Volterra-type Variational Inequalities

The variational inequalities studied in this chapter involve the Volterra operator, (4.42), and therefore are history dependent. The term containing this operator appears as a perturbation of the bilinear form a and it is not involved in the function j; even if various other cases may be considered, we made this choice since it is suggested by the structure of variational models that describe the frictional contact of viscoelastic materials with long memory. We consider two types of variational inequalities with a Volterra integral term: the .rst one is elliptic, and the second one is evolutionary. For both inequalities, we provide existence and uniqueness results. Finally, we study the behavior of the solution with respect to the integral term and with respect to the nondi.erentiable function and provide convergence results. The results presented in this chapter will be applied in the study of antiplane frictional contact problems involving viscoelastic materials with long memory. Everywhere in this chapter, X is a real Hilbert space with the inner product (·,·)X, and the norm $||\cdot||\ X$, and [0, T] denotes the time interval of interest, where T > 0.