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<title>Monotoner Operator</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Monotoner Operator</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <b>monotoner Operator</b> ist ein Begriff aus der <a href="Mathematik" title="Mathematik">Mathematik</a> aus dem Teilgebiet der nichtlinearen <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>. Sie sind besondere (nicht lineare) Operatoren und eine Verallgemeinerung der <a href="Monotone_Funktion" class="mw-redirect" title="Monotone Funktion">monotonen reellen Funktionen</a> einer Variable.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Es seien <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> ein <a href="Normierter_Raum" title="Normierter Raum">normierter Raum</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\subset V}">

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<annotation encoding="application/x-tex">{\displaystyle M\subset V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/096e9772f3d4b1bba3a4e6a5b38aac791ab226ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.328ex; height:2.176ex;" alt="{\displaystyle M\subset V}" loading="lazy"></span> eine <a href="Konvexe_Teilmenge" class="mw-redirect" title="Konvexe Teilmenge">konvexe Teilmenge</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>. Ein (nicht linearer) <a href="Operator_(Mathematik)" title="Operator (Mathematik)">Operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon M\rightarrow V^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle F\colon M\rightarrow V^{*}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a023a0ffcb67c522677e9705538e08f03b32d486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.802ex; height:2.343ex;" alt="{\displaystyle F\colon M\rightarrow V^{*}}" loading="lazy"></span> heißt monoton, falls für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in M}">

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<annotation encoding="application/x-tex">{\displaystyle x,y\in M}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea304ca242a255b620d3dd16ec47f19efc2e7ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.802ex; height:2.509ex;" alt="{\displaystyle x,y\in M}" loading="lazy"></span> die Ungleichung

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<annotation encoding="application/x-tex">{\displaystyle \langle x-y,F(x)-F(y)\rangle \geq 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c95e5844c198f16948d779b9fab95ae9aed1d38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.855ex; height:2.843ex;" alt="{\displaystyle \langle x-y,F(x)-F(y)\rangle \geq 0}" loading="lazy"></span></dd></dl>

<p>gilt. Hierbei bezeichnet <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}" loading="lazy"></span> den <a href="Topologischer_Dualraum" class="mw-redirect" title="Topologischer Dualraum">topologischen Dualraum</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"></span> die <a href="Duale_Paarung" title="Duale Paarung">duale Paarung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\times V^{*}\rightarrow \mathbb {K} ,\,(x,f)\mapsto f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle V\times V^{*}\rightarrow \mathbb {K} ,\,(x,f)\mapsto f(x)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1af212eea7a10cc8c4f251c1970a12574cba0ebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.925ex; height:2.843ex;" alt="{\displaystyle V\times V^{*}\rightarrow \mathbb {K} ,\,(x,f)\mapsto f(x)}" loading="lazy"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Diesen Begriff kann man wörtlich auf allgemeinere Raumklassen, insbesondere auf <a href="Lokalkonvexer_Raum" title="Lokalkonvexer Raum">lokalkonvexe Räume</a>, übertragen. Weiter kann dieser Begriff auf mengenwertige Funktionen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon M\rightarrow {\mathcal {P}}(V^{*})}">

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle F\colon M\rightarrow {\mathcal {P}}(V^{*})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66c54be347db9bb64cc61ed57af6cd09dba7846d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.315ex; height:2.843ex;" alt="{\displaystyle F\colon M\rightarrow {\mathcal {P}}(V^{*})}" loading="lazy"></span> ausgedehnt werden. Eine solche Funktion heißt dann monoton, falls für alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in M}">

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<annotation encoding="application/x-tex">{\displaystyle x,y\in M}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea304ca242a255b620d3dd16ec47f19efc2e7ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.802ex; height:2.509ex;" alt="{\displaystyle x,y\in M}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in F(x),g\in F(y)}">

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<annotation encoding="application/x-tex">{\displaystyle f\in F(x),g\in F(y)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e79d950d7d45d5b9a0e0196876d3e7e6a7d7825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.695ex; height:2.843ex;" alt="{\displaystyle f\in F(x),g\in F(y)}" loading="lazy"></span> die Ungleichung

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<annotation encoding="application/x-tex">{\displaystyle \langle x-y,f-g\rangle \geq 0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbe20fb10866a579eb657802066658b414404085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.665ex; height:2.843ex;" alt="{\displaystyle \langle x-y,f-g\rangle \geq 0}" loading="lazy"></span></dd></dl>

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<div class="mw-heading mw-heading2"><h2 id="Anwendung">Anwendung</h2></div>

<p>Der Begriff des monotonen Operators hat viele Anwendungen in der nichtlinearen <a href="Funktionalanalysis" title="Funktionalanalysis">Funktionalanalysis</a>, insbesondere bei nichtlinearen <a href="Partielle_Differentialgleichung" title="Partielle Differentialgleichung">partiellen Differentialgleichungen</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Heinz H. Bauschke & Patrick L. Combettes: <cite style="font-style:italic">Convex Analysis and Monotone Operator Theory in Hilbert Spaces</cite>. Springer New York, New York, NY 2011, ISBN 978-1-4419-9466-0.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Monotoner+Operator&rft.au=Heinz+H.+Bauschke+%26+Patrick+L.+Combettes&rft.btitle=Convex+Analysis+and+Monotone+Operator+Theory+in+Hilbert+Spaces&rft.date=2011&rft.genre=book&rft.isbn=9781441994660&rft.place=New+York%2C+NY&rft.pub=Springer+New+York" style="display:none"> </span></li>

<li>R. E. Showalter: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Monotone Operators in Banach Space and Nonlinear Partial Differential Equations</cite>. American Mathematical Society, 2014, ISBN 978-1-4704-1280-7 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Monotoner+Operator&rft.au=R.+E.+Showalter&rft.btitle=Monotone+Operators+in+Banach+Space+and+Nonlinear+Partial+Differential+Equations&rft.date=2014&rft.genre=book&rft.isbn=9781470412807&rft.pub=American+Mathematical+Society" style="display:none"> </span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">R. E. Showalter: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Monotone Operators in Banach Space and Nonlinear Partial Differential Equations</cite>. American Mathematical Society, 2014, ISBN 978-1-4704-1280-7, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>37</span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Monotoner+Operator&rft.au=R.+E.+Showalter&rft.btitle=Monotone+Operators+in+Banach+Space+and+Nonlinear+Partial+Differential+Equations&rft.date=2014&rft.genre=book&rft.isbn=9781470412807&rft.pages=37&rft.pub=American+Mathematical+Society" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Regina S. Burachik, Alfredo N. Iusem: <i>Set-Valued Mappings and Enlargements of Monotone Operators</i>, Springer-Verlag (2008), ISBN 978-0-387-69755-0</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Klaus Deimling: <i>Nonlinear Functional Analysis.</i> 1. Auflage. Springer-Verlag, Berlin/Heidelberg 1985, ISBN 3-540-13928-1, Kap 3.</span>

</li>

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