Schauder fixed-point theorem for integral equations

Remarks

Let $B$ be the set of continuous functions on $[a, b]$ with values in $[-R, R]$; it is closed, bounded and convex in $C[a, b]$ with the sup norm. Let $T\varphi$ be the right side of the equation. The hypothesis gives $T(B) \subseteq B$, and the uniform continuity of $K$ on its compact domain shows that $T$ is continuous and that $T(B)$ is equicontinuous. By the Arzelà–Ascoli theorem, $T(B)$ is relatively compact. The Schauder fixed-point theorem, in its form for continuous maps of a closed convex set into a compact subset, gives a fixed point.

The theorem gives existence but not uniqueness. Example: on $[0, 1]$ take $f = 0$, $R = 1$ and $K(x, t, u) = \sqrt{\abs{u}}$, so $M = 1$ and the hypothesis $0 + 1 \le 1$ holds. Both $\varphi = 0$ and $\varphi = 1$ are solutions, since $\int_0^1 \sqrt{1}\, dt = 1$. Here $K$ is not Lipschitz in $u$ near $0$, so the Banach fixed point theorem does not apply.

The size condition cannot be dropped: for $f = 0$ and $K(x, t, u) = 1 + u^2$ on $[0, 1]$, any solution would be a constant $c = \int_0^1 (1 + c^2)\, dt = 1 + c^2$, which has no real root.

Going deeper

For a student

This is the model for existence proofs with compactness, as opposed to proofs with contractions. Compactness comes from the smoothing effect of integration and the Arzelà–Ascoli theorem; continuity of the kernel is all that is needed. Peano's existence theorem for ordinary differential equations is the Volterra analogue. Sometimes no invariant ball is easy to find. Then Schaefer's fixed point theorem or the Leray–Schauder method replaces it by an a priori bound for a family of equations.