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Abstrakty
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.
Wydawca
Czasopismo
Rocznik
Tom
Strony
327--334
Opis fizyczny
Bibliogr. 26 poz.
Twórcy
autor
- Department of Mathematics Bengal Engineering and Science University Shibpur, West Bengal, India
autor
- Department of Mathematics Siliguri Institute of Technology West Bengal, India
Bibliografia
- [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.
- [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.
- [3] I. Altun, V. Rakočević, Ordered cone metric spaces and fixed point results, Comput. Math. Appl. 60 (2010), 1145–1151
- [4] B. S. Choudhury, P. Maity, Coupled fixed point results in generalized metric spaces, Math. Comput. Modelling 54 (2011), 73–79.
- [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.
- [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.
- [7] Lj. B. Ciric, D. Mihet, R. Saadati, Monotone generalized contractions in partially ordered probabilistic metric spaces, Topology Appl. 156 (2009), 2838–2844.
- [8] E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc. 10 (1959), 974–979.
- [9] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604–608.
- [10] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65(7) (2006), 1379–1393.
- [11] J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. TMA 71 (2009), 3403–3410.
- [12] J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear Anal. 74 (2011), 768–774.
- [13] L. Janos, On mappings contractive in the sense of Kannan, Proc. Amer. Math. Soc. 61(1) (1976), 171–175.
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- [16] M. Kikkaw, T. Suzuki, Some similarity between contractions and Kannan mappings, Fixed Point Theory and Appl. 2008 (2008), Article ID 649749.
- [17] V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. TMA 70 (2009), 4341–4349.
- [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.
- [19] J. J. Nieto, R. Rodriguez-Lopez, Applications of contractive-like mapping principles to fuzzy equations, Rev. Mat. Comp. 19(2) (2006), 361–383.
- [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.
- [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.
- [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.
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- [26] Y. Wu, New fixed point theorems and applications of mixed monotone operator, J. Math. Anal. Appl. 341(2) (2008), 883–893.
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