# Weak $\theta-\phi-$contraction and discontinuity

Volume 10, Issue 5, pp 2318--2323

Publication Date: 2017-05-22

http://dx.doi.org/10.22436/jnsa.010.05.04

### Authors

Dingwei Zheng - College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, P. R. China.
Pei Wang - School of Mathematics and Information Science, Yulin Normal University, Yulin, Guangxi 537000, P. R. China.

### Abstract

In this paper, we introduce the notion of weak $\theta-\phi-$contraction ensuring a convergence of successive approximations but does not force the mapping to be continuous at the fixed point. Thus, we answer one more solution to the open question raised by Rhoades in [B. E. Rhoades, Fixed point theory Appl, Berkeley, CA, (1986), Contemp. Math., Amer. Math. Soc., Providence, RI, 72 (1988), 233–245].

### Keywords

Fixed point, discontinuity, weak $\theta-\phi-$contraction.

### References

[1] R. K. Bisht, R. P. Pant, A remark on discontinuity at fixed point, J. Math. Anal. Appl., 445 (2017), 1239–1241.
[2] F. Bojor, Fixed point theorems for Reich type contractions on metric spaces with a graph, Nonlinear Anal., 75 (2012), 3895–3901.
[3] L. Ćirić, M. Abbas, R. Saadati, N. Hussain, Common fixed points of almost generalized contractive mappings in ordered metric spaces, Appl. Math. Comput., 217 (2011), 5784–5789.
[4] M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 8 pages.
[5] R. Kannan, Some results on fixed points, II, Amer. Math. Monthly, 76 (1969), 405–408.
[6] R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl., 240 (1999), 284–289.
[7] H. Piri, P. Kumam, Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Point Theory Appl., 2014 (2014), 11 pages.
[8] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2003), 1435–1443.
[9] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), 257– 290.
[10] B. E. Rhoades, Contractive definitions and continuity, Fixed point theory and its applications, Berkeley, CA, (1986), Contemp. Math., Amer. Math. Soc., Providence, RI, 72 (1988), 233–245.
[11] A. F. Roldán-López de Hierro, N. Shahzad, New fixed point theorem under R-contractions, Fixed Point Theory Appl., 2015 (2015), 18 pages.
[12] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $\alpha\psi$ -contractive type mappings, Nonlinear Anal., 75 (2012), 2154–2165.
[13] T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317.
[14] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 6 pages.
[15] D.-W. Zheng, Z.-Y. Cai, P. Wang, New fixed point theorems for $\theta-\phi-$contraction in complete metric spaces, (Preprint).