Fixed Point Theory in Fuzzy Normed Linear Spaces: A General View
DOI:
https://doi.org/10.15837/ijccc.2021.6.4587Keywords:
fuzzy normed linear space, fuzzy Banach spaces, completeness, fixed point theoryAbstract
In this paper we have presented, firstly, an evolution of the concept of fuzzy normed linear spaces, different definitions, approaches as well as generalizations. A special section is dedicated to fuzzy Banach spaces. In the case of fuzzy normed linear spaces, researchers have been working, until now, with a definition of completeness inspired by M. Grabiec’s work in the context of fuzzy metric spaces. We propose another definition and we prove that it is much more adequate, inspired by the work of A.George and P. Veeramani. Finally, some important results in fuzzy fixed point theory were highlighted.
References
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[25] Huang, H.; Caric, B.; Došenovic, T.; Rakic, D.; Brdar, M. (2021). Fixed-Point Theorems in Fuzzy Metric Spaces via Fuzzy F-Contraction,Mathematics, 9, 641. https://doi.org/10.3390/math9060641
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[27] Kaleva, O.; Seikkala, S. (1984). On fuzzy metric spaces, Fuzzy Sets and Systems, 12(3), 215-229. https://doi.org/10.1016/0165-0114(84)90069-1
[28] Katsaras, A.K. (1984). Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12(2), 143- 154. https://doi.org/10.1016/0165-0114(84)90034-4
[29] Katsaras, A.K.; Liu, D.B. (1977). Fuzzy vector spaces and fuzzy topological vector spaces, Journal of Mathematical Analysis and Applications, 58, 135-146. https://doi.org/10.1016/0022-247X(77)90233-5
[30] Kramosil, I.; Michálek, J. (1975). Fuzzy metric and statistical metric spaces, Kybernetica, 11, 326-334.
[31] Al-Mezel, S.A.; Ahmad, J.; De La Sen, M. (2020). Some New Fuzzy Fixed Point Results with Applications, Mathematics, 8, 995. https://doi.org/10.3390/math8060995
[32] Mohsenialhosseini, S.A.M.; Mazaheri, H. (2013). Approximate fixed point theorems in fuzzy norm spaces for an operator, Advaces in Fuzzy Systems, Article ID 613604. https://doi.org/10.1155/2013/613604
[33] Mohsenialhosseini, S.A.M.; Mazaheri, H.; Dehghan, M.A. (2013). Approximate fixed point in fuzzy normed spaces for nonlinear maps, Iranian Journal of Fuzzy Systems, 10(1), 135-142.
[34] Nadaban, S. (2015). Fuzzy continuous mapping in fuzzy normed linear spaces, International Journal of Computers Communications & Control, 10(6), 836-844. https://doi.org/10.15837/ijccc.2015.6.2074
[35] Nadaban, S. (2016). Fuzzy pseudo-norms and fuzzy F-spaces, Fuzzy Sets and Systems, 282, 99-114. https://doi.org/10.1016/j.fss.2014.12.010
[36] Nadaban, S; Bí®nzar, T; Pater, F. (2017). Some fixed point theorems for '-contractive mappings in fuzzy normed linear spaces, J. Nonlinear Sci. Appl., 10, 5668-5676. https://doi.org/10.22436/jnsa.010.11.05
[37] Nadaban, S; Dzitac, I. (2014). Atomic decompositions of fuzzy normed linear spaces for wavelet applications, Informatica, 25(4), 643-662. https://doi.org/10.15388/Informatica.2014.33
[38] Rana, A.M.; Buthainah, A.A.A.; Fadhel, F.S. (2015). On fixed point theorem in fuzzy normed space, Journal of Al-Nahrain University, 18(4), 138-143. https://doi.org/10.22401/JNUS.18.4.19
[39] Romaguera, S.; Sapena, A.; Tirado, P. (2007). The Banach fixed point theorem in fuzzy quasimetric spaces with application to the domain of words, Topology and its Applications, 154(10), 2196-2203. https://doi.org/10.1016/j.topol.2006.09.018
[40] Secelean, N.A.; Mathew, S.; Wardowski, D. (2019). New fixed-point results in quasi-metric spaces and applications in fractals theory, Adv. Differ. Equ., 2019, 177. https://doi.org/10.1186/s13662-019-2119-z
[41] Saadati, R; Park, J.H. (2006). Intuitionistic fuzzy euclidean normed spaces, Communications in Mathematical Analysis, 1(2), 85-90.
