The purpose of this paper is to introduce a new class of quasi-contractive operators and to show that the most used fixed point iterative methods, that is, the Picard and Mann iterations, are convergent to the unique fixed point. The comparison of these methods with respect to their convergence rate is obtained.

Let ∆ be the ball |x| < 1 in the complex vector space C, let f : ∆ → C be a holomorphic mapping and let M be a positive integer. Assume that the origin 0 = (0, . . . , 0) is an isolated fixed point of both f and the M -th iteration f of f . Then for each factor m of M, the origin is again an isolated fixed point of f and the fixed point index μfm(0) of f m at the origin is well defined, and so ...

In this lecture, we discuss splitting algorithms for convex minimization problems with objective given by the sum of two nonsmooth functions. We start with the fixed point property of such problems and derive a general scheme of splitting algorithm based on fixed point iteration. This covers Douglas-Rachford splitting and Peaceman-Rachford splitting algorithms. We also discuss the convergence r...

In this paper, we prove the stability in random normed spaces via fixed point method for the functional equation f(x+ 2y)− 2f(x+ y) + 2f(x− y)− f(x− 2y) = 0.