Contractive definitions and discontinuity at fixed point

Ravindra K Bisht, R. P. Pant

Abstract

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.

 


Keywords

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

Subject classification

47H09; 47H10.

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References

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

https://doi.org/10.1016/j.jmaa.2016.02.053

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

https://doi.org/10.1090/S0002-9939-1969-0239559-9

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.

https://doi.org/10.2307/2040075

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.

https://doi.org/10.1006/jmaa.1995.1299

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

https://doi.org/10.2307/2316437

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.

https://doi.org/10.1017/cbo9781139086639

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.

https://doi.org/10.1016/0022-247X(69)90031-6

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

https://doi.org/10.1006/jmaa.1999.6560

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.

https://doi.org/10.4153/CMB-1971-024-9

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

https://doi.org/10.1090/S0002-9947-1977-0433430-4

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

https://doi.org/10.1090/conm/072/956495

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