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Periodic systems with time dependent maximal monotone operators

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Abstract

We consider a first order periodic system in ℝN, involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation. We prove the existence theorems for both the convex and nonconvex problems. We also show the existence of extremal periodic solutions and provide a strong relaxation theorem. Finally, we provide an application to nonlinear periodic control systems.

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References

  1. Bressan A, Colombo G. Extensions and selections of maps with decomposable values. Studia Math, 1988, 90: 69–86

    Article  MathSciNet  Google Scholar 

  2. Egghe L. Stopping Time Techniques for Analysts and Probabilists. Cambridge: Cambridge Univ Press, 1984

    Book  Google Scholar 

  3. Frigon M. Systems of first order differential inclusions with maximal monotne terms. Nonlinear Anal, 2007, 66: 2064–2077

    Article  MathSciNet  Google Scholar 

  4. Gasiński L, Papageorgiou N S. Nonlinear Analysis. Boca Raton: Chapman & Hall/CRC, 2006

    Google Scholar 

  5. Gasiński L, Papageorgiou N S. Nonlinear multivalued periodic systems. J Dyn Control Syst, 2019, 25: 219–243

    Article  MathSciNet  Google Scholar 

  6. Hale J. Ordinary Differential Equations. New York: Wiley-Interscience, 1969

    Google Scholar 

  7. Hartman P. On boundary value problems for systems of ordinary nonlinear second differential equations. Trans Amer Math Soc, 1960, 96: 493–509

    Article  MathSciNet  Google Scholar 

  8. Hartman P. Ordinary Differential Equations. New York: Wiley, 1964

    Google Scholar 

  9. Hu S, Papageorgiou N S. On the existence of periodic solutions for nonconvex valued differential inclusions in ℝN. Proc Amer Math Soc, 1995, 123: 3043–3050

    MathSciNet  Google Scholar 

  10. Hu S, Kandilakis D, Papageorgiou N S. Periodic solutions for nonconvex differential inclusions. Proc Amer Math Soc, 1999, 127: 89–94

    Article  MathSciNet  Google Scholar 

  11. Liu J, Liu Z H. On the existence of anti-periodic solutions for implicit differential equations. Acta Math Hungar, 2011, 132(3): 294–305

    Article  MathSciNet  Google Scholar 

  12. Liu Z H. Anti-periodic solutions to nonlinear evolution equations. Journal of Functional Analysis, 2010, 58(6): 2026–2033

    MathSciNet  Google Scholar 

  13. Hu S, Papageorgiou N S. Handbook of Multivalued Analysis, Vol I: Theory. Dordrecht: Kluwer Academic Publishers, 1997

    Book  Google Scholar 

  14. Hu S, Papageorgiou N S. Handbook of Multivalued Analysis, Vol II: Applications. Dordrecht: Kluwer Academic Publishers, 2000

    Book  Google Scholar 

  15. Hu S, Papageorgiou N S. Research Topics in Analysis. Volume I: Grounding Theory. Boston: Birkhäuser, 2022

    Book  Google Scholar 

  16. Knobloch H W. On the existence of periodic solutions for second order vector differential equations. J Differential Equ, 1971, 9: 67–85

    Article  MathSciNet  Google Scholar 

  17. Mawhin J. Some boundary value problems for Hartman-type perturbations of the ordinary p-Laplacian. Nonlinear Anal, 2000, 40: 497–503

    Article  MathSciNet  Google Scholar 

  18. Moreau J. Evolution problem associated with a moving convex set in a Hilbert space. J Differential Equ, 1977, 26: 347–374

    Article  MathSciNet  Google Scholar 

  19. Papageorgiou N S, Kyritsi-Yiallourou S. Handbook of Applied Analysis. Dordrecht: Springer, 2009

    Google Scholar 

  20. Papageorgiou N S, Rădulescu V D, Repovš D. Nonlinear Analysis-Theory and Methods. Swizerland: Springer, 2019

    Book  Google Scholar 

  21. Papageorgiou N S, Rădulescu V D. Multivalued periodic systems with maximal monotone terms. Pure Appl Funct Anal, 2018, 3: 179–192

    MathSciNet  Google Scholar 

  22. Papageorgiou N S, Winkert P. Applied Nonlinear Functional Analyss. Berlin: De Gruyter, 2018

    Book  Google Scholar 

  23. Papageorgiou N S, Winkert P. Nonlinear systems with Hartman-type perturbations. Monats für Math, 2019, 190: 389–404

    Article  MathSciNet  Google Scholar 

  24. Pavel N. Differential equations associated with compact evolution generators. Diff Integral Equ, 1988, 1: 117–123

    MathSciNet  Google Scholar 

  25. Selamnia F, Azzam Laouir D, Monteiro Marques M. Evolution problems involving state-dependent maximal monotone operators. Appl Anal, 2022, 101: 297–313

    Article  MathSciNet  Google Scholar 

  26. Vladimirov A. Nonstationary dissipative evolution equations in a Hilbert space. Nonlinear Anal, 1991, 17: 499–518

    Article  MathSciNet  Google Scholar 

  27. Vilches E, Nguyen B T. Evolution inclusions governed by time-dependent maximal monoone operators with full domain. Set Valued Var Anal 2020, 28: 569–581

    Article  MathSciNet  Google Scholar 

  28. Zhao J, Gan C M, Liu Z H. Differential Evolution Hemivariational Inequalities with Anti-periodic Conditions. Acta Mathematica Sinica, English Ser, 2023. DOI: https://doi.org/10.1007/s10114-023-2065-2

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Authors and Affiliations

  1. Center for Applied Mathematics of Guangxi, Yulin Normal University, Yulin, 537000, China

    Zhenhai Liu

  2. Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, College of Mathematics and Physics, Guangxi Minzu University, Nanning, 530006, China

    Zhenhai Liu

  3. Department of Mathematics, National Technical University, Zografou Campus, 15780, Athens, Greece

    Nikolaos S. Papageorgiou

Authors
  1. Zhenhai Liu
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  2. Nikolaos S. Papageorgiou
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Correspondence to Zhenhai Liu.

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Conflict of Interest The authors declare that they have no conflict of interest.

Additional information

The work was supported by the NSFC (12071413), the Guangxi Natural Science Foundation (2023GXNSFAA026085) and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH.

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Liu, Z., Papageorgiou, N.S. Periodic systems with time dependent maximal monotone operators. Acta Math Sci 44, 1280–1300 (2024). https://doi.org/10.1007/s10473-024-0406-6

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  • DOI: https://doi.org/10.1007/s10473-024-0406-6

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