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  "body": "\n## Statement\n\nLet $a \u003c b$, let $f \\colon [a, b] \\to \\R$ be continuous, and let $H \\colon [a, b] \\times [a, b] \\times \\R \\to \\R$ be continuous.\nAssume that $H$ satisfies a [[Lipschitz condition|0AW1E6YTJY]] in its third variable: there is $L \\ge 0$ with\n$$\\abs{H(x, s, u) - H(x, s, v)} \\le L \\abs{u - v} \\quad \\text{for all } x, s \\in [a, b] \\text{ and } u, v \\in \\R.$$\nLet $\\lambda \\in \\R$ with $\\abs{\\lambda} L (b - a) \u003c 1$.\nThen the [[nonlinear integral equation|BSCYBP6AS6]]\n$$\\varphi(x) = f(x) + \\lambda \\int_a^b H\\left(x, s, \\varphi(s)\\right) ds, \\quad x \\in [a, b],$$\nhas exactly one continuous solution $\\varphi \\colon [a, b] \\to \\R$.\nFor every continuous $\\varphi_0$, the functions $\\varphi_{k+1}(x) = f(x) + \\lambda \\int_a^b H(x, s, \\varphi_k(s)) \\, ds$ converge to $\\varphi$ uniformly on $[a, b]$.\n\n## Remarks\n\nProof idea: let $T$ be the right side, viewed as a map of $C[a, b]$ with the maximum norm.\nThis space is a complete metric space, by the [[completeness of the space of bounded continuous functions|09GRCZVR5E]].\nThe map $T$ sends continuous functions to continuous functions, since $H$ is uniformly continuous on compact sets.\nThe Lipschitz condition gives $\\max \\abs{T \\varphi - T \\psi} \\le \\abs{\\lambda} L (b - a) \\max \\abs{\\varphi - \\psi}$, so $T$ is a [[contraction|0QN69YA357]].\nThe [[Banach fixed point theorem|0VK6VEH345]] gives the unique fixed point and the convergence of the [[fixed-point iteration|K6ZF91051B]].\n\nExample: the linear case $H(x, s, u) = K(x, s) u$ with a continuous kernel $K$ has $L = \\max \\abs{K}$, and the theorem gives unique solvability of $\\varphi = f + \\lambda \\int_a^b K \\varphi$ for $\\abs{\\lambda} \\max \\abs{K} (b - a) \u003c 1$.\n\nThe smallness condition matters: for $H(x, s, u) = u$ on $[0, 1]$ and $\\lambda = 1$, so $\\abs{\\lambda} L (b - a) = 1$, the equation $\\varphi = f + \\int_0^1 \\varphi$ has no solution when $\\int_0^1 f \\ne 0$.\nFor the Volterra form, with $\\int_a^x$ in place of $\\int_a^b$, the conclusion holds for every $\\lambda$, because some power of $T$ is a contraction.\nThe same proof works for complex-valued functions.\n\n## Explanations\n\n### L1 human\n\nIf the integral term reacts only weakly to changes of the unknown, the equation has exactly one solution.\nOne finds it by guessing any function and feeding it into the right side again and again.\nThe guesses get closer to each other at a fixed rate and converge to the solution.\n\n### L2 human\n\nThis is the standard first existence theorem for nonlinear Fredholm-type equations of Urysohn form.\nThe error after $k$ steps is at most $q^k (1 - q)^{-1} \\max \\abs{\\varphi_1 - \\varphi_0}$ with $q = \\abs{\\lambda} L (b - a)$, so the method also gives numerical bounds.\nThe same argument, applied on a short time interval, proves the Picard–Lindelöf theorem for ordinary differential equations through their integral form.\nFor larger $\\lambda$ one needs other tools, such as Schauder's fixed point theorem or Fredholm theory in the linear case.\n",
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