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

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

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