Weak Solutions to a System of Nonlinear Fractional Boundary Value Problems Via Variational Form

  • H. Goudarzi
  • E. ShivanianEmail author
  • S. J. Hosseini Ghoncheh


In this paper, the weak solutions to a class of nonlinear system of fractional boundary value problems are studied. Each equation of the system does have two kinds of nonlinear terms; one of them is in terms of gradient, and the other one is a nonlinear source term with respect to dependent variable. It is proved that there exists at least three distinct weak solutions by using the critical point theory and the variational method. To this aim, we apply the well-known theorem on the construction of the critical set of functionals with a weak compactness condition. Moreover, the main result is demonstrated by some examples to show its validity.


System of fractional differential equations Dirichlet condition Weak solution Critical point theory Variational method 

Mathematics Subject Classification




The authors are grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper. The second author is thankful to his wife, Fatemeh Abdolrazaghi, for her spiritual supporting and suggesting to study in this research line


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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  • H. Goudarzi
    • 1
  • E. Shivanian
    • 2
    Email author
  • S. J. Hosseini Ghoncheh
    • 3
  1. 1.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Department of Applied MathematicsImam Khomeini International UniversityQazvinIran
  3. 3.Department of Mathematics, Takestan BranchIslamic Azad UniversityTakestanIran

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