Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorems. (English) Zbl 1381.47040
Summary: In this paper, we introduce a multi-valued cyclic generalized contraction by extending the Mizoguchi and Takahashi’s contraction for non-self mappings. We also establish a best proximity point for such type contraction mappings in the context of metric spaces. Later, we characterize this result to investigate the existence of best proximity point theorems in uniformly convex Banach spaces. We state some illustrative examples to support our main theorems. Our results extend, improve and enrich some celebrated results in the literature, such as Nadler’s fixed point theorem, Mizoguchi and Takahashi’s fixed point theorem.
MSC:
| 47H10 | Fixed-point theorems |
| 46B20 | Geometry and structure of normed linear spaces |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
Keywords:
best proximity points; multi-valued contraction; cyclic contraction; \(\mathcal{MT}\)-function (or \(\mathcal{R}\)-function)References:
| [1] | Banach S: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales.Fundam. Math. 1922, 3:133-181. · JFM 48.0201.01 |
| [2] | Arvanitakis AD: A proof of the generalized Banach contraction conjecture.Proc. Am. Math. Soc. 2003,131(12):3647-3656. · Zbl 1053.54047 · doi:10.1090/S0002-9939-03-06937-5 |
| [3] | Boyd DW, Wong JSW: On nonlinear contractions.Proc. Am. Math. Soc. 1969, 20:458-464. · Zbl 0175.44903 · doi:10.1090/S0002-9939-1969-0239559-9 |
| [4] | Merryfield J, Rothschild B, Stein JD Jr.: An application of Ramsey’s theorem to the Banach contraction principle.Proc. Am. Math. Soc. 2002,130(4):927-933. · Zbl 1001.47042 · doi:10.1090/S0002-9939-01-06169-X |
| [5] | Suzuki T: A generalized Banach contraction principle that characterizes metric completeness.Proc. Am. Math. Soc. 2008,136(5):1861-1869. · Zbl 1145.54026 · doi:10.1090/S0002-9939-07-09055-7 |
| [6] | Fan K: Extensions of two fixed point theorems of FE Browder.Math. Z. 1969, 112:234-240. · Zbl 0185.39503 · doi:10.1007/BF01110225 |
| [7] | Di Bari C, Suzuki T, Vetro C: Best proximity points for cyclic Meir-Keeler contractions.Nonlinear Anal. 2008, 69:3790-3794. · Zbl 1169.54021 · doi:10.1016/j.na.2007.10.014 |
| [8] | Mongkolkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces.J. Optim. Theory Appl. 2012,155(1):215-226. · Zbl 1257.54041 · doi:10.1007/s10957-012-9991-y |
| [9] | Mongkolkeha, C.; Cho, YJ; Kumam, P., Best proximity points for generalized proximal C-contraction mappings in metric spaces with partial orders, No. 2013 (2013) · Zbl 1417.47016 |
| [10] | Mongkolkeha, C.; Cho, YJ; Kumam, P., Best proximity points for Geraghty’s proximal contraction mapping mappings, No. 2013 (2013) · Zbl 1469.54158 |
| [11] | Nashine, HK; Vetro, C.; Kumam, P., Best proximity point theorems for rational proximal contractions, No. 2013 (2013) · Zbl 1295.41038 |
| [12] | Sanhan, W.; Mongkolkeha, C.; Kumam, P., Generalized proximal ψ-contraction mappings and Best proximity points, No. 2012 (2012) · Zbl 1387.54033 |
| [13] | Sintunavarat, W.; Kumam, P., The existence theorems of an optimal approximate solution for generalized proximal contraction mappings, No. 2013 (2013) · Zbl 1297.54106 |
| [14] | Vetro C: Best proximity points: convergence and existence theorems forp-cyclic mappings.Nonlinear Anal. 2010,73(7):2283-2291. · Zbl 1229.54066 · doi:10.1016/j.na.2010.06.008 |
| [15] | Nadler SB Jr.: Multivalued contraction mappings.Pac. J. Math. 1969, 30:475-488. · Zbl 0187.45002 · doi:10.2140/pjm.1969.30.475 |
| [16] | Daffer PZ, Kaneko H: Fixed points of generalized contractive multi-valued mappings.J. Math. Anal. Appl. 1995, 192:655-666. · Zbl 0835.54028 · doi:10.1006/jmaa.1995.1194 |
| [17] | Mizoguchi N, Takahashi W: Fixed point theorems for multi-valued mappings on complete metric space.J. Math. Anal. Appl. 1989, 141:177-188. · Zbl 0688.54028 · doi:10.1016/0022-247X(89)90214-X |
| [18] | Petrusel A: On Frigon-Granas-type multifunctions.Nonlinear Anal. Forum 2002, 7:113-121. · Zbl 1043.47036 |
