The asymptotic behavior of the solution of a Volterra equation

While our interest here is with the behavior of the solutions of (1.1) as t-* oo, the following observations on existence and uniqueness are relevant. Suppose g(x) also satisfies a Lipschitz condition on every interval x < X < oo. The usual method of successive approximations, together with certain a priori bounds obtained in the present proof, then easily imply that for each initial value (1.1) has a unique solution and that this solution exists on 0< t < oo. Even under the present hypothesis it follows readily from these a priori bounds and recent results of Nohel [2] that every solution of (1.1) can be continued (though not necessarily uniquely) over 0 < t < oo. Differentiating (1.1) yields