We consider here a type of pseudo-monotone parametric variational inequalities on a class of Banach spaces and show that such problems admit continuous (with respect to the parameter) solutions, as long as generic existence and uniqueness conditions for these solutions are satisfied. In particular, we show that such results are valid on a class of Banach spaces whenever we deal with strong pseudo-monotonicity, while others are valid in Hilbert spaces, whenever strict monotonicity is present. We also provide examples to illustrate the new results.
In this paper a global existence and uniqueness theorem for the Cauchy problem related to the class of hyperbolic operators with double characteristics P =x 2 , depending on the parameter λ in the half-space = R 2 ×]0, +∞[ is proved. In the first part of the paper, a priori estimates in Sobolev spaces W r,2 , with r negative, have been established. Then, the regularity is studied by using these spaces and by constructing a Green's function in the half-plane for the operator of the waves.
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