Set-Valued and Variational Analysis

, Volume 27, Issue 4, pp 949–970 | Cite as

A Class of Generalized Evolutionary Problems Driven by Variational Inequalities and Fractional Operators

  • Stanisław Migórski
  • Shengda ZengEmail author
Open Access


This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution set of the mixed quasi-variational inequality involved in system is nonempty, closed and convex. Next, the measurability and upper semicontinuity for the mixed quasi-variational inequality with respect to the time variable and state variable are established. Finally, the existence of mild solutions for the system is delivered. The approach is based on the theory of operator semigroups, the Bohnenblust-Karlin fixed point principle for multivalued mappings, and theory of fractional operators.


Fractional partial differential variational inequalities Caputo derivative Knaster-Kuratowski-Mazurkiewicz theorem Bohnenblust-Karlin fixed point principle ϕ-pseudomonotonicity Mixed quasi-variational inequalities 

Mathematics Subject Classification (2010)

47J20 49J40 35J88 26A33 34A08 



The authors would like to express their thanks to the Editors and the Reviewers for their helpful comments and advices.


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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of SciencesQinzhou UniversityQinzhouPeople’s Republic of China
  2. 2.Chair of Optimization and ControlJagiellonian University in KrakowKrakowPoland
  3. 3.Faculty of Mathematics and Computer ScienceJagiellonian University in KrakowKrakowPoland

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