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One of the notions in [[Non-linear functional analysis|non-linear functional analysis]].
 
One of the notions in [[Non-linear functional analysis|non-linear functional analysis]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m0648501.png" /> be a [[Banach space|Banach space]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m0648502.png" /> its dual, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m0648503.png" /> be the value of a linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m0648504.png" /> at an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m0648505.png" />. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m0648506.png" />, in general non-linear and acting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m0648507.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m0648508.png" />, is called monotone if
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Let $E$ be a [[Banach space|Banach space]], $E^*$ its dual, and let $(y,x)$ be the value of a linear functional $y\in E^*$ at an element $x\in E$. An operator $A$, in general non-linear and acting from $E$ into $E^*$, is called monotone if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m0648509.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$\operatorname{Re}(Ax_1-Ax_2,x_1-x_2)\geq0\tag{1}$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485010.png" />. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485011.png" /> is called semi-continuous if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485012.png" /> the numerical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485013.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485014.png" />. An example of a semi-continuous monotone operator is the gradient of a convex Gâteaux-differentiable functional. Many functionals in variational calculus are convex and hence generate monotone operators; they are useful in the solution of non-linear integral equations and were in fact first applied there.
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for any $x_1,x_2\in E$. An operator $A$ is called semi-continuous if for any $u,v,w\in E$ the numerical function $(A(u+tv),w)$ is continuous in $t$. An example of a semi-continuous monotone operator is the gradient of a convex Gâteaux-differentiable functional. Many functionals in variational calculus are convex and hence generate monotone operators; they are useful in the solution of non-linear integral equations and were in fact first applied there.
  
Various applications of monotone operators in questions regarding the solvability of non-linear equations are based on the following theorem (see [[#References|[1]]], [[#References|[2]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485015.png" /> be a reflexive Banach space (cf. [[Reflexive space|Reflexive space]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485016.png" /> be a semi-continuous monotone operator with the property of coerciveness:
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Various applications of monotone operators in questions regarding the solvability of non-linear equations are based on the following theorem (see [[#References|[1]]], [[#References|[2]]]). Let $E$ be a reflexive Banach space (cf. [[Reflexive space|Reflexive space]]) and let $A$ be a semi-continuous monotone operator with the property of coerciveness:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485017.png" /></td> </tr></table>
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$$\lim_{\|u\|\to\infty}\frac{\operatorname{Re}(Au,u)}{\|u\|}=\infty.$$
  
Then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485018.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485019.png" /> has at least one solution.
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Then for any $f\in E$ the equation $Au=f$ has at least one solution.
  
An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485020.png" /> defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485021.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485022.png" /> is called monotone on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485023.png" /> if (1) holds for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485024.png" />, and it is called maximal monotone if it is monotone on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485025.png" /> and has no monotone proper (strict) extension.
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An operator $A$ defined on a set $D\subset E$ with values in $E^*$ is called monotone on $D$ if \ref{1} holds for any $x_1,x_2\in D$, and it is called maximal monotone if it is monotone on $D$ and has no monotone proper (strict) extension.
  
 
Research into equations with monotone operators has been stimulated to a large extent by problems in the theory of quasi-linear elliptic and parabolic equations. For example, boundary value problems for quasi-linear parabolic equations lead to equations of the form
 
Research into equations with monotone operators has been stimulated to a large extent by problems in the theory of quasi-linear elliptic and parabolic equations. For example, boundary value problems for quasi-linear parabolic equations lead to equations of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$\Lambda x+Ax=f\tag{2}$$
  
in a suitable Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485027.png" />. The same equation also arises naturally in the investigation of the [[Cauchy problem|Cauchy problem]] for an abstract [[Evolution equation|evolution equation]] with a non-linear operator in Banach spaces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485028.png" /> is reflexive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485029.png" /> is a bounded, semi-continuous and coercive operator with dense domain of definition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485030.png" />, then (2) is solvable for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064850/m06485031.png" />. The idea of monotonicity has also been applied in the problem of almost-periodic solutions of non-linear parabolic equations.
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in a suitable Banach space $E$. The same equation also arises naturally in the investigation of the [[Cauchy problem|Cauchy problem]] for an abstract [[Evolution equation|evolution equation]] with a non-linear operator in Banach spaces. If $E$ is reflexive and $A$ is a bounded, semi-continuous and coercive operator with dense domain of definition in $E$, then \ref{2} is solvable for any $f\in E^*$. The idea of monotonicity has also been applied in the problem of almost-periodic solutions of non-linear parabolic equations.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Browder,  "Non-linear parabolic boundary value problems of arbitrary order"  ''Bull. Amer. Math. Soc.'' , '''69'''  (1963)  pp. 858–861</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.J. Minty,  "On a  "monotonicity"  method for the solution of non-linear problems in Banach spaces"  ''Proc. Nat. Acad. Sci. USA'' , '''50'''  (1963)  pp. 1038–1041</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Vainberg,  R.I. Kachurovskii,  "On the variational theory of non-linear operators and equations"  ''Dokl. Akad. Nauk SSSR'' , '''129''' :  6  (1959)  pp. 1199–1202  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Vainberg,  "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.-L. Lions,  "Quelques méthodes de résolution des problèmes aux limites nonlineaires" , Dunod  (1969)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.M. Levitan,  V.V. Zhikov,  "Almost-periodic functions and differential equations" , Cambridge Univ. Press  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R.I. Kachurovskii,  "Nonlinear monotone operators in Banach spaces"  ''Russian Math. Surveys'' , '''23''' :  2  (1968)  pp. 117–165  ''Uspekhi Mat. Nauk'' , '''23''' :  2  (1968)  pp. 121–168</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Browder,  "Non-linear parabolic boundary value problems of arbitrary order"  ''Bull. Amer. Math. Soc.'' , '''69'''  (1963)  pp. 858–861</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.J. Minty,  "On a  "monotonicity"  method for the solution of non-linear problems in Banach spaces"  ''Proc. Nat. Acad. Sci. USA'' , '''50'''  (1963)  pp. 1038–1041</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Vainberg,  R.I. Kachurovskii,  "On the variational theory of non-linear operators and equations"  ''Dokl. Akad. Nauk SSSR'' , '''129''' :  6  (1959)  pp. 1199–1202  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.M. Vainberg,  "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.-L. Lions,  "Quelques méthodes de résolution des problèmes aux limites nonlineaires" , Dunod  (1969)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.M. Levitan,  V.V. Zhikov,  "Almost-periodic functions and differential equations" , Cambridge Univ. Press  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R.I. Kachurovskii,  "Nonlinear monotone operators in Banach spaces"  ''Russian Math. Surveys'' , '''23''' :  2  (1968)  pp. 117–165  ''Uspekhi Mat. Nauk'' , '''23''' :  2  (1968)  pp. 121–168</TD></TR></table>

