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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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Wilber Berryhill is the title his parents gave him and he completely digs that title. He works as a bookkeeper. I've always cherished residing in Alaska. It's not a typical thing but what she likes doing is to play domino but she doesn't have the time recently.
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