In the early seventies A. BENSOUWAN and J. L. LIONS (cf.e.g. [2]) introduced the notion of a quasi variational inequality (q.v.i.) in connection with a problem of impulse control. C. BAIOCCHI and others (cf. e.g. [l]) succeeded in treating some free boundary problems (concerning earth dams separating water reservoirs) as q.v.i. Besides of these deep but special investigations a more general treatment of q.v.i. in abstract spaces was initiated by U. Mosco (cf. [S] and further papers) who formulated a q.v.i. as a fixed point problem and gave existence and regularity results. Also the work of R. KLUGE (cf. e.g. [S]) and others (cf. [3] and [4]) goes towards a unified theory in abstract spaces and intends mainly to develop approximation methods -as it has been carried out for variational inequalities in [ti] and for identification problems in [6, 71 and further papers.
In this contribution a q.v.i. will be considered as a problem of parameter identification (or you can say: of optimal control) in a parametric variational inequality where the defect of the solution and the parameter itself is to be minimized. This way, a lot of material is wait,ing to be applied t o the theory of q.v.i.
Here we will only take into account the work of R. KLUGE (especially his book [7]) concerning identification theory. After the introduction ($ 1) we will give in $ 2 existence theorems and a TYCHONOFF regularization procedure and in $3 an elliptic regularizat,ion procedure. Other approximation procedures such as projection and penalty methods will be investigated in a forthcoming paper.
Let Y be a reflexive BANACH space, 1(*11 its norm, U c Y weskly closed, c: U-2Y1 C(v) =i= 0, convex, closed for every vC U. Let further be A a mapping from Y x U into Y*, h an admissible functional on Y X 77, and y*E Y*. We consider t,he following 14 Math. Nachr. Bd. 104