Existence results for evolutionary inclusions and variational–hemivariational inequalities

Abstract

We consider an abstract first-order evolutionary inclusion in a reflexive Banach space. The inclusion contains the sum of L-pseudomonotone operator and a maximal monotone operator. We provide an existence theorem which is a generalization of former results known in the literature. Next, we apply our result to the case of nonlinear variational–hemivariational inequalities considered in the setting of an evolution triple of spaces. We specify the multivalued operators in the problem and obtain existence results for several classes of variational–hemivariational inequality problems. Finally, we illustrate our existence result and treat a class of quasilinear parabolic problems under nonmonotone and multivalued flux boundary conditions.

Keywords:

• surjectivity result
• evolutionary inclusion
• pseudomonotone operator
• variational–hemivariational inequality
• Clarke subdifferential
• nonmonotone flux parabolic problem

AMS Subject Classifications:

• 35K85
• 47J20
• 35K20
• 35K60
• 35K90
• 34G25
• 35R70

Notes

1 The research was supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme [grant agreement number 295118]; the National Science Center of Poland [grant number N N201 604640]; the International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland [grant number W111/7.PR/2012]; the National Science Center of Poland [Maestro Advanced Project number DEC-2012/06/A/ST1/00262]; the project Polonium “Mathematical and Numerical Analysis for Contact Problems with Friction” 2014/15 between the Jagiellonian University and Université de Perpignan Via Domitia.