On solvability of autonomous delay differential equations on the real axis

The existence and structure of solutions of autonomous functional differential equations on the real axis are studied. A detailed survey of papers on the subject is given at the beginning of the article. It is established that the space of solutions to homogeneous equations whose solutions are in the given space of integrally bounded functions is of finite dimensionality. The basis of the space is formed by the exponential type solutions generated by roots of the characteristic equation. The key point in the proof of the main result is the expansion of the solution in series by the exponential system. On this basis the equation is transformed so that the Fourier transform can be used. The results obtained are used for the study of the boundary problem for the singular equation with delay.