[42] Saadati, R.; Park, C.; O'Regan, D.; Nadaban, S. (2021). n-Expansively super-homogeneous and (n, k)-contractively sub-homogeneous fuzzy control functions and stability results with numerical examples, Advances in Difference Equations, 2021:153. https://doi.org/10.1186/s13662-021-03287-y
[43] Saadati, R.; Vaezpour, S.M. (2005). Some results on fuzzy Banach spaces, Journal of Applied Mathematics and Computing, 17(1-2), 475-484. https://doi.org/10.1007/BF02936069
[44] Saadati, R.; Vaezpour, S.M.; Cho, Y.J. (2009). Quicksort algorithm: Application of a fixed point theorem in intuitionistic fuzzy quasi-metric spaces at a domain of words, Journal of Computational and Applied Mathematics, 228(1), 219-225. https://doi.org/10.1016/j.cam.2008.09.013
[45] Sadeqi, I.; Solaty kia, F. (2009). Fuzzy normed linear space and its topological structure, Chaos, Solitons and Fractals, 40(5), 2576-2589. https://doi.org/10.1016/j.chaos.2007.10.051
[46] Sadeqi, I.; Solaty kia, F. (2009). Some fixed point theorems in fuzzy reflexive Banach spaces, Chaos, Solitons and Fractals, 41, 2606-2612. https://doi.org/10.1016/j.chaos.2008.09.050
[47] Saheli, M. (2016). A contractive mapping on fuzzy normed linear spaces, Iranian Journal of Numerical Analysis and Optimization, 6(1), 121-136.
[48] Schweizer, A; Sklar, A. (1960). Statistical metric spaces, Pacific Journal of Mathematics, 10, 314-334. https://doi.org/10.2140/pjm.1960.10.313
[49] Xiao, J.-z.; Zhu, X.-h. (2004). Some basic theorems for linear operators between fuzzy normed spaces, Journal of University of Science and Technology of China, 34(4), 414-430.
[50] Xiao, J.-z.; Zhu, X.-h. (2004). Topological degree theory and fixed point theorems in fuzzy normed linear spaces, Fuzzy Sets and Systems, 147(3), 437-452. https://doi.org/10.1016/j.fss.2004.01.003
[51] Zadeh, L. A. (1965). Fuzzy Sets, Information and Control, 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
[2] Alegre, C.; Romaguera, S. (2010). Characterizations of fuzzy metrizable topological vector spaces and their asymmetric generalization in terms of fuzzy (quasi-)norms, Fuzzy Sets and Systems, 161(16), 2181-2192. https://doi.org/10.1016/j.fss.2010.04.002
[3] Andres, J.; Rypka, M. (2019). On a topological fuzzy fixed point theorem and its application to non-ejective fuzzy fractals II. Fuzzy Sets and Systems, 370, 79-90. https://doi.org/10.1016/j.fss.2018.09.013
[4] Bag, T. (2011). Some fundamental theorems in Felbin's type fuzzy normed linear spaces, International Journal of Mathematics and Scientific Computing, 1(2), 44-49.
[5] Bag, T.; Samanta, S.K. (2003). Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11(3), 687-705.
[6] Bag, T.; Samanta, S.K. (2005). Fuzzy bounded linear operators, Fuzzy Sets and Systems, 151, 513-547. https://doi.org/10.1016/j.fss.2004.05.004
[7] Bag, T.; Samanta, S.K. (2006). Fixed point theorems on fuzzy normed linear spaces, Information Sciences, 176, 2910-2931. https://doi.org/10.1016/j.ins.2005.07.013
[8] Bag, T.; Samanta, S.K. (2007). Some fixed point theorems in fuzzy normed linear spaces, Information Sciences, 177, 3271-3289. https://doi.org/10.1016/j.ins.2007.01.027
[9] Bag, T.; Samanta, S.K. (2008). A comparative study of fuzzy norms on a linear space, Fuzzy Sets and Systems, 159, 670-684. https://doi.org/10.1016/j.fss.2007.09.011
[10] Bag, T.; Samanta, S.K. (2008). Fuzzy bounded linear operators in Felbin's type fuzzy normed linear spaces, Fuzzy Sets and Systems, 159, 685-707. https://doi.org/10.1016/j.fss.2007.09.006
[11] Berinde, M. (2006). Approximate fixed point theorems, Studia Univ. Babes-Bolyai, Mathematica, 51(1), 11-25.
