Math 2.0

Operator theory Nonlinear operators and fixed points Theorem

Browder–Minty theorem

Builds on Banach spaceReflexive spaceContinuous dual spaceMonotone operator

Statement

Let XX be a real separable reflexive Banach space with norm ∥⋅∥\norm{\cdot}, and let X∗X^* be its continuous dual space. Write ⟨g,u⟩\langle g, u \rangle for the value of g∈X∗g \in X^* at u∈Xu \in X. Let T ⁣:X→X∗T \colon X \to X^* be a map with these three properties.

  1. TT is monotone: ⟨T(u)−T(v),u−v⟩≥0\langle T(u) - T(v), u - v \rangle \ge 0 for all u,v∈Xu, v \in X.
  2. TT is hemicontinuous: for all u,v,w∈Xu, v, w \in X, the map t↦⟨T(u+tv),w⟩t \mapsto \langle T(u + t v), w \rangle is continuous on [0,1][0, 1].
  3. TT is coercive: ⟨T(u),u⟩/∥u∥→+∞\langle T(u), u \rangle / \norm{u} \to +\infty as ∥u∥→∞\norm{u} \to \infty.

Then for every g∈X∗g \in X^* the equation T(u)=gT(u) = g has a solution u∈Xu \in X. If moreover ⟨T(u)−T(v),u−v⟩>0\langle T(u) - T(v), u - v \rangle > 0 for all u≠vu \ne v, then the solution is unique.

Remarks

A standard proof uses the Galerkin method. On an increasing sequence of finite-dimensional subspaces whose union is dense, the Brouwer fixed point theorem gives approximate solutions. Coercivity bounds them, reflexivity gives a weakly convergent subsequence, and Minty's trick passes to the limit. The trick: for a monotone hemicontinuous TT, if ⟨g−T(v),u−v⟩≥0\langle g - T(v), u - v \rangle \ge 0 for all v∈Xv \in X, then T(u)=gT(u) = g. For X=ℝX = \R the theorem says that a nondecreasing continuous function TT with T(u)→±∞T(u) \to \pm\infty as u→±∞u \to \pm\infty takes every real value. Coercivity cannot be dropped: T(u)=arctan⁡uT(u) = \arctan u on ℝ\R is monotone and continuous but never takes the value 22.

Going deeper

For a student

The theorem is the nonlinear analogue of the Lax–Milgram theorem. A bounded linear operator given by a coercive bilinear form is a special case of a coercive monotone operator. Its main use is in quasilinear elliptic equations such as −div⁡(∣∇u∣p−2∇u)=f-\operatorname{div}(\abs{\nabla u}^{p-2} \nabla u) = f, whose weak form defines a monotone coercive operator from the Sobolev space W01,pW_0^{1,p} to its dual for 1<p<∞1 < p < \infty.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Banach space, concept, no evidence levelBanach spaceconceptContinuous dual space, concept, no evidence levelContinuous dual spaceconceptMonotone operator, concept, no evidence levelMonotone operatorconceptReflexive space, concept, no evidence levelReflexive spaceconceptBrowder–Minty theorem, statement, no evidence levelBrowder–Minty theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Banach space concept
  2. Continuous dual space concept
  3. Monotone operator concept
  4. Reflexive space concept
  5. Browder–Minty theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Browder–Minty_theorem
About this page
TagNWK9Z80BT2
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Kindstatement, theorem
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MSC47H05, 47J05
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AboutMonotone operator
UsesBanach space, Reflexive space, Continuous dual space, Monotone operator
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Changed2 November 2026

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