Evolutionary Variational Inequalities with Volterra Type Operators

Abstract

In this paper, we consider the initial-value problem for parabolic variational inequalities (subdifferential inclusions) with Volterra type operators. We prove the existence and the uniqueness of the solution. Furthermore, the estimates of the solution are obtained. The results are achieved using the Banach's fixed point theorem (the principle of compression mappings). The motivation for this work comes from the evolutionary variational inequalities arising in the study of frictionless contact problems for linear viscoelastic materials with long-term memory. Also, such kind of problems have their application in constructing different models of the injection molding processes.

Optimal Control Problems for Evolutionary Variational Inequalities with Volterra type Operators

“…The main goal was achieved by satisfying necessary assumptions on input data. The result of our paper extends results studied in [24], where the existence and uniqueness of the weak solution for the initial value problem of evolution inclusions with Volterra type operators was proved.…”

“…Furthermore, the stated problem is considered in the framework of an evolution triple of spaces. The existence and uniqueness results for the initial value problem of such inclusions have been proved in [24] As our main result, we establish the existence of a solution to the stated optimal control problem under some hypothesis on data-in. Based on the preceding papers, mentioned above, we can conclude that optimal control problems for subdifferential inclusions with Volterra type operators, considered here, have not been investigated yet.…”

Unilateral Problem for a Viscoelastic Beam Equation Type p-Laplacian with Strong Damping and Logarithmic Source

“…By using results from the theory of monotone operators, was established the existence of weak solutions. In [8] was studied the problem for parabolic variational inequalities with Volterra type operators. The authors proved the existence and the uniqueness of the solution.…”