An observation on F-weak contractions and discontinuity at the fixed point with an application

In this paper, we have some observations on F-weak contractions due to Wardowski and Van Dung (Demonstr Math 47(1):146-155, 2014). Our observations lead us to introduce the notion of F*-weak contractions and utilize the same to prove some fixed point results. The proven results give an affirmative answer to certain open questions raised by Kannan (Bull Calcutta Math Soc 60:71-76, 1968) and Rhoades (Contemp. Math. 72:233-245, 1988) on the existence of contractive definitions not forcing the continuity at the fixed point. Some illustrative examples are also given. As an application, we investigate the existence and uniqueness of the solution of an integral equation of Volterra type.