On coupled generalised Banach and Kannan type contractions
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Authors
B. S. Choudhury
- Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, West Bengal, India.
Amaresh Kundu
- Department of Mathematics, Siliguri Institute of Technology, Sukna, Darjeeling-734009, West Bengal, India.
Abstract
In this paper we have proved two theorems in which we have established the existence of coupled fixed
point results in partially ordered complete metric spaces for generalised coupled Banach and Kannan type
mappings. The generalisation has been accomplished by following the line of argument given by Geraghty
[Proc. Amer. Math. Soc., 40 (1973), 604-608] . Here the mapping are assumed to satisfy certain contractive
type inequalities. We have illustrated our result with two examples. First example is presented to show that
our result is a proper generalizations of the corresponding results of Bhaskar et al [Nonlinear Anal. TMA,
65 (7) (2006), 1379-1393].
Share and Cite
ISRP Style
B. S. Choudhury, Amaresh Kundu, On coupled generalised Banach and Kannan type contractions, Journal of Nonlinear Sciences and Applications, 5 (2012), no. 4, 259--270
AMA Style
Choudhury B. S., Kundu Amaresh, On coupled generalised Banach and Kannan type contractions. J. Nonlinear Sci. Appl. (2012); 5(4):259--270
Chicago/Turabian Style
Choudhury, B. S., Kundu, Amaresh. "On coupled generalised Banach and Kannan type contractions." Journal of Nonlinear Sciences and Applications, 5, no. 4 (2012): 259--270
Keywords
- Partially ordered set
- Contractive-type mapping
- Mixed monotone property
- Coupled fixed point.
MSC
References
-
[1]
R. P. Agarwal, M. A. El-Gebeily, D. O'Regan, Generalized contractions in partially ordered metric spaces , Appl. 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]
H. Aydi , Some coupled fixed point results on partial metric spaces, Int. J. Math. Math. Sci., Article ID 647091, doi:10.1155/2011/647091. , 2001 (2011), 11 pages
-
[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]
B. S. Choudhury, N. Metiya, A. Kundu , Coupled coincidence point theorems in ordered metric Spaces, Ann. Univ. Ferrara, 57 (2011), 1-16.
-
[7]
B. S. Choudhury, A. Kundu , A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal. TMA, 73 (2010), 2524-2531.
-
[8]
B. S. Choudhury, K. Das, S. K. Bhandari, A fixed point theorem for Kannan type mappings in 2-menger space using a control function, Bull. Math. Anal. Appl., 3 (2011), 141-148.
-
[9]
B. S. Choudhury, A. Kundu, A Kannan-like contraction in partially ordered spaces, Demonstratio Mathematica, (to appear),
-
[10]
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., Article ID 916064, doi:10.1155/2010/916064. , (2010), 14 pages
-
[11]
Lj. B. Ćirić, V. Lakshmikantham , Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces, Stoch. Anal. Appl., 27 (6) (2009), 1246-1259.
-
[12]
Lj. B. Ćirić, D. Mihet, R. Saadati, Monotone generalized contractions in partially ordered probabilistic metric spaces, Topol. Appl., 156 (2009), 2838-2844.
-
[13]
L. Ćirić, N. Cakić, M. Rajović, J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory and Appl., Article ID 131294, 11 pages (2008)
-
[14]
E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc., 10 (1959), 974-979.
-
[15]
M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604-608.
-
[16]
T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA, 65 (7) (2006), 1379-1393.
-
[17]
X. Hu, Common Coupled fixed point theorems for contractive mappings in fuzzy metric spaces, Fixed Point Theory and Appl., Article ID 363716, doi:10.1155/2011/363716., (2011), 14 pages
-
[18]
J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear Anal., 74 (2011), 768-774.
-
[19]
L. Janos, On mappings contractive in the sense of Kannan, Proc. Amer. Math. Soc., 61(1) (1976), 171-175.
-
[20]
R. Kannan, Some results on fixed points , Bull. Cal. Math. Soc., 60 (1968), 71-76.
-
[21]
R. Kannan, Some results on fixed points-II, Amer. Math. Monthly, 76 (1969), 405-408.
-
[22]
M. Kikkaw, T. Suzuki , Some similarity between contractions and Kannan mappings, Fixed Point Theory and Appl., Article ID 649749. , 2008 (2008), 8 pages
-
[23]
E. Karapinar, Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl., 59 (2010), 3656-3668.
-
[24]
V. Lakshmikantham, L. Ciric , Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. TMA, 70 (2009), 4341-4349.
-
[25]
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.
-
[26]
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.
-
[27]
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.
-
[28]
D. A. Ruiz, A. J. Melado, A continuation method for weakly Kannan maps, Fixed Point Theory and Appl., Article ID 321594, doi:10.1155/2010/321594. , 2010 (2010), 12 pages
-
[29]
B. Samet, H. Yazidi , Coupled fixed point theorems in partially ordered ''-chainable Metric spaces, TJMCS, 1 (3) (2010), 142-151.
-
[30]
N. Shioji, T. Suzuki and W. Takahashi, Contractive mappings, Kannan mappings and metric completeness, Proc. Amer. Math. Soc., 126 (1998), 3117-3124.
-
[31]
Y. Enjouji, M. Nakanishi, T. Suzuki, A generalization of Kannan's fixed point theorem, Fixed Point Theory and Appl., Article ID 192872, doi:10.1155/2009/192872. , 2009 (2009), 10 pages.
-
[32]
P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math., 80 (1975), 325-330.