In this paper, we are concerned with the existence of positive solutions for boundary value problems of nonlinear fourth-order differential equations

$$\begin{array}{rcl}& & u^{(4)}+c(x)u=\lambda a(x)f(u),x\in (0,1),\\ & & u(0)=u(1)=u^{″}(0)=u^{″}(1)=0,\end{array}$$

where $a(x)$ may change signs. The proof of main results is based on Leray-Schauder's fixed point theorem and the properties of Green's function of the fourth-order differential operator $L_{c}u=u^{(4)}+c(x)u$.

In this paper, we primarily focused on the existence and uniqueness of the initial value problem for fractional order fuzzy ordinary differential equations in a fuzzy metric space. First, definitions and relevant properties of the Gamma function and Beta function within a fuzzy metric space were provided. Second, by employing the principle of fuzzy compression mapping and Choquet integral of fuzzy numerical functions, we established the existence and uniqueness of solutions to initial value problems for fuzzy ordinary differential equations. Finally, several examples were presented to demonstrate the validity of our obtained results.