| Title:
|
Topological degree theory in fuzzy metric spaces (English) |
| Author:
|
Rashid, M.H.M. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
55 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
83-96 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The aim of this paper is to modify the theory to fuzzy metric spaces, a natural extension of probabilistic ones. More precisely, the modification concerns fuzzily normed linear spaces, and, after defining a fuzzy concept of completeness, fuzzy Banach spaces. After discussing some properties of mappings with compact images, we define the (Leray-Schauder) degree by a sort of colimit extension of (already assumed) finite dimensional ones. Then, several properties of thus defined concept are proved. As an application, a fixed point theorem in the given context is presented. (English) |
| Keyword:
|
fuzzy metric space |
| Keyword:
|
$t$-norm of $h$-type |
| Keyword:
|
topological degree theory |
| MSC:
|
47H05 |
| MSC:
|
47H09 |
| MSC:
|
47H10 |
| MSC:
|
54H25 |
| idZBL:
|
Zbl 07088760 |
| idMR:
|
MR3964436 |
| DOI:
|
10.5817/AM2019-2-83 |
| . |
| Date available:
|
2019-06-07T14:48:37Z |
| Last updated:
|
2020-02-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147748 |
| . |
| Reference:
|
[1] Akhmerov, R.R., Kamenskii, M.I., Potapov, A.S., Rodkina, A.E., Sadovskii, B.N.: Measures of noncompactness and condensing operators.Birkhäuser-Verlag, Basel-Boston-Berlin, 1992. MR 1153247 |
| Reference:
|
[2] Amann, H.: A note on degree theory for gradient maps.Proc. Amer. Math. Soc. 85 (1982), 591–595. MR 0660610, 10.1090/S0002-9939-1982-0660610-2 |
| Reference:
|
[3] Amann, H., Weiss, S.: On the uniqueness of the topological degree.Math. Z. 130 (1973), 39–54. MR 0346601, 10.1007/BF01178975 |
| Reference:
|
[4] Bag, T., Samanta, S. K.: Finite dimensional fuzzy normed linear spaces.J. Fuzzy Math. 11 (3) (2003), 687–705. MR 2005663 |
| Reference:
|
[5] Blasi, F.S. De, Myjak, J.: A remark on the definition of topological degree for set-valued mappings.J. Math. Anal. Appl. 92 (1983), 445–451. MR 0697030, 10.1016/0022-247X(83)90261-5 |
| Reference:
|
[6] Browder, F.E.: Nonlinear operators and nonlinear equations of evolution in Banach spaces.Proc. Symp. Pura Math., vol. 18, Amer. Math. Soc. Providence, 1976. Zbl 0327.47022, MR 0405188 |
| Reference:
|
[7] Browder, F.E.: Fixed point theory and nonlinear problems.Bull. Amer. Math. Soc. 9 (1) (1983), 1–41. MR 0699315, 10.1090/S0273-0979-1983-15153-4 |
| Reference:
|
[8] Browder, F.E., Nussbaum, R.D.: The topological degree for noncompact nonlinear mappings in Banach spaces.Bull. Amer. Math. Soc. 74 (1968), 671–676. MR 0232257, 10.1090/S0002-9904-1968-11988-3 |
| Reference:
|
[9] Cellina, A., Lasota, A.: A new approach to the definition of topological degree for multi valued mappings.Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 47 (1969), 434–440. MR 0276937 |
| Reference:
|
[10] Cho, Yeol Je, Chen, Yu-Qing: Topological Degree Theory and Applications.CRC Press, 2006. MR 2223854 |
| Reference:
|
[11] Cronin, J.: Fixed points and topological degree in nonlinear analysis.Mathematical Surveys, no. 11, American Mathematical Society, Providence, R.I, 1964, pp. xii+198 pp. MR 0164101 |
| Reference:
|
[12] Deimling, K.: Nonlinear functional analysis.Springer-Verlag, Berlin, 1985. MR 0787404 |
| Reference:
|
[13] Diestel, J.: Geometry of Banach spaces, Selected Topics.Lecture Notes in Math, vol. 485, Springer-Verlag, Berlin-New York, 1975, pp. xi+282 pp. MR 0461094 |
| Reference:
|
[14] Fitzpatrick, .M.: A generalized degree for uniform limit of A-proper mappings.J. Math. Anal. Appl. 35 (1971), 536–552. MR 0281069, 10.1016/0022-247X(71)90201-0 |
| Reference:
|
[15] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order.Springer-Verlag, Berlin, 1983. Zbl 0562.35001, MR 0737190 |
| Reference:
|
[16] Gossez, J.-P.: On the subdifferential of saddle functions.J. Funct. Anal. 11 (1972), 220–230. MR 0350416, 10.1016/0022-1236(72)90092-4 |
| Reference:
|
[17] Leray, J.: Les problemes nonlineaires.Enseign. Math. 30 (1936), 141. |
| Reference:
|
[18] Leray, J., Schauder, J.: Topologie et équations fonctionnelles.Ann. Sci. Ecole. Norm. Sup. 51 (1934), 45–78. MR 1509338, 10.24033/asens.836 |
| Reference:
|
[19] Mawhin, J.: Topological Degree Methods in Nonlinear Boundary Value Problems.Amer. Math. Soc., vol. 40, Providence, RI, 1979. Zbl 0414.34025, MR 0525202 |
| Reference:
|
[20] Nǎdǎban, S., Dzitac, I.: Atomic Decomposition of fuzzy normed linear spaces for wavelet applications.Informatica 25 (4) (2014), 643–662. MR 3301468, 10.15388/Informatica.2014.33 |
| Reference:
|
[21] Nirenberg, L.: Variational and topological methods in nonlinear problems.Bull. Amer. Math. Soc. 4 (1981), 267–302. MR 0609039, 10.1090/S0273-0979-1981-14888-6 |
| Reference:
|
[22] Roldán, A., Martínez-Moreno, J., Roldán, C.: On interrelationships between fuzzy metric structures.Iran. J. Fuzzy Syst. 10 (2) (2013), 133–150. MR 3098998 |
| Reference:
|
[23] Schweizer, B., Sklar, A.: Statistical metric spaces.Pacific J. Math. 10 (1960), 313–334. Zbl 0096.33203, MR 0115153, 10.2140/pjm.1960.10.313 |
| Reference:
|
[24] Schweizer, B., Sklar, A.: Probabilistical Metric Spaces.Dover Publications, New York, 2005. MR 0790314 |
| Reference:
|
[25] Sherwood, H.: On the completion of probabilistic metric spaces.Z. Wahrsch. Verw. Gebiete 6 (1966), 62–64. MR 0212844, 10.1007/BF00531809 |
| Reference:
|
[26] Wardowski, D.: Fuzzy contractive mappings and fixed points in fuzzy metric spaces.Fuzzy Sets and Systems 222 (2013), 108–114. MR 3053895 |
| . |
