Multi-valued monotone nonlinear mappings and duality mappings in Banach spaces
Felix E. Browder · Transactions of the American Mathematical Society · 1965
Introduction.Let I bea reflexive real Banach space, X* its conjugate space, (w, u) the pairing between w in X* and u in X.We consider multi-valued mappings T of X into X* (i.e, mappings in the ordinary sense of X into 2**) which are monotone, i.e., if veT(u), Vy eT(ux) for u and ux in X, then (V -Vy, U -Uy) ^ 0.It is our object in the present paper to generalize to the multi-valued case the results obtained in a number of recent papers by the author and G. J. Minty for single-valued mappings T (cf.[2]-[14]).The first results for multi-valued mappings for X a Hubert space have been obtained in an unpublished paper of Minty [15].The methods of [15] are not directly extendable to more general spaces, but our discussion of the finite-dimensional case (Lemma 2.1) has been very much influenced by the manuscript of [15] which Minty has recently transmitted to the author.(The basic result of [15] is stated at the end of §2 below.)Our results for general multi-valued monotone mappings have an interesting specific application given in §3 below to the generalization of a theorem of Beurling and Livingston [1] on duality mappings in Banach spaces.In a previous paper [12], we showed that for strictly convex reflexive spaces, this theorem could be obtained from results on single-valued monotone mappings.In §3 below we give a generalization of this theorem to general reflexive Banach spaces which runs as follows: Let X be a reflexive Banach space, ej)(r) a non-negative nondecreasing function on P1 with ej)(0) =0.The duality map Tof X with respect to c¡> is defined by T{u) m \v\veX*> M-*M>.> \(V,U) = \\v\\ • || M I .