In this paper, we investigate the existence results for fractional differential equations of the form 0.1$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T)\left( 0<T\le \infty \right) , \quad q \in (1,2),\\ x(0)=a_{0},\quad x^{'}(0)=a_{1}, \end{array}\right. } \end{aligned}$$ Dcqx(t)=f(t,x(t))t∈[0,T)0<T≤∞,q∈(1,2),x(0)=a0,x′(0)=a1, and 0.2$$\begin{aligned} {\left\{ \begin{array}{ll} D_{c}^{q}x(t)=f(t,x(t)) \quad t\in [0, T), \quad q \in (0,1),\\ x(0)=a_{0}, \end{array}\right. } \end{aligned}$$ Dcqx(t)=f(t,x(t))t∈[0,T),q∈(0,1),x(0)=a0, where $$D_{c}^{q}$$ Dcq is the Caputo fractional derivative. We prove the above equations have solutions in C[0, T). Particularly, we present the existence and uniqueness results for the above equations on $$[0,+\infty )$$ [0,+∞) .