Contraction mapping and asymptotic stability in delay differential equations

In this paper we study asymptotic stability in delay differential equations by means of contraction mappings. The paper is motivated by a number difficulties encountered in the study of stability by means of Liapunov's direct method. We notice that most of these difficulties vanish when applying fixed point theory. While Liapunov's direct method usually asks pointwise conditions on the functions governing the system, the stability results we offer ask conditions of an averaging nature. The reader will also see how very complete, simple, and rigorous analysis on a highly sophisticated classical problem can be achieved using only a contraction mapping on a complete metric space. New stability criteria are derived for Volterra and general delay differential equations.