# American Institute of Mathematical Sciences

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

 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

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.
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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