Global existence and uniqueness of solutions of integral equations with delay: progressive contractions

In the theory of progressive contractions an equation such as x(t) = L(t) + integral(t)(0) A(t - s)[f(s, x(s)) + g(s, x(s - r(s))]ds, with initial function omega with omega(0) = L(0) defined by t <= 0 double right arrow x(t) = omega(t) is studied on an interval [0, E] with r(t) >= alpha > 0. The interval [0, E] is divided into parts by 0 = T-0 < T-1 < . . . < T-n = E with T-i Ti-1 < alpha. It is assumed that f satisfies a Lipschitz condition, but there is no growth condition on g. When we try for a contraction on [0, T-1] the terms with g add to zero and we get a unique solution xi(1) on [0, T-1]. Then we get a complete metric space on [0, T-2] with all functions equal to xi(1) on [0, T-1] enabling us to get a contraction. In n steps we have obtained a solution on [0, E]. When r(t) > 0 on [0, infinity) we obtain a unique solution on that interval as follows. As we let E = 1, 2, . . . we obtain a sequence of solutions on [0, n] which we extend to [0, infinity) by a horizontal line, thereby obtaining functions converging uniformly on compact sets to a solution on [0, infinity). Lemma 2.1 extends progressive contractions to delay equations.