Revision as of 17:19, 28 October 2014

One of the notions in non-linear functional analysis.

Let $E$ be a Banach space, $E^*$ its dual, and let $(y,x)$ be the value of a linear functional $y\in E^*$ at an element $x\in E$. An operator $A$, in general non-linear and acting from $E$ into $E^*$, is called monotone if

$$\operatorname{Re}(Ax_1-Ax_2,x_1-x_2)\geq0\tag{1}$$

for any $x_1,x_2\in E$. An operator $A$ is called semi-continuous if for any $u,v,w\in E$ the numerical function $(A(u+tv),w)$ is continuous in $t$. An example of a semi-continuous monotone operator is the gradient of a convex Gâteaux-differentiable functional. Many functionals in variational calculus are convex and hence generate monotone operators; they are useful in the solution of non-linear integral equations and were in fact first applied there.

Various applications of monotone operators in questions regarding the solvability of non-linear equations are based on the following theorem (see [1], [2]). Let $E$ be a reflexive Banach space (cf. Reflexive space) and let $A$ be a semi-continuous monotone operator with the property of coerciveness:

$$\lim_{\|u\|\to\infty}\frac{\operatorname{Re}(Au,u)}{\|u\|}=\infty.$$

Then for any $f\in E$ the equation $Au=f$ has at least one solution.

An operator $A$ defined on a set $D\subset E$ with values in $E^*$ is called monotone on $D$ if \ref{1} holds for any $x_1,x_2\in D$, and it is called maximal monotone if it is monotone on $D$ and has no monotone proper (strict) extension.

Research into equations with monotone operators has been stimulated to a large extent by problems in the theory of quasi-linear elliptic and parabolic equations. For example, boundary value problems for quasi-linear parabolic equations lead to equations of the form

$$\Lambda x+Ax=f\tag{2}$$

in a suitable Banach space $E$. The same equation also arises naturally in the investigation of the Cauchy problem for an abstract evolution equation with a non-linear operator in Banach spaces. If $E$ is reflexive and $A$ is a bounded, semi-continuous and coercive operator with dense domain of definition in $E$, then \ref{2} is solvable for any $f\in E^*$. The idea of monotonicity has also been applied in the problem of almost-periodic solutions of non-linear parabolic equations.

References

[1] F. Browder, "Non-linear parabolic boundary value problems of arbitrary order" Bull. Amer. Math. Soc. , 69 (1963) pp. 858–861
[2] G.J. Minty, "On a "monotonicity" method for the solution of non-linear problems in Banach spaces" Proc. Nat. Acad. Sci. USA , 50 (1963) pp. 1038–1041
[3] M.M. Vainberg, R.I. Kachurovskii, "On the variational theory of non-linear operators and equations" Dokl. Akad. Nauk SSSR , 129 : 6 (1959) pp. 1199–1202 (In Russian)
[4] M.M. Vainberg, "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley (1973) (Translated from Russian)
[5] J.-L. Lions, "Quelques méthodes de résolution des problèmes aux limites nonlineaires" , Dunod (1969)
[6] B.M. Levitan, V.V. Zhikov, "Almost-periodic functions and differential equations" , Cambridge Univ. Press (1982) (Translated from Russian)
[7] R.I. Kachurovskii, "Nonlinear monotone operators in Banach spaces" Russian Math. Surveys , 23 : 2 (1968) pp. 117–165 Uspekhi Mat. Nauk , 23 : 2 (1968) pp. 121–168
How to Cite This Entry:
Monotone operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monotone_operator&oldid=34108
This article was adapted from an original article by V.V. Zhikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article