Nonlinear Fractal Interpolation Functions Under Integral-Type Contractive Conditions

Given a finite set of interpolation data {(xi,yi)∈I×R,i=0,1,…,N}, I=[x0,xN], we construct a class of nonlinear fractal interpolation functions whose graphs are realized as attractors of appropriately defined iterated function systems. In contrast to the classical framework based on uniform contraction mappings, the present approach is built upon an integral-type contraction condition, which extends the standard Banach setting to a more general and flexible context. By applying Branciari’s fixed point theorem, we prove the existence and uniqueness of continuous fractal interpolants associated with these systems. This generalized formulation contains the classical Barnsley fractal interpolation functions as a particular case, while allowing greater adaptability in the modeling of complex and irregular phenomena. As an application, the proposed methodology is implemented on real time-series data describing vaccination dynamics in four different countries, illustrating the effectiveness of the constructed fractal interpolation functions in approximating highly irregular real-world signals.