Tykhonov triples and convergence results for hemivariational inequalities

Abstract

Consider an abstract Problem P in a metric space (X; d) assumed to have a unique solution u. The aim of this paper is to compare two convergence results u'n → u and u''n → u, both in X, and to construct a relevant example of convergence result un → u such that the two convergences above represent particular cases of this third convergence. To this end, we use the concept of Tykhonov triple. We illustrate the use of this new and nonstandard mathematical tool in the particular case of hemivariational inequalities in reflexive Banach space. This allows us to obtain and to compare various convergence results for such inequalities. We also specify these convergences in the study of a mathematical model, which describes the contact of an elastic body with a foundation and provide the corresponding mechanical interpretations.

On the well-posedness of variational-hemivariational inequalities and associated fixed point problems

“…The unique solvability of the problem was obtained in [5] by using a minimization principle and in [25] by using a fixed point argument associated to the resolvent of a maximal monotone operator which governs the variationalhemivariational inequality. Finally, recall that well-posedness results in the study of Problem P were obtained in [24].…”

“…A general well-posedness concept for abstract problems in metric spaces was introduced in the recent paper [23]. Based on the notion of Tykhonov triple, this concept was used in [24] in the study of hemivariational inequalities.…”

On some variational inequality-constrained control problems

“…The notion of well-posedness for unconstrained optimization problems was defined by Tykhonov [34]. Following this concept (see, for instance, [12,28]), many types of well-posedness for variational problems were introduced, namely: wellposedness of Levitin-Polyak type [11,17,19,20], and extended well-posedness (for instance, [5,6,9,14,15,22,[25][26][27]38]), α-well-posedness [23,36], and L-well-posedness [21]. Also, this tool can be useful to investigate the connected problems, namely: fixedpoint problems [3], hemivariational inequality [37], variational inequality [2,7,18], equilibrium problems [4,8], Nash equilibrium [24], complementary problems [10], etc.…”

Duality arguments for well-posedness of history-dependent variational inequalities

“…The concept of well-posedness with respect to a Tykhonov triple was introduced in [14]. It extends the concept of well-posedness for a minimization problem, introduced in the pioneering work [12] as well as the concepts of well-posedness used in [1,8,15] for various optimization problems and [2,3,4,5,6,7,13] for various classes of inequalities. This abstract concept was applied in [10] in the study of variational inequalities governed by a history-dependent operator, the so-called history-dependent variational inequalities.…”