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  "body": "\n## Statement\n\nLet $a \u003c b$ and $R \u003e 0$.\nLet $K \\colon [a, b] \\times [a, b] \\times [-R, R] \\to \\R$ and $f \\colon [a, b] \\to \\R$ be [[continuous|0X3WGX95FE]], and let $M = \\max \\abs{K}$.\nAssume that\n$$\\max_{x \\in [a, b]} \\abs{f(x)} + (b - a)\\, M \\le R.$$\nThen the [[Urysohn equation|WZ4Z26XSVC]]\n$$\\varphi(x) = f(x) + \\int_a^b K\\bigl(x, t, \\varphi(t)\\bigr)\\, dt, \\quad x \\in [a, b],$$\nhas a continuous solution $\\varphi$ with $\\abs{\\varphi(x)} \\le R$ for all $x \\in [a, b]$.\n\n## Remarks\n\nLet $B$ 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.\nLet $T\\varphi$ be the right side of the equation.\nThe 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.\nBy the [[Arzelà–Ascoli theorem|0381S8Z5P0]], $T(B)$ is relatively compact.\nThe [[Schauder fixed-point theorem|DCPFVVZZ54]], in its form for continuous maps of a closed convex set into a compact subset, gives a fixed point.\n\nThe theorem gives existence but not uniqueness.\nExample: 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.\nBoth $\\varphi = 0$ and $\\varphi = 1$ are solutions, since $\\int_0^1 \\sqrt{1}\\, dt = 1$.\nHere $K$ is not Lipschitz in $u$ near $0$, so the [[Banach fixed point theorem|0VK6VEH345]] does not apply.\n\nThe 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.\n\n## Explanations\n\n### L1 human\n\nTo solve an integral equation, one looks for a function that the right side of the equation leaves unchanged.\nIf the right side maps a ball of continuous functions into itself and squeezes it into a compact set, Schauder's theorem says that such a function exists.\nThe size condition makes sure the ball is mapped into itself.\nThe theorem says nothing about how many solutions there are, and there can be several.\n\n### L2 human\n\nThis is the model for existence proofs with compactness, as opposed to proofs with contractions.\nCompactness comes from the smoothing effect of integration and the Arzelà–Ascoli theorem; continuity of the kernel is all that is needed.\nPeano's existence theorem for ordinary differential equations is the Volterra analogue.\nSometimes no invariant ball is easy to find.\nThen [[Schaefer's fixed point theorem|7H3YSSDEBK]] or the [[Leray–Schauder method|QGRBR04JJN]] replaces it by an a priori bound for a family of equations.\n",
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