Weak convergence under nonlinearities

Anais da Academia Brasileira de Ciências

Diego R. Moreira¹  Eduardo V. O. Teixeira¹ 

[1] ,Universidade Federal do Ceará Departamento de MatemáticaCE ,Brazil

关键词: weak continuity;    nonlinearities;    Nemytskii operator;    continuidade fraca;    não linearidades;    operador de Nemytskii;   

DOI  :  10.1590/S0001-37652003000100002

来源: SciELO

【 摘 要 】

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.

【 授权许可】

CC BY   
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