Soft Numerical Computing in Uncertain Dynamic Systems

3.6.1.8: Proposition—Composition of gH-differentiability

Suppose that $x:\left[a,b\right]\to {\mathbb{F}}_R$ is a fuzzy number valued function and it is gH-differentiable at y(t) on the interval (a, b) (Allahviranloo et al., 2015). Also, let us suppose the function y : [a, b] → R is a differentiable real function in the same open interval. Then:

If y′(t) ≥ 0:

${\left(xoy\right)}_{i-\mathit{gH}}^{\prime }\left(t\right)=y^{\prime }\left(t\right)\odot x_{i-\mathit{gH}}^{\prime }\left(y\left(t\right)\right):$

${\left(xoy\right)}_{\mathit{ii}-\mathit{gH}}^{\prime }\left(t\right)=y^{\prime }\left(t\right)\odot x_{\mathit{ii}-\mathit{gH}}^{\prime }\left(y\left(t\right)\right).$

(if) y′(t) < 0,

${\left(xoy\right)}_{i-\mathit{gH}}^{\prime }\left(t\right)=y^{\prime }\left(t\right)\odot x_{\mathit{ii}-\mathit{gH}}^{\prime }\left(y\left(t\right)\right),$

${\left(xoy\right)}_{\mathit{ii}-\mathit{gH}}^{\prime }\left(t\right)=y^{\prime }\left(t\right)\odot x_{i-\mathit{gH}}^{}$