Picard-Tikhonov-Mann (PTM) iteration process with applications

In mathematics, a fixed point of a function is the process of finding a solution to an equation that can be written as Υx = x for a suitable function Υ. Fixed point theory has many applications. For instance, in compilers (computer programs), fixed point computations are used for program analysis. An example of this is the data-flow analysis that is often required to optimize code. The vector of PageRank values of web pages is the fixed point of a derived linear transformation of the world wide web connectivity structure. Our main focus in this paper is to consider a novel algorithm for finding the fixed point in some abstract spaces. Some applications are presented regarding split feasibility/constrained minimization problems/signal enhancement. Furthermore, we perform the numerical illustrations to investigate the basic techniques using Matlab R2016a. The proposed novel algorithm demonstrates strong convergence to fixed points of the considered mapping, with practical applications in solving split feasibility and constrained minimization problems. Numerical results confirm the effectiveness of the method in signal enhancement tasks. This approach offers promising potential for further research in abstract metric spaces and iterative solution techniques.