## Abstract and Applied Analysis

### Fixed Points for Multivalued Mappings in $b$-Metric Spaces

#### Abstract

In 2012, Samet et al. introduced the notion of α-ψ-contractive mapping and gave sufficient conditions for the existence of fixed points for this class of mappings. The purpose of our paper is to study the existence of fixed points for multivalued mappings, under an α-ψ-contractive condition of Ćirić type, in the setting of complete b-metric spaces. An application to integral equation is given.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2014), Article ID 718074, 7 pages.

Dates
First available in Project Euclid: 15 April 2015

https://projecteuclid.org/euclid.aaa/1429104836

Digital Object Identifier
doi:10.1155/2015/718074

Mathematical Reviews number (MathSciNet)
MR3332068

Zentralblatt MATH identifier
1351.54024

#### Citation

Jleli, Mohamed; Samet, Bessem; Vetro, Calogero; Vetro, Francesca. Fixed Points for Multivalued Mappings in $b$ -Metric Spaces. Abstr. Appl. Anal. 2015, Special Issue (2014), Article ID 718074, 7 pages. doi:10.1155/2015/718074. https://projecteuclid.org/euclid.aaa/1429104836

#### References

• I. A. Rus, A. Petruşel, and G. Petruşel, Fixed Point Theory, Cluj University Press, Cluj-Napoca, Romania, 2008.
• J. Nadler, “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.
• M. Abbas, B. Ali, and C. Vetro, “A Suzuki type fixed point theorem for a generalized multivalued mapping on partial Hausdorff metric spaces,” Topology and Its Applications, vol. 160, no. 3, pp. 553–563, 2013.
• H. Alikhani, D. Gopal, M. A. Miandaragh, S. Rezapour, and N. Shahzad, “Some endpoint results for $\beta$-generalized weak contractive multifunctions,” The Scientific World Journal, vol. 2013, Article ID 948472, 7 pages, 2013.
• A. Amini-Harandi, “Fixed point theory for set-valued quasi-contraction maps in metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1791–1794, 2011.
• C. Chifu and G. Petruşel, “Existence and data dependence of fixed points and strict fixed points for contractive-type multivalued operators,” Fixed Point Theory and Applications, vol. 2007, Article ID 34248, 8 pages, 2007.
• P. Z. Daffer and H. Kaneko, “Fixed points of generalized contractive multi-valued mappings,” Journal of Mathematical Analysis and Applications, vol. 192, no. 2, pp. 655–666, 1995.
• S. G. Matthews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728 of Annals of the New York Academy of Sciences, pp. 183–197, 1994.
• I. A. Bakhtin, “The contraction mapping principle in quasimetric spaces,” Functional Analysis, vol. 30, pp. 26–37, 1989 (Russian).
• S. Czerwick, “Nonlinear set-valued contraction mappings in b-metric spaces,” Atti del Seminario Matematico e Fisico dell'Università di Modena, vol. 46, pp. 263–276, 1998.
• V. Berinde, “Generalized contractions in quasimetric spaces,” in Seminar on Fixed Point Theory, pp. 3–9, 1993.
• M. Boriceanu, “Fixed point theory for multivalued generalized contraction on a set with two $b$-metrics,” Studia Universitatis Babeş-Bolyai–-Series Mathematica, vol. 54, no. 3, pp. 3–14, 2009.
• S. Czerwick, K. Dlutek, and S. L. Singh, “Round-off stability of iteration procedures for set-valued operators in b-metric spaces,” Journal of Natural & Physical Sciences, vol. 11, pp. 87–94, 2007.
• D. Paesano and P. Vetro, “Fixed point theorems for $\alpha$-set-valued quasi-contractions in b-metric spacesčommentComment on ref. [19?]: Please update the information of this reference, if possible.,” Journal of Nonlinear and Convex Analysis. In press.
• 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.
• 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.
• H. Aydi, M.-F. Bota, E. Karapinar, 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.
• S. Czerwick, “Contraction mappings in b-metric spaces,” Acta Mathematica et Informatica Universitatis Ostraviensis, vol. 1, pp. 5–11, 1993.
• J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.
• J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205–2212, 2007.
• A. C. M. Ran and M. C. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004.
• M. Cosentino, P. Salimi, and P. Vetro, “Fixed point results on metric-type spaces,” Acta Mathematica Scientia. Series B. English Edition, vol. 34, no. 4, pp. 1237–1253, 2014. \endinput