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Keywords:
fuzzy metric space; $t$-norm of $h$-type; topological degree theory
fuzzy metric space; $t$-norm of $h$-type; topological degree theory
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.
References:
[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
[2] Amann, H.: A note on degree theory for gradient maps. Proc. Amer. Math. Soc. 85 (1982), 591–595. DOI 10.1090/S0002-9939-1982-0660610-2 | MR 0660610
[3] Amann, H., Weiss, S.: On the uniqueness of the topological degree. Math. Z. 130 (1973), 39–54. DOI 10.1007/BF01178975 | MR 0346601
[4] Bag, T., Samanta, S. K.: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11 (3) (2003), 687–705. MR 2005663
[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. DOI 10.1016/0022-247X(83)90261-5 | MR 0697030
[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. MR 0405188 | Zbl 0327.47022
[7] Browder, F.E.: Fixed point theory and nonlinear problems. Bull. Amer. Math. Soc. 9 (1) (1983), 1–41. DOI 10.1090/S0273-0979-1983-15153-4 | MR 0699315
[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. DOI 10.1090/S0002-9904-1968-11988-3 | MR 0232257
[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
[10] Cho, Yeol Je, Chen, Yu-Qing: Topological Degree Theory and Applications. CRC Press, 2006. MR 2223854
[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
[12] Deimling, K.: Nonlinear functional analysis. Springer-Verlag, Berlin, 1985. MR 0787404
[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
[14] Fitzpatrick, .M.: A generalized degree for uniform limit of A-proper mappings. J. Math. Anal. Appl. 35 (1971), 536–552. DOI 10.1016/0022-247X(71)90201-0 | MR 0281069
[15] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, 1983. MR 0737190 | Zbl 0562.35001
[16] Gossez, J.-P.: On the subdifferential of saddle functions. J. Funct. Anal. 11 (1972), 220–230. DOI 10.1016/0022-1236(72)90092-4 | MR 0350416
[17] Leray, J.: Les problemes nonlineaires. Enseign. Math. 30 (1936), 141.
[18] Leray, J., Schauder, J.: Topologie et équations fonctionnelles. Ann. Sci. Ecole. Norm. Sup. 51 (1934), 45–78. DOI 10.24033/asens.836 | MR 1509338
[19] Mawhin, J.: Topological Degree Methods in Nonlinear Boundary Value Problems. Amer. Math. Soc., vol. 40, Providence, RI, 1979. MR 0525202 | Zbl 0414.34025
[20] Nǎdǎban, S., Dzitac, I.: Atomic Decomposition of fuzzy normed linear spaces for wavelet applications. Informatica 25 (4) (2014), 643–662. DOI 10.15388/Informatica.2014.33 | MR 3301468
[21] Nirenberg, L.: Variational and topological methods in nonlinear problems. Bull. Amer. Math. Soc. 4 (1981), 267–302. DOI 10.1090/S0273-0979-1981-14888-6 | MR 0609039
[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
[23] Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10 (1960), 313–334. DOI 10.2140/pjm.1960.10.313 | MR 0115153 | Zbl 0096.33203
[24] Schweizer, B., Sklar, A.: Probabilistical Metric Spaces. Dover Publications, New York, 2005. MR 0790314
[25] Sherwood, H.: On the completion of probabilistic metric spaces. Z. Wahrsch. Verw. Gebiete 6 (1966), 62–64. DOI 10.1007/BF00531809 | MR 0212844
[26] Wardowski, D.: Fuzzy contractive mappings and fixed points in fuzzy metric spaces. Fuzzy Sets and Systems 222 (2013), 108–114. MR 3053895
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