×

Fixed point theorems for nonlinear weakly-contractive mappings in metric spaces. (English) Zbl 1235.54054

Summary: The purpose of this paper is to present some fixed point and coupled fixed point theorems for a nonlinear weakly \(C\)-contraction type mapping in metric and ordered metric spaces. Also, an example is given to support our results. Our results generalize several well-known results from the current literature.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54E40 Special maps on metric spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Banach, S., Surles operations dans les ensembles et leur application aux equation sitegrales, Fund. Math., 3, 133-181 (1922)
[2] Agarwal, R. P.; El-Gebeily, M. A.; O’regan, D., Generalized contractions in partially ordered metric spaces, Appl. Anal., 87, 1, 109-116 (2008) · Zbl 1140.47042
[3] Altun, I.; Simsek, H., Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl., 2010 (2010), Article ID 621492, 17 pages · Zbl 1197.54053
[4] Bari, C. Di.; Vetro, P., \( \phi \)-paris and common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo, 57, 279-285 (2008) · Zbl 1164.54031
[5] Dutta, P. N.; Choudhury, B. S., A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. (2008), Article ID 406368, 8 pages · Zbl 1177.54024
[6] Harjani, J.; Sadarangani, K., Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal., 71, 7-8, 3403-3410 (2008) · Zbl 1221.54058
[7] Harjani, J.; Sadarangani, K., Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal., 72, 3-4, 1188-1197 (2010) · Zbl 1220.54025
[8] Nieto, J. J.; Rodŕiguez-López, R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 223-239 (2005) · Zbl 1095.47013
[9] Nieto, J. J.; Rodŕiguez-López, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica, 23, 12, 2205-2212 (2007) · Zbl 1140.47045
[10] Nieto, J. J.; Pouso, R. L.; Rodŕiguez-López, R., Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc., 135, 2505-2517 (2007) · Zbl 1126.47045
[11] Nashine, H. K.; Samet, B., Fixed point results for mappings satisfying \((\psi, \phi)\)-weakly contractive condition in partially ordered metric spaces, Nonlinear Anal., 74, 2201-2209 (2010) · Zbl 1208.41014
[12] Ran, A. C.M.; Reurings, M. C.B., A fixed point theorem in partially ordered sets and some applications to metrix equations, Proc. Amer. Math. Soc., 132, 5, 1435-1443 (2004) · Zbl 1060.47056
[13] Rhoades, B. E., Some theorems on weakly contractive maps, Nonlinear Anal., 47, 2683-2693 (2001) · Zbl 1042.47521
[14] W. Shatanawi, Fixed point theory for contractive mappings satisfying \(\PhiG\) doi:10.1155/2010/181650; W. Shatanawi, Fixed point theory for contractive mappings satisfying \(\PhiG\) doi:10.1155/2010/181650 · Zbl 1204.54039
[15] Chatterjea, S. K., Fixed point theorems, C. R. Acad. Bulgare Sci., 25, 727-730 (1972) · Zbl 0274.54033
[16] Alber, Ya. I.; Guerre-Delabriere, S., Principles of weakly contractive maps in Hilbert spaces, (Gohberg, I.; Lyubich, Yu., New Results in Operator Theory. New Results in Operator Theory, Advances and Appl., vol. 98 (1997), Birkhäuser: Birkhäuser Basel), 7-22 · Zbl 0897.47044
[17] Doric, D., Common fixed point for generalized \((\psi, \phi)\)-weak contraction, Appl. Math. Lett., 22, 1896-1900 (2009) · Zbl 1203.54040
[18] Boyd, D. W.; Wong, T. S.W., On nonlinear contractions, Proc. Amer. Math. Soc., 20, 458-464 (1969) · Zbl 0175.44903
[19] Popescu, O., Fixed points for \((\psi, \phi)\)-weak contractions, Appl. Math. Lett., 24, 1-4 (2011) · Zbl 1202.54038
[20] Zhang, Q.; Song, Y., Fixed point theory for generalized \(\phi \)-weak contraction, Appl. Math. Lett., 22, 75-78 (2009) · Zbl 1163.47304
[21] Bhaskar, T. G.; Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65, 1379-1393 (2006) · Zbl 1106.47047
[22] Choudhury, B. S.; Kundu, A., A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal., 73, 2524-2531 (2010) · Zbl 1229.54051
[23] Lakshmikantham, V.; Ćirić, Lj. B., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70, 4341-4349 (2009) · Zbl 1176.54032
[24] Samet, B., Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal., 72, 4508-4517 (2010) · Zbl 1264.54068
[25] Saadati, R.; Vaezpour, S. M.; Vetro, P.; Rhoades, B. E., Fixed point theorems in generalized partially ordered \(G\)-metric spaces, Math. Comput. Modelling, 52, 797-801 (2010) · Zbl 1202.54042
[26] Shatanawi, W., Partially ordered cone metric spaces and coupled fixed point results, Comput. Math. Appl., 60, 2508-2515 (2010) · Zbl 1205.54044
[27] Shatanawi, W., Some fixed point theorems in ordered G-metric spaces and applications, Abstr. Appl. Anal., 2011 (2011), Article ID 126205, 11 pages doi:10.1155/2011/126205 · Zbl 1217.54057
[28] Choudhury, B. S., Unique fixed point theorem for weak \(C\)-contractive mappings, Kathmandu Univ. J. Sci. Eng. Tech., 5, 1, 6-13 (2009)
[29] Harjani, J.; López, B.; Sadarangani, K., Fixed point theorems for weakly \(C\)-contractive mappings in ordered metric spaces, Comput. Math. Appl., 61, 790-796 (2011) · Zbl 1217.54046
[30] Abbas, M.; Khan, A. R.; Nazir, T., Coupled common fixed point results in two generalized metric spaces, Appl. Math. Comput., 217, 6328-6336 (2011) · Zbl 1210.54048
[31] Aydi, H.; Damjanovic, B.; Samet, B.; Shatanawi, W., Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces, Math. Comput. Modelling (2011) · Zbl 1237.54043
[32] Nashine, H. K.; Shatanawi, W., Coupled common fixed point theorems for a pair of commuting mappings in partially ordered complete metric spaces, Comput. Math. Appl. (2011) · Zbl 1231.65100
[33] Shatanawi, W., Some common coupled fixed point results in cone metric spaces, Int. J. Math. Anal., 4, 2381-2388 (2010) · Zbl 1227.54056
[34] Khan, M. S.; Swaleh, M.; Sessa, S., Fixed point theorems by altering distancces between the points, Bull. Aust. Math. Soc., 30, 1-9 (1984) · Zbl 0553.54023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.