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On Uniform h-stability of Non-autonomous Evolution Equations in Banach Spaces

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Abstract

In this paper, we are concerned with the global existence of mild solutions as well as uniform h-stability for non-autonomous evolution equations in a Banach space, where the operators in linear part (possibly unbounded) depend on time t and generate an evolution family. Under some estimates on the perturbation terms, we discuss the global uniform h-stability, the uniform h-stability and the global practical uniform h-stability using Gronwall integral inequalities. Some examples are provided to show the validity of the obtained results.

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References

  1. Acquistapace, P., Terreni, B.: A unified approach to abstract linear parabolic equations. Rendiconti del Seminario matematico della universita di Padova. 78, 47–107 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Acquistapace, P.: Evolution operators and strong solution of abstract linear parabolic equations. Differ. Integral Equ. 1, 433–457 (1988)

    MathSciNet  MATH  Google Scholar 

  3. Bensoussan, A., Prato, G.. Da.., Delfour, M.C., Mitter, S.K.: