Convergence and Optimization Results for a History-Dependent Variational Problem
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Convergence and Optimization Results for a History-Dependent Variational Problem Mircea Sofonea1 · Andaluzia Matei2
Received: 25 June 2019 / Accepted: 17 October 2019 © Springer Nature B.V. 2019
Abstract We consider a mixed variational problem in real Hilbert spaces, defined on the unbounded interval of time [0, +∞) and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2015). Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis of a large class of nonlinear boundary value problems which, in a weak formulation, lead to history-dependent mixed variational problems. To provide an example, we illustrate our abstract results in the study of a frictional contact problem for viscoelastic materials with long memory. Keywords History-dependent operator · Mixed variational problem · Lagrange multiplier · Mosco convergence · Pointwise convergence · Optimization problem · Viscoelastic material · Frictional contact Mathematics Subject Classification (2000) 35M86 · 35M87 · 49J40 · 74M15 · 74M10
1 Introduction Mixed variational problems involving Lagrange multipliers provide a useful framework in which a large number of problems with or without unilateral constraints can be cast and can be solved numerically. They are intensively used in Solid and Contact Mechanics as well as in many engineering applications. Existence and uniqueness results in the study of stationary mixed variational problems can be found in [4, 5, 7, 8, 14], for instance. Recently,
B M. Sofonea
[email protected]
1
Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France
2
Department of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585, Craiova, Romania
M. Sofonea, A. Matei
there is an interest in the study of time-dependent mixed variational problems involving a special case of operators, the so-called history-dependent operators. Such kind of operators arise in Solid and Contact Mechanics and describe memory effects in both the constitutive law and the interface boundary conditions. Reference in the field are [23–25]. The analysis of various mixed variational problems associated to mathematical models which describe the contact between a deformable body and a foundation can be found in [9–12, 15–17] and, more recently, in [1–3, 27]. The current paper represents a continuation of our previous paper [26]. There, we considered a new class of mixed variational problems with history-dependent operators and used arguments of saddle point and fixed point in order to prove its unique solvability. Here, we consider a special case of the hi
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