Nonlinear Volterra equation

Remarks

If $K(t, s, u) = k(t, s)\, u$, this is the linear Volterra integral equation of the second kind. If $K$ satisfies a Lipschitz condition $\abs{K(t, s, u) - K(t, s, v)} \le L \abs{u - v}$ on all of $\Delta \times \R$, the equation has exactly one continuous solution on $[a, b]$, and the successive approximations converge to it. If the Lipschitz condition holds only for $u, v$ in bounded sets, there is still a unique solution on some interval $[a, c]$ with $c > a$.

Example: $\varphi(t) = 1 + \int_0^t \varphi(s)^2\, ds$ on $[0, 1)$, with $K(t, s, u) = u^2$. The function $\varphi(t) = 1/(1 - t)$ solves it, since $1 + \int_0^t (1 - s)^{-2}\, ds = 1 + \frac{1}{1 - t} - 1 = \frac{1}{1 - t}$. It is the integral form of $y' = y^2$, $y(0) = 1$, and the solution does not extend past $t = 1$.

Non-example: $\varphi(t) = f(t) + \int_a^b K(t, s, \varphi(s))\, ds$ has the fixed upper limit $b$; it is a Urysohn equation, not a Volterra one.