[12] Bí®nzar, T; Pater, F.; Nadaban, S. (2020). Fuzzy bounded operators with application to Radon transform, Chaos, Solitons and Fractals, 141, 110359. https://doi.org/10.1016/j.chaos.2020.110359
[13] Cancan, M. (2010). Browder's fixed point theorem and some interesting results in intuitionistic fuzzy normed spaces, Fixed Point Theory and Applications, 642303. https://doi.org/10.1155/2010/642303
[14] Chandok, S.; Huang, H.; Radenovic, S. (2018). Some fixed-point results for the generalized F-suzuki type contractions in b-metric spaces, Sahand Commun. Math. Anal., 11, 81-89.
[15] Chatterjee, S.; Bag, T.; Lee, J.-G. (2020). Schauder fixed point theorem in generalized fuzzy normed linear spaces, Mathematics, 8, 1643. https://doi.org/10.3390/math8101643
[16] Cheng, S.C.; Mordeson, J.N. (1994). Fuzzy linear operators and fuzzy normed linear spaces, Bull. Cal. Math. Soc., 86, 429-436.
[17] Das, N.R.; Saha, M.L. (2015). On fixed points in complete fuzzy normed linear spaces, Annals of Fuzzy Mathematics and Informatics, 10(4), 515-524.
[18] Elagan, S.K.; Rahmat, M.R.S. (2010). Some fixed point theorems in fuzzy n-normed spaces, International J. Math. Combin., 3, 45-56.
[19] Fang, J.X. (1992). On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 46, 107-113. https://doi.org/10.1016/0165-0114(92)90271-5
[20] Felbin, C. (1992). Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48(2), 239-248. https://doi.org/10.1016/0165-0114(92)90338-5
[21] George, A.; Veeramani, P. (1994). On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64(3), 395-399. https://doi.org/10.1016/0165-0114(94)90162-7
[22] Grabiec, M. (1998). Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27, 385-389. https://doi.org/10.1016/0165-0114(88)90064-4
[23] Golet, I. (2010). On generalized fuzzy normed spaces and coincidence point theorems, Fuzzy Sets and Systems, 161(8), 1138-1144. https://doi.org/10.1016/j.fss.2009.10.004
[24] Gordji, M.E.; Baghani, H.; Cho, Y.J. (2011). Coupled fixed point theorems for contractions in intuitionistic fuzzy normed spaces, Mathematical and Computer Modelling, 54, 1897-1906. https://doi.org/10.1016/j.mcm.2011.04.014
[25] Huang, H.; Caric, B.; Došenovic, T.; Rakic, D.; Brdar, M. (2021). Fixed-Point Theorems in Fuzzy Metric Spaces via Fuzzy F-Contraction,Mathematics, 9, 641. https://doi.org/10.3390/math9060641
[26] Janfada, M.; Baghani, H.; Baghani, O. (2011). On Felbin's-type fuzzy normed linear spaces and fuzzy bounded operators,Iranian Journal of Fuzzy Systems, 8(5), 117-130.
[27] Kaleva, O.; Seikkala, S. (1984). On fuzzy metric spaces, Fuzzy Sets and Systems, 12(3), 215-229. https://doi.org/10.1016/0165-0114(84)90069-1
[28] Katsaras, A.K. (1984). Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12(2), 143- 154. https://doi.org/10.1016/0165-0114(84)90034-4
[29] Katsaras, A.K.; Liu, D.B. (1977). Fuzzy vector spaces and fuzzy topological vector spaces, Journal of Mathematical Analysis and Applications, 58, 135-146. https://doi.org/10.1016/0022-247X(77)90233-5
[30] Kramosil, I.; Michálek, J. (1975). Fuzzy metric and statistical metric spaces, Kybernetica, 11, 326-334.
[31] Al-Mezel, S.A.; Ahmad, J.; De La Sen, M. (2020). Some New Fuzzy Fixed Point Results with Applications, Mathematics, 8, 995. https://doi.org/10.3390/math8060995
[32] Mohsenialhosseini, S.A.M.; Mazaheri, H. (2013). Approximate fixed point theorems in fuzzy norm spaces for an operator, Advaces in Fuzzy Systems, Article ID 613604. https://doi.org/10.1155/2013/613604
[33] Mohsenialhosseini, S.A.M.; Mazaheri, H.; Dehghan, M.A. (2013). Approximate fixed point in fuzzy normed spaces for nonlinear maps, Iranian Journal of Fuzzy Systems, 10(1), 135-142.
