Abstract
In this note, we show that the assumption of continuity considered in the recent result of Miculescu and Mihail (J Fixed Point Theory Appl, doi:10.1007/s11784-017-0411-7, 2017) can be relaxed further. We also observe that the power convex contraction introduced by Miculescu and Mihail (see condition (1.1)) provides one more solution to the open question of Rhoades (Contemp Math 72:233–245, 1988] regarding existence of a contractive definition which is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point.
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Miculescu, R., Mihail, A.: A generalization of Matkowski s fixed point theorem and Istrăţescu s fixed point theorem concerning convex contractions. J. Fixed Point Theory Appl. (2017). doi:10.1007/s11784-017-0411-7
Pant, R.P.: Discontinuity and fixed points, J. Math. Anal. Appl. \(\bf {240}\), 284–289 (1999)
Rhoades,B.E.: Contractive definitions and continuity, Contemp. Math. \(\bf {72}\), 233–245 (1988)
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Bisht, R.K. A remark on the result of Radu Miculescu and Alexandru Mihail. J. Fixed Point Theory Appl. 19, 2437–2439 (2017). https://doi.org/10.1007/s11784-017-0433-1
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DOI: https://doi.org/10.1007/s11784-017-0433-1