Fractional-Order Delay Differential Equation with Separated Conditions

In this paper, we prove the existence and uniqueness solution of fractional delay differential equation and separated condition of the from: D-C(0+)q u(t) = f (t, u(t), u(g(t))), t is an element of [0,T], 1 < q < 2, a(1)u(0) + b(1C )D(0+)(p) u(T) = eta(1), a(2)u(0) + b(2C) D-0+(p) u(T) = eta(2), 0 < p <= 1, u(t) = psi(t), t is an element of [-h,0) where D-C(0+)q ,(C) D-0+(p) are the Caputo fraction derivative of order q, p and we consider a(1), a(2), b(1), b(2), eta(1), eta(2) is an element of R, the functions f is an element of C([0, T]x R x R, R), g is an element of C ((0, T], [- h, T]) with g(t) <= t and h > 0, psi(t) is continuous and bounded via fixed point theorem of Schaefers and Boyd-Wong nonlinear contraction. Also we give example as an application to illustrate the results obtained.