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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...

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

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

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Multivalued monotone mappings are almost everywhere single-valued

Multivalued solutions of a linear functional equation

Currently displaying 1 – 13 of 13