Soft Numerical Computing in Uncertain Dynamic Systems

6.4.1.3: Theorem—Existence and Uniqueness in Real Fractional Differential Equation

Consider the initial value problem as follows:

${\mathit{D}}_{\mathit{CK}}^{\alpha }\mathcal{u}\left(t\right)=g\left(t,\mathcal{u}\left(t\right)\right),\mathcal{u}\left(t_0\right)={\mathcal{u}}_0=0,t\in \left[t_0,T\right]$

Let η > 0 be a given constant and $B\left({\mathcal{u}}_0,\eta \right)=\left\{\mathcal{u}\in R;\left|\mathcal{u}-{\mathcal{u}}_0\right|\le \eta \right\}$ is a ball around ${\mathcal{u}}_0$ with radius η. Also assume a real valued function g : [t₀, T] × [0, η] → R⁺ satisfies the following conditions:

1. 1. $g\in C\left(\left[t_0,T\right]\times \left[0,\eta \right],R^{+}\right),g\left(t,0\right)\equiv 0,\exists M_g\ge 0,0\le g\left(t,\mathcal{u}\right)\le M_g,$ for all $\left(t,\mathcal{u}\right)\in \left[t_0,T\right]\times \left[0,\eta \right]$.
2. 2. $g\left(t,\mathcal{u}\right)$ is a nondecreasing function with respect to x for ...