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  "body": "\n## Definition\n\nLet $X$ be a real [[Banach space|06GZ956657]].\nA [[continuous|0X3WGX95FE]] map $T \\colon [0, 1] \\times X \\to X$ is *compact* if it maps bounded sets to sets with [[compact|0R6EWFKRCF]] closure.\nThe *Leray–Schauder method* for an equation $u = F(u)$ in $X$ consists of the following steps.\n\n1. Choose a compact map $T \\colon [0, 1] \\times X \\to X$ with $T(1, \\cdot) = F$ and $T(0, \\cdot) = 0$.\n2. Prove an a priori bound: there is $R \u003e 0$ such that every solution of $u = T(\\lambda, u)$ with $\\lambda \\in [0, 1]$ satisfies $\\norm{u} \u003c R$.\n3. Conclude by the homotopy invariance of the [[Leray–Schauder degree|3VJC5Q9MYA]] on the ball $B_R = \\set{u : \\norm{u} \u003c R}$:\n$$\\deg\\bigl(I - T(1, \\cdot), B_R, 0\\bigr) = \\deg\\bigl(I - T(0, \\cdot), B_R, 0\\bigr) = \\deg(I, B_R, 0) = 1,$$\nso $u = F(u)$ has a solution in $B_R$.\n\n## Remarks\n\nStep 2 guarantees that no solution lies on the boundary of $B_R$ for any $\\lambda$, which is what homotopy invariance needs.\nThe form with $T(\\lambda, u) = \\lambda F(u)$ is the Leray–Schauder alternative, also known as [[Schaefer's fixed point theorem|7H3YSSDEBK]]: either the set of solutions of $u = \\lambda F(u)$, $0 \\le \\lambda \\le 1$, is unbounded, or $u = F(u)$ has a solution.\n\nExample: in $X = C[0, 1]$, let $F(u)(x) = \\int_0^1 k(x, t)\\, \\arctan u(t)\\, dt$ with $k$ continuous.\nThen $F$ is compact by the Arzelà–Ascoli theorem.\nAny solution of $u = \\lambda F(u)$ with $0 \\le \\lambda \\le 1$ satisfies $\\norm{u}_\\infty \\le \\frac{\\pi}{2} \\max \\abs{k}$, since $\\abs{\\arctan} \u003c \\frac{\\pi}{2}$.\nSo $R = \\frac{\\pi}{2} \\max \\abs{k} + 1$ works, and $u = F(u)$ has a continuous solution.\n\nNon-example: in an infinite-dimensional space, without compactness, step 3 has no meaning.\nFor 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.\n",
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