On the Convergence of Successive Approximations in the Theory of Ordinary Differential Equations

Let R denote the rectangle: |t-t0| ≤ a, | x-x0| ≤ b (a,b > 0) in the (t,x) plane and let f(t, x) be a function of two real variables t and x, defined and continuous on R. If I is the interval |t—t0| ≤ d with d = min(a,b/M), where M = max(|f(t, x)|, (t, x) ϵ R), then every solution x = x (t) of the differential equation x' = f(t, x) defined on I and which satisfies the initial condition x(t0) = x0, satisfies the integral equation 1.1 and conversely. In some cases, in order to prove the existence and uniqueness of the solutions of (1.1) on I, one forms the successive approximations 1.2