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

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May 2008, 7(3): 645-658. doi: 10.3934/cpaa.2008.7.645

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

Mircea Sofonea ^1, , Cezar Avramescu ^2, and Andaluzia Matei ^3,

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

Received March 2007 Revised December 2007 Published February 2008

Abstract
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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.

Keywords: weak solution., Fréchet space, frictionless contact, viscoplastic body, Fixed point, normal compliance, nonlinear operator.

Mathematics Subject Classification: Primary: 47H10, 46T20; Secondary: 74M15, 74C1.

Citation: Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure & Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645

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