Iteration of Operators with Contractive Mutual Relations of Kannan Type

Abstract

Inspired by the ideas of R. Kannan, we define the new concepts of mutual Kannan contractivity and mutual contractivity between two self-maps on a metric space that generalize the concepts of the Kannan map and contraction. We give some examples and deduce the properties of the operators satisfying this type of condition; in particular, we study the case where the space is normed, and the maps are linear. Then we generalize some theorems proposed by this author on the existence of a fixed point of one operator or a common fixed point for two operators. Our results first prove the existence of a common fixed point of a set of self-maps of any cardinal number (countable or uncountable) satisfying the conditions of Kannan type in metric spaces. The same is proved for a set of maps satisfying the mutual relations of classical contractivity. We prove in both cases the convergence of iterative schemes involving operators with mutual relations of contractivity, proposing sufficient conditions for the iteration of the operators on any element of the space to converge to the common fixed point when a different operator is taken in each step. The results obtained are applied to operators acting on real functions, coming from the fractal convolution with the null function.