Lower and Upper Approximations of Real-Valued Functions and Their Applications to Differential Equations

Abstract This paper investigates the use of fuzzy set theory in approximating solutions to differential equations under uncertainty. By defining lower and upper bounds, a robust framework is developed for modeling transitions between function extrema, extending classical methods to fuzzy initial value problems. The study demonstrates universal approximation properties and proposes numerical techniques that ensure stability and convergence. Applications highlight the practical utility of this approach in engineering and computational mathematics, solidifying fuzzy set theory as a powerful tool for addressing uncertainty in mathematical modeling.