Fuzzy Variational Calculus in Linearly Correlated Space: Part I

This article is the first part of series of articles that aim to present the foundations for fuzzy variational calculus for functions taking values in the space of linearly correlated fuzzy numbers $\mathbb{R}_{\mathcal{F}(A)}$. Recall that the space $\mathbb{R}_{\mathcal{F}(A)}$ is composed by all sums of a real number $r$ and a fuzzy number $qA$, where $A$ is a given asymmetric fuzzy number. Two advantages of this space are that it can be equipped with a Banach space structure and, similar to additive models in Statistics, its elements can be interpreted as the sum of a deterministic expected/predictable value with an uncertain/noise component. The foundation of variational calculus theory requires the definition and establishement of many concepts and results. This aritcle presents a total order relation on $\mathbb{R}_{\mathcal{F}(A)}$ for wihch the notions of local minimal and maximal of a $\mathbb{R}_{\mathcal{F}(A)}$-valued function $f$ can be derived. We present a fuzzy version of the first and second optimality conditions in terms of derivatives of $f$. Finally, we present a generalized fuzzy version of du Bois-Reymond lemma which is essential in variational calculus theory.