Abstract. In this paper, we study the observability inequality on a kind of linear parabolic equation.To show the observability estimate, we first derive the inequalities about the norms of the solution and its gradient, then reduce a lemma of inequality from a corollary directly [1]. At last, combing these inequalities with Nash inequality and Poincare inequality, we give the proof of the observability estimate.

In this paper, we introduced a new class of generalized mixed quasi-variational inequality in Hilbert space. We get some gap functions for the generalized mixed quasi-variational inequality in term of regularized gap function and D-gap function. Further, we obtain global error bounds for the solution of the generalized mixed quasi-variational inequality.