Equations d’evolution du second ordre associees a des operateurs monotones

This paper extends some recent results of V. Barbu. It is concerned with bounded solutions of the problem:u″∈Au, u′(0)∈ϖj(u(0)−a) whereA is a maximal monotone operator in a Hilbert spaceH, a∈D(A) andj is a strictly convex l.s.c. function fromH to 0,+∞]. An existence and uniqueness theorem for this problem is proved. Takingj to be the indicator function of a pointu ₀∈D(A), one obtains a bounded solutionu(t) of the initial value problem:u″∈Au, u(0)=u ₀. Denotingu(t)=S _1/2(t)u₀ one obtains a semi-group of contractions onD(A). The generator of this semigroup is denoted byA _1/2. Further properties ofS _1/2(t) andA _1/2 are studied.