Journal of Operators

Volume 2013 (2013), Article ID 186910, 11 pages

http://dx.doi.org/10.1155/2013/186910

Research Article

## Unified Metrical Common Fixed Point Theorems in 2-Metric Spaces via an Implicit Relation

^{1}Department of Mathematics, R.H. Government Postgraduate College, Kashipur 244 713, U.S. Nagar, Uttarakhand, India^{2}Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India^{3}Università degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34, 90123 Palermo, Italy

Received 29 March 2013; Accepted 22 May 2013

Academic Editor: Lingju Kong

Copyright © 2013 Sunny Chauhan et al. 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

- S. Gähler, “2-metrische Räume und ihre topologische Struktur,”
*Mathematische Nachrichten*, vol. 26, pp. 115–148, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Gähler, “Über die Uniformisierbarkeit $(A)$-metrischer Räume,”
*Mathematische Nachrichten*, vol. 28, pp. 235–244, 1964-1965. View at Publisher · View at Google Scholar · View at MathSciNet - S. Gähler, “Zur Geometrie 2-metrischer Räume,”
*Revue Roumaine de Mathématiques Pures et Appliquées*, vol. 11, pp. 665–667, 1966. View at Google Scholar · View at MathSciNet - K. Iséki, “Fixed point theorem in 2-metric spaces,”
*Kobe University. Mathematics Seminar Notes*, vol. 3, no. 1, p. 4, 1975. View at Google Scholar · View at MathSciNet - M. E. Abd El-Monsef, H. M. Abu-Donia, and Kh. Abd-Rabou, “New types of common fixed point theorems in 2-metric spaces,”
*Chaos, Solitons & Fractals*, vol. 41, no. 3, pp. 1435–1441, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Abd-Rabou, “Common fixed point theorems for six mappings with some weaker conditions in 2-metric spaces,”
*Kuwait Journal of Science & Engineering A*, vol. 39, no. 1, pp. 99–112, 2012. View at Google Scholar · View at MathSciNet - K. P. Chi and H. T. Thuy, “A fixed point theorem in 2-metric spaces for a class of maps that satisfy a contractive condition dependent on an another function,”
*Lobachevskii Journal of Mathematics*, vol. 31, no. 4, pp. 338–346, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. J. Cho, M. S. Khan, and S. L. Singh, “Common fixed points of weakly commuting mappings,”
*Univerzitet u Novom Sadu. Zbornik Radova Prirodno-Matematičkog Fakulteta. Serija za Matemati*, vol. 18, no. 1, pp. 129–142, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Constantin, “Common fixed points of weakly commuting mappings in 2-metric spaces,”
*Mathematica Japonica*, vol. 36, no. 3, pp. 507–514, 1991. View at Google Scholar · View at MathSciNet - L. Gajić, “On common fixed point for compatible mappings of $(A)$ type on metric and 2-metric spaces,”
*Filomat*, no. 10, pp. 177–186, 1996. View at Google Scholar · View at MathSciNet - M. Imdad, M. S. Khan, and M. D. Khan, “A common fixed point theorem in $(A)$-metric spaces,”
*Mathematica Japonica*, vol. 36, no. 5, pp. 907–914, 1991. View at Google Scholar · View at MathSciNet - K. Iséki, P. L. Sharma, and B. K. Sharma, “Contraction type mapping on $(A)$-metric space,”
*Mathematica Japonica*, vol. 21, no. 1, pp. 67–70, 1976. View at Google Scholar · View at MathSciNet - M. S. Khan and M. Swaleh, “Results concerning fixed points in $(A)$-metric spaces,”
*Mathematica Japonica*, vol. 29, no. 4, pp. 519–525, 1984. View at Google Scholar · View at MathSciNet - S. V. R. Naidu, “Some fixed point theorems in metric and 2-metric spaces,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 28, no. 11, pp. 625–636, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. V. R. Naidu and J. Rajendra Prasad, “Fixed point theorems in $(A)$-metric spaces,”
*Indian Journal of Pure and Applied Mathematics*, vol. 17, no. 8, pp. 974–993, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Piao, “Unique common fixed point of a family of self-maps with same type contractive condition in 2-metric space,”
*Analysis in Theory and Applications*, vol. 24, no. 4, pp. 316–320, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. L. Singh, B. M. L. Tiwari, and V. K. Gupta, “Common fixed points of commuting mappings in $(A)$-metric spaces and an application,”
*Mathematische Nachrichten*, vol. 95, pp. 293–297, 1980. View at Publisher · View at Google Scholar · View at MathSciNet - D. Tan, Z. Liu, and J. K. Kim, “Common fixed points for compatible mappings of type (
*P*) in 2-metric spaces,”*Nonlinear Functional Analysis and Applications*, vol. 8, no. 2, pp. 215–232, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W.-Z. Wang, “Common fixed points for compatible mappings of type$(A)$ in 2-metric spaces,”
*Honam Mathematical Journal*, vol. 22, no. 1, pp. 91–97, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Popa, “Some fixed point theorems for compatible mappings satisfying an implicit relation,”
*Demonstratio Mathematica*, vol. 32, no. 1, pp. 157–163, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Popa, M. Imdad, and J. Ali, “Using implicit relations to prove unified fixed point theorems in metric and 2-metric spaces,”
*Bulletin of the Malaysian Mathematical Sciences Society*, vol. 33, no. 1, pp. 105–120, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Popa, M. Imdad, and J. Ali, “Fixed point theorems for a class of mappings governed by strictly contractive implicit function,”
*Southeast Asian Bulletin of Mathematics*, vol. 34, no. 5, pp. 941–952, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jungck, “Compatible mappings and common fixed points. II,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 11, no. 2, pp. 285–288, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,”
*Publications de l'Institut Mathématique*, vol. 32, pp. 149–153, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Pant, “Common fixed points of four mappings,”
