Hybrid Picard-Mann Algorithm for Fixed Points of Kannan Contraction Mappings: Convergence Analysis and Numerical Tests

Abstract

This study examines a four-step hybrid Picard-Mann iteration for Kannan contraction mappings. Unlike Banach contractions, Kannan maps need not be continuous. The method is tested on a discontinuous example on $[0,1]$ with constant parameters $\alpha=0.5$, $\beta=0.33$, $\gamma=0.25$ and stopping tolerance $10^{-7}$. Starting from $x_0=0.8$, the hybrid scheme finds the fixed point $x^*=0$ in 9 iterations, while Picard requires 12 iterations, Mann 25, and Ishikawa 23. Thus the hybrid method is about 25\% faster than Picard and more than 60\% faster than Mann or Ishikawa. Stability is further examined by changing the initial guess to $0.5$ and $0.2$, where the hybrid method converges in 6 iterations. These results demonstrate that the hybrid Picard-Mann iteration performs well for a discontinuous Kannan contraction.