×

On optimal fuzzy best proximity coincidence points of proximal contractions involving cyclic mappings in non-Archimedean fuzzy metric spaces. (English) Zbl 1367.54025

Summary: The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings \(T\) integrated in a pair \((g,T)\) of mappings. In particular, \(g\) is a non-contractive fuzzy self-mapping, in the framework of non-Archimedean ordered fuzzy complete metric spaces and \(T\) is a \(p\)-cyclic proximal contraction. Two types of such contractions (so called of type I and of type II) are dealt with. In particular, the existence, uniqueness and limit properties for sequences to optimal fuzzy best proximity coincidence points are investigated for such pairs of mappings.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54A40 Fuzzy topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Schweizer, B.; Sklar, A.; Statistical metric spaces; Pac. J. Math.: 1960; Volume 10 ,314-334. · Zbl 0091.29801
[2] George, A.; Veeramani, P.; On some results of analysis for fuzzy metric spaces; Fuzzy Sets Syst.: 1997; Volume 90 ,365-368. · Zbl 0917.54010
[3] George, A.; Veeramani, P.; On some results in fuzzy metric spaces; Fuzzy Sets Syst.: 1994; Volume 64 ,395-399. · Zbl 0843.54014
[4] Zadeh, L.A.; Fuzzy sets; Inf. Control: 1965; Volume 8 ,103-112.
[5] Grabiec, M.; Fixed points in fuzzy metric spaces; Fuzzy Sets Syst.: 1983; Volume 27 ,385-389. · Zbl 0664.54032
[6] Heilpern, S.; Fuzzy mappings and fixed point theorem; J. Math. Anal. Appl.: 1981; Volume 83 ,566-569. · Zbl 0486.54006
[7] Vetro, C.; Salimi, P.; Best proximity point results in non-Archimedean fuzzy metric spaces; Fuzzy Inf. Eng.: 2013; Volume 5 ,417-429. · Zbl 1430.54014
[8] Gregori, V.; Sapena, A.; On fixed-point theorems in fuzzy metric spaces; Kybernetica: 1975; Volume 11 ,326-334. · Zbl 0995.54046
[9] Sadiq Basha, S.; Common best proximity points: Global minimization of multi-objective functions; J. Glob. Optim.: 2012; Volume 54 ,367-373. · Zbl 1281.90058
[10] Sadiq Basha, S.; Best proximity points: Optimal fuzzy solutions; J. Optim. Fuzzy Theory Appl.: 2011; Volume 151 ,210-216. · Zbl 1226.90135
[11] Sadiq Basha, S.; Best proximity point theorems on partially ordered sets; Optim. Lett.: 2012; Volume 7 ,1035-1043. · Zbl 1267.90104
[12] Mongkolkeha, C.; Cho, Y.J.; Kumam, P.; Best proximity points for generalized proximal contraction mappings in metric spaces with partial orders; J. Inequal. Appl.: 2013; Volume 2013 ,94. · Zbl 1417.47016
[13] Vetro, C.; Best proximity points: Convergence and existence theorems for p-cyclic mappings; Nonlinear Anal. Theory Methods Appl.: 2010; Volume 73 ,2283-2291. · Zbl 1229.54066
[14] Chitra, A.; Subrahmanyam, P.V.; Fuzzy sets and fixed points; J. Math. Anal. Appl.: 1987; Volume 124 ,584-590. · Zbl 0628.47035
[15] Azam, A.; Beg, I.; Common fuzzy fixed points for fuzzy mappings; Fixed Point Theory Appl.: 2013; Volume 2013 ,14. · Zbl 1409.54015
[16] Cho, Y.J.; Pathak, H.K.; Kang, S.M.; Jung, J.S.; Common fixed points of compatible mappings of type (β) on fuzzy metric spaces; Fuzzy Sets Syst.: 1998; Volume 93 ,99-111. · Zbl 0915.54004
[17] Abbas, M.; Altun, I.; Gopal, D.; Common fixed point theorems for non compatible mappings in fuzzy metric spaces; Bull. Math. Anal. Appl.: 2009; Volume 1 ,47-56. · Zbl 1175.54048
[18] Phiangsungnoen, S.; Sintunavarat, W.; Kumam, P.; Common α-fuzzy fixed point theorems for fuzzy mappings via βF-admissible pair; J. Intell. Fuzzy Syst.: 2014; Volume 27 ,2463-2472. · Zbl 1338.54053
[19] De la Sen, M.; Agarwal, R.P.; Ibeas, A.; Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets; Fixed Point Theory Appl.: 2014; Volume 2014 ,169. · Zbl 1469.54086
[20] Gabeleh, M.; Best proximity point theorems via proximal non self-mapping; J. Optim. Theory Appl.: 2015; Volume 164 ,565-576. · Zbl 1312.47067
[21] De la Sen, M.; Roldan, A.; Agarwal, R.P.; On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points; Fixed Point Theory Appl.: 2015; Volume 2015 ,103. · Zbl 1347.54114
[22] Rashid, M.; Mehmood, N.; Azam, A.; Radenović, S.; Fuzzy fixed point theorems in ordered cone metric spaces; Filomat: 2015; Volume 29 ,887-896. · Zbl 1460.54035
