Weak \(\theta-\phi-\)contraction and discontinuity


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.


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].