Fuzzy fixed point theorems for multivalued fuzzy contractions in \(b\)-metric spaces
-
1897
Downloads
-
3843
Views
Authors
Supak Phiangsungnoen
- Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bang Mod, Thrung Kru, Bangkok 10140, Thailand.
- Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand.
Poom Kumam
- Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bang Mod, Thrung Kru, Bangkok 10140, Thailand.
- Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand.
Abstract
In this paper, we introduce the new concept of multivalued fuzzy contraction mappings in \(b\)-metric spaces
and establish the existence of \(\alpha\)-fuzzy fixed point theorems in \(b\)-metric spaces which can be utilized to derive
Nadler's fixed point theorem in the framework of b-metric spaces. Moreover, we provide examples to support
our main result.
Share and Cite
ISRP Style
Supak Phiangsungnoen, Poom Kumam, Fuzzy fixed point theorems for multivalued fuzzy contractions in \(b\)-metric spaces, Journal of Nonlinear Sciences and Applications, 8 (2015), no. 1, 55--63
AMA Style
Phiangsungnoen Supak, Kumam Poom, Fuzzy fixed point theorems for multivalued fuzzy contractions in \(b\)-metric spaces. J. Nonlinear Sci. Appl. (2015); 8(1):55--63
Chicago/Turabian Style
Phiangsungnoen, Supak, Kumam, Poom. "Fuzzy fixed point theorems for multivalued fuzzy contractions in \(b\)-metric spaces." Journal of Nonlinear Sciences and Applications, 8, no. 1 (2015): 55--63
Keywords
- \(b\)-metric spaces
- fuzzy fixed point
- fuzzy mappings
- fuzzy set.
MSC
References
-
[1]
M. Abbas, B. Damjanović, R. Lazović, Fuzzy common fixed point theorems for generalized contractive mappings, Appl. Math. Lett., 23 (2010), 1326-1330.
-
[2]
J. Ahmad, A. Azam, S. Romaguera , On locally contractive fuzzy set valued mappings, J. Inequalities Appl., 2014 (2014), 10 pages.
-
[3]
H. Aydi, M-F. Bota, E. Karapinar, S. Mitrović, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl, 2012 (2012), 8 pages.
-
[4]
H. Aydi, M-F. Bota, E. Karapinar, S. Moradi , A common fixed point for weak -contractions on b-metric spaces, Fixed Point Theory, 13 (2012), 337-346.
-
[5]
A. Azam, M. Arshad, I. Beg, Fixed points of fuzzy contractive and fuzzy locally contractive maps, Chaos, Solitons and Fractals, 4 (2009), 2836-2841.
-
[6]
A. Azam, I. Beg, Common fixed points of fuzzy maps, Mathematical and Computer Modelling, 49 (2009), 1331- 1336.
-
[7]
I. A. Bakhtin, The contraction mapping principle in quasimetric spaces , Funct. Anal., Unianowsk Gos. Ped. Inst., 30 (1989), 26-37.
-
[8]
V. Berinde , Generalized contractions in quasimetric spaces, Seminar on Fixed Point Theory, Preprint, 3 (1993), 3-9.
-
[9]
M. Boriceanu, M. Bota, A.Petru, Multivalued fractals in b-metric spaces , Central European J. Math., 8 (2010), 367-377.
-
[10]
M. Boriceanu, A. Petruşel, I. A. Rus , Fixed point theorems for some multivalued generalized contractions in b-metric spaces, Int. J. Math. Stat., 6 (2010), 65-76.
-
[11]
M. Boriceanu, Strict fixed point theorems for multivalued operators in b-metric spaces, Int. J. Mod. Math., 4 (2009), 285-301.
-
[12]
M. Boriceanu , Fixed point theory for multivalued generalized contraction on a set with two b-metrics, Studia Univ. Babeş-Bolyai, Mathematica, 3 (2009), 3-14.
-
[13]
R. K. Bose, M. K. Roychowdhury, Fixed point theorems for generalized weakly contractive mappings, Surv. Math. Appl., 4 (2009), 215-238.
-
[14]
M. Bota, E. Karapnar, O.Mleşniţe, Ulam-Hyers stability results for fixed point problems via \(\alpha-\psi\) -contractive mapping in (b)-metric space, Abstract and Applied Analysis, Article ID 825293 , 2013 (2013), 6 pages.
-
[15]
S. Czerwik, Contraction meppings in b-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1 (1993), 5-11.
-
[16]
S. Czerwik, K. Dlutek, S. L. Singh, Round-off stability of iteration procedures for operators in b-metric spaces, J. Natur. Phys. Sci., 11 (1997), 87-94.
-
[17]
S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Univ. Modena, 46 (1998), 263-276.
-
[18]
V. D. Estruch, A. Vidal , A note on fixed fuzzy points for fuzzy mappings, Rend Istit Univ Trieste, 32 (2001), 39-45.
-
[19]
M. Frigon, D. O'Regan, Fuzzy contractive maps and fuzzy fixed points, Fuzzy Sets and Systems, 129 (2002), 39-45.
-
[20]
S. Heilpern, Fuzzy mappings and fixed point theorems, J. Math. Anal. Appl., 83 (1981), 566-569.
-
[21]
M. A. Kutbi, J. Ahmad, A. Azam, N. Hussain, On fuzzy fixed points for fuzzy maps with generalized weak property, J. Appl. Math., Article ID 549504, 2014 (2014), 12 pages.
-
[22]
M. A. Kutbi, E. Karapinar, J. Ahmad, A. Azam, Some fixed point results for multi-valued mappings in b-metric spaces, J. Inequalities Appl., 2014 (2014), 11 pages.
-
[23]
B. S. Lee, S. J. Cho, A fixed point theorem for contractive type fuzzy mappings, Fuzzy Sets and Systems, 61 (1994), 309-312.
-
[24]
S. B. Nadler , Multivalued contraction mapping, Pacific J. Math., 30 (1969), 475-488.
-
[25]
J. von Neumann , Zur theorie der gesellschaftsspiele, Math. Annalen, 100 (1928), 295-320.
-
[26]
S. Phiangsungnoen, W. Sintunavarat, P. Kumam, Common \(\alpha\)-fuzzy fixed point theorems for fuzzy mappings via \(\beta_ \mathcal{F}\)-admissible pair, J. Intelligent and Fuzzy Systems, 27 (2014), 2463-2472.
-
[27]
S. Phiangsungnoen, W. Sintunavarat, P. Kumam, Fuzzy fixed point theorems in Hausdorff fuzzy metric spaces, J. Inequalities Appl., 2014 (2014), 10 pages.
-
[28]
S. Phiangsungnoen, W. Sintunavarat, P. Kumam, Fuzzy fixed point theorems for fuzzy mappings via \(\beta\)-admissible with applications, J. Uncertainty Anal, Appl, 2 (2014), 17 pages.
-
[29]
B. Singh, M. S. Chauhan, Fixed points of associated multimaps of fuzzy maps, Fuzzy Sets and Systems, 110 (2000), 131-134.
-
[30]
S. L. Singh, B. Prasad, Some coincidence theorems and stability of iterative procedures, Comput. Math. Appl., 55 (2008), 2512-2520.
-
[31]
W. Sintunavarat, S. Plubtieng, P. Katchang, Fixed point result and applications on b-metric space endowed with an arbitrary binary relation, Fixed Point Theory and Applications, 2013 (2013), 13 pages.
-
[32]
D. Turkoglu, B. E. Rhoades, A fixed fuzzy point for fuzzy mapping in complete metric spaces, Math. Commun., 10 (2005), 115-121.