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We consider an inclusion in a real Hilbert space governed by a time-dependent set of constraints and a history-dependent operator. We introduce the concept of T - well-posedness for this inclusion, associated to a given Tykhonov triple T . Next, we provide a T -well-posedness result that we use in order to deduce the continuous dependence of the solution with respect to the data. Then, we state and prove a convergence criterion to the solution of the inclusion that we use to prove a convergence result for an associate penalty problem. Moreover, we show that this criterion allows us to construct a Tykhonov triple T -which give rise to an optimal well-posedness concept for the corresponding inclusion. Finally, we use these abstract results in the study of a nonlinear viscoelastic constitutive law with long memory term and unilateral constraints. [ABSTRACT FROM AUTHOR]

Abstract We consider a fixed point problem S u = u $\mathcal {S}u=u$ where S : C ( R + ; X ) → C ( R + ; X ) ${\mathcal {S}:C(\mathbb{R}_{+};X)\to C(\mathbb{R}_{+};X)}$ is an almost history-dependent operator. First, we recall the unique solvability of the problem. Then, we introduce the concept of Tykhonov triple, provide several relevant examples, and prove the corresponding well-posedness results for the considered fixed point problem. This allows us to deduce various consequences which illustrate the stability of the solution with respect to perturbations of the operator S $\mathcal {S}$ . Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in solid and contact mechanics. To give some examples, we consider two mathematical models which describe the equilibrium of a viscoelastic body in frictionless contact with a foundation. We state the mechanical problems, list the assumptions on the data, and derive their associated fixed point formulation. Then, we illustrate the use of our abstract results in order to deduce the continuous dependence of the solution with respect to the data and parameters. We also provide the corresponding mechanical interpretations.

We consider an abstract problem P in a metric space X which has a unique solution u ∈ X. Our aim in this current paper is two folds: first, to provide a convergence criterion to the solution of Problem P, that is, to give necessary and sufficient conditions on a sequence {un} ⊂ X which guarantee the convergence un → u in the space X; second, to find a Tyknonov triple T such that a sequence {un} ⊂ X is a T-approximating sequence if and only if it converges to u. The two problems stated above, associated to the original Problem P, are closely related. We illustrate how they can be solved in three particular cases of Problem P: a variational inequality in a Hilbert space, a fixed point problem in a metric space and a minimization problem in a reflexive Banach space. For each of these problems we state and prove a convergence criterion that we use to define a convenient Tykhonov triple T which requires the condition stated above. We also show how the convergence criterion and the corresponding T -well posedness concept can be used to deduce convergence and classical well-posedness results, respectively. [ABSTRACT FROM AUTHOR]

We consider a fixed point problem S u = u where S : C (R + ; X) → C (R + ; X) is an almost history-dependent operator. First, we recall the unique solvability of the problem. Then, we introduce the concept of Tykhonov triple, provide several relevant examples, and prove the corresponding well-posedness results for the considered fixed point problem. This allows us to deduce various consequences which illustrate the stability of the solution with respect to perturbations of the operator S . Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in solid and contact mechanics. To give some examples, we consider two mathematical models which describe the equilibrium of a viscoelastic body in frictionless contact with a foundation. We state the mechanical problems, list the assumptions on the data, and derive their associated fixed point formulation. Then, we illustrate the use of our abstract results in order to deduce the continuous dependence of the solution with respect to the data and parameters. We also provide the corresponding mechanical interpretations. [ABSTRACT FROM AUTHOR]

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.

In this paper we present a unified theory of convergence results in the study of abstract problems. To this end we introduce a new mathematical object, the so-called Tykhonov triple T = (I; C), constructed by using a set of parameters I, a multivalued function and a set of sequences C. Given a problem P and a Tykhonov triple T, we introduce the notion of well-posedness of problem P with respect to T and provide several preliminary results, in the framework of metric spaces. Then we show how these abstract results can be used to obtain various convergences in the study of a nonlinear equation in reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

This work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem with unilateral constraints for elastic materials, which is governed by three parameters. A weak formulation of the problem is derived, which is in the form of an elliptic variational inequality, and the Tykhonov well-posedness of the problem is established, under appropriate assumptions on the data and parameters, with respect to a special Tykhonov triple. The proof is based on arguments on coercivity, compactness, and lower-semicontinuity. This abstract result leads to different convergence results, which establish the continuous dependence of the weak solution on the data and the parameters. Moreover, these results elucidate the links among the weak solutions of the different models. Finally, the corresponding mechanical interpretations of the conditions and the results are provided. The novelty in this work is the application of the Tykhonov well-posedness concept, which allows a unified and elegant framework for this class of static contact problems.

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.

We consider an abstract problem P K in the framework of a normed space, associated to a set of constraints K. We use the concept of T -well-posedness to deduce various convergence results to the unique solution of P K. We illustrate our abstract results in the study two representative nonlinear problems: a minimization problem and a second kind elliptic variational inequality, both in the framework of reflexive Banach spaces. Finally, we consider a static contact model with elastic materials which leads to a weak formulation in terms of a variational inequality for the displacement field. We specify the corresponding convergence results, give their numerical validation and provide various mechanical interpretations. [ABSTRACT FROM AUTHOR]

This work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem with unilateral constraints for elastic materials, which is governed by three parameters. A weak formulation of the problem is derived, which is in the form of an elliptic variational inequality, and the Tykhonov well-posedness of the problem is established, under appropriate assumptions on the data and parameters, with respect to a special Tykhonov triple. The proof is based on arguments on coercivity, compactness, and lower-semicontinuity. This abstract result leads to different convergence results, which establish the continuous dependence of the weak solution on the data and the parameters. Moreover, these results elucidate the links among the weak solutions of the different models. Finally, the corresponding mechanical interpretations of the conditions and the results are provided. The novelty in this work is the application of the Tykhonov well-posedness concept, which allows a unified and elegant framework for this class of static contact problems. [ABSTRACT FROM AUTHOR]