Teorema Titik Tetap Kontraksi Tipe Banach–Kannan dan Konvergensi Iterasi General Picard–Mann

Abstract

Fixed point theory plays a crucial role in solving problems modeled by differential
equations and nonlinear equations. In many problems, finding exact fixed points analytically
is difficult, thus numerical algorithms are needed to approximate them. This study investigates
the convergence of the General Picard-Mann (GPM) iteration scheme for Banach-Kannan type
contraction mappings in Banach spaces. The GPM iteration scheme combines Picard and Mann
iterations, generalized by performing k iterations. The main results show that Banach-Kannan type
contraction mappings on complete metric spaces have a unique fixed point. Moreover, it is proved
that sequences generated by the GPM iteration scheme converge to this fixed point in Banach spaces
under the assumption α + 2β < 1. This study complements previous works by extending the class of
mappings that can be approximated using the GPM iteration.