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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access May 10, 2017

A Kannan-like contraction in partially ordered spaces

  • Binayak S. Choudhury EMAIL logo and Amaresh Kundu
From the journal Demonstratio Mathematica

Abstract

In this paper, we have introduced a generalised Kannan type contraction. It has been established that such mappings necessarily have fixed points in a complete partially ordered metric space. The fixed point is unique under some additional conditions. The result is illustrated with an example. The work is in the line of research in fixed point theory on ordered metric structures.

MSC 2010: 47H10; 54H25

References

[1] R. P. Agarwal, M. A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Applicable Anal. 87(1) (2008), 109–116.10.1080/00036810701556151Search in Google Scholar

[2] A. Amini-Harandi, H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. TMA 72(5) (2010), 2238–2242.10.1016/j.na.2009.10.023Search in Google Scholar

[3] I. Altun, V. Rakočević, Ordered cone metric spaces and fixed point results, Comput. Math. Appl. 60 (2010), 1145–115110.1016/j.camwa.2010.05.038Search in Google Scholar

[4] B. S. Choudhury, P. Maity, Coupled fixed point results in generalized metric spaces, Math. Comput. Modelling 54 (2011), 73–79.10.1016/j.mcm.2011.01.036Search in Google Scholar

[5] B. S. Choudhury, K. Das, Fixed points of generalized Kannan type mappings in generalized Menger spaces, Commun. Korean Math. Soc. 24 (2009), 529–537.10.4134/CKMS.2009.24.4.529Search in Google Scholar

[6] J. Caballero, J. Harjani, K. Sadarangani, Contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations, Fixed Point Theory and Appl. 2010, Article ID 916064, doi:10.1155/2010/916064.10.1155/2010/916064Search in Google Scholar

[7] Lj. B. Ciric, D. Mihet, R. Saadati, Monotone generalized contractions in partially ordered probabilistic metric spaces, Topology Appl. 156 (2009), 2838–2844.10.1016/j.topol.2009.08.029Search in Google Scholar

[8] E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10 (1959), 974–979.10.1090/S0002-9939-1959-0110093-3Search in Google Scholar

[9] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604–608.10.1090/S0002-9939-1973-0334176-5Search in Google Scholar

[10] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65(7) (2006), 1379–1393.10.1016/j.na.2005.10.017Search in Google Scholar

[11] J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. TMA 71 (2009), 3403–3410.10.1016/j.na.2009.01.240Search in Google Scholar

[12] J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear Anal. 74 (2011), 768–774.10.1016/j.na.2010.09.025Search in Google Scholar

[13] L. Janos, On mappings contractive in the sense of Kannan, Proc. Amer. Math. Soc. 61(1) (1976), 171–175.10.1090/S0002-9939-1976-0425936-3Search in Google Scholar

[14] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76.Search in Google Scholar

[15] R. Kannan, Some results of fixed points-II, Amer. Math. Monthly 76 (1969), 405–408.Search in Google Scholar

[16] M. Kikkaw, T. Suzuki, Some similarity between contractions and Kannan mappings, Fixed Point Theory and Appl. 2008 (2008), Article ID 649749.Search in Google Scholar

[17] V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. TMA 70 (2009), 4341–4349.10.1016/j.na.2008.09.020Search in Google Scholar

[18] J. J. Nieto, R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22(3) (2005), 223–239.10.1007/s11083-005-9018-5Search in Google Scholar

[19] J. J. Nieto, R. Rodriguez-Lopez, Applications of contractive-like mapping principles to fuzzy equations, Rev. Mat. Comp. 19(2) (2006), 361–383.10.5209/rev_REMA.2006.v19.n2.16599Search in Google Scholar

[20] J. J. Nieto, R. R. Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta. Math. Sin. (Engl. Ser.) 23(12) (2007), 2205–2212.10.1007/s10114-005-0769-0Search in Google Scholar

[21] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132(5) (2004), 1435–1443.10.1090/S0002-9939-03-07220-4Search in Google Scholar

[22] D. O’Regan, A. Petruşel, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341(2) (2008), 1241–1252.10.1016/j.jmaa.2007.11.026Search in Google Scholar

[23] N. Shioji, T. Suzuki, W. Takahashi, Contractive mappings, Kannan mappings and metric completeness, Proc. Amer. Math. Soc. 126 (1998), 3117–3124.10.1090/S0002-9939-98-04605-XSearch in Google Scholar

[24] P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math. 80 (1975), 325–330.10.1007/BF01472580Search in Google Scholar

[25] M. Turinici, Abstract comparison principles and multivariable Gronwall–Bellman inequalities, J. Math. Anal. Appl. 117 (1986), 100–127.10.1016/0022-247X(86)90251-9Search in Google Scholar

[26] Y. Wu, New fixed point theorems and applications of mixed monotone operator, J. Math. Anal. Appl. 341(2) (2008), 883–893.10.1016/j.jmaa.2007.10.063Search in Google Scholar

Received: 2011-6-23
Published Online: 2017-5-10
Published in Print: 2013-6-1

© 2013 Binayak S. Choudhury et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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