Mircea Sofonea
This paper is dedicated to the memory of Haim Brezis, a prominent mathematician whose work has impacted the research of thousands of scientists all over the world. I have had the privilege to be in contact with him concerning the organization of the 6th French-Romanian Colloquium on Applied Mathematics held in Perpignan, in August 2002.
Abstract. We consider a variational inequality in a reflexive Banach space X, governed by a history-dependent operator. The existence of a unique solution to the inequality can be proven by using a fixed point argu- ment. Based on this fixed point property, we provide necessary and sufficient conditions which guarantee the uniform convergence of a se- quence of functions to the solution of the variational inequality. We then exploit this result in the study of both a penalty method and the well-posedness analysis of the problem. Moreover, we present an example which arises in Contact Mechanics. It concerns the study of a mathematical model which describes the contact of a viscoelastic membrane with a foundation.
Keywords: variational inequality, history-dependent operator, conver- gence criterion, penalty method, well-posedness result, viscoelastic mem- brane, contact problem.
MSC: 49J40, 47J20, 47G10, 74K15, 74M15.
DOI 10.56082/annalsarscimath.2025.1.143
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics on Its Application, Volume 17 no 1, 2025
