Convergence of a Time Discretization for a Nonlinear Second-Order Inclusion

Bartosz, Krzysztof; Gasiński, Leszek; Liu, Zhenhai; Szafraniec, Paweł

doi:10.1017/s0013091516000560

Cited by 3 publications

(9 citation statements)

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“…In this way, we will prove Theorem 4.9. The technique presented below is referred as the Rothe method and was already used in many references, including [5,7,8]. For a given τ > 0 we formulate the following Rothe problem.…”

Section: The Rothe Methodsmentioning

confidence: 99%

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Modeling and analysis of a contact problem for a viscoelastic rod

Bartosz¹,

Sofonea²

2016

Z. Angew. Math. Phys.

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Abstract. We consider a nonlinear viscoelastic rod which is in contact with a foundation along its length and is in contact with an obstacle at its end. The rod is acted up by body forces and, as a result, its mechanical state evolves. Our aim in this paper is twofold. The first one is to construct an appropriate mathematical model which describes the evolution of the rod. The second one is to prove the weak solvability of the problem. To this end, we use arguments on second-order inclusions with multivalued pseudomonotone operators.Mathematics Subject Classification. 49J40, 47J20, 47J22.

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Section: The Rothe Methodsmentioning

confidence: 99%

“…4. It is based on technique used recently in [5]. Here we mention that the main difficulty in the analysis of Problem P arises in the nonlinearities of this problem, which appear both in the second-order Eq.…”

Section: The Modelmentioning

confidence: 99%

Modeling and analysis of a contact problem for a viscoelastic rod

Bartosz¹,

Sofonea²

2016

Z. Angew. Math. Phys.

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“…[25,26,33] and the references therein). Other aspects including evolution inclusions or boundary conditions which are multivalued, nonmonotone, and of subdifferential form can be found in [4,5,[16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Multivalued nonmonotone dynamic boundary condition

et al. 2021

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In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

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“…In [4,5], Rothe method has been applied for more general parabolic problem, namely variational-hemivariational inequalities. The techniques introduced in [12,13] has been successfully adapted to a second order evolutionary inclusion in Chapter 5 of [10] and in [2,3]. We also refer to [11,14,17,18,20] for more result concerning Rothe method in analysis of various kinds of evolutionary hemivariational inequalities.…”

Section: Introductionmentioning

confidence: 99%

“…Such inclusions describe a dynamic viscoelastic contact problem in mechanics involving nonmonotone, multivalued frictional contact law. The detailed description of the above mechanical process can be found, for instance, in Chapter 5.6 of [9] and in Sections 6-7 of [2]. It is worth to mention that such processes are typically modeled by means of hemivariational inequalities (HVIs), which represent a weak formulation of a mechanical problems, as specified in Section 6 of [2].…”

Section: Introductionmentioning

confidence: 99%