Some fixed point results via γ-contraction in non-Archimedean fuzzy metric spaces

Year 2022, , 659 - 666, 01.06.2022

Müzeyyen Sangurlu Sezen

https://doi.org/10.35378/gujs.828180

Abstract

As other authors have been very interested in the topic of fixed points, we have obtained some results in this study that emphasize the importance of the fixed point theory. Kannan described a more general contraction than the Banach contraction that took its name and later Reich generalized this contraction further in metric spaces. In this paper, we have introduced some new contractions called Reich type γ-contraction and Kannan type γ-contraction which are generalization of γ-contraction and we have obtained some fixed point results for Reich type γ-contraction in non-Archimedean fuzzy metric spaces. We have presented a result about Kannan type-contraction. Furtermore, we have established an example about our main result.

Keywords

Reich type γ-contraction, Kannan type γ-contraction, Fuzzy metric

References

• [1] Banach, S., “Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales”, Fundamenta Mathematicae, 3: 133-181, (1922).
• [2] Kannan, R., “Some results on fixed points”, Bulletin of Calcutta Mathematical Society, 60: 71-76, (1968).
• [3] Reich, R., “Some remarks concernin contraction mappings”, Canadian Mathematical Bulletin, 14: 121–124, (1971).
• [4] Deng, Z., “Fuzzy pseudometric spaces”, Journal of Mathematical Analysis and Applications, 86: 74-95, (1922).
• [5] Salimi, P., Vetro, C. and Vetro, P., “Some new fixed point results in non-Archimedean fuzzy metric spaces”, Nonlinear Analysis: Modelling and Control, 18(3): 344–358, (2013).
• [6] George, A. and Veeramani P., “On some results in fuzzy metric spaces”, Fuzzy Sets and Systems, 64, (1994), 395-399.
• [7] Istrăţescu, V., “An introduction to theory of probabilistic metric spaces with applications”, Ed, Tehnică, Bucureşti, in Romanian, (1974).
• [8] Grabiec, M., “Fixed points in fuzzy metric spaces”, Fuzzy Sets and Systems, 27: 385-389, (1988).
• [9] Gregori, V. and Sapena, A., “On fixed-point theorems in fuzzy metric spaces”, Fuzzy Sets Systems, 125: 245-252, (2002).
• [10] Vetro, C. and Vetro, P., “Common fixed points for discontinuous mappings in fuzzy metric spaces”, Rendiconti del Circolo Matematico di Palermo, 57: 295-303, (2008).
• [11] Kramosil, I. and Michalek, J., “Fuzzy metric and statistical metric spaces”, Kybernetika, 11: 336-344, (1975).
• [12] Altun, I. and Mihet, D., “Ordered non-Archimedean fuzzy metric spaces and some fixed point results”, Fixed Point Theory and Applications, 2010: (2010).
• [13] Hussain, N., Hezarjaribi M. and Salimi, P., “Suzuki type theorems in triangular and non-Archimedean fuzzy metric spaces with application”, Fixed Point Theory and Applications, 2015: 134, (2015). [14] Cho, Y. J., “Fixed points in fuzzy metric spaces”, Journal of Fuzzy Mathematics, 5(4): 949-962, (1997).
• [15] Schweizer, B. and Sklar, A., “Statistical metric spaces”, Pacific Journal of Mathematics, 10: 385-389, (1960). [16] Schweizer, B. and Sklar, A., “Probabilistic Metric Spaces”, North-Holland, Amsterdam, (1983).
• [17] Özgür, N. Y. and Taş, N., “Some fixed-circle theorems on metric spaces”, Bulletin of the Malaysian Mathematical Sciences Society, 42: 1433-1449, (2019).
• [18] Taş, N., “Bilateral type solutions to the fixed-circle problem with recti-ed linear units application”, Turkish Journal of Mathematics, 44: 1330- 1344, (2020).
• [19] Aydi, H., Taş, N., Özgür, N. Y. and Mlaiki, N., “Fixed-discs in rectangular metric spaces”, Symmetry, 11(2): 294, (2019).
• [20] Sahin, H., “Best proximity point theory on vector metric spaces”, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70 (1): 130-142, (2021).
• [21] Altun, I., Sahin, H. and Turkoglu, D., “Caristi-Type fixed point theorems and some generalizations on M-metric space”, Bulletin of the Malaysian Mathematical Sciences Society, 43: 2647-2657, (2020).
• [22] Sezen, M. S., “Fixed point theorems for new type contractive mappings”, Journal of Function Spaces, Article ID: 2153563, (2019).