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

Abstract

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) D0+α u(t)= f(t, u(t)), t ∈ ( h), t2-αu(t) |t=0= b D0+α-1 u(t) |t=0= b2, is discussed, where f ∈ C([ h]×R,R), D0+αu(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.