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Banach's contraction principle is a fundamental result in pure and applied mathematics, and all the fields of physics. In this paper a new type of contraction, called in the text partial contractivity, is presented in the framework of b-metric spaces (an extension of the usual metric spaces). The new mappings generalize the classical Banach contractions, that appear as a particular case. The properties of the new self-maps are studied, giving sufficient conditions for the existence and uniqueness of their fixed points. Afterwords three different iterative algorithms for the approximation of critical points (Picard, Ishikawa and Karakaya) are considered, concerning their convergence and stability. These findings are applied to the approximation of fractal functions, coming from contractive and non-contractive operators. • Definition of a new type of contraction. • Existence and uniqueness of fixed points for this self-map. • Study of Ishikawa and Karakaya algorithms for the approximation of critical points. • Analysis of convergence and stability of these procedures. • Application to the approximation of fractal functions. [ABSTRACT FROM AUTHOR]

This article aims to present a family of fractal functions that provides solutions to a (0, 4) lacunary interpolation problem based on methodology of fractal perturbation with function scaling factors. Approximation orders for the proposed class of fractal interpolants and their derivatives are provided. With an eye towards “optimal” perturbation, a strategy to select suitable constant scaling factors is outlined. [ABSTRACT FROM AUTHOR]

In the framework of non-autonomous discrete dynamical systems in metric spaces, we propose new equilibrium points, called quasi-fixed points, and prove that they play a role similar to that of fixed points in autonomous discrete dynamical systems. In this way some sufficient conditions for the convergence of iterative schemes of type x k + 1 = T k x k in metric spaces are presented, where the maps T k are contractivities with different fixed points. The results include any reordering of the maps, even with repetitions, and forward and backward directions. We also prove generalizations of the Banach fixed point theorems when the self-map is substituted by a sequence of contractivities with different fixed points. The theory presented links the field of dynamical systems with the theory of iterated function systems. We prove that in some cases the set of quasi-fixed points is an invariant fractal set. The hypotheses relax the usual conditions on the underlying space for the existence of invariant sets in countable iterated function systems. [ABSTRACT FROM AUTHOR]

The present note aims to establish the notion of non-stationary bivariate α -fractal functions and discusses some of their approximation properties. We see that using a sequence of iterated function systems generalizes the existing stationary fractal interpolation function (FIF). Also, we show the existence of Borel probability fractal measures supported on the graph of the non-stationary fractal function. Further, we define a fractal operator associated with the constructed non-stationary FIFs, and many applications of this operator such as fractal approximation and the existence of fractal Schauder bases are observed. In the end, we study the constrained approximation with the proposed interpolant. • The existence of Borel probability fractal measures supported on the graph of the non-stationary fractal interpolation function (FIF) is shown. • Some approximation properties of non-stationary bivariate α -fractal functions are studied. • A fractal operator associated with the constructed non-stationary FIFs is defined. • Some applications of the fractal operator such as fractal approximation and the existence of fractal Schauder bases are observed. • Some constrained approximation aspects with the proposed interpolant are also considered. [ABSTRACT FROM AUTHOR]

The box dimension of the graph of non-affine α -fractal interpolation function f α with variable scaling factors is estimated in the interval [0, 1]. Due to the non-affinity of f α , the behavior of the graph is non-uniform in the subintervals. An attempt is made to estimate the box dimension of the graph of f α in the subintervals as well and it is compared with the box dimension of f α on the whole interval. [ABSTRACT FROM AUTHOR]