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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]