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  "body": "\n## Statement\n\nLet $H$ be a real [[Hilbert space|HM2HHE3J51]] with inner product $\\ip{\\cdot}{\\cdot}$.\nLet $K \\colon H \\to H$ be a bounded linear [[self-adjoint operator|0NEGN5CRSC]] with $\\ip{K v}{v} \\ge 0$ for all $v \\in H$.\nLet $F \\colon H \\to H$ be [[monotone|Y1M03JQKKR]], that is $\\ip{F(u) - F(v)}{u - v} \\ge 0$ for all $u, v \\in H$, and [[hemicontinuous|XNMKX7H4FD]], that is $t \\mapsto \\ip{F(u + t v)}{w}$ is continuous on $[0, 1]$ for all $u, v, w \\in H$.\nThen for every $f \\in H$ the abstract [[Hammerstein equation|T7E2EW2T0G]]\n$$u + K F(u) = f$$\nhas exactly one solution $u \\in H$.\n\n## Remarks\n\nLet $S$ be the positive [[square root|3JY5HNFRMF]] of $K$, so $K = S^2$ with $S$ self-adjoint.\nLook for $u$ of the form $u = f - S v$, and put $G(v) = v - S F(f - S v)$.\nFor $a = f - S v$ and $b = f - S w$ one finds $\\ip{G(v) - G(w)}{v - w} = \\norm{v - w}^2 + \\ip{F(a) - F(b)}{a - b} \\ge \\norm{v - w}^2$.\nSo $G$ is strongly monotone, hemicontinuous and coercive, and Minty's theorem in Hilbert space, the case of the [[Browder–Minty theorem|NWK9Z80BT2]], gives a unique $v$ with $G(v) = 0$.\nThen $u = f - S v$ solves the equation, and each solution $u$ gives a zero $v = S F(u)$ of $G$, which yields uniqueness.\n\nApplication: let $\\Omega \\subseteq \\R^m$ be measurable with finite measure and $H = L^2(\\Omega)$.\nLet $k$ be a symmetric square-integrable kernel with $\\int\\!\\!\\int k(x, t)\\, v(x) v(t)\\, dx\\, dt \\ge 0$ for all $v$, and let $g(x, u)$ be a Carathéodory function that is nondecreasing in $u$ with $\\abs{g(x, u)} \\le a(x) + b \\abs{u}$, $a \\in L^2(\\Omega)$.\nThen the [[Nemytskii operator|W8G3AZ0DNG]] $F = N_g$ is continuous and monotone on $L^2(\\Omega)$, and $\\varphi(x) + \\int_\\Omega k(x, t)\\, g(t, \\varphi(t))\\, dt = f(x)$ has exactly one solution in $L^2(\\Omega)$ for each $f \\in L^2(\\Omega)$.\n\nThe signs matter.\nIn $H = \\R$ with $K = 1$ and $F(u) = -u$, which is not monotone, the equation reads $0 = f$ and has no solution for $f \\ne 0$.\n\n## Explanations\n\n### L1 human\n\nA Hammerstein equation mixes a linear smoothing step with a pointwise nonlinear rule.\nIf the linear part is \"positive\" and the nonlinear rule is \"increasing\", the two effects push in the same direction.\nThen the equation behaves like a strictly increasing function on the real line: it takes every value exactly once.\n\n### L2 human\n\nThe theorem is one of the main uses of [[monotone operator|Y1M03JQKKR]] theory outside partial differential equations.\nThe trick with the square root $S$ moves the problem to one where the operator is strongly monotone, which Browder–Minty theory handles directly.\nPositive kernels arise as Green's functions of self-adjoint boundary value problems, so the theorem gives existence and uniqueness for $-u'' + g(x, u) = f$ with Dirichlet conditions when $g$ is nondecreasing in $u$.\nVersions without symmetry of $K$ and in Banach spaces exist, where $K$ maps $X^*$ to $X$ and $F$ maps $X$ to $X^*$.\n",
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