Contractive definitions and discontinuity at fixed point

Ravindra K. Bisht; R. P. Pant

doi:10.4995/agt.2017.6713

Authors

Ravindra K. Bisht National Defence Academy
R. P. Pant Kumaun University Nainital

DOI:

https://doi.org/10.4995/agt.2017.6713

Keywords:

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

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.

Downloads

Download data is not yet available.

Author Biographies

Ravindra K. Bisht, National Defence Academy

Department of Mathematics

R. P. Pant, Kumaun University Nainital

Department of Mathematics

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'Â’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