On the existence of blow up solutions for a class of fractional differential equations

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{gathered} D_{0 + }^\alpha u(t) = f(t,u(t)),t \in (0,h), \hfill \\ t^{2 - \alpha } u(t)|_{t = 0} = b_1 D_{0 + }^{\alpha - 1} u(t) = |_{t = 0} = b_2 , \hfill \\ \end{gathered} $ is discussed, where f ∈ C([0, h]×R,R), D 0+α u(t) is the standard Riemann-Liouville fractional derivative, 1 < α < 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.