Demicontinuity, hemicontinuity and monotonicity

Recently the notions of monotone, demicontinuous and hemicontinuous functions have been introduced in connection with nonlinear problems in functional analysis (Browder [l; 2; 3; 4; 5], Minty [6; 7; 8]).The object of the present note is to show that under rather general conditions, hemicontinuity is equivalent to demicontinuity for monotone functions.Let X be a (real or complex) Banach space and X* its adjoint space as the set of all bounded conjugate-linear functionals on X.The value of f (EX* at uÇiX is denoted by (ƒ, u).We use the notations -> and -^ for strong convergence in X (or in X* or in the set of real numbers) and weak* convergence in X*, respectively.Let G be a function from X to X* with domain D~D(G) C.X. G is said to be demicontinuous if u n (E.D, n -1, 2, 3, • • -, uÇ^D and H/ji 'U imply Gun-^Gu.G is hemicontinuous if uÇzD, vÇîX and u+t n v£D, where t n is a sequence of positive numbers such that t n -*0, imply G(u+t n v)-*Gu.We shall say that G is locally bounded if u n Ç.D } uÇ^D and u n -*u imply that Gu n is bounded.Obviously a demicontinuous function is hemicontinuous and locally bounded.G is said to be monotone if Re (Gu -Gv, u -v) ^0 for w, VÇE.D.These definitions may be void if D is too arbitrary.In what follows we shall assume that D is quasi-dense.By this we mean that for each uÇiD there is a dense subset M u of X such that for each vCzM u , u-\-tvÇzD for sufficiently small />0 (the smallness of / depending on v).Thus any open subset of X as well as any dense linear manifold of X is quasi-dense.