---
tag: NF0F8M86W4
kind: statement
role: theorem
name: Monotone operator method for Hammerstein equations
msc: [45G10, 47H05, 47J05]
uses: [T7E2EW2T0G, Y1M03JQKKR, NWK9Z80BT2, HM2HHE3J51, 0NEGN5CRSC, 3JY5HNFRMF, XNMKX7H4FD, W8G3AZ0DNG]
about: [T7E2EW2T0G]
sources:
  - url: https://en.wikipedia.org/wiki/Browder%E2%80%93Minty_theorem
---

## Statement

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] with inner product $\ip{\cdot}{\cdot}$.
Let $K \colon H \to H$ be a bounded linear [[self-adjoint operator|0NEGN5CRSC]] with $\ip{K v}{v} \ge 0$ for all $v \in H$.
Let $F \colon H \to H$ 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 $u, v, w \in H$.
Then for every $f \in H$ the abstract [[Hammerstein equation|T7E2EW2T0G]]
$$u + K F(u) = f$$
has exactly one solution $u \in H$.

## Remarks

Let $S$ be the positive [[square root|3JY5HNFRMF]] of $K$, so $K = S^2$ with $S$ self-adjoint.
Look for $u$ of the form $u = f - S v$, and put $G(v) = v - S F(f - S v)$.
For $a = f - S v$ and $b = f - S w$ one finds $\ip{G(v) - G(w)}{v - w} = \norm{v - w}^2 + \ip{F(a) - F(b)}{a - b} \ge \norm{v - w}^2$.
So $G$ 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 $G(v) = 0$.
Then $u = f - S v$ solves the equation, and each solution $u$ gives a zero $v = S F(u)$ of $G$, which yields uniqueness.

Application: let $\Omega \subseteq \R^m$ be measurable with finite measure and $H = L^2(\Omega)$.
Let $k$ 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 $\abs{g(x, u)} \le a(x) + b \abs{u}$, $a \in L^2(\Omega)$.
Then 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 $f \in L^2(\Omega)$.

The signs matter.
In $H = \R$ with $K = 1$ and $F(u) = -u$, which is not monotone, the equation reads $0 = f$ and has no solution for $f \ne 0$.

## Explanations

### L1 human

A Hammerstein equation mixes a linear smoothing step with a pointwise nonlinear rule.
If the linear part is "positive" and the nonlinear rule is "increasing", the two effects push in the same direction.
Then the equation behaves like a strictly increasing function on the real line: it takes every value exactly once.

### L2 human

The theorem is one of the main uses of [[monotone operator|Y1M03JQKKR]] theory outside partial differential equations.
The trick with the square root $S$ moves the problem to one where the operator is strongly monotone, which Browder–Minty theory handles directly.
Positive 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 $u$.
Versions without symmetry of $K$ and in Banach spaces exist, where $K$ maps $X^*$ to $X$ and $F$ maps $X$ to $X^*$.