Leray–Schauder method

Remarks

Step 2 guarantees that no solution lies on the boundary of $B_R$ for any $\lambda$, which is what homotopy invariance needs. The form with $T(\lambda, u) = \lambda F(u)$ is the Leray–Schauder alternative, also known as Schaefer's fixed point theorem: either the set of solutions of $u = \lambda F(u)$, $0 \le \lambda \le 1$, is unbounded, or $u = F(u)$ has a solution.

Example: in $X = C[0, 1]$, let $F(u)(x) = \int_0^1 k(x, t)\, \arctan u(t)\, dt$ with $k$ continuous. Then $F$ is compact by the Arzelà–Ascoli theorem. Any solution of $u = \lambda F(u)$ with $0 \le \lambda \le 1$ satisfies $\norm{u}_\infty \le \frac{\pi}{2} \max \abs{k}$, since $\abs{\arctan} < \frac{\pi}{2}$. So $R = \frac{\pi}{2} \max \abs{k} + 1$ works, and $u = F(u)$ has a continuous solution.

Non-example: in an infinite-dimensional space, without compactness, step 3 has no meaning. For instance, $T(\lambda, u) = \lambda u$ on the unit ball of $\ell^2$ is not compact, and the Leray–Schauder degree of $I - T(\lambda, \cdot)$ is not defined.