Solution for Boundary Value Problem of Non-Integer Order in L^2 -Space

In this paper, we shall prove the existence and uniqueness of a square - integrable solution in L2-space, for the boundary value problem of non- integer order which has the form: cDαx y(x) = f(x, y(x)), 1 α ≤2 y(a) = ya, y(b) = yb Where cDα is the Caputo fractional derivative and a, b are positive constants with b not equal a. The contraction mapping principle has been used in establishing our main results.