Document Type : Research Paper
Authors
Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.
Abstract
In this paper, we generalize the concepts of weakly Kannan, weakly Chatterjea and weakly Zamfirescu for fuzzy metric spaces. Also, we investigate Banach's fixed point theorem for the mentioned classes of functions in these spaces. Moreover, we show that the class of weakly Kannan and weakly Chatterjea maps are subclasses of the class of weakly Zamfirescu maps.
Keywords
[1] A.H. Ansari and A. Razani, Some fixed point theorems for C-class functions in b-metric spaces, Sahand Commun. Math. Anal., 10 (2018), pp. 85-96.
[2] D. Ariza-Ruiz and A. Jiménez-Melado, A continuation method for weakly Kannan maps, Fixed Point Theory Appl., 2010 (2010), pp. 1-12,
[3] D. Ariza-Ruiza, A. Jiménez-Meladob and G. Lopez-Acedo, A fixed point theorem for weakly Zamfirescu mappings, Nonlinear Analysis, 74 (2011), pp. 1628-1640.
[4] S.K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci., 25 (1972), pp. 727-730.
[5] L.B. Ciric, Generalized contractions and fixed point theorem, Publ. Inst. Math., 12 (1971), pp. 19-26.
[6] J. Dugundji and A. Granas, Weakly contractive maps and elementary domain invariance theorem, Bull. Soc. Math. Grece (N.S.), 19 (1978), pp. 141-151.
[7] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), pp. 395-399.
[8] M.B. Ghaemi and A. Razani, Fixed and periodic points in the probabilistic normed and metric spaces, Chaos Solitons Fractals, 28 (2006), pp. 1181-1187.
[9] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst., 27 (1988), pp. 385-389.
[10] V. Gregori, J. Minana and S. Morillas, On completable fuzzy metric spaces, Fuzzy Sets and Systems, 267 (2015), pp. 133-139
[11] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), pp. 71-76.
[12] E. Rakotch, A note on contractive mappings, Proceedings of the American Mathematical Society, 13 (1962), pp. 459-465.
[13] A. Razani, A contraction theorem in fuzzy metric spaces, Fixed Point Theory Appl., 3 (2005), pp. 257-265.
[14] A. Razani, A fixed point theorem in the Menger probabilistic metric space, New Zealand J. Math., 35 (2006), pp. 109-114.
[15] A. Razani, Existence of fixed point for the nonexpansive mapping of intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 30 (2006), pp. 367-373.
[16] A. Razani, Results in fixed point theory, Andisheh Zarin publisher, Qazvin, August 2010.
[17] A. Razani and R. Moradi, Fixed point theory in modular space, Saieh Ghostar publisher, Qazvin, April 2006.
[18] A. Razani and M. Shirdaryazdi, A common fixed point theorem of compatible maps in Menger space, Chaos Solitons Fractals, 37 (2007), pp. 26-34.
[19] A. Razani and M. Shirdaryazdi, Some results on fixed points in the fuzzy metric space, J. Appl. Math. Comput., 20 (2006), pp. 401-408.
[20] R. Saadati, A. Razani and H. Adibi, A Common Fixed Point Theorem in L-fuzzy metric spaces, Chaos Solitons Fractals, 33 (2007), pp. 358-363.
[21] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math., 23 (1972), pp. 292-298.
[2] D. Ariza-Ruiz and A. Jiménez-Melado, A continuation method for weakly Kannan maps, Fixed Point Theory Appl., 2010 (2010), pp. 1-12,
[3] D. Ariza-Ruiza, A. Jiménez-Meladob and G. Lopez-Acedo, A fixed point theorem for weakly Zamfirescu mappings, Nonlinear Analysis, 74 (2011), pp. 1628-1640.
[4] S.K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci., 25 (1972), pp. 727-730.
[5] L.B. Ciric, Generalized contractions and fixed point theorem, Publ. Inst. Math., 12 (1971), pp. 19-26.
[6] J. Dugundji and A. Granas, Weakly contractive maps and elementary domain invariance theorem, Bull. Soc. Math. Grece (N.S.), 19 (1978), pp. 141-151.
[7] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), pp. 395-399.
[8] M.B. Ghaemi and A. Razani, Fixed and periodic points in the probabilistic normed and metric spaces, Chaos Solitons Fractals, 28 (2006), pp. 1181-1187.
[9] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst., 27 (1988), pp. 385-389.
[10] V. Gregori, J. Minana and S. Morillas, On completable fuzzy metric spaces, Fuzzy Sets and Systems, 267 (2015), pp. 133-139
[11] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), pp. 71-76.
[12] E. Rakotch, A note on contractive mappings, Proceedings of the American Mathematical Society, 13 (1962), pp. 459-465.
[13] A. Razani, A contraction theorem in fuzzy metric spaces, Fixed Point Theory Appl., 3 (2005), pp. 257-265.
[14] A. Razani, A fixed point theorem in the Menger probabilistic metric space, New Zealand J. Math., 35 (2006), pp. 109-114.
[15] A. Razani, Existence of fixed point for the nonexpansive mapping of intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 30 (2006), pp. 367-373.
[16] A. Razani, Results in fixed point theory, Andisheh Zarin publisher, Qazvin, August 2010.
[17] A. Razani and R. Moradi, Fixed point theory in modular space, Saieh Ghostar publisher, Qazvin, April 2006.
[18] A. Razani and M. Shirdaryazdi, A common fixed point theorem of compatible maps in Menger space, Chaos Solitons Fractals, 37 (2007), pp. 26-34.
[19] A. Razani and M. Shirdaryazdi, Some results on fixed points in the fuzzy metric space, J. Appl. Math. Comput., 20 (2006), pp. 401-408.
[20] R. Saadati, A. Razani and H. Adibi, A Common Fixed Point Theorem in L-fuzzy metric spaces, Chaos Solitons Fractals, 33 (2007), pp. 358-363.
[21] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math., 23 (1972), pp. 292-298.
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