On Fuzzy Sobolev Spaces W ̃1,p(Ω): Analysis of Dot Product and Fuzzy Norm
Grace Nkwese Mazoni *
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, National Pedagogical University, Kinshasa, DRC.
Clara Paluku Kasoki
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, National Pedagogical University, Kinshasa, DRC.
Fidèle Muaku Mvunzi
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, National Pedagogical University, Kinshasa, DRC.
Gérard Tawaba Musian Ta-yen
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, National Pedagogical University, Kinshasa, DRC.
Fernand Mamanya Tapasa
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, National Pedagogical University, Kinshasa, DRC.
Cédric Kabeya Tshiseba
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, National Pedagogical University, Kinshasa, DRC.
Rostin Mabela Makengo Matendo
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, University of Kinshasa, Kinshasa, DRC.
Samuel Diangitukulu Ndimba
Department of Mathematics, Faculty of Science and Technology, Statistics and Computer Science, University of Kinshasa, Kinshasa, DRC.
Jonathan Opfointshi Engombangi
Higher Institute of Medical Techniques of Bandundu (ISTM BDD), DRC.
Emilien Loranu Londjiringa
Department of Mathematics and Physics, Exact Sciences Section, ISP BUNIA, Ituri -DRC.
*Author to whom correspondence should be addressed.
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.
Keywords: Fuzzy W ̃1;p (Ω)sobolev spaces, fuzzy spaces L ̃p (Ω), fuzzy scalar products, fuzzy norms, fuzzy derivatives, fuzzy integrations, α-cuts and fuzzy functions