[34] Nadaban, S. (2015). Fuzzy continuous mapping in fuzzy normed linear spaces, International Journal of Computers Communications & Control, 10(6), 836-844. https://doi.org/10.15837/ijccc.2015.6.2074
[35] Nadaban, S. (2016). Fuzzy pseudo-norms and fuzzy F-spaces, Fuzzy Sets and Systems, 282, 99-114. https://doi.org/10.1016/j.fss.2014.12.010
[36] Nadaban, S; Bí®nzar, T; Pater, F. (2017). Some fixed point theorems for '-contractive mappings in fuzzy normed linear spaces, J. Nonlinear Sci. Appl., 10, 5668-5676. https://doi.org/10.22436/jnsa.010.11.05
[37] Nadaban, S; Dzitac, I. (2014). Atomic decompositions of fuzzy normed linear spaces for wavelet applications, Informatica, 25(4), 643-662. https://doi.org/10.15388/Informatica.2014.33
[38] Rana, A.M.; Buthainah, A.A.A.; Fadhel, F.S. (2015). On fixed point theorem in fuzzy normed space, Journal of Al-Nahrain University, 18(4), 138-143. https://doi.org/10.22401/JNUS.18.4.19
[39] Romaguera, S.; Sapena, A.; Tirado, P. (2007). The Banach fixed point theorem in fuzzy quasimetric spaces with application to the domain of words, Topology and its Applications, 154(10), 2196-2203. https://doi.org/10.1016/j.topol.2006.09.018
[40] Secelean, N.A.; Mathew, S.; Wardowski, D. (2019). New fixed-point results in quasi-metric spaces and applications in fractals theory, Adv. Differ. Equ., 2019, 177. https://doi.org/10.1186/s13662-019-2119-z
[41] Saadati, R; Park, J.H. (2006). Intuitionistic fuzzy euclidean normed spaces, Communications in Mathematical Analysis, 1(2), 85-90.
[42] Saadati, R.; Park, C.; O'Regan, D.; Nadaban, S. (2021). n-Expansively super-homogeneous and (n, k)-contractively sub-homogeneous fuzzy control functions and stability results with numerical examples, Advances in Difference Equations, 2021:153. https://doi.org/10.1186/s13662-021-03287-y
[43] Saadati, R.; Vaezpour, S.M. (2005). Some results on fuzzy Banach spaces, Journal of Applied Mathematics and Computing, 17(1-2), 475-484. https://doi.org/10.1007/BF02936069
[44] Saadati, R.; Vaezpour, S.M.; Cho, Y.J. (2009). Quicksort algorithm: Application of a fixed point theorem in intuitionistic fuzzy quasi-metric spaces at a domain of words, Journal of Computational and Applied Mathematics, 228(1), 219-225. https://doi.org/10.1016/j.cam.2008.09.013
[45] Sadeqi, I.; Solaty kia, F. (2009). Fuzzy normed linear space and its topological structure, Chaos, Solitons and Fractals, 40(5), 2576-2589. https://doi.org/10.1016/j.chaos.2007.10.051
[46] Sadeqi, I.; Solaty kia, F. (2009). Some fixed point theorems in fuzzy reflexive Banach spaces, Chaos, Solitons and Fractals, 41, 2606-2612. https://doi.org/10.1016/j.chaos.2008.09.050
[47] Saheli, M. (2016). A contractive mapping on fuzzy normed linear spaces, Iranian Journal of Numerical Analysis and Optimization, 6(1), 121-136.
[48] Schweizer, A; Sklar, A. (1960). Statistical metric spaces, Pacific Journal of Mathematics, 10, 314-334. https://doi.org/10.2140/pjm.1960.10.313
[49] Xiao, J.-z.; Zhu, X.-h. (2004). Some basic theorems for linear operators between fuzzy normed spaces, Journal of University of Science and Technology of China, 34(4), 414-430.
[50] Xiao, J.-z.; Zhu, X.-h. (2004). Topological degree theory and fixed point theorems in fuzzy normed linear spaces, Fuzzy Sets and Systems, 147(3), 437-452. https://doi.org/10.1016/j.fss.2004.01.003
[51] Zadeh, L. A. (1965). Fuzzy Sets, Information and Control, 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
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