The main motivation for this paper is to investigate the fixed point property for non-expansive mappings defined on $ b $-fuzzy metric spaces. First, following the idea of S. Ješić's result from 2009, we introduce convex, strictly convex and normal structures for sets in $ b $-fuzzy metric spaces. By using topological methods and these notions, we prove the existence of fixed points for self-mappings defined on $ b $-fuzzy metric spaces satisfying a nonlinear type condition. This result generalizes and improves many previously known results, such as W. Takahashi's result on metric spaces from 1970. A representative example illustrating the main result is provided.
Citation: Siniša N. Ješić, Nataša A. Ćirović, Rale M. Nikolić, Branislav M. Ranƌelović. A fixed point theorem in strictly convex $ b $-fuzzy metric spaces[J]. AIMS Mathematics, 2023, 8(9): 20989-21000. doi: 10.3934/math.20231068
The main motivation for this paper is to investigate the fixed point property for non-expansive mappings defined on $ b $-fuzzy metric spaces. First, following the idea of S. Ješić's result from 2009, we introduce convex, strictly convex and normal structures for sets in $ b $-fuzzy metric spaces. By using topological methods and these notions, we prove the existence of fixed points for self-mappings defined on $ b $-fuzzy metric spaces satisfying a nonlinear type condition. This result generalizes and improves many previously known results, such as W. Takahashi's result on metric spaces from 1970. A representative example illustrating the main result is provided.
| [1] | B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math., 10 (1960), 313–334. |
| [2] | O. Hadžić, E. Pap, Fixed point theory in probabilistic metric spaces, Dordrecht: Springer, 2001. https://doi.org/10.1007/978-94-017-1560-7 |
| [3] |
L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
|
| [4] | I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975), 336–344. |
| [5] |
A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), 395–399. https://doi.org/10.1016/0165-0114(94)90162-7 doi: 10.1016/0165-0114(94)90162-7
|
| [6] | I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., 30 (1989), 26–37. |
| [7] | S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), 5–11. |
| [8] | S. Czerwik, Nonlinear set-valued contraction mappings in $b$-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263–276. |
| [9] |
T. Suzuki, Basic inequality on a $b$-metric space and its applications, J. Inequal. Appl., 2017 (2017), 256. https://doi.org/10.1186/s13660-017-1528-3 doi: 10.1186/s13660-017-1528-3
|
| [10] | S. Sedghi, N. Shobe, J. H. Park, Common fixed point theorem in $b$-fuzzy metric space, Nonlinear Funct. Anal. Appl., 17 (2012), 349–359. |
| [11] |
T. Došenović, A. Javaheri, S. Sedghi, N. Shobe, Coupled fixed point theorem in $b$-fuzzy metric spaces, Novi Sad J. Math., 47 (2017), 77–88. https://doi.org/10.30755/NSJOM.04361 doi: 10.30755/NSJOM.04361
|
| [12] |
Badshah-e-Rome, M. Sarwar, T. Abdeljawad, $\mu$-extended fuzzy $b$-metric spaces and related fixed point results, AIMS Math., 5 (2020), 5184–5192. https://doi.org/10.3934/math.2020333 doi: 10.3934/math.2020333
|
| [13] |
R. Mecheraoui, Z. D. Mitrović, V. Parvaneh, H. Aydi, N. Saleem, On some fixed point results in E-fuzzy metric spaces, J. Math., 2021 (2021), 1–6. https://doi.org/10.1155/2021/9196642 doi: 10.1155/2021/9196642
|
| [14] |
Humaira, M. Sarwar, T. Abdeljawad, Existence of solutions for nonlinear impulsive fractional differential equations via common fixed-point techniques in complex valued fuzzy metric spaces, Math. Probl. Eng., 2020 (2020), 1–14. https://doi.org/10.1155/2020/7042715 doi: 10.1155/2020/7042715
|
| [15] | S. Sedghi, N. Shobe, Common fixed point theorem for $R$-weakly commuting maps in $b$-fuzzy metric space, Nonlinear Funct. Anal. Appl., 19 (2014), 285–295. |
| [16] |
B. M. Ranƌ elović, N. A. Ćirović, S. N. Ješić, A characterization of completeness of $b$-fuzzy metric spaces and nonlinear contractions, Appl. Anal. Discrete Math., 15 (2021), 233–242. https://doi.org/10.2298/AADM200911057R doi: 10.2298/AADM200911057R
|
| [17] | M. S. Brodskii, D. P. Milman, On the center of a convex set, Dokl. Akad. Nauk. SSSR, 59 (1948), 837–840. |
| [18] |
W. Takahashi, A convexity in metric space and nonexpansive mappings, Kodai Math. Sem. Rep., 22 (1970), 142–149. https://doi.org/10.2996/kmj/1138846111 doi: 10.2996/kmj/1138846111
|
| [19] | O. Hadžić, Common fixed point theorems in probabilistic metric spaces with a convex structure, Zb. Rad. Prirod. Mat. Fak. Ser. Mat., 18 (1987), 165–178. |
| [20] |
S. N. Ješić, Convex structure, normal structure and a fixed point theorem in intuitionistic fuzzy metric spaces, Chaos Solitons Fract., 41 (2009), 292–301. https://doi.org/10.1016/j.chaos.2007.12.002 doi: 10.1016/j.chaos.2007.12.002
|
| [21] |
M. Gabeleh, E. U. Ekici, M. De La Sen, Noncyclic contractions and relatively nonexpansive mappings in strictly convex fuzzy metric spaces, AIMS Math., 7 (2022), 20230–20246. https://doi.org/10.3934/math.20221107 doi: 10.3934/math.20221107
|
| [22] |
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006. https://doi.org/10.2307/2313345 doi: 10.2307/2313345
|
| [23] |
F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA, 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041 doi: 10.1073/pnas.54.4.1041
|
Siniša N. Ješić, Nataša A. Ćirović, Rale M. Nikolić, Branislav M. Ranƌelović. A fixed point theorem in strictly convex $ b $-fuzzy metric spaces[J]. AIMS Mathematics, 2023, 8(9): 20989-21000. doi: 10.3934/math.20231068
DownLoad: