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The paper deals with the existence of solutions of the form $A u = b$ with operators monotone in a broader sense, including pseudomonotone operators and operators satisfying conditions $S$ and $M$ . The first part of the paper which has a methodical character is concluded by the proof of an existence theorem for the equation on a reflexive separable Banach space with a bounded demicontinuous coercive operator satisfying condition ${(M)}_{0}$ . The second part which has a character of a survey compares various types of...

Monotone operators in divergence form with $x$ -dependent multivalued graphs

Gilles Francfort, François Murat, Luc Tartar (2004)

Bollettino dell'Unione Matematica Italiana

We prove the existence of solutions to $- div a (x, grad u) = f$ , together with appropriate boundary conditions, whenever $a (x, e)$ is a maximal monotone graph in $e$ , for every fixed $x$ . We propose an adequate setting for this problem, in particular as far as measurability is concerned. It consists in looking at the graph after a $45^{\circ}$ rotation, for every fixed $x$ ; in other words, the graph $d \in a (x, e)$ is defined through $d - e = φ (x, d + e)$ , where $φ$ is a Carathéodory contraction in $R^{N}$ . This definition is shown to be equivalent to the fact that $a (x, \cdot)$ is pointwise monotone...

Monotone operators with linear range

Jens Frehse (1983)

Banach Center Publications

Multivalued monotone mappings are almost everywhere single-valued

P. Kenderov (1976)

Studia Mathematica

Multivalued solutions of a linear functional equation

Wilhelmina Smajdor (1985)

Annales Polonici Mathematici

Non-convex perturbations of evolution equations with $m$ -dissipative operators in Banach spaces

Evgenios P. Avgerinos, Nikolaos S. Papageorgiou (1989)

Commentationes Mathematicae Universitatis Carolinae

Nonlinear diffusion equations with perturbation terms on unbounded domains

Kurima, Shunsuke (2017)

Proceedings of Equadiff 14

This paper considers the initial-boundary value problem for the nonlinear diffusion equation with the perturbation term $u_{t} + (- Δ + 1) β (u) + G (u) = g in Ω \times (0, T)$ in an unbounded domain $Ω \subset ℝ^{N}$ with smooth bounded boundary, where $N \in ℕ$ , $T > 0$ , $β$ , is a single-valued maximal monotone function on $ℝ$ , e.g., $β (r) = {| r |}^{q - 1} r (q > 0, q \neq 1)$ and $G$ is a function on $ℝ$ which can be regarded as a Lipschitz continuous operator from ${(H^{1} (Ω))}^{*}$ to ${(H^{1} (Ω))}^{*}$ . The present work establishes existence and estimates for the above problem.

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Lectures on maximal monotone operators.

Limit behaviour of trajectories involving subgradients of convex functions

Maximal elements and equilibria of generalized games for $𝒰$ -majorized and condensing correspondences.

Maximal monotone mappings in banach spaces

Maximal monotone relations and the second derivatives of nonsmooth functions

Maximal Monotonicity of Operators with Sufficiently Large Domain and Application to the Hartree Problem.

Mean value theorem for convex functionals

Méthode de descente sur un fermé non convexe

Minimal convex-valued weak $^{*}$ USCO correspondences and the Radon-Nikodým property

Monotone mappings and flows of viscous media.

Monotone operators. A survey directed to applications to differential equations