Contractive definitions and discontinuity at fixed point

Ravindra K Bisht, R. P. Pant


In this paper, we investigate some contractive definitions which are strong enough to generate a fixed point that do not force the mapping to be continuous at the fixed point. Finally, we obtain a fixed point theorem for generalized nonexpansive mappings in metric spaces by employing Meir-Keeler type conditions.



Fixed point; $(\epsilon-\delta)$ contractions; power contraction; orbital continuity

Subject classification

47H09; 47H10.

Full Text:



R. K. Bisht and R. P. Pant, A remark on discontinuity at fixed point, J. Math. Anal. Appl. 445 (2017), 1239-1242.

D. W. Boyd and J. S. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464.

Lj. B. Ciric, On contraction type mappings, Math. Balkanica 1 (1971), 52-57.

Lj. Ciric, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc. 45, no. 2 (1974), 267-273.

Lj. B. Ciric, Fixed points of weakly contraction mappings, Publications de L'Institut Mathematique 20 (34) (1976), 79-84.

Lj. B. Ciric, A new fixed-point theorem for contractive mapping, Publications de l'Institut Mathematique 30 (44) (1981), 25-27.

J. Jachymski, Common fixed point theorems for some families of maps, Indian J. Pure Appl. Math. 25 (1994), 925-937.

J. Jachymski, Equivalent conditions and Meir-Keeler type theorems, J. Math. Anal. Appl. 194 (1995), 293-303.

R. Kannan, Some results on fixed points-II, Amer. Math. Mon. 76 (1969) 405-408.

M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, in: Encyclopedia of Mathematics and its Applications, Vol. 32, Cambridge Univ. Press, Cambridge, UK, 1990.

J. Matkowski, Integrable solutions of functional equations, Diss. Math. 127 (1975) 1-68.

A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329.

R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1999), 284-289.

R. P. Pant, Non-expansive mappings and Meir-Keeler type conditions, J. Indian Math. Soc. 71 (2004), 239-244.

M. Pacurar, Iterative Methods for Fixed Point Approximation. Risoprint, Cluj-Napoca, 2010.

S. Reich, Some remarks concerning contraction mappings. Canad. Math. Bull. 14 (1971), 121-124.

B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 257-290.

B. E. Rhoades, Contractive definitions and continuity, Contemporary Mathematics 72 (1988), 233-245.

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