## Abstract and Applied Analysis

### Discussion on $\alpha -\psi$ Contractions on Generalized Metric Spaces

Erdal Karapınar

#### Abstract

We discuss the existence and uniqueness of fixed points of $\alpha -\psi$ contractive mappings in complete generalized metric spaces, introduced by Branciari. Our results generalize and improve several results in the literature.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 962784, 7 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606986

Digital Object Identifier
doi:10.1155/2014/962784

Mathematical Reviews number (MathSciNet)
MR3173299

Zentralblatt MATH identifier
07023407

#### Citation

Karapınar, Erdal. Discussion on $\alpha -\psi$ Contractions on Generalized Metric Spaces. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 962784, 7 pages. doi:10.1155/2014/962784. https://projecteuclid.org/euclid.aaa/1412606986

#### References

• A. Branciari, “A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces,” Publicationes Mathematicae Debrecen, vol. 57, no. 1-2, pp. 31–37, 2000.
• B. Samet, “Discussion on “a fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces” by A. Branciari,” Publicationes Mathematicae Debrecen, vol. 76, no. 4, pp. 493–494, 2010.
• M. Jleli and B. Samet, “The Kannan's fixed point theorem in a cone rectangular metric space,” The Journal of Nonlinear Sciences and Applications, vol. 2, no. 3, pp. 161–167, 2009.
• W. A. Kirk and N. Shahzad, “Generalized metrics and Caristis theorem,” Fixed Point Theory and Applications, vol. 2013, article 129, 2013.
• L. Kikina and K. Kikina, “A fixed point theorem in generalized metric spaces,” Demonstratio Mathematica, vol. 46, no. 1, pp. 181–190, 2013.
• W. A. Wilson, “On semi-metric spaces,” The American Journal of Mathematics, vol. 53, no. 2, pp. 361–373, 1931.
• P. Das and L. K. Dey, “Fixed point of contractive mappings in generalized metric spaces,” Mathematica Slovaca, vol. 59, no. 4, pp. 499–504, 2009.
• H. Lakzian and B. Samet, “Fixed points for $(\psi ,\varphi )$-weakly contractive mappings in generalized metric spaces,” Applied Mathematics Letters, vol. 25, no. 5, pp. 902–906, 2012.
• H. Aydi, E. Karapinar, and H. Lakzian, “Fixed point results on a class of generalized metric spaces,” Mathematical Sciences, vol. 6, no. 1, article 46, pp. 1–6, 2012.
• N. Bilgili, E. Karapinar, and D. Turkoglu, “A note on common fixed points for $(\psi ,\alpha ,\beta )$-weakly contractive mappings in generalized metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 287, 2013.
• C.-M. Chen and W. Y. Sun, “Periodic points for the weak contraction mappings in complete generalized metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 79, 2012.
• I. Erhan, E. Karapinar, and T. Sekulić, “Fixed points of $(\psi ,\phi )$ contractions on rectangular metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 138, 2012.
• D. Miheţ, “On Kannan fixed point principle in generalized metric spaces,” The Journal of Nonlinear Science and Its Applications, vol. 2, no. 2, pp. 92–96, 2009.
• 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.
• M. U. Ali, T. Kamran, and E. Karapinar, “$(\alpha ,\psi ,\xi )$-contractive multivalued mappings,” Fixed Point Theory and Applications, vol. 2014, article 7, 2014.
• M. Jleli, E. Karap\inar, and B. Samet, “Best proximity points for generalized $(\alpha ,\psi )$-proximal contractive type mappings,” Journal of Applied Mathematics, vol. 2013, Article ID 534127, 10 pages, 2013.
• M. Jleli, E. Karap\inar, and B. Samet, “Fixed point results for $\alpha \text{-}{\psi }_{\lambda }$contractions on gauge spaces and applications,” Abstract and Applied Analysis, vol. 2013, Article ID 730825, 7 pages, 2013.
• E. Karap\inar and B. Samet, “Generalized $\alpha$-$\psi$-contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 793486, 17 pages, 2012.
• B. Mohammadi, S. Rezapour, and N. Shahzad, “Some results on fixed points of $\alpha$-$\psi$-Ciric generalized multifunctions,” Fixed Point Theory and Applications, vol. 2013, article 24, 2013.
• R. M. Bianchini and M. Grandolfi, “Trasformazioni di tipo contrattivo generalizzato in uno spazio metrico,” ATTI Della Accademia Nazionale Dei Lincei, vol. 45, pp. 212–216, 1968.
• P. D. Proinov, “A generalization of the Banach contraction principle with high order of convergence of successive approximations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 67, no. 8, pp. 2361–2369, 2007.
• P. D. Proinov, “New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems,” Journal of Complexity, vol. 26, no. 1, pp. 3–42, 2010.
• I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, Romania, 2001.
• M. Berzig and M. Rus, “Fixed point theorems for $\alpha$-contractive mappings of Meir-Keeler type and applications,” http://arxiv.org/abs/1303.5798.
• M. Berzig and E. Karap\inar, “Fixed point results for $(\alpha \psi -\alpha \phi )$-contractive mappings for a generalized altering distance,” Fixed Point Theory and Applications, vol. 2013, article 205, 2013. \endinput