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| In [[mathematics]], a '''pseudo-monotone operator''' from a [[reflexive space|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.
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| ==Definition==
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| Let (''X'', || ||) be a reflexive Banach space. A map ''T'' : ''X'' → ''X''<sup>∗</sup> from ''X'' into its continuous dual space ''X''<sup>∗</sup> is said to be '''pseudo-monotone''' if ''T'' is a [[bounded operator]] (not necessarily continuous) and if whenever
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| :<math>u_{j} \rightharpoonup u \mbox{ in } X \mbox{ as } j \to \infty</math>
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| (i.e. ''u''<sub>''j''</sub> [[weak topology|converges weakly]] to ''u'') and
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| :<math>\limsup_{j \to \infty} \langle T(u_{j}), u_{j} - u \rangle \leq 0,</math>
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| it follows that, for all ''v'' ∈ ''X'',
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| :<math>\liminf_{j \to \infty} \langle T(u_{j}), u_{j} - v \rangle \geq \langle T(u), u - v \rangle.</math>
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| ==Properties of pseudo-monotone operators==
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| Using a very similar proof to that of the [[Browder-Minty theorem]], one can show the following:
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| Let (''X'', || ||) be a [[real number|real]], reflexive Banach space and suppose that ''T'' : ''X'' → ''X''<sup>∗</sup> is [[continuous function|continuous]], [[coercive function|coercive]] and pseudo-monotone. Then, for each [[continuous linear functional]] ''g'' ∈ ''X''<sup>∗</sup>, there exists a solution ''u'' ∈ ''X'' of the equation ''T''(''u'') = ''g''.
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| ==References==
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| * {{cite book
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| | author = Renardy, Michael and Rogers, Robert C.
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| | title = An introduction to partial differential equations
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| | series = Texts in Applied Mathematics 13
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| | edition = Second edition
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| |publisher = Springer-Verlag
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| | location = New York
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| | year = 2004
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| | pages = 367
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| | isbn = 0-387-00444-0
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| }} (Definition 9.56, Theorem 9.57)
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| [[Category:Banach spaces]]
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| [[Category:Calculus of variations]]
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| [[Category:Operator theory]]
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Andrew Simcox is the name his parents gave him and he totally loves this title. One of the issues she enjoys most is canoeing and she's been doing it for fairly a while. My wife and I live in Kentucky. Invoicing is my occupation.
Have a look at my page; clairvoyants