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On variational–hemivariational inequalities in Banach spaces
Journal article   Peer reviewed

On variational–hemivariational inequalities in Banach spaces

Weimin Han and M.Z. Nashed
Communications in nonlinear science & numerical simulation, Vol.124, 107309
09/2023
DOI: 10.1016/j.cnsns.2023.107309

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Abstract

This paper is devoted to a well-posedness analysis of elliptic variational–vemivariational inequalities in Banach spaces. The differential operator associated with the variational–vemivariational inequality is assumed to be strongly monotone of a general order, in contrast to that in the majority of existing references on this subject where the differential operator is assumed to be strongly monotone of order 2. Moreover, the solution existence is proved with an approach more accessible to applied mathematicians and engineers, instead of through an abstract surjectivity result for pseudomonotone operators in existing references. Equivalent minimization principles are established for certain variational–vemivariational inequalities, which are valuable for developing efficient numerical algorithms. The theoretical results are applied to the analysis of a mixed hemivariational inequality in the study of a generalized Newtonian fluid flow problem involving a nonsmooth slip boundary condition of friction type. Existence and uniqueness of both the velocity and pressure unknowns are shown for the mixed hemivariational inequalities. •Well-posedness analysis of elliptic variational–vemivariational inequalities in Banach spaces where the associated differential operator is assumed to be strongly monotone of a general order, in contrast to that in the majority of existing references on this subject where the differential operator is assumed to be strongly monotone of order 2.•The solution existence is proved with an approach more accessible to applied mathematicians and engineers.•Equivalent minimization principles are established for certain variationalhemivariational inequalities.•Existence and uniqueness of both the velocity and pressure unknowns are shown for the mixed hemivariational inequalities.
Existence Generalized Newtonian fluid flow Hemivariational inequality Minimization principle Slip condition of friction type Uniqueness Variational–vemivariational inequality

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