felicia Bota, Erdal Karapınar, and Oana Mleşniţe

Full-text: Open access

Abstract

We will investigate some existence, uniqueness, and Ulam-Hyers stability results for fixed point problems via α - ψ -contractive mapping of type-( b ) in the framework of b -metric spaces. The presented theorems extend, generalize, and unify several results in the literature, involving the results of Samet et al. (2012).

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 825293, 6 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393448679

Digital Object Identifier
doi:10.1155/2013/825293

Mathematical Reviews number (MathSciNet)
MR3095362

Zentralblatt MATH identifier
07095399

Citation

Bota, Monica-Felicia; Karapınar, Erdal; Mleşniţe, Oana. Ulam-Hyers Stability Results for Fixed Point Problems via $\alpha $ - $\psi $ -Contractive Mapping in ( $b$ )-Metric Space. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 825293, 6 pages. doi:10.1155/2013/825293. https://projecteuclid.org/euclid.aaa/1393448679


Export citation

References

  • B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for $\alpha $-$\psi $-contractive type mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 4, pp. 2154–2165, 2012.
  • S. Czerwik, “Nonlinear set-valued contraction mappings in b-metric spaces,” Atti del Seminario Matematico e Fisico dell'Università di Modena, vol. 46, no. 2, pp. 263–276, 1998.
  • S. Czerwik, “Contraction mappings in b-metric spaces,” Acta Mathematica et Informatica Universitatis Ostraviensis, vol. 1, pp. 5–11, 1993.
  • N. Bourbaki, Topologie Générale, Herman, Paris, France, 1974.
  • I. A. Bakhtin, “The contraction mapping principle in quasimetric spaces,” Functional Analysis, vol. 30, pp. 26–37, 1989.
  • J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York, NY, USA, 2001.
  • M. Boriceanu, A. Petruşel, and I. A. Rus, “Fixed point theorems for some multivalued generalized contractions in b-metric spaces,” International Journal of Mathematics and Statistics, vol. 6, no. 10, pp. 65–76, 2010.
  • M. Boriceanu, “Strict fixed point theorems for multivalued operators in b-metric spaces,” International Journal of Modern Mathematics, vol. 4, no. 3, pp. 285–301, 2009.
  • M. Boriceanu, “Fixed point theory for multivalued generalized contraction on a set with two b-metrics,” Mathematica, vol. 54, no. 3, pp. 3–14, 2009.
  • M. Bota, Dynamical Aspects in the Theory of Multivalued Operators, Cluj University Press, 2010.
  • H. Aydi, M.-F. Bota, E. Karap\inar, and S. Mitrović, “A fixed point theorem for set-valued quasi-contractions in b-metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 88, 2012.
  • H. Aydi, M.-F. Bota, E. Karapinar, and S. Moradi, “A common fixed point for weak $\phi $-contractions on b-metric spaces,” Fixed Point Theory, vol. 13, no. 2, pp. 337–346, 2012.
  • S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964.
  • 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, pp. 222–224, 1941.
  • M. F. Bota-Boriceanu and A. Petruşel, “Ulam-Hyers stability for operatorial equations,” Analele Stiintifice ale Universitatii, vol. 57, no. 1, pp. 65–74, 2011.
  • V. L. Lazăr, “Ulam-Hyers stability for partial differential inclusions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 21, pp. 1–19, 2012.
  • I. A. Rus, “The theory of a metrical fixed point theoremml: theoretical and applicative relevances,” Fixed Point Theory, vol. 9, no. 2, pp. 541–559, 2008.
  • I. A. Rus, “Remarks on Ulam stability of the operatorial equations,” Fixed Point Theory, vol. 10, no. 2, pp. 305–320, 2009.
  • F. A. Tişe and I. C. Tişe, “Ulam-Hyers-Rassias stability for set integral equations,” Fixed Point Theory, vol. 13, no. 2, pp. 659–667, 2012.
  • J. Brzd\kek, J. Chudziak, and Z. Páles, “A fixed point approach to stability of functional equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6728–6732, 2011.
  • J. Brzd\kek and K. Ciepliński, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 18, pp. 6861–6867, 2011.
  • J. Brzd\kek 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.
  • L. Cadariu, L. Gavruta, and P. Gavruta, “Fixed points and generalized Hyers-Ulam stability,” Abstract and Applied Analysis, vol. 2012, Article ID 712743, 10 pages, 2012.
  • V. Berinde, “Generalized contractions in quasimetric spaces,” in Seminar on Fixed Point Theory, vol. 93 of Preprint 3, pp. 3–9, Babeş-Bolyai University, Cluj-Napoca, Romania, 1993.
  • I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, Romania, 2001.
  • V. Berinde, Contractii Generalizate şi Aplicatii, vol. 2, Editura Cub Press, Baia Mare, Romania, 1997.
  • V. Berinde, “Sequences of operators and fixed points in quasimetric spaces,” Mathematica, vol. 41, no. 4, pp. 23–27, 1996.
  • N. Hussain, Z. Kadelburg, S. Radenović, and F. Al-Solamy, “Comparison functions and fixed point results in partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 605781, 15 pages, 2012.
  • T. P. Petru, A. Petruşel, and J.-C. Yao, “Ulam-Hyers stability for operatorial equations and inclusions via nonself operators,” Taiwanese Journal of Mathematics, vol. 15, no. 5, pp. 2195–2212, 2011.