The perturbed Tikhonov's algorithm and some of its applications

Tikhonov regularization of metrically regular inclusions

“…One can also observe that relation (1.7) agrees with the nonrecursive rule with error introduced by Tossings in [29] when T is a maximal monotone operator acting in a Hilbert space. So far, we can already note that any converging sequence x n satisfying (1.5) converges to a solution to the inclusion (1.4) whenever the mapping T has closed graph.…”

“…It is well known that if argmin f = ∅ and t goes to 0 then prox 1 t fx0 strongly converges to the unique minimizer of the function x → x −x 0 2 over the set argminf (see [17,28]). In [29] Tossings extended in a very natural way this regularization method to inclusion problems. When T is a maximal monotone operator on H, Tossings proposed the following regularized method of the inclusion T (x) 0 :…”

“…Positivity Tx 0,λn (x n ) 0, (1.1) where Tx 0,λn = T + λ n (· −x 0 ) withx 0 ∈ H and λ n being a sequence of positive numbers having to go to zero. Note that in [29], Tossings only studied the case whenx 0 = 0 but it obviously doesn't affect the theory. Since Tx 0,λn is maximal monotone and strongly monotone it has a unique zero given by the resolvent of T with parameter 1/λ n .…”

On ill-posedness and stability of tensor variational inequalities: application to an economic equilibrium

“…For what concerns the variational inequalities and their approximations, there is an extensive literature. More precisely, existence and approximations of solutions to variational inequalities for various classes of operators in Hilbert and Banach spaces have been considered by Browder [11], Stampacchia [43], Mosco [30,31], Alber [2], Bakushinskii [7], Doktor and Kucera [16], Liskovets [23], Alber and Rjazantseva [3], Rjazantseva [37,38], Liskovets [24,25], McLinden [29], Tossings [46], Gwinner [19], see also Liu [26], Liu and Nashed [27,34], the related references cited in [34], and the monographs by Tikhonov [44,45], Kaplan and Tichartschke [20], Bakushinskii and Goncharskii [8,9], Vasin and Ageev [47] and others. Browder [11] and Stampacchia [43] investigated on the convergence of solutions to variational inequalities when there is no perturbation of the convex set.…”

Stability properties of the Tikhonov regularization for nonmonotone inclusions

“…This process of regularization consists in replacing an ill-posed problem with a family or a sequence of well-posed ones. In the nineteen nineties, Tossings [13] extends the Tikhonov regularization to monotone inclusions on Hilbert spaces, i.e., problems of the form T (x) 0 where T is a maximal monotone operator. Lately, Moudafi [10] deals with the regularization of Tikhonov, in the case when the operator T is γ -hypomonotone, on a Hilbert space.…”