Proximal and dynamical approaches to equilibrium problems
Résumé
In the paper the following mixed equilibrium problem in a Hilbert space is considered: find x 2 C such that F(x, x)+hG(x), x−xi 0, 8x 2 C, where C is a nonempty convex closed subset of a real Hilbert space. For its solution a version of the proximal approach is studied. The general proximal method always includes a quadratic term of the kind |x−xn|2, which provides a process of transition from a point xn to a point xn+1. The authors replace this quadratic term by its linearization. Thus, they obtain a method of the kind F(xn+1, x)+hG(xn)+ −1(h 0(xn+1)−h 0(xk)), x−xn+1i 0, 8x 2 C, where h0 is the derivative of a given function on C. It is proved that if the function F(x, y) is monotone and convex in y for any x, F(x, x) = 0, and the operator G(x) is co-coercive, then the method weakly converges in a real Hilbert space to an equilibrium solution. Note that this result will also be true if the co-coercive condition is replaced by a monotonicity property. The Tikhonov regularization method (in the absence of disturbances and calculating auxiliary solutions approximately) is justified and its weak convergence in a Hilbert space is proved. It is known that the proximal method can always be interpreted as an implicit discret approximation of some differential equation or inclusion. Therefore the authors consider a differential inclusion for the extreme mapping. Permitting "-approximation for the function F(x, y) by means of a parametric family F(x, y, ") and assuming the strong monotonicity of the operator @F(x, y, ") (it is a very strong condition), the authors prove the convergence of the trajectory of the differential inclusion to an equilibrium solution.