Analysis of Functional Properties of Fuzzy \(\tilde{L}\) \(^p\Omega\) Spaces

Abstract

This paper proposes a study of fuzzy \(\tilde{L}\)p (Ω) spaces, incorporating functions with triangular fuzzy coefficients. These fuzzy functional spaces provide a better adaptation to fuzzy or imprecise functions. We establish the theoretical foundations of these spaces by examining key functional properties, such as the fuzzy scalar product and the fuzzy norm. To do this, we checked the bilinearity, symmetry, positivity, homogeneity and triangular inequality in a fuzzy environment and in the presence of functions whose coefficients are triangular fuzzy numbers, by the \(\alpha\) -cut Dubois and Prade approach. The aim of this paper is to address observations identified in the existing literature, where some functional properties of \(\tilde{L}\)p (Ω) fuzzy spaces are often addressed in a too general manner, without specifying the types of fuzzy functions. This study aims to provide a more detailed and rigorous analysis, thus enriching mathematical understanding and paving the way for practical applications in diverse fields such as: fuzzy differential equations, artificial intelligence, information processing, and decision making in uncertain environments.