Weak convergence under nonlinearities

• Autor(a) principal: MOREIRA,DIEGO R.
• Data de Publicação: 2003
• Outros Autores: TEIXEIRA,EDUARDO V. O.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Anais da Academia Brasileira de Ciências (Online)
• Texto Completo: http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652003000100002
• topic: weak continuity; nonlinearities; Nemytskii operator

In this paper, we prove that if a Nemytskii operator maps Lp(omega, E) into Lq(omega, F), for p, q greater than 1, E, F separable Banach spaces and F reflexive, then a sequence that converge weakly and a.e. is sent to a weakly convergent sequence. We give a counterexample proving that if q = 1 and p is greater than 1 we may not have weak sequential continuity of such operator. However, we prove that if p = q = 1, then a weakly convergent sequence that converges a.e. is mapped into a weakly convergent sequence by a Nemytskii operator. We show an application of the weak continuity of the Nemytskii operators by solving a nonlinear functional equation on W1,p(omega), providing the weak continuity of some kind of resolvent operator associated to it and getting a regularity result for such solution.