SOLUTIONS OF TWO-TERM TIME FRACTIONAL ORDER DIFFERENTIAL EQUATIONS WITH NONLOCAL INITIAL CONDITIONS

We study the existence of mild solutions for the two-term fractional order abstract differential equation D(t)(alpha+1)u(t) + mu D(t)(beta)u( ) - Au(t) = D(t)(alpha-1)f (t, u(t)), t is an element of [0, 1], 0 < alpha <= beta <= 1, mu >= 0, with nonlocal initial conditions and where A is a linear operator of sectorial type. To achieve our goal, we use a new mixed method, which combines a generalization of the theory of C-0-semigroups, Hausdorff measure of noncompactness and a fixed point argument.