Further generalized contraction mapping principle and best proximity theorem in metric spaces. (English) Zbl 1505.54094
Summary: The aim of this paper is to prove a more generalized contraction mapping principle. By using this more generalized contraction mapping principle, a further generalized best proximity theorem was established. Some concrete results have been derived by using the above two theorems. The results of this paper improve many important results published recently in the literature.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 47H10 | Fixed-point theorems |
| 54E40 | Special maps on metric spaces |
References:
| [1] | Yan, F, Su, Y, Feng, Q: A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory Appl. 2012, 152 (2012) · Zbl 1308.54040 · doi:10.1186/1687-1812-2012-152 |
| [2] | Browder, FE: On the convergence of successive approximations for nonlinear functional equations. Proc. K. Ned. Akad. Wet., Ser. A, Indag. Math. 30, 27-35 (1968) · Zbl 0155.19401 · doi:10.1016/S1385-7258(68)50004-0 |
| [3] | Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458-464 (1969) · Zbl 0175.44903 · doi:10.1090/S0002-9939-1969-0239559-9 |
| [4] | Jachymski, J: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125, 2327-2335 (1997) · Zbl 0887.47039 · doi:10.1090/S0002-9939-97-03853-7 |
| [5] | Jachymski, J.; Jóźwik, I., Nonlinear contractive conditions: a comparison and related problems, No. 77, 123-146 (2007), Warsaw · Zbl 1149.47044 · doi:10.4064/bc77-0-10 |
| [6] | Geraghty, M: On contractive mappings. Proc. Am. Math. Soc. 40, 604-608 (1973) · Zbl 0245.54027 · doi:10.1090/S0002-9939-1973-0334176-5 |
| [7] | Samet, B, Vetro, C, Vetro, P: Fixed point theorem for α-ψ-contractive type mappings. Nonlinear Anal. 75, 2154-2165 (2012) · Zbl 1242.54027 · doi:10.1016/j.na.2011.10.014 |
| [8] | Dhutta, P, Choudhury, B: A generalization of contractions in partially ordered metric spaces. Appl. Anal. 87, 109-116 (2008) · Zbl 1140.47042 · doi:10.1080/00036810701556151 |
| [9] | Sastry, K, Babu, G: Some fixed point theorems by altering distance between the points. Indian J. Pure Appl. Math. 30, 641-647 (1999) · Zbl 0938.47044 |
| [10] | Amini-Harandi, A, Emami, H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 72, 2238-2242 (2010) · Zbl 1197.54054 · doi:10.1016/j.na.2009.10.023 |
| [11] | Harjani, J, Sadarangni, K: Fixed point theorems for weakly contraction mappings in partially ordered sets. Nonlinear Anal. 71, 3403-3410 (2009) · Zbl 1221.54058 · doi:10.1016/j.na.2009.01.240 |
| [12] | Burgic, D, Kalabusic, S, Kulenovic, M: Global attractivity results for mixed monotone mappings in partially ordered complete metric spaces. Fixed Point Theory Appl. 2009, Article ID 762478 (2009) · Zbl 1168.54327 · doi:10.1155/2009/762478 |
| [13] | Ciric, L, Cakid, N, Rajovic, M, Uma, J: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008, Article ID 131294 (2008) · Zbl 1158.54019 · doi:10.1155/2008/131294 |
| [14] | Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379-1393 (2006) · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017 |
| [15] | Lakshmikantham, V, Ciric, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70, 4341-4349 (2009) · Zbl 1176.54032 · doi:10.1016/j.na.2008.09.020 |
| [16] | Nieto, JJ, Rodriguez-López, 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 |
| [17] | Nieto, JJ, Rodriguez-López, R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. 23, 2205-2212 (2007) · Zbl 1140.47045 · doi:10.1007/s10114-005-0769-0 |
| [18] | O’Regan, D, Petrusel, A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 341, 1241-1252 (2008) · Zbl 1142.47033 · doi:10.1016/j.jmaa.2007.11.026 |
| [19] | Harjani, J, Sadarangni, K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 72, 1188-1197 (2010) · Zbl 1220.54025 · doi:10.1016/j.na.2009.08.003 |
| [20] | Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30(1), 1-9 (1984) · Zbl 0553.54023 · doi:10.1017/S0004972700001659 |
| [21] | Sadiq Basha, S: Best proximity point theorems generalizing the contraction principle. Nonlinear Anal. 74, 5844-5850 (2011) · Zbl 1238.54021 · doi:10.1016/j.na.2011.04.017 |
| [22] | Mongkolkeha, C, Cho, YJ, Kumam, P: Best proximity points for Geraghty’s proximal contraction mappings. Fixed Point Theory Appl. 2013, 180 (2013) · Zbl 1469.54158 · doi:10.1186/1687-1812-2013-180 |
| [23] | Dugundji, J, Granas, A: Weakly contractive mappings and elementary domain invariance theorem. Bull. Greek Math. Soc. 19, 141-151 (1978) · Zbl 0417.54010 |
| [24] | Alghamdi, MA, Alghamdi, MA, Shahzad, S: Best proximity point results in geodesic metric spaces. Fixed Point Theory Appl. 2013, 164 (2013) · Zbl 1297.47061 · doi:10.1186/1687-1812-2013-164 |
| [25] | Nashine, HK, Kumam, P, Vetro, C: Best proximity point theorems for rational proximal contractions. Fixed Point Theory Appl. 2013, 95 (2013) · Zbl 1295.41038 · doi:10.1186/1687-1812-2013-95 |
| [26] | Karapinar, E: On best proximity point of ψ-Geraghty contractions. Fixed Point Theory Appl. 2013, 200 (2013) · Zbl 1295.41037 · doi:10.1186/1687-1812-2013-200 |
| [27] | Zhang, J, Su, Y, Cheng, Q: Best proximity point theorems for generalized contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2013, 83 (2013) · Zbl 1423.54105 · doi:10.1186/1687-1812-2013-83 |
| [28] | Zhang, J, Su, Y, Cheng, Q: A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl. 2013, 99 (2013) · Zbl 1423.54104 · doi:10.1186/1687-1812-2013-99 |
| [29] | Zhang, J, Su, Y: Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces. Fixed Point Theory Appl. 2014, 50 (2014) · Zbl 1333.54052 · doi:10.1186/1687-1812-2014-50 |
| [30] | Amini-Harandi, A, Hussain, N, Akbar, F: Best proximity point results for generalized contractions in metric spaces. Fixed Point Theory Appl. 2013, 164 (2013) · Zbl 1297.47061 · doi:10.1186/1687-1812-2013-164 |
| [31] | Kannan, R: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71-76 (1968) · Zbl 0209.27104 |
| [32] | Caballero, J, Harjani, J, Sadarangani, K: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231 · Zbl 1281.54021 · doi:10.1186/1687-1812-2012-231 |
| [33] | Kirk, WA, Reich, S, Veeramani, P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851-862 (2003) · Zbl 1054.47040 · doi:10.1081/NFA-120026380 |
| [34] | Sun, Y, Su, Y, Zhang, J: A new method for the research of best proximity point theorems of nonlinear mappings. Fixed Point Theory Appl. 2014, 116 (2014) · Zbl 1328.54053 · doi:10.1186/1687-1812-2014-116 |
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.
