Best Proximity Point Theorems in Non-Archimedean Menger Probabilistic Spaces
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Authors: A. A. KARAASLAN AND V. KARAKAYA
DOI: 10.46793/KgJMat2403.331K
Abstract:
In this work, we prove best proximity point theorems for γ-contractions with conditions the weak P-property in non-Archimedean Menger probabilistic metric spaces. We give the notion of γ- proximal contractions of first and second type in non-Archimedean Menger probabilistic metric spaces and also we establish best proximity point theorems for these proximal contractions. Lastly, we complete our study by giving examples that support our results.
Keywords:
Fixed point, best proximity point, γ-contraction, non-Archimedean Menger probabilistic metric space
References:
[1] F. Ali, J. Ali and I. Uddin, A novel approach for the solution of BVPs via Green’s function and fixed point iterative method, J. Appl. Math. Comput. 60 (2020), 1–15. https://doi.org/10.1007/s12190-020-01431-7
[2] J. Ali, M. Imdad and D. Bahuguna, Common fixed point theorems in Menger spaces with common property (E.A), Comput. Math. Appl. 60(12) (2010), 3152–3159. https://doi.org/10.1016/j.camwa.2010.10.020
[3] G. A. Anastassiou, Y. J. Cho, R. Saadati and Y. O. Yang, Common best proximity points for proximally commuting mappings in non-Archimedean pm-spaces, J. Comput. Anal. Appl. 20(6) (2016), 1021–1030.
[4] M. Asim, I. Uddin and M. Imdad, Fixed point results in Mv-metrices with an application, J. Inequal. Appl. 280 (2019), 1–19. https://doi.org/10.1186/s13660-019-2223-3
[5] S. S. Chang, On some fixed point theorems in probabilistic metric space and its application, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 63 (1983), 463–474.
[6] S. S. Chang, On some fixed point theorems for generalized Meir-Keeler type mappings, Journal of Sichuan University (Natural Science Edition) (1983), 17–20.
[7] L. Ćirić, Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces, Nonlinear Anal. 72(3-4) (2010), 2009–2018. https://doi.org/10.1016/j.na.2009.10.001
[8] P. Das, S. Som, S. Ghosal and V. Karakaya, A notion of αβ-statistical convergence of order γ in probability, Kragujevac J. Math. 42(1) (2018), 51–67.
[9] K. Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240.
[10] O. Hadžić and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
[11] M. Imdad, J. Ali and M. Tanveer, Coincidence and common fixed point theorems for nonlinear contractions in Menger pm spaces, Chaos Solitons Fractals 42(5) (2009), 3121–3129. https://doi.org/10.1016/j.chaos.2009.04.017
[12] J. Jachymski, On probabilistic ϕ-contractions on Menger spaces, Nonlinear Anal. 73(7) (2010), 2199–2203. https://doi.org/10.1016/j.na.2010.05.046
[13] M. Jamali and S. M. Vaezpour, Best proximity point for certain nonlinear contractions in Menger probabilistic metric spaces, J. Adv. Math. Stud. 9(2) (2016), 338–347.
[14] M. Jamali and S. M. Vaezpour, Existence and convergence of best proximity points in new type of contractions in Menger probabilistic metric spaces, J. Nonlinear Funct. Anal. 2016 (2016), Article ID 36.
[15] K. Menger, Statistical metrics, Proc. Amer. Math. Soc. 28(12) (1942), 535–537. https://doi.org/10.1073/pnas.28.12.535
[16] D. Mihet, Fixed point theorems in probabilistic metric spaces, Chaos Solitions Fractals 41(2) (2009), 1014–1019. https://doi.org/10.1016/j.chaos.2008.04.030
[17] D. Mihet, Altering distances in probabilistic Menger spaces, Nonlinear Anal. 71(7–8) (2009), 2734–2738. https://doi.org/10.1016/j.na.2009.01.107
[18] D. O’Regan and R. Saadati, Nonlinear contraction theorems in probabilistic spaces, Appl. Math. Comput. 195(1) (2008), 86–93. https://doi.org/10.1016/j.amc.2007.04.070
[19] R. Saadati, Best proximity point theorems for probabilistic proximal cyclic contraction with applications in nonlinear programming, Fixed Point Theory and Algorithms for Sciences and Engineering 2015 (2015), Article ID 79. https://doi.org/10.1186/s13663-015-0330-5
[20] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland series in Probability and Applied Mathematics, North-Holland Publishing Co., New York, 1983.
[21] V. M. Sehgal, Some fixed point theorems in functional analysis and probability, Ph.D. Thesis, Wayne State University, 1966.
[22] V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Mathematical Systems Theory 6 (1972), 97–102.
[23] M. Sangurlu Sezen, Fixed point theorems for new type contractive mappings, J. Funct. Spaces 2019 (2019). https://doi.org/10.1155/2019/2153563
[24] Y. Su and J. Zhang, Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces, Fixed Point Theory and Algorithms for Sciences and Engineering 2014 (2014), Article ID 170.
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