Summary: In this paper, by using fixed-point theorems, and lower and upper solution method, the existence for a class of fractional initial value problem (FIVP) \[ \begin{aligned} D_{0 +}^\alpha u(t) = f(t,u(t)),t \in (0,h), \\ t^{2-\alpha} u(t)|_{t=0} = b_1 \,\,D_{0+}^{\alpha-1} u(t)=|_{t=0} = b_2, \end{aligned} \] is discussed, where \(f \in C([0, h]\times \mathbb{R},\mathbb{R})\), \(D_{0+}^\alpha u(t)\) is the standard Riemann-Liouville fractional derivative, \(1 < \alpha < 2\). Some hidden confusion and fallacy in the literature are commented. A new condition on the nonlinear term is given to guarantee the equivalence between the solution of the FIVP and the fixed-point of the operator. Based on the new condition, some new existence results are obtained and presented as example.