Abstract
The paper deals with second order nonlinear evolution inclusions and their applications. First, we study an evolution inclusion involving Volterra-type integral operator which is considered within the framework of an evolution triple of spaces. We provide a result on the unique solvability of the Cauchy problem for the inclusion. Next, we examine a dynamic frictional contact problem of viscoelasticity for materials with long memory and derive a weak formulation of the model in the form of a hemivariational inequality. Then, we embed the hemivariational inequality into a class of second order evolution inclusions involving Volterra-type integral operator and indicate how the result on evolution inclusion is applicable to the model of the contact problem. We conclude with examples of the subdifferential boundary conditions for different types of frictional contact.
Highlights
An important number of problems arising in Mechanics, Physics and Engineering Science lead to mathematical models expressed in terms of nonlinear inclusions and hemivariational inequalities
The purpose of this paper is to use a recent result on unique solvability of the following second order evolution inclusion t u (t) + A t, u (t) + B t, u(t) + C(t − s)u(s) ds + F t, u(t), u (t) f (t) which is considered on a finite time interval in the framework of evolution triple of spaces (V, H, V ∗) and show how the result on the evolution inclusion is applicable to the model of the contact problem
In order to formulate Problem (HVI) in the form of evolution inclusion, we extend the pointwise relations (6) and (7) to relations involving multifunctions
Summary
An important number of problems arising in Mechanics, Physics and Engineering Science lead to mathematical models expressed in terms of nonlinear inclusions and hemivariational inequalities. For this reason the mathematical literature dedicated to this field is extensive and the progress made in the last decades is impressive. In order to illustrate the cross fertilization between rigorous mathematical description and Nonlinear Analysis on one hand, and modeling and applications on the other hand, we provide several examples of contact and friction subdifferential boundary conditions. We recall a result on the existence and uniqueness of solutions to the Cauchy problem for the second order nonlinear evolution inclusion involving a Volterratype integral operator.
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