In this paper, we introduce the new concept of mutual Reich contraction that involves a pair of operators acting on a distance space. We chose the framework of strong b-metric spaces (generalizing the standard metric spaces) in order to add a more extended underlying structure. We provide sufficient conditions for two mutually Reich contractive maps in order to have a common fixed point. The result is extended to a family of operators of any cardinality. The dynamics of iterative discrete systems involving this type of self-maps is studied. In the case of normed spaces, we establish some relations between mutual Reich contractivity and classical contractivity for linear operators. Then, we introduce the new concept of mutual functional contractivity that generalizes the concept of classical Banach contraction, and perform a similar study to the Reich case. We also establish some relations between mutual functional contractions and Banach contractivity in the framework of quasinormed spaces and linear mappings. Lastly, we apply the obtained results to convolutional operators that had been defined by the first author acting on Bochner spaces of integrable Banach-valued curves.

Fractal interpolation provides an efficient way to describe a smooth or non-smooth structure associated with nature and scientific data. The aim of this paper is to introduce a bivariate Hermite fractal interpolation formula that generalizes the classical Hermite interpolation formula for two variables. It is shown here that the proposed Hermite fractal interpolation function and its derivatives of all orders are good approximations of original function even if the partial derivatives of original function are non-smooth in nature.

This paper deals with (1) the construction of a recurrent fractal interpolation function (RFIF) corresponding to a weak iterated function system (IFS), and (2) the estimation of the lower and upper bounds of its box dimension. RFIF is a generalization of fractal interpolation functions, which provides a more efficient way to reconstruct data that presents self-affinity in a piecewise form. Here, we present a general framework to construct the recurrent attractor and the functions using the Edelstein contraction, which broadens the class of functions. En route, we calculate the upper and lower box dimensions of the graph of non-linear RFIFs, extending the technique of Barnsley, Elton, and Hardin. Finally, we provide graphical and numerical examples for illustration.