[23] Chauan, S.; Shatanawi, W.; Kumar, S.; Radenović, S.; Existence and uniquenmess of fixed points in modified intuitionistic fuzzy metric spaces; J. Nonlinear Sci. Appl.: 2017; Volume 7 ,28-41. · Zbl 1477.54066
[24] Chauan, S.; Radenović, S.; Imdad, M.; Vetro, C.; Some integral type fixed point theorems in non-Archimedean Menger PM-spaces with common property (EA) and applications of functional equations in dynamic programming; Rev. Real Acad. Cienc. Exact. Fís. Nat. Ser. A Mat. (RACSAM): 2014; Volume 108 ,795-810. · Zbl 1395.54040
[25] Chauan, S.; Radenović, S.; Bhatnagar, S.; Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces; Mathematiche: 2013; Volume 68 ,87-98. · Zbl 1282.54035
[26] Rezapour, S.; Derafshpour, M.; Shahzad, N.; On the existence of best proximity points of cyclic contractions; Adv. Dyn. Syst. Appl.: 2011; Volume 6 ,33-40. · Zbl 1227.54046
[27] Al-Thagafi, M.A.; Shahzad, N.; Convergence and existence results for best proximity points; Nonlinear Anal. Theory Methods Appl.: 2009; Volume 70 ,3665-3671. · Zbl 1197.47067
[28] Derafshpour, M.; Rezapour, S.; Shahzad, N.; Best proximity points of cyclicφ-contractions on reflexive Banach spaces; Fixed Point Theory Appl.: 2010; Volume 2010 ,946178. · Zbl 1197.47071
[29] De la Sen, M.; Karapinar, E.; Some results on best proximity points of cyclic contractions in probabilistic metric spaces; J. Funct. Spaces: 2015; Volume 2015 ,470574. · Zbl 1349.54093
[30] De la Sen, M.; Ibeas, A.; On the global stability of an iterative scheme in a probabilistic Menger space; J. Inequal. Appl.: 2015; Volume 2015 ,243. · Zbl 1391.41020
[31] Khojasteh, F.; Shukla, S.; Radenović, S.; A new approach to the study of fixed point theory for simulation functions; Filomat: 2015; Volume 29 ,1189-1194. · Zbl 1462.54072
[32] De la Sen, M.; Ibeas, A.; Properties of convergence of a class of iterative processes generated by sequences of self-mappings with applications to switched Systems; J. Inequal. Appl.: 2014; Volume 2014 ,498. · Zbl 1346.47081
[33] De la Sen, M.; Alonso-Quesada, S.; Ibeas, A.; On the asymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraints; IMA J. Math. Control Inf.: 2015; Volume 32 ,359-386. · Zbl 1328.93223
[34] De la Sen, M.; Agarwal, R.P.; Ibeas, A.; Alonso-Quesada, S.; On a generalized time-varying SEIR epidemic model with mixed point and distributed time-varying delays and combined regular and impulsive vaccination controls; Adv. Differ. Equ.: 2010; Volume 2010 ,281612. · Zbl 1219.34104
[35] De la Sen, M.; On the robust adaptive stabilization of a class of nominally stable first-order systems; IEEE Trans. Autom. Control,: 1986; Volume 44 ,597-602. · Zbl 1056.93616
[36] Marchenko, V.M.; Hybrid discrete-continuous systems with control: II. State-space method; Differ. Equ.: 2015; Volume 51 ,54-68. · Zbl 1311.93010
[37] Marchenko, V.M.; Hybrid discrete-continuous control systems: I. Representation of solutions; Differ. Equ.: 2015; Volume 50 ,1526-1540. · Zbl 1311.93046
[38] Abbas, M.; Saleem, N.; de la Sen, M.; Optimal coincidence point results in partially ordered non-Archimedean fuzzy metric spaces; Fixed Point Theory Appl.: 2016; Volume 2016 ,534. · Zbl 1387.54025
[39] Saleem, N.; Ali, B.; Abbas, M.; Raza, Z.; Fixed points of Suzuki type generalized multivalued mappings in fuzzy metric space with applications; Fixed Point Theory Appl.: 2015; Volume 2015 ,36. · Zbl 1310.54065
[40] Shehu, Y.; Cai, G.; Iyiola, O.S.; Iterative approximation of solutions for proximal split feasibility problems; Fixed Point Theory Appl.: 2015; Volume 2015 ,123. · Zbl 1336.49023
[41] Cai, G.; Shehu, Y.; Iyiola, O.S.; Iterative algorithms for solving variational inequalities and fixed point problems for asymptotically nonexpansive mappings in Banach spaces; Numer. Algorithms: 2016; Volume 73 ,869-906. · Zbl 1352.49030
[42] Gaba, Y.U.; Iyiola, O.S.; Advances in the study of metric type spaces; Appl. Math. Sci.: 2015; Volume 9 ,4179-4190.
[43] Cai, G.; Shehu, Y.; Iyiola, O.S.; Viscosity iterative algorithms for fixed point problems of asymptotically nonexpansive mappings in the intermediate sense and variational inequality problems in Banach spaces; Numer. Algorithms: 2017; . · Zbl 1422.47066
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.