Common fixed points for generalized (v,o)-weak contractions in ordered cone metric spaces

Authors

  • Hemant Kumar Nashine Disha Institute of Management and Technology
  • Hassen Aydi Université de Sousse

DOI:

https://doi.org/10.4995/agt.2012.1626

Keywords:

Coincidence point, Common fixed point, weakly contractive condition, Dominating map, Dominated map, Ordered set, one metric space

Abstract

The purpose of this paper is to establish coincidence point and common fixed point results for four maps satisfying generalized (v,o)-weak contractions in partially ordered cone metric spaces. Also, some illustrative examples are presented.

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Author Biographies

Hemant Kumar Nashine, Disha Institute of Management and Technology

Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur-492101(Chhattisgarh), India.

hemantnashine@rediffmail.com

Hassen Aydi, Université de Sousse

Université de Sousse, Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Route GP1-4011, H. Sousse, Tunisie.

References

M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl. 341 (2008), 416–420. http://dx.doi.org/10.1016/j.jmaa.2007.09.070

M. Abbas, T. Nazir and S. Radenovic, Common fixed point of four maps in partially ordered metric spaces, Appl. Math. Lett. 24 (2011), 1520–1526. http://dx.doi.org/10.1016/j.aml.2011.03.038

M. Abbas and B. E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl. Math. Lett. 22 (2009), 511–515. http://dx.doi.org/10.1016/j.aml.2008.07.001

R. P. Agarwal, Contraction and approximate contraction with an application to multipoint boundary value problems, J. Comput. Appl. Appl. Math. 9 (1983), 315–325. http://dx.doi.org/10.1016/0377-0427(83)90003-1

R. P. Agarwal, M. A. El-Gebeily and D. O’Regan, Generalized contractions in partially ordered metric spaces, Applicable Anal. 87 (2008), 109–116.

I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl. 2010(2010) Article ID 621492, 17 pages.

H. Aydi, H. K. Nashine, B. Samet and H. Yazidi, Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations, Nonlinear Anal. 74, no. 17 (2011), 6814–6825. http://dx.doi.org/10.1016/j.na.2011.07.006

I. Beg and A.R. Butt, Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal. 71 (2009), 3699–3704. http://dx.doi.org/10.1016/j.na.2009.02.027

B. S. Choudhury and N. Metiya, The point of coincidence and common fixed point for a pair of mappings in cone metric spaces, Comput. Math. Appl. 60 (2010), 1686–1695. http://dx.doi.org/10.1016/j.camwa.2010.06.048

Lj. B. Ciric, N. Cakic, M. Rajovic and J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. 2008 (2008), Article ID 131294, 11 pages.

Lj. B. Ciric, B. Samet, N. Cakic and B. Damjanovic, Coincidence and fixed point theorems for generalized ( , _)-weak nonlinear contraction in ordered K-metric spaces, Comput. Math. Appl. 62 (2011), 3305–3316. http://dx.doi.org/10.1016/j.camwa.2011.07.061

W.-S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259–2261. http://dx.doi.org/10.1016/j.na.2009.10.026

L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468–1476. http://dx.doi.org/10.1016/j.jmaa.2005.03.087

D. Ilic and V. Rakoèevic, Common fixed points for maps on cone metric space, J. Math. Anal. Appl. 341 (2008), 876–882. http://dx.doi.org/10.1016/j.jmaa.2007.10.065

E. Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. 59, no. 12 (2010), 3656–3668. http://dx.doi.org/10.1016/j.camwa.2010.03.062

E.M. Mukhamadiev and V.J. Stetsenko, Fixed point principle in generalized metric space, Izvestija AN Tadzh. SSR, fiz.-mat. igeol.-chem. nauki. 10 (4) (1969), 8-19 (in Russian).

H.K. Nashine and I. Altun, Fixed point theorems for generalized weakly contractive condition in ordered metric spaces, Fixed Point Theory Appl. 2011 (2011), Article ID 132367, 20 pages.

H. K. Nashine and M. Abbas, Common fixed point point of mappings satisfying implicit contractive conditions in TVS-valued ordered cone metric spaces, preprint.

H. K. Nashine and B. Samet, Fixed point results for mappings satisfying ( , ')-weakly contractive condition in partially ordered metric spaces, Nonlinear Anal. 74 (2011), 2201–2209. http://dx.doi.org/10.1016/j.na.2010.11.024

H. K. Nashine, B. Samet and C. Vetro, Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces, Math. Comput. Modelling, to appear. http://dx.doi.org/10.1016/j.mcm.2011.03.014

H .K. Nashine and W. Shatanawi, Coupled common fixed point theorems for pair of commuting mappings in partially ordered complete metric spaces, Comput. Math. Appl. 62 (2011), 1984–1993. http://dx.doi.org/10.1016/j.camwa.2011.06.042

J. J. Nieto and R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordianry differential equations, Order 22 (2005), 223–239. http://dx.doi.org/10.1007/s11083-005-9018-5

J. O. Olaleru, Some generalizations of fixed point theorems in cone metric spaces, Fixed Point Theory Appl. 2009 (2009), Article ID 657914, 10 pages.

A. I. Perov, The Cauchy problem for systems of ordinary differential equations, in: Approximate Methods of Solving Differential Equations, Kiev, Naukova Dumka, 1964, pp. 115–134 (in Russian).

A. I. Perov and A.V. Kibenko, An approach to studying boundary value problems, Izvestija AN SSSR, Ser. Math. 30, no. 2 (1966), 249–264 (in Russian).

A. C. M. Ran and M. C. B. Reurings, A fixed point thm in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435–1443. http://dx.doi.org/10.1090/S0002-9939-03-07220-4

D. O’regan and A. Petrusel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008), 241–1252. http://dx.doi.org/10.1016/j.jmaa.2007.11.026

Sh. Rezapour and R. Hamlbarani, Some notes on the paper: cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345 (2008), 719–724. http://dx.doi.org/10.1016/j.jmaa.2008.04.049

B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508–4517. http://dx.doi.org/10.1016/j.na.2010.02.026

B. Samet, Common fixed point theorems involving two pairs of weakly compatible mappings in K-metric spaces, Appl. Math. Lett. 24 (2011), 1245–1250. http://dx.doi.org/10.1016/j.aml.2011.02.016

W. Shatanawi, Partially ordered cone metric spaces and coupled fixed point results, Comput. Math. Appl. 60 (2010), 2508–2515. http://dx.doi.org/10.1016/j.camwa.2010.08.074

W. Shatanawi and B. Samet, On ( , _)-weakly contractive condition in partially ordered metric spaces, Comput. Math. Appl. 62 (2011), 3204–3214 http://dx.doi.org/10.1016/j.camwa.2011.08.033

J. S. Vandergraft, Newton’s method for convex operators in partially ordered spaces, SIAM J. Numer. Anal. 4 (1967), 406–432. http://dx.doi.org/10.1137/0704037

P. P. Zabrejko, K-metric and K-normed linear spaces: survey, Collect. Math. 48 (1997), 825–859.

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How to Cite

[1]
H. K. Nashine and H. Aydi, “Common fixed points for generalized (v,o)-weak contractions in ordered cone metric spaces”, Appl. Gen. Topol., vol. 13, no. 2, pp. 151–166, Oct. 2012.

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Section

Regular Articles