Pseudo-monotone operator

In mathematics, a pseudo-monotone operator from a reflexive Banach space into its continuous dual space is one that is, in some sense, almost as well-behaved as a monotone operator.Many problems in the calculus of variations can be expressed using operators that are pseudo-monotone, and pseudo-monotonicity in turn implies the existence of solutions to these problems.Let (X, || ||) be a reflexive Banach space.A map T : X → X∗ from X into its continuous dual space X∗ is said to be pseudo-monotone if T is a bounded operator (not necessarily continuous) and if whenever (i.e. uj converges weakly to u) and it follows that, for all v ∈ X, Using a very similar proof to that of the Browder–Minty theorem, one can show the following: Let (X, || ||) be a real, reflexive Banach space and suppose that T : X → X∗ is bounded, coercive and pseudo-monotone.Then, for each continuous linear functional g ∈ X∗, there exists a solution u ∈ X of the equation T(u) = g.
mathematicsreflexiveBanach spacecontinuous dual spacewell-behavedmonotone operatorcalculus of variationsbounded operatorconverges weaklyBrowder–Minty theoremboundedcoercivecontinuous linear functionalFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory