Abstract
We deal with the Tykhonov well-posedness of a time-dependent variational inequality defined on the unbounded interval of time ℝ+= [0, +∞ ), governed by a history-dependent operator. To this end we introduce the concept of Tykhonov triple, provide three relevant examples, then we state and prove the corresponding well-posedness results. This allows us to deduce various corollaries which illustrate the continuous dependence of the solution with respect to the data. Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in Mechanics, Physics and Engineering Sciences. To give an example, we consider a mathematical model which describes the equilibrium of a viscoelastic body in frictionless contact with a rigid foundation.
Highlights
The concept of well-posedness in the sense of Tykhonov was introduced in [23] for a minimization problem and it has been generalized for different optimization problems
{un}, we are looking for an appropriate Tykhonov triple T with whom problem P is well posed and, {un} is a T -approximating sequence
There, we prove various convergence results which show the continuous dependence of the solution with respect to the data
Summary
The concept of well-posedness in the sense of Tykhonov was introduced in [23] for a minimization problem and it has been generalized for different optimization problems. Following the arguments presented there, the well-posedness of a problem P with respect to the Tykhonov triple T implicitely provides a convergence result, namely the convergence of the approximating sequences. Examples and appropriate reference can be found in our recent papers [8, 22] This represents only a theoretical principle since the choice of such an appropriate Tykhonov triple remains an open question which depends on the problem we consider and the convergence result we are interested in, as well. The first one is to study the well-posedness of the inequality (1.2) with respect to various Tykhonov triples This will allow us to deduce various convergence results, which represents our second aim.
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