Startseite Best approximation and fixed points for rational-type contraction mappings
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Best approximation and fixed points for rational-type contraction mappings

  • Sumit Chandok ORCID logo EMAIL logo
Veröffentlicht/Copyright: 20. November 2019

Abstract

In this paper, we prove a fixed point theorem for a rational type contraction mapping in the frame work of metric spaces. Also, we extend Brosowski–Meinardus type results on invariant approximation for such class of contraction mappings. The results proved extend some of the known results existing in the literature.

MSC 2010: 41A50; 47H10; 54H25

Funding statement: The author is thankful to Thapar Institute of Engineering & Technology for SEED grant.

Acknowledgements

The author is thankful to the learned referees for valuable suggestions.

References

[1] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 29 (2002), no. 9, 531–536. 10.1155/S0161171202007524Suche in Google Scholar

[2] B. Brosowski, Fixpunktsätze in der Approximationstheorie, Mathematica (Cluj) 11(34) (1969), 195–220. Suche in Google Scholar

[3] S. Chandok, Some common fixed point theorems for generalized nonlinear contractive mappings, Comput. Math. Appl. 62 (2011), no. 10, 3692–3699. 10.1016/j.camwa.2011.09.009Suche in Google Scholar

[4] S. Chandok, Common fixed points, invariant approximation and generalized weak contractions, Int. J. Math. Math. Sci. (2012), Article ID 102980. 10.1155/2012/102980Suche in Google Scholar

[5] S. Chandok, Common fixed points and invariant approximation for noncommuting asymptotic weak contractions, J. Adv. Math. Stud. 6 (2013), no. 1, 12–18. Suche in Google Scholar

[6] S. Chandok, J. Liang and D. O’Regan, Common fixed points and invariant approximations for noncommuting contraction mappings in strongly convex metric spaces, J. Nonlinear Convex Anal. 15 (2014), no. 6, 1113–1123. Suche in Google Scholar

[7] S. Chandok and T. D. Narang, Common fixed points of nonexpansive mappings with applications to best and best simultaneous approximation, J. Appl. Anal. 18 (2012), no. 1, 33–46. 10.1515/jaa-2012-0002Suche in Google Scholar

[8] S. Chandok and T. D. Narang, Common fixed points with applications to best simultaneous approximations, Anal. Theory Appl. 28 (2012), 1–12. 10.4208/ata.2012.v28.n1.1Suche in Google Scholar

[9] S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 727–730. 10.1501/Commua1_0000000548Suche in Google Scholar

[10] B. Fisher, A fixed-point theorem for compact metric spaces, Publ. Math. Debrecen 25 (1978), no. 3–4, 193–194. 10.5486/PMD.1978.25.3-4.01Suche in Google Scholar

[11] C. Franchetti and M. Furi, Some characteristic properties of real Hilbert spaces, Rev. Roumaine Math. Pures. Appl. 17 (1972), 1045–1048. Suche in Google Scholar

[12] M. I. Ganzburg, Invariance theorems in approximation theory and their applications, Constr. Approx. 27 (2008), no. 3, 289–321. 10.1007/s00365-006-0670-3Suche in Google Scholar

[13] M. I. Ganzburg and S. A. Pichugov, Invariance of the elements of best approximation and a theorem of Glaeser, Ukrainian Math. J. 33 (1982), 508–510. 10.1007/BF01085888Suche in Google Scholar

[14] D. S. Jaggi, Some unique fixed point theorems, Indian J. Pure Appl. Math. 8 (1977), no. 2, 223–230. Suche in Google Scholar

[15] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76. Suche in Google Scholar

[16] G. Meinardus, Invarianz bei linearen Approximationen, Arch. Ration. Mech. Anal. 14 (1963), 301–303. 10.1007/BF00250708Suche in Google Scholar

[17] R. N. Mukherjee and T. Som, A note on an application of a fixed point theorem in approximation theory, Indian J. Pure Appl. Math. 16 (1985), no. 3, 243–244. Suche in Google Scholar

[18] R. N. Mukherjee and V. Varma, Best approximations and fixed points of nonexpansive maps, Bull. Calcutta Math. Soc. 81 (1989), no. 3, 191–196. Suche in Google Scholar

[19] T. D. Narang, On best coapproximation in normed linear spaces, Rocky Mountain J. Math. 22 (1992), no. 1, 265–287. 10.1216/rmjm/1181072810Suche in Google Scholar

[20] T. D. Narang and S. Chandok, Fixed points and best approximation in metric spaces, Indian J. Math. 51 (2009), no. 2, 293–303. Suche in Google Scholar

[21] T. D. Narang and S. Chandok, Fixed points of quasi-nonexpansive mappings and best approximation, Selçuk J. Appl. Math. 10 (2009), no. 2, 75–80. Suche in Google Scholar

[22] T. D. Narang and S. Chandok, On ϵ-approximation and fixed points of nonexpansive mappings in metric spaces, Mat. Vesnik 61 (2009), no. 2, 165–171. Suche in Google Scholar

[23] G. S. Rao and S. A. Mariadoss, Applications of fixed point theorems to best approximations, Serdica 9 (1983), no. 3, 244–248. Suche in Google Scholar

[24] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14 (1971), 121–124. 10.4153/CMB-1971-024-9Suche in Google Scholar

[25] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Translated from the Romanian by Radu Georgescu, Grundlehren Math. Wiss. 171, Springer, Berlin, 1970. 10.1007/978-3-662-41583-2Suche in Google Scholar

[26] S. P. Singh, An application of a fixed-point theorem to approximation theory, J. Approx. Theory 25 (1979), no. 1, 89–90. 10.1016/0021-9045(79)90036-4Suche in Google Scholar

[27] S. P. Singh, Application of fixed point theorems in approximation theory, Applied Nonlinear Analyis, Academic Press, New York (1979), 389–397. 10.1016/B978-0-12-434180-7.50038-3Suche in Google Scholar

Received: 2018-01-28
Accepted: 2019-03-14
Published Online: 2019-11-20
Published in Print: 2019-12-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 21.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jaa-2019-0021/html
Button zum nach oben scrollen