On the existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations

In this paper, we study the existence and uniqueness of solutions for the nonlinear fractional differential equation initial value problem D^αu(t) = f(t, u(t)), t ∈ (0, T], t^2-αu(t)|_t=0 = b₁, t^2-αu´(t)|_t=0 = b₂, where 1 <; a <; 2 is a real number, D^α is the Riemann-Liouville fractional derivative. Using the monotone iterative method and the lower and upper solution method, some existence and uniqueness criteria are established, which extend the given results in recent literary works.