<i>θ</i>*-Weak Contractions and Discontinuity at the Fixed Point with Applications to Matrix and Integral Equations

Abstract

In this paper, the notion of θ*-weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan and Rhoades on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of non-linear matrix equations and integral equations of Volterra type as well.

Keywords

• θ*-weak contraction
• fixed point
• discontinuity at the fixed point
• property P
• matrix equation
• integral equation