ON CLOSED GRAPH AND IMPLICIT FUNCTION THEOREMS FOR MULTIFUNCTIONS

We give several general implicit function and closed graph theorems fo r set-valued functions. Let Z be a normed space, X, Y metric spaces wi th X complete. Let f : X paired right arrows Z, F : X x Y paired right arrows Z be multifunctions with z(0) is an element of f(x(0)) boolean AND F(x(0), y(0)) such that f is open at (x(0), y(0)) and f 'approxim ates' F in an appropriate sense. Suppose that f(-1)(z) is closed, F(x, y) is compact for each x, y and z and suppose that F(x(0),.) is lower semi-continuous at y(0). Then F(., y) is of closed graph 'locally', i s open at x(0), and there exists a function x(.) with x(y) --> x(0) fo r y --> y(0) such that z(0) is an element of F(x(y), (y)) for all y ne ar y(0). A more general form dealing with the non-linear rate situatio n is also established.