Let the nonlinear operator $ F:H\to H$ be weakly continuous on the separable Hilbert space $ H$ , i.e., if $ x_n\to x_0$ weakly then $ F(x_n)\to F(x_0)$ weakly in $ H$ . This nonlinear operator also admits local Lipschitz continuity in the strong sense.
Now, consider a sequence $ x_n(t)\in C(0,T;H)$ weakly convergent in $ C(0,T;H)$ . Is the image of this sequence, $ F(x_n(t))$ , weakly convergent in $ L^p(0,T;H)$ ?
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