| [19] | Reich S: A fixed point theorem for locally contractive multi-valued functions.Rev. Roum. Math. Pures Appl. 1972, 17:569-572. · Zbl 0239.54033 |
| [20] | Alesina A, Massa S, Roux D: Punti uniti di multifunzioni con condizioni di tipo Boyd-Wong.Boll. Unione Mat. Ital. 1973,4(8):29-34. · Zbl 0274.54036 |
| [21] | Reich S: Some problems and results in fixed point theory.Contemp. Math. 1983, 21:179-187. · Zbl 0531.47048 · doi:10.1090/conm/021/729515 |
| [22] | Daffer PZ, Kaneko H, Li W: On a conjecture of S. Reich.Proc. Am. Math. Soc. 1996, 124:3159-3162. · Zbl 0866.47040 · doi:10.1090/S0002-9939-96-03659-3 |
| [23] | Jachymski J: On Reich’s question concerning fixed points of multimaps.Boll. Unione Mat. Ital. 1995,7(9):453-460. · Zbl 0863.54042 |
| [24] | Semenov PV: Fixed points of multivalued contractions.Funct. Anal. Appl. 2002, 36:159-161. · Zbl 1026.47040 · doi:10.1023/A:1015682926496 |
| [25] | Eldred AA, Anuradha J, Veeramani P: On equivalence of generalized multi-valued contractions and Nadler’s fixed point theorem.J. Math. Anal. Appl. 2007, 336:751-757. · Zbl 1128.47051 · doi:10.1016/j.jmaa.2007.01.087 |
| [26] | Suzuki T: Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s.J. Math. Anal. Appl. 2008, 340:752-755. · Zbl 1137.54026 · doi:10.1016/j.jmaa.2007.08.022 |
| [27] | Kirk WA, Srinavasan PS, Veeramani P: Fixed points for mapping satisfying cyclical contractive conditions.Fixed Point Theory 2003, 4:79-89. · Zbl 1052.54032 |
| [28] | Agarwal, RP; Alghamdi, MA; Shahzad, N., Fixed point theory for cyclic generalized contractions in partial metric spaces, No. 2012 (2012) · Zbl 1477.54033 |
| [29] | Eldered AA, Veeramani P: Convergence and existence for best proximity points.J. Math. Anal. Appl. 2006, 323:1001-1006. · Zbl 1105.54021 · doi:10.1016/j.jmaa.2005.10.081 |
| [30] | Eldered AA, Veeramani P: Proximal pointwise contraction.Topol. Appl. 2009, 156:2942-2948. · Zbl 1180.47035 · doi:10.1016/j.topol.2009.01.017 |
| [31] | Chen, CM, Fixed point theorems for cyclic Meir-Keeler type mappings in complete metric spaces, No. 2012 (2012) · Zbl 1273.54047 |
| [32] | Karapinar E: Fixed point theory for cyclic weakϕ-contraction.Appl. Math. Lett. 2011, 24:822-825. · Zbl 1256.54073 · doi:10.1016/j.aml.2010.12.016 |
| [33] | Karapinar, E.; Sadarangani, K., Fixed point theory for cyclic [InlineEquation not available: see fulltext.] contractions, No. 2011 (2011) |
| [34] | Karapinar E, Erhan İM, Ulus AY: Fixed point theorem for cyclic maps on partial metric spaces.Appl. Math. Inform. Sci. 2012, 6:239-244. |
| [35] | Karpagam S, Agrawal S: Best proximity points theorems for cyclic Meir-Keeler contraction maps.Nonlinear Anal. 2011, 74:1040-1046. · Zbl 1206.54047 · doi:10.1016/j.na.2010.07.026 |
| [36] | Kosuru, GSR, Veeramani, P: Cyclic contractions and best proximity pair theorems (29 May 2011).. e-printatarXiv.org [math.FA] · Zbl 1145.54026 |
| [37] | Petruşel G: Cyclic representations and periodic points.Stud. Univ. Babeş-Bolyai, Math. 2005, 50:107-112. · Zbl 1117.47046 |
| [38] | Rezapour S, Derafshpour M, Shahzad N: Best proximity point of cyclicφ-contractions in ordered metric spaces.Topol. Methods Nonlinear Anal. 2011, 37:193-202. · Zbl 1227.54046 |
| [39] | Rus IA: Cyclic representations and fixed points.Ann. “Tiberiu Popoviciu” Sem. Funct. Equ. Approx. Convexity 2005, 3:171-178. · Zbl 1463.54126 |
| [40] | Sintunavarat, W.; Kumam, P., Coupled best proximity point theorem in metric spaces, No. 2012 (2012) · Zbl 1308.41036 |
| [41] | Suzuki T, Kikkawa M, Vetro C: The existence of best proximity points in metric spaces with the property UC.Nonlinear Anal., Theory Methods Appl. 2009,71(7-8):2918-2926. · Zbl 1178.54029 · doi:10.1016/j.na.2009.01.173 |
| [42] | Eldred AA, Veeramani P: Existence and convergence of best proximity points.J. Math. Anal. Appl. 2006, 323:1001-1006. · Zbl 1105.54021 · doi:10.1016/j.jmaa.2005.10.081 |
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.
