On the Sequential Strong-Weak Closedness of the Nemytskij Multivalued Operator

Abstract

Abstract. Let Q be a measure space, and E, F be separable Banach spaces. Given a multifunction / : fi x E -> 2 F , denote by Nf(x) the set of all measurable selections of the multifunction /(•, x(s)), for a function x : £1 -> E. In this note we obtain a general theorem on the sequential strong-weak closedness for the Nemytskij multivalued superposition operator Nf acting into a Banach space of measurable F-valued functions in the infinite-dimensional case dim F = +oo, via discovering a new relation between the Q-upper limit and the M-upper limit of a sequence of subsets of F. IntroductionClosedness-type theorems play important roles in many problems of the theories of differential / integral inclusions and optimal control. The first results of this kind were obtained in the work of C. Olech, A. Lasota, L. Cesari, C. Castaing, C. Castaing and M. Valadier, and others in 1960's decade (see references e.g. in [7,9]).The present note devotes the sequential strong-weak closedness problem for the Nemytskij multivalued superposition operator Nf generated by a multifunction / : f2 x E -> 2 F (fi is a measure space, and E, F are separable Banach spaces) and which acts into a Banach space Y of measurable F-valued functions. In the finite-dimensional case dim F < +oo the general sequential strong-weak closedness result for Nf, at least in the case of the L p -type space Y, can be immediately deduced from the above mentioned work. In the infinite-dimensional case dim F = +oo, various results on the sequential strong-weak closedness for Nf acting into the L p -type space Y 2000 Mathematics Subject Classification: Primary 47H04, 47H30; Secondary 28B20, 46E30.Key words and phrases: multivalued Nemytskij superposition operator, sequential strong-weak closedness, Q-upper limit / M-upper limit of subsets, Q-upper / M-upper semicontinuities, Banach space of measurable functions, Kôthe-Bochner space.