A fixed point result with applications in the study of viscoplastic frictionless contact problems

Mircea Sofonea; Cezar Avramescu; Andaluzia Matei

doi:10.3934/cpaa.2008.7.645

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2008, Volume 7, Issue 3: 645-658. Doi: 10.3934/cpaa.2008.7.645

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A fixed point result with applications in the study of viscoplastic frictionless contact problems

1.
Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan
2.
Departement of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585, Craiova
3.
Departement of Mathematics, University of Craiov, A.I. Cuza Street 13, 200585, Craiova

Received: March 2007

Revised: December 2007

Published: May 2008

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Abstract

Let $C(\mathbb R_+;X)$ denote the Fréchet space of continuous functions defined on $\mathbb R_+=[0,\infty)$ with values on a real Banach space $(X,$ ||$\cdot$||$_X)$. We prove a fixed point theorem for operators $\Lambda:C(\mathbb R_+;X)\to C(\mathbb R_+;X)$ which satisfy a sequence of inequalities involving an integral term. Then we consider a mathematical model which describes the frictionless contact between a viscoplastic body and a deformable foundation. The process is quasistatic and is studied on the unbounded interval of time $[0,\infty)$. We provide the variational formulation of the problem, then we use the abstract fixed point theorem to prove the existence of a unique weak solution to the model. We complete our study with a regularity result.

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Mathematics Subject Classification: Primary: 47H10, 46T20; Secondary: 74M15, 74C10.

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