On the maximality of the sum of two maximal monotone operators

Publicacions Ma emá iques, Vol 34 (1990), 269-271 . A bs ac ON THE MAXIMALITY OF THE SUM OF TWO MAXIMAL MONOTONE OPERATORS RIAHI HASSAN In his pape we deal wi h he maximal mono onici y o A -1- B when he wo maximal mono one ope a o s A and B de ined in a Hilbe space X a e sa is ying he condi ion :  U A (domB-domA) is a closed linea A>o subspace o X . In his no e we s udy he maximal mono onici y o he sum o wo maximal mono one ope a o s by in oducing a new weakened condi ion . The classical heo em o Rocka ella [5] and B ezis [3] ell us ha A + B is a maximal mono one ope a o whene e A and B a e so and domA l (in (domB)) z~ ~~ . A ouch [1] deal wi h he same~p oblem wi h he condi ion : 0 Ein (domA - domB) . Ou idea is o use A ouch and B ezis assump ion kind, see [2] : U A(domB - domA) is a closed linea subspace . »0 Le X be a eal Hilbe space wi h he no m 11 - 11, and scala p oduc (-, .) . De ini ion 4 .1 . A mul i alued ope a o A in X is said o be mono one i o e e y XI, x2 E X and e e y yl E Ax 1 and y2 E Ax 2 one has (yi - y2, XI - x2) >- 0- A is maximal mono one i i is maximal, ela i ely o he inclusion, in he se o all mono one ope a o s . Gi en A a maximal mono one ope a o in X, we shall deno e by domain (Le . x E domA i Ax :~ 0), espec i ely by AA = (1p)(I - Já) and Já = (I +,U) - ' o a > 0, domA i s i s Yosida app oxima ion and esol an e and .b y A ° i s minimal sec ion (Le . A ° x is he p ojec ion o ze o on Ax) . See B ezis [3] o mo e de ails . 270  R . HASSAN Theo em 4 .2 . Le X be a Hilbe space, A and B be wo maximalmono one ope a o s such ha domAndomB 7É 0 and R+ (domB-domA) is a closed linea subspace o X . Then A+B is maximal mono one . P oo : Wi hou loss o gene ali y we can assume ha 0 E domA n domB . Indeed, since he e exis s x o E domA n domB, hen he ope a o s A  and B . , de ined by A  (x) = A(xo -x) and B, :, (x) = B(xo -x) a e maximal mono one ope a o s and sa is ying : 0 E domA,, n domB  and R + (domB,, - domA  ) = IR+(domB - domA) . Le x E X and A > 0, hen, c . [4], p oposi ion 2 .6 and lemma 2 .6, A + BA is a maximal mono one ope a o . Le ua be a solu ion o he inclusion x E ua + Aua + Baua . F om [4], lemma 2 .5 and he ac ha domA n domB 7É 0, we deduce ha he se {ua ;, > 0} is bounded and included in R+ (domB-domA) . Now om [4], heo em 2 .4, we shall conclude ha A+ B is maximal mono one p o ided we show ha sup »o IIBauaIl < +oo . Indeed, le us ix some y E X, and p o e ha sup »o (Baua, y) < +oo . I yE IR + (domB - domA), he e exis s a > 0, a EdomA and b E domB such ha y = a(b - a) . Hence (Baua, y) = ,((Baua, b) - (Baua, a)) . Since B is mono one hen (B ) ,ua - Bab, u, - b) > 0 and consequen ly (Baua, y) < , ((Baua, ua - a) + IjB°bil - llua - bil) He e we use he a,c sup»o IIBabli = IIB°bll, see [4, p op . 2 .6] . On he o he hand, since x E ua + Aua + Baua, he e exis s ya E Aua such ha Baua = x - ua - ya . I ollows ha : (Baua, y) < a((x - ua - ya, u a - a) + IIB°bll - Ilua - al¡) . F om he mono onici y o A we de i e (y>, - A°a, ua - a) > 0 . Hence (Baua, y) < a(Ilu, - all(llx - uall + IIA'aJI) + Ilua - bll - IIB'bil) = (y) Thus (Baua, y) < (y) < + oo o e e y y E R+(domB- domA) . *I y 1 R+(domB- domA), we shall ha e (Baua, y) = 0 .  Indeed, since H = R+(domB-domA) is a closed linea subspace o X, hen X= H®H 1 (Le . HnHl = {0} and X = H+H1) . On he o he hand we ha e Baua E B(JBu a ), hen JBua E domD C H, and since ua E H we ge B, U, = ~( ua-JBua) E H . Hence (Baua, y) = 0, since Baua E H and. y E H 1 . We hen ha e o e e y y E X, sup a>o (Bau>,, y) < +oo, and om he Banach-S einhaus heo em, we de i e ha {Baua ; A > 0} is bounded in X, which comple es he p oo o he heo em . Rema k 4 .3 . When domA and domB a e con ex, we can omi he assump- iom domA n domB í 0, since R+(domB - domA) is a closed linea subspace o X p o ided 0 E (domB - domA) . ON THE MAXIMAL MONOTONICITY OF THESUM  271 Theo em 4 .4 . Unde he assump ions o heo em 1, .8, i we assume ha R+(co(domB) - co(do nA)) is a closed linea subspace, whe e co(do nA) is he con ex hull o domA, hen A+ B is s ill maximal mono one . The p oo o his heo em is simila o ha o heo em 4 .2 . Rema k 4 .5 . I is cleax ha he assump ion in heo em 4 .4 is weake han he condi ion o Rocka ella [5], B ezis [3], in (domA) l domB :~ 0 and he condi ion o A ouch [1] ha is : 0 E in (domB - domA) . Mo e gene ally we can ob ain he same esul when 0 E i(co(domB - co(do nA» ( he ela i e in e io ) since his condi ion implies ha o heo em 4 .4 . Re e ences 1 .  H . ATTOUCH, "On maximali y o ¡he sum o wo maximal mono one op- e a o s . Nonlinea Analysis, heo y me hods and applica ions," 5, 2, 1981, pp . 143-147 . 2 .  H . ATTOUCH AND H . BREZIS, Duali y o he sum o con ex unc ions in gene al Banach spaces, Aspec s o Ma h . and i s Appl . ( o appea ), a olume in hono o L . Nachbin, J . Ba oso, ed . No h Holland (1984) . 3 .  H . BREZIS, "Opé a eu s maximaux mono ones e semig oupes de con ac- ions dans les espaces de Hilbe ," No h -Holland, Ams e dam, 1971 . 4 .  H . BREZIS, M . CRANDALL AND A . PAZY, Pe u ba ions o maximal mono one se s, Comm . Pu e Appl . Ma h . 23 (1970), 123-144 . 5 .  ROCKAFELLAR, R .T ., On he maximali y o sums o nonlinea mono one ope a o s, T ans . Ame . Ma h . Soe . 14 9 (1970), 75-88 . 6 .  E . KRAUSS, On he maximali y o he sum o mono one ope a o s, Ma h . Nach . (1981), 199-209 . Keywo ds . Maximal mono one ope a o , Yosida app oxima ion . U .S .T .L . Mon pellie II Place Eugéne Ba aillon 34 000 Mon pellie FRANCE Rebu el 19 de Juliol de 1989