Two coupled weak contraction theorems in partially ordered metric spaces. (English) Zbl 1295.54048
Summary: In this work, we establish two general weak coupled contraction mapping theorems in partially ordered metric spaces. Weak contractions are extensions of contractive mappings which are intermediate to Banach’s contractions and nonexpansive mappings. They have been considered elaborately in recent literatures. Our work generalises the coupled contraction mapping theorem established by T. Gnana Bhaskar and V. Lakshmikantham [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 65, No. 7, 1379–1393 (2006; Zbl 1106.47047)] to weak coupled contractions. We use certain control functions in our theorems. Two illustrative examples are given.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E40 | Special maps on metric spaces |
| 54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |
Keywords:
partial order; coupled fixed point; mixed monotone property; weak contraction; control functionCitations:
Zbl 1106.47047References:
| [1] | Alber, Ya.I., Guerre-Delabriere S.: Principles of weakly contractive maps in Hilbert spaces, new results in operator theory. In: Gohberg, I., Lyubich, Y. (eds.) Advances and Applications, vol. 98, pp. 7-22. Birkhäuser, Basel (1997) · Zbl 0897.47044 |
| [2] | Arvanitakis, A.D.: A proof of the generalized Banach contraction conjecture. Proc. Am. Math. Soc. 131(12), 3647-3656 (2003) · Zbl 1053.54047 · doi:10.1090/S0002-9939-03-06937-5 |
| [3] | Babu, G.V.R., Vara Prasad, K.N.V.V.: Common fixed point theorems of different compatible type mappings using Ciric’s contraction type condition. Math. Commun. 11, 87-102 (2006) · Zbl 1120.47045 |
| [4] | Choudhury, B.S., Kundu, A.: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. TMA 73, 2524-2531 (2010) · Zbl 1229.54051 · doi:10.1016/j.na.2010.06.025 |
| [5] | Choudhury, B.S., Metiya, N.: Fixed points of weak contractions in cone metric spaces. Nonlinear Anal. TMA 72, 1589-1593 (2010) · Zbl 1191.54036 · doi:10.1016/j.na.2009.08.040 |
| [6] | Chidume, C.E., Zegeye, H., Aneke, S.J.: Approximation of fixed points of weakly contractive non self maps in Banach spaces. J. Math. Anal. Appl. 270(1), 189-199 (2002) · Zbl 1005.47053 · doi:10.1016/S0022-247X(02)00063-X |
| [7] | Ciric, L., Cakic, N., Rajovic, M., Ume, J.S.: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications, vol. 2008. Article ID 131294 · Zbl 1158.54019 |
| [8] | Cirić, L., Lakshmikantham, V.: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stoch. Anal. Appl. 27(6), 1246-1259 (2009) · Zbl 1176.54030 · doi:10.1080/07362990903259967 |
| [9] | Saadati, R., Mihect, D., Ciric, Lj.B.: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topol. Appl. 156(17), 2838-2844 (2009) · Zbl 1206.54039 |
| [10] | Doric, D.: Common fixed point for generalized \[( \phi , \psi )\]-weak contractions. Appl. Math. Lett. 22, 1896-1900 (2009) · Zbl 1203.54040 · doi:10.1016/j.aml.2009.08.001 |
| [11] | Dutta P.N., Choudhury B.S.: A generalisation of contraction principle in metric spaces. Fixed Point Theory and Applications, vol. 2008. Article ID 406368 · Zbl 1177.54024 |
| [12] | Lakshmikantham, V., Gnana Bhaskar, T.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. TMA 65, 1379-1393 (2006) · Zbl 1106.47047 |
| [13] | Harjani, J., Sadarangani, K.: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. TMA 72, 1188-1197 (2010) · Zbl 1220.54025 · doi:10.1016/j.na.2009.08.003 |
| [14] | Harjani, J., Sadarangani, K.: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. TMA 71, 3403-3410 (2009) · Zbl 1221.54058 · doi:10.1016/j.na.2009.01.240 |
| [15] | Jachymski, J.: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 74, 768-774 (2011) · Zbl 1201.54034 · doi:10.1016/j.na.2010.09.025 |
| [16] | Khan, M.S., Swaleh, M., Sessa, S.: Fixed points theorems by altering distances between the points. Bull. Aust. Math. Soc. 30, 1-9 (1984) · Zbl 0553.54023 · doi:10.1017/S0004972700001659 |
| [17] | Lakshmikantham, V., Ciric, L.: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. TMA 70(12), 4341-4349 (2009) · Zbl 1176.54032 · doi:10.1016/j.na.2008.09.020 |
| [18] | Merryfield, J., Rothschild, B., J.D. Stein Jr.: An application of Ramsey’s theorem to the Banach contraction principle. Proc. Am. Math. Soc. 130(4), 927-933 (2002) · Zbl 1001.47042 |
| [19] | Nieto, J.J., Rodriguez-Lopez, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223-239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5 |
| [20] | Nieto, J.J., Lopez, R.R.: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta. Math. Sinica (Engl. Ser.) 23(12), 2205-2212 (2007) · Zbl 1140.47045 |
| [21] | Popescu, O.: Fixed points for \[(\psi,\phi )-\] weak contractions. Appl. Math. Lett. 24, 1-4 (2011) · Zbl 1202.54038 · doi:10.1016/j.aml.2010.06.024 |
| [22] | Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Pror. Am. Math. Soc. 132, 1435-1443 (2004) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4 |
| [23] | Rhoades, B.E.: Some theorems on weakly contractive maps. Nonlinear Anal. TMA 47(4), 2683-2693 (2001) · Zbl 1042.47521 |
| [24] | Rouhani B.D., Moradi S.: Common fixed point of multivalued generalized \[\varphi \]-weak contractive mappings (2010). Article ID-708984 · Zbl 1202.54041 |
| [25] | Sastry, K.P.R., Naidu, S.V.R., Babu, G.V.R., Naidu, G.A.: Generalization of common fixed point theorems for weakly commuting maps by altering distances. Tamkang J. Math. 31(3), 243-250 (2000) · Zbl 0995.47035 |
| [26] | Samet, B.: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. TMA 72(12), 4508-4517 (2010) · Zbl 1264.54068 · doi:10.1016/j.na.2010.02.026 |
| [27] | Suzuki, T.: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 136(5), 1861-1869 (2008) · Zbl 1145.54026 · doi:10.1090/S0002-9939-07-09055-7 |
| [28] | Zhang, Q., Song, Y.: Fixed point theory for generalized \[\phi -\] weak contractions. Appl. Math. Lett. 22(1), 75-78 (2009) · Zbl 1163.47304 · doi:10.1016/j.aml.2008.02.007 |
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.
