On Fuzzy Sobolev Spaces W 1,p(Ω): Analysis of Dot Product and Fuzzy Norm

Abstract

This paper proposes a study of fuzzy Sobolev spaces W ̃1,p(Ω), integrating functions with triangular fuzzy coefficients. These fuzzy functional spaces aim to better model fuzzy functions, thus generalizing classical Sobolev spaces. We establish the theoretical foundations of these spaces by analyzing the fuzzy scalar product and the fuzzy norm. We focus on verifying essential properties such as bilinearity, symmetry, positivity, homogeneity, and triangle inequality, using the Dubois and Prade α-cut approach to formalize the notion of uncertainty. This paper addresses observations identified in the literature, where the lack of suitable fuzzy functional spaces for solving fuzzy differential equations, in particular fuzzy Sobolev spaces W ̃1,p(Ω) , is often not taken into account. Moreover, the analysis of fuzzy scalar product and norm properties is often presented vaguely. We thus propose a detailed approach to the functional properties of these spaces, extending classical Sobolev spaces to a fuzzy framework. This research opens up prospects for practical applications in areas such as fuzzy differential equations and decision-making in medicine, economics, artificial intelligence, information processing and various other fields.