Time scales calculus for $\mathcal{S}$-linearly correlated fuzzy number-valued functions and application to fuzzy functional dynamic equations

Based on the existing results, this paper systematically reviews and extends the calculus theory of $\mathcal{S}$-linearly correlated fuzzy number-valued functions on time scales. As an application of the theory, the local existence and uniqueness of the solution of the initial value problem of the fuzzy functional dynamic equation is established in the space of $\mathcal{S}$-linearly correlated fuzzy numbers. Moreover, a theorem for the existence of solutions to fuzzy functional dynamics equations in a wider range is presented via Arzela-Ascoli’s theorem. Finally, the continuous dependence of the solution on the initial conditions is also considered. Several illustrative examples are provided in order to explain the main results presented in this paper.