On parametric optimization in hilbert spaces

We consider parametric optimization problems of the following type; Minimize p(x,t) subject to x?C, for all t?T, where C is a weakly locally compact, closed, and convex subset of a Hilbert space H, T is a topological space, and p is a real-valued function on H×T lower semicontinuous if H is provided with the weak topology, and continuous in the second variable for all x?H. We investigate how the optimal value, and the set of optimal solutions depend on tET, We apply our results to show that the discrete eigenvalues, and the corresponding normed eigenelements of certain self-adjoint operators in Hilbert spaces depend (the latter strongly) continuous on parameters under reasonable conditions.