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State-Dependent Implicit Sweeping Process in the Framework of Quasistatic Evolution Quasi-Variational Inequalities

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Abstract

This paper deals with the existence and uniqueness of solutions for a class of state-dependent sweeping processes with constrained velocity in Hilbert spaces without using any compactness assumption, which is known to be an open problem. To overcome the difficulty, we introduce a new notion called hypomonotonicity-like of the normal cone to the moving set, which is satisfied by many important cases. Combining this latter notion with an adapted Moreau’s catching-up algorithm and a Cauchy technique, we obtain the strong convergence of approximate solutions to the unique solution, which is a fundamental property. Using standard tools from convex analysis, we show the equivalence between this implicit state-dependent sweeping processes and quasistatic evolution quasi-variational inequalities. As an application, we study the state-dependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics.

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Authors and Affiliations

  1. XLIM UMR-CNRS 7252, Université de Limoges, 87060, Limoges, France

    Samir Adly

  2. Laboratoire LMPEA, Université de Jijel, 18000, Jijel, Algeria

    Tahar Haddad

  3. Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Rancagua, Chile

    Ba Khiet Le

Authors
  1. Samir Adly
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  2. Tahar Haddad
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  3. Ba Khiet Le
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Correspondence to Samir Adly.

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Adly, S., Haddad, T. & Le, B.K. State-Dependent Implicit Sweeping Process in the Framework of Quasistatic Evolution Quasi-Variational Inequalities. J Optim Theory Appl 182, 473–493 (2019). https://doi.org/10.1007/s10957-018-1427-x

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  • DOI: https://doi.org/10.1007/s10957-018-1427-x

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