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Actes Du Congres International Des Mathematiciens: 1/10 by Comité d'Organisation du Congrès

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1 to prove a fundamental result in theory of maximal monotone operators due to G. Minty and F. Browder. 2. Let X and X * be refexive and strictly convex. Let A c X X X * be a monotone subset of X X X * and let J : X + X * be the duality mapping of X . Then A is maximal monotone if and only if, for any A > 0 (equivalently, for some A > O), R( A + A J ) = X * . Proof: “If” part. Assume that R ( A + A J ) = X * for some A > 0. We suppose that A is not maximal monotone, and argue from this to a contradiction.

1. 1. Definitions and Basic Results If X and Y are two linear spaces, we will denote by X X Y their Cartesian product. The elements of X X Y will be written as [ x , y ] where x E X and y E Y. If A is a multivalued operator from X to Y , we may identify it with its graph in X X Y : {[x,yIE X X y ; y E W. 1) Conversely, if A c X X Y , then we define Ax = R(A) = D ( A ) = { x E X ; Ax { y E Y ;[ x , y ] € A } , u X€D(A) Ax, A-' = ( [ y , ~ [] x;, Y l € A } . 3) 36 2. Nonlinear Operators of Monotone Type In this way here and in the following, we shall identify the operators from X to Y with their graphs in X X Y and so we shall equivalently speak of subsets of X X Y instead of operators from X to Y.

20), ( x - u , B u A ) I1imsup(xan - u , B x a n ) I n-m ( U - X , U ) V[u,u] € A . 1. 1. 1 to prove a fundamental result in theory of maximal monotone operators due to G. Minty and F. Browder. 2. Let X and X * be refexive and strictly convex. Let A c X X X * be a monotone subset of X X X * and let J : X + X * be the duality mapping of X . Then A is maximal monotone if and only if, for any A > 0 (equivalently, for some A > O), R( A + A J ) = X * . Proof: “If” part. Assume that R ( A + A J ) = X * for some A > 0.

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