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By using inequalities, fixed point theorems, and lower and upper solution method, the existence and uniqueness of a class of fractional initial value problems, D0+qx(t)=f(t,x(t),  D0+q-1x(t)),  t∈(0,T),  x(0)=0,  D0+q-1x(0)=x0,  are discussed, where f∈C([0,T]×R2,R),D0+qx(t) is the standard Riemann-Liouville fractional derivative, 1<q<2. Some mistakes in the literature are pointed out and some new inequalities and existence and uniqueness results are obtained.