Construction of solutions for the initial value problem of a functional-differential equation of advanced type

Let lambda > 1 and a is an element of R, a not equal 0. We give a method to construct solutions for the initial value problem of the functional-differential equation f'(x) = af(lambda x) for x is an element of R and the initial condition f(0) - 0. Several authors studied relevant equations. Our method is different from all of them and gives numerical data readily. We give graphs of several solutions. We use the Fourier transform mainly to prove the results.