ON THE MAXIMALITY OF THE SUM OF TWO MAXIMAL MONOTONE OPERATORS

In this note we study the maximal monotonicity of the sum of two maximal monotone operators by introducing a new weakened condition . The classical theorem of Rockafellar [5] and Brezis [3] tell us that A + B is a maximal monotone operator whenever A and B are so and domA fl (int(domB)) z~ ~~ . Attouch [1] dealt with the same~problem with the condition : 0 E int(domA domB). Our idea is to use Attouch and Brezis assumption kind, see [2] :


A bstrac t ON THE MAXIMALITY OF THE SUM OF TWO MAXIMAL MONOTONE OPERATORS RIAHI HASSAN
In this paper we deal with the maximal monotonicity of A -1-B when the two maximal monotone operators A and B defined in a Hilbert space X are satisfying the condition : U A (domB-domA) is a closed linear A>o subspace of X .
In this note we study the maximal monotonicity of the sum of two maximal monotone operators by introducing a new weakened condition .The classical theorem of Rockafellar [5] and Brezis [3] tell us that A + B is a maximal monotone operator whenever A and B are so and domA fl (int(domB)) z~~~.Attouch [1] dealt with the same~problem with the condition : 0 E int(domA -domB) .
Our idea is to use Attouch and Brezis assumption kind, see [2] : U A(domB -domA) is a closed linear subspace. »0 Let X be a real Hilbert space with the norm 11 -11, and scalar product (-, .) .Definition 4.1.A multivalued operator A in X is said to be monotone if for every XI, x2 E X and every yl E Ax 1 and y2 E Ax2 one has (yi -y2, XI -x2) >-0-A is maximal monotone if it is maximal, relatively to the inclusion, in the set of all monotone operators .
Given A a maximal monotone operator in X, we shall denote by domain (Le .x E domA if Ax :~0), respectively by AA = (1p)(I -Já) and Já = (I +,U)-' for a > 0, domA its its Yosida approximation and resolvante and .byA°its minimal section (Le .A°x is the projection of zero on Ax).See Brezis [3] for more details.Theorem 4 .2.Let X be a Hilbert space, A and B be two maximal monotone operators such that domAndomB 7É 0 and R+ (domB-domA) is a closed linear subspace of X.Then A+B is maximal monotone .
Let x E X and A > 0, then, cf.[4], proposition 2.6 and lemma 2 .6,A + BA is a maximal monotone operator .Let ua be a solution of the inclusion x E ua + Aua + Baua.
Indeed, let us fix some y E X, and prove that sup »o (Baua, y) < +oo .
The proof of this theorem is similar to that of theorem 4.2.Remark 4.5.It is cleax that the assumption in theorem 4.4 is weaker than the condition of Rockafellar [5], Brezis [3], int(domA) fl domB :~0 and the condition of Attouch [1] that is : 0 E int(domB -domA).More generally we can obtain the same result when 0 E ri(co(domBco(dornA» (the relative interior) since this condition implies that of theorem 4.4.