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  "body": "\n## Statement\n\nLet $X$ be a real [[Banach space|06GZ956657]] and let $A \\colon X \\to X$ be a [[continuous|0X3WGX95FE]] map.\nAssume that $A$ is compact: it maps each bounded subset of $X$ into a relatively compact set.\nAssume that the set\n$$\\set{u \\in X : u = \\lambda A(u) \\text{ for some } \\lambda \\in [0, 1]}$$\nis bounded.\nThen $A$ has a fixed point: there is $u \\in X$ with $A(u) = u$.\n\n## Remarks\n\nThe proof uses the [[Schauder fixed-point theorem|DCPFVVZZ54]].\nChoose $M$ larger than the norm of every $u$ in the bounded set of the hypothesis, and let $B$ be the closed ball of radius $M$.\nOn $B$, define $\\tilde A(u) = A(u)$ if $\\norm{A(u)} \\le M$ and $\\tilde A(u) = M A(u)/\\norm{A(u)}$ otherwise.\nThen $\\tilde A$ maps $B$ continuously into a compact subset of $B$, so it has a fixed point $u$.\nIf $\\norm{A(u)} \u003e M$, then $u = \\lambda A(u)$ with $\\lambda = M/\\norm{A(u)} \u003c 1$ and $\\norm{u} = M$, which contradicts the choice of $M$; so $A(u) = u$.\n\nThe hypothesis is an a priori bound: one shows that all solutions of the family of problems $u = \\lambda A(u)$ lie in a fixed ball, without knowing in advance that any exists.\n\nExample: on $X = \\R$, let $A(u) = 1 + \\frac{1}{2}\\sin u$.\nIf $u = \\lambda A(u)$ with $\\lambda \\in [0, 1]$, then $\\abs{u} \\le 3/2$, so the set is bounded, and $A$ has a fixed point.\n\nThe a priori bound is necessary: $A(u) = u + 1$ on $\\R$ is continuous and compact, but has no fixed point, and the solutions $u = \\lambda/(1 - \\lambda)$ of $u = \\lambda A(u)$ are unbounded as $\\lambda \\to 1$.\n\n## Explanations\n\n### L1 human\n\nTo show that an equation $u = A(u)$ has a solution, it is enough to deform it into a trivial problem, $u = 0$, through the family $u = \\lambda A(u)$.\nIf all the solutions along the way stay in a bounded region, a solution survives at the end.\nThis turns an existence question into the often easier task of proving estimates.\n\n### L2 human\n\nThe theorem is a standard tool for nonlinear elliptic and parabolic equations, as in Evans's book on partial differential equations.\nOne writes the equation as $u = A(u)$, where $A$ solves a linear problem with the nonlinearity frozen, and gets compactness of $A$ from elliptic regularity and a compact Sobolev embedding.\nThe a priori bound is then proved by energy estimates or the maximum principle.\nIt is a special case of the Leray–Schauder continuation principle, which is based on degree theory.\n",
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  "name": "Schaefer's fixed point theorem",
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