In my research I need to show that the set $$M \mathrel{:=} \{X \in \mathbb{R}^4,X\ge0\}$$ where $$X(t)=(x_1(t),x_2(t),x_3(t),x_4(t))^T$$ is positively invariant with respect to the following system of fractional ordinary differential equations $$D^{\alpha}(x(t))=f(t,x(t))$$ with initial non-negative condition $x(0)=x_0$, where $f$ is nonlinear and continuous.
My question is: how do I show that $M$ is positively invariant with respect to the system given? Any ideas, references are appreciated.
Let $\alpha\in(0,1]$. Rewrite the equation in an integral form $$x(t)=x_0+\frac1{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}f(s,x(s))ds.$$ The classical method to prove the existence of a solution $x(t)$ is to define a sequence $(x^n)_{n\ge0}$ by $x^0\equiv x_0$ and $$x^{n+1}(t)=x_0+\frac1{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}f(s,x^n(s))ds,$$ and to prove that it converges to some $x$.
If your assumption on $f$ is that $f(t,M)\subset M$, then every $x^n$ takes values in $M$, and therefore the limit $x$ takes values in $M$.
