Nonlinear evolution equations in Banach lattices
Bruce Calvert · Bulletin of the American Mathematical Society · 1970
1. Nonlinear operators in a Banach lattice.We recall a Banach lattice is a Banach space X over the real numbers R, which is a lattice under the ordering â, satisfying for x, y, z in X and a^O in R, (1) x^y implies x+z^y+z, (2) xSy implies ax^ay, and (3) |*| â M implies J|*||s||y||.Following [12] we write # + = sup(#, 0) and ar~ = sup(--x, 0), giving x = x+-x~ and \x\ =x + +x~.A positive duality map / is a function from X to the dual X* with (1) (Jx, x)=\\x\\\ (2) ||J*HI*IL (3) (Jx,3/)è0ifxê0andyè0,and (4) (/*,y)=0if*±y(i.e.inf(|*| l M)=0).This was introduced in [l0].PROPOSITION 1.1.A Banach lattice has a positive duality map.If g is a convex real valued function on X, then the subgradient dg:X-» subsets of X* is defined by: w is a dg(x) iff for all u in X, g(u) *zg(x) + (w, u-x).A selection of a function F:X-+ subsets of Y is a function ƒ :X-» Y with ƒ (*) in ^(x) for * in X.PROPOSITION 1.2.If X is a Banach lattice with positive duality map J then y->2J(y + ) is a selection of the subgradient ofy-*\\y + \\ 2 .