# Monotone operator

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

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

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 , ). 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 \eqref{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

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

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 \eqref{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.

How to Cite This Entry:
Monotone operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monotone_operator&oldid=44748
This article was adapted from an original article by V.V. Zhikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article