• Sumit Chandok School of Mathematics, Thapar Institute of Engineering Technology, Patiala-147004, India
  • T. D. Narang Department of Mathematics, Guru Nanak Dev University, Amritsar-143005, India
Keywords: ε-simultaneous approximatively compact set, Starshaped set, Best approximation, Best simultaneous approximation, ε-simultaneous approximation


For the set of ε-simultaneous approximation and ε-simultaneous coapproximation, we derive certain BrosowskiMeinardus type invariant point results in this paper. As a consequence, some results on ε-approximation, εcoapproximation, best approximation, and best coapproximation are also deduced.


Brosowski, B. 1969, Fixpunktsätze in der Approximationstheorie, Mathematica (Cluj), 11, pp. 195-200.

Browder, F. E. & Petryshyn, W. V. 1966, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc., 72, pp. 571-5775.

Chandok, S. 2019, Best approximation and fixed points for rational-type contraction mappings, J. Appl. Anal., 25(2), pp. 205-209. https://doi.org/10.1515/jaa-2019-0021

Chandok, S. & Narang, T. 2011a, Invariant points and ε-simultaneous approximation, Internat. J. Math. Math. Sci., 2011(579819). doi:10.1155/2011/579819

Chandok, S. & Narang, T. 2011b, ε-simultaneous approximation and invariant points, Bull. Belgian Math. Soc., 18, pp. 821-834. https://doi.org/10.36045/bbms/1323787169

Chandok, S. & Narang, T. D. 2012a, Common fixed points of nonexpansive mappings with applications to best and best simultaneous approximation, J. Appl. Anal., 18, pp. 33-46.

Chandok, S. & Narang, T. D. 2012b, Common fixed points with applications to best simultaneous approximations, Anal. Theory Appl., 28(1), pp. 1-12. https://doi.org/10.1515/jaa-2012-0002

Chandok, S. & Narang, T. D. 2013, Some fixed point theorem for generalized asymptotically nonexpansive mapping, Tamkang J. Math., 44(1), pp. 23-29. https://doi.org/10.5556/j.tkjm.44.2013.898

Guay, M. D., Singh, K. L. & Whitfield, J. H. M. 1982, Fixed point theorems for nonexpansive mappings in convex metric spaces, Proc. Conference on nonlinear analysis (Ed. S.P. Singh and J.H. Bury) Marcel Dekker, 80, pp. 179-189.

Khan, A. R. & Akbar, F. 2009a, Best simultaneous approximations, asymptotically nonexpansive mappings and variational inequalities in Banach spaces, J. Math. Anal. Appl., 354, pp. 469-477. https://doi.org/10.1016/j.jmaa.2009.01.007

Khan, A. R. & Akbar, F. 2009b, Common fixed points from best simultaneous approximation, Taiwanese J. Math., 13, pp. 1379-1386. doi:10.11650/twjm/1500405546

Meinardus, G. 1963, Invarianz bei linearen Approximationen, Arch. Rational Mech. Anal., 14, pp. 301-303.

Mukherjee, R. N. & Som, T. 1985, A note on application of a fixed point theorem in approximation theory, Indian J. Pure Appl. Math., 16, pp. 243-244.

Mukherjee, R. N. & Verma, V. 1989, Best approximations and fixed points of nonexpansive maps, Bull. Cal. Math. Soc., 81, pp. 191-196.

Narang, T. D. & Chandok, S. 2009a, Fixed points and best approximation in metric spaces, Indian J. Math., 51, pp. 293-303.

Narang, T. D. & Chandok, S. 2009b, Fixed points of quasinonexpansive mappings and best approximation, Selcuk J. Appl. Math., 10, pp. 75-80.

Narang, T. D. & Chandok, S. 2009c, On ε-approximation and fixed points of nonexpansive mappings in metric spaces, Mat. Vesnik, 61, pp. 165-171.

Rao, G. S. & Mariadoss, S. A. 1983, Applications of fixed point theorems to best approximations, Serdica-Bulgaricae Math. Publ., 9, pp. 244-248.

Singh, S. P. 1979a, An application of a fixed-point theorem to approximation theory, J. Approx. Theory, 25, pp. 89-90.

Singh, S. P. 1979b, Application of fixed point theorems in approximation theory, Appl. Nonlinear Anal. (Ed. V. Lakshmikantham), Academic Press, New York, pp. 389-397.

Takahashi, W. 1970, A convexity in metric space and nonexpansive mappings I, Kodai Math. Sem. Rep., 22, pp. 142-149.

Original Scientific Paper