Approximate best proximity pair results on metric spaces using contraction operators
Main Article Content
Abstract
The aim of this paper is to prove some new approximate best proximity pair theorems on metric spaces using contraction mappings such as $P$-Bianchini contraction, $P-B$ contraction and so on. A few examples are provided to exemplify our findings. Finally, we discuss some applications that are related to the main results.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.
Supporting Agencies
References
[1] Balraj, D., Marudai, M., Mitrovic, Z. D., Ege, O., Piramanantham, V., Existence of best proximity points satisfying two constraint inequalities, Electranic Research Archive 28 (1) (2020), 549–557. Google Scholar
[2] Banach, S., Surles operations dans les ensembles abstract et leur application aux equation integrals, Fund.Math. 3 (1922), 133–181. Google Scholar
[3] Beer, G., Pai, D., Proximal maps, prox maps and coincidence points, Numerical Functional Analysis and Optimization 11 (5-6) (1990), 429–448. https://doi.org/10.1080/01630569008816382 Google Scholar
[4] Berinde, M., Approximate fixed point theorems, Stud. Univ. ”Babes-Bolyai”, Mathematica LI 1 (2006), 11–25. Google Scholar
[5] Berinde, V. , Iterative approximation of fixed points, Editura Efemeride, Baia Mare, 2002. Google Scholar
[6] Berinde, V., On the approximation of fixed points of weak contractive mappings, Carpathian J. Math. 19 (1) (2003), 7–22. Google Scholar
[7] Bianchini, R. M. T., Su un problema di S. Reich riguardante la teoria dei punti fissi, Bolletino U.M.I. 4 (5) (1972), 103–106. Google Scholar
[8] Chatterjea, S.K., Fixed point theorems , C.R. Acad.Bulg. Sci. 25 (1972), 727–730. Google Scholar
[9] Ciric, L. B., Generalized contractions and fixed point theorems, Publ.Inst.Math.(Bulgr). 12 (26) (1971), 19–26. Google Scholar
[10] Dass, B. K., Gupta, S., An extension of Banach contraction principle through rational expression, Communicated by F.C. Auluck, FNA., (1975). Google Scholar
[11] Dey, D., Saha, M., Approximate fixed point of Reich operator, Acta Mathematica Universitatis Comenianae, (2012). Google Scholar
[12] Eldred, A. A., Veeramani, P., Existence and convergence of best proximity points, Journal of Mathematical Analysis and Applications, 323 (2) (2006), 1001–1006. Google Scholar
[13] Fallahi, K., Rad, G. S., Fulga, A., Best proximity points for (φψ)-weak contractions and some applications, Filomat, 37 (6), (2023), 18351842. Google Scholar
[14] Gardasevic-Filipovic, M., Kukic, K., Gardasevic, D., Mitrovic, D., Some best proximity point results in the orthogonal O-complete b-metric-like spaces, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 58 (2023), 105115. Google Scholar
[15] Hardy, G. E., Rogers, T. D., A generalization of fixed point theorem of Reich, Can. Math. Bull. 16 (1973), 201–206. Google Scholar
[16] Iqbal, M., Batool, A., Ege, O., de la Sen, M., Fixed point of generalized weak contraction in b-metric spaces, Journal of Function Spaces, (2021), 8 pages, Article ID 2042162. Google Scholar
[17] Iqbal, M., Batool, A., Ege, O., de la Sen, M., Fixed point of almost contraction in b-metric spaces, Journal of Mathematics (2020), 3218134, 1–6. Google Scholar
[18] Jaggi, D. S., Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics, 8, (1977), 223–230. Google Scholar
[19] Kannan, R., Some results on fixed points, Bull Calcutta Math. Soc 60 (1968), 71–76. Google Scholar
[20] Kannan, R., Some results on fixed points, II. Am. Math.Mon. 76 (1969), 405–408. Google Scholar
[21] Khan, M. S., A fixed point theorems for metric spaces, Rendiconti Dell ’istituto di mathematica dell’ Universtia di tresti, 8, (1976), 69–72. Google Scholar
[22] Khuri, S. A., Louhichi, I., A novel Ishikawa-Green’s fixed point scheme for the solution of BVPs, Appl. Math. Lett. 82 (2018), 50–57. Google Scholar
[23] Kim, W. K., Lee, K. H., Corrigendum to existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl. 316 (2006), 433–446. Google Scholar
[24] Kirk, W. A., Reich, S., Veeramani, P., Proximinal retracts and best proximity pair theorems, Numerical Functional Analysis and Optimization 24 (7-8), (2003). Google Scholar
[25] Marudai, M., Bright V. S., Unique fixed point theorem weakly B-contractive mappings, Far East journal of Mathematical Sciences(FJMS) 98 (7) (2015), 897–914. Google Scholar
[26] Mohsenalhosseini, S.A.M., Approximate best proximate pairs in metric space for contraction maps, Fixed Point Theory 4 (2) (2014), 310–324. Google Scholar
[27] Mohsenalhosseini, S. A. M., Saheli. M., Some of family of contractive type maps and approximate common fixed point, J.Fixed Point Theory (2021), 2021:2. Google Scholar
[28] Mohsenalhosseini, S.A.M., Mzaheri, H.,Dehghan, M. A., Approximate best proximity pairs in metric space, Abstr. Appl. Anal., (2011), Article ID 596971. Google Scholar
[29] Nallaselli, G., Gnanaprakasam, A. J., Mani, G., Ege, O., Solving integral equations via admissible contraction mappings, Filomat, 36 (14), (2022), 4947–4961. Google Scholar
[30] Reich, S., Some remarks connecting contraction mappings, Can. Math. Bull., 14, (1971), 121– 124. Google Scholar
[31] Singer, I., Best approximation in normed linear spaces by elements of linear subspaces, Translated from the Romanian by Radu Georgescu. Die Grundlehren der mathematischen Wis- senschaften, Band 171 Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. Google Scholar
[32] Tis, S., Torre, A., Branzei, R., Approximate fixed point theorems, Libertas Mathematica 23 (2003), 35–39. Google Scholar
[33] Vetrivel, V., Veeramani, P., Bhattacharyya, P., Some extensions of Fan’s best approximation theorem, Numerical Functional Analysis and Optimization 13 (3-4), (1992), 397–402. Google Scholar
[34] Wlodarczyk, K., Plebaniak, R., Banach, A., Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spacand es, Nonlinear Analysis: Theory, Methods and Applications, 2009, 3332–3341. Google Scholar
[35] Wlodarczyk, K., Plebaniak, R., Obczyn ́ski, C., Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces, Nonlinear Analysis: Theory, Methods and Applications 72 (2) (2010), 794–805. Google Scholar
[36] Xu, X. B., A result on best proximity pair of two sets, Journal of Approximation Theory 54 (3) (1988), 322–325. Google Scholar
[37] Zamfirescu, T., Fixed Point theorems in metric spaces, Arch. Math.(Basel) 23 (1972), 292–298. Google Scholar