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Structural Stability of Autonomous Semilinear Nonlocal Evolution Equations and the Related Semi-dynamical Systems

Abstract

Our work firstly investigates the unique existence and the continuous dependence (on the singular kernel and initial data) of solutions to nonlocal evolution equations on Hilbert spaces. Secondly, we prove the well-definedness of a related semi-dynamical system consisting of Lipschitz continuous mappings in the space of continuous functions by constructing a metric utilizing the kernel of nonlocal derivative. Our results extend and generalize the existing results on Caputo fractional differential equations, namely the stability and structural stability results in Diethelm and Ford (J. Math. Anal. Appl. 265, 229–248, 2002), the related semi-dynamical systems in Son and Kloeden (Vietnam J. Math. 49, 1305–1315, 2021), to the case of nonlocal differential equations.

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Acknowledgements

The authors would like to thank anonymous reviewers for their constructive comments that lead to an improvement of the paper. A part of this research was complete while the second author visited the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank VIASM for the support and hospitality. The work is partially supported by the Vietnam Ministry of Education and Training under grant number B2021-SPH-15, and by Hanoi National University of Education under grant number SPHN19-03.

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Correspondence to Nhu Thang Nguyen.

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Nguyen, T.T.H., Nguyen, N.T. & Pham, A.T. Structural Stability of Autonomous Semilinear Nonlocal Evolution Equations and the Related Semi-dynamical Systems. Vietnam J. Math. (2022). https://doi.org/10.1007/s10013-022-00572-5

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  • DOI: https://doi.org/10.1007/s10013-022-00572-5

Keywords

  • Existence and uniqueness solutions
  • Continuous dependence on the initial condition
  • Semi-dynamical systems
  • Volterra integral equations

Mathematics Subject Classification (2010)

  • 34A08
  • 34A12
  • 34B10
  • 34C11