Weak convergence under nonlinearities

Abstract

In this paper, we prove that if a Nemytskii operator maps Lp( , E) into Lq( , 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( ), providing the weak continuity of some kind of resolvent operator associated to it and getting a regularity result for such solution.

On the Brezis–Lieb lemma without pointwise convergence

“…There have been some modifications of Brezis-Lieb lemma, in literature, namely [3,4], but we could not find any related results without the assumption of the a.e. convergence.…”

Existence Results for Quasi-Variational Inequalities with Multivalued Perturbations of Maximal Monotone Mappings

“…(b) Another (simpler) example, also based on sequences of fast oscillating functions, is suggested by [28,29]. Let us consider now f : (0, 1) × R × R → R given by f (x, u, ξ) = ξ + = max{ξ, 0}, for x ∈ 0, π 2 , u, ξ ∈ R. Again, f is a continuous, and in particular, a Carathéodory function.…”

Existence of Solutions of a Nonlinear Hammerstein Integral Equation

“…It should be noticed that, in general, the Nemytskii operator N f is not weakly continuous. In fact, even in the scalar case, only linear functions generate weakly continuous Nemytskii operators in L 1 spaces (see, e.g., [1,24]). …”