*Bulletin of the Calcutta Mathematical Society*, vol. 90, no. 4, pp. 281–286, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Pant, “A common fixed point theorem under a new condition,”
*Indian Journal of Pure and Applied Mathematics*, vol. 30, no. 2, pp. 147–152, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jungck, “Common fixed points for noncontinuous nonself maps on nonmetric spaces,”
*Far East Journal of Mathematical Sciences*, vol. 4, no. 2, pp. 199–215, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. P. Murthy, S. S. Chang, Y. J. Cho, and B. K. Sharma, “Compatible mappings of type (
*A*) and common fixed point theorems,”*Kyungpook Mathematical Journal*, vol. 32, no. 2, pp. 203–216, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. Abu-Donia and M. A. Atia, “Common fixed points theorem in 2-metric spaces,”
*Arabian Journal for Science and Engineering*, 2007-2008. View at Google Scholar - M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 181–188, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Liu, J. Wu, and Z. Li, “Common fixed points of single-valued and multivalued maps,”
*International Journal of Mathematics and Mathematical Sciences*, no. 19, pp. 3045–3055, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Sintunavarat and P. Kumam, “Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces,”
*Journal of Applied Mathematics*, vol. 2011, Article ID 637958, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. L. Singh, B. D. Pant, and S. Chauhan, “Fixed point theorems in non-Archimedean Menger PM-spaces,”
*Journal of Nonlinear Analysis and Optimization. Theory and Applications*, vol. 3, no. 2, pp. 153–160, 2012. View at Google Scholar · View at MathSciNet - W. Sintunavarat and P. Kumam, “Common fixed points for
*R*-weakly commuting in fuzzy metric spaces,”*Annali dell'Universitá di Ferrara. Sezione VII. Scienze Matematiche*, vol. 58, no. 2, pp. 389–406, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - R. K. Verma and H. K. Pathak, “Common fixed point theorems using property (E.A) in complex-valued metric spaces,”
*Thai Journal of Mathematics*. In press. - M. Imdad, B. D. Pant, and S. Chauhan, “Fixed point theorems in Menger spaces using the ($CL{R}_{ST}$) property and applications,”
*Journal of Nonlinear Analysis and Optimization. Theory and Applications*, vol. 3, no. 2, pp. 225–237, 2012. View at Google Scholar · View at MathSciNet - M. Imdad, S. Chauhan, and Z. Kadelburg, “Fixed point theorems for mappings with common limit range property satisfying generalized (
*ψ*;*ϕ*)-weak contractive conditions,”*Mathematical Sciences*, vol. 7, article 16, 2013. View at Publisher · View at Google Scholar - M. Imdad, J. Ali, and M. Tanveer, “Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces,”
*Chaos, Solitons & Fractals*, vol. 42, no. 5, pp. 3121–3129, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. K. Pathak, R. Rodríguez-López, and R. K. Verma, “A common fixed point theorem using implicit relation and property (E.A) in metric spaces,”
*Filomat*, vol. 21, no. 2, pp. 211–234, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Husain and V. M. Sehgal, “On common fixed points for a family of mappings,”
*Bulletin of the Australian Mathematical Society*, vol. 13, no. 2, pp. 261–267, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Imdad, S. Kumar, and M. S. Khan, “Remarks on some fixed point theorems satisfying implicit relations,”
*Radovi Matematički*, vol. 11, no. 1, pp. 135–143, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. S. Khan and B. Fisher, “Some fixed point theorems for commuting mappings,”
*Mathematische Nachrichten*, vol. 106, pp. 323–326, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. S. Khan and M. Imdad, “A common fixed point theorem for a class of mappings,”
*Indian Journal of Pure and Applied Mathematics*, vol. 14, no. 10, pp. 1220–1227, 1983. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. V. R. Naidu and J. Rajendra Prasad, “Common fixed points for four self-maps on a metric space,”
*Indian Journal of Pure and Applied Mathematics*, vol. 16, no. 10, pp. 1089–1103, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Chugh and S. Kumar, “Common fixed points for weakly compatible maps,”
*Proceedings of The Indian Academy of Sciences-Mathematical Sciences*, vol. 111, no. 2, pp. 241–247, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Ali and M. Imdad, “An implicit function implies several contraction conditions,”
*Sarajevo Journal of Mathematics*, vol. 4, no. 2, pp. 269–285, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. S. Jeong and B. E. Rhoades, “Some remarks for improving fixed point theorems for more than two maps,”
*Indian Journal of Pure and Applied Mathematics*, vol. 28, no. 9, pp. 1177–1196, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. E. Hardy and T. D. Rogers, “A generalization of a fixed point theorem of Reich,”
*Canadian Mathematical Bulletin*, vol. 16, pp. 201–206, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. N. Lal, P. P. Murthy, and Y. J. Cho, “An extension of Telci, Tas and Fisher's theorem,”
*Journal of the Korean Mathematical Society*, vol. 33, no. 4, pp. 891–908, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. W. Bryant, “A remark on a fixed-point theorem for iterated mappings,”
*The American Mathematical Monthly*, vol. 75, pp. 399–400, 1968. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. K. Pathak, M. S. Khan, and R. Tiwari, “A common fixed point theorem and its application to nonlinear integral equations,”
*Computers & Mathematics with Applications*, vol. 53, no. 6, pp. 961–971, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet