EXISTENCE AND CONTINUOUS-DISCRETE ASYMPTOTIC BEHAVIOUR FOR TIKHONOV-LIKE DYNAMICAL EQUILIBRIUM SYSTEMS

We consider the regularized Tikhonov-like dynamical equilibrium problem: find u : [0, +infinity[-> H such that for a.e. t >= 0 and every y is an element of K, < u(over dot)(t),y - u(t)) F(u(t),y) + epsilon(t)< u(t),y - u(t)> >= 0, where F:K x K -> R is a monotone bifunction, K is a closed convex set in Hilbert space H and the control function epsilon(t) is assumed to tend to 0 as t -> +infinity. We first establish that the corresponding Cauchy problem admits a unique absolutely continuous solution. Under the hypothesis that integral(+infinity)(0) epsilon(t)dt < infinity, we obtain weak ergodic convergence of u(t) to x is an element of K solution of the following equilibrium problem F(x, y) >= 0, for all y is an element of K. If in addition the bifunction is assumed demipositive, we show weak convergence of u(t) to the same solution. By using a slow control integral(+infinity)(0) epsilon(t)dt = infinity and assuming that the bifunction F is 3-monotone, we show that the term epsilon(t)u(t) asymptotically acts as a Tikhonov regularization, which forces all the trajectories to converge strongly towards the element of minimal norm of the closed convex set of equilibrium points of F. Also, in the case where epsilon has a slow control property and integral(+infinity)(0) vertical bar epsilon(over dot)(t)vertical bar dt < +infinity, we show that the strong convergence property of u(t) is satisfied. As applications, we propose a dynamical system to solve saddle-point problem and a neural dynamical model to handle a convex programming problem. In the last section, we propose two Tikhonov regularization methods for the proximal algorithm. We firstly use the prox-penalization algorithm (ProxPA) by iteration x(n+1) = J(lambda n)(Fn)(x(n)) where F-n (x, y) = F(x, y) + epsilon(n) < x, y - x), and epsilon(n) is the Liapunov parameter; afterwards, we propose the descent-proximal (forward-backward) algorithm (DProxA): x(n+1) = J(lambda n)(F)((1- lambda(n)epsilon(n))x(n)). We provide low conditions that guarantee a strong convergence of these algorithms to least norm element of the set of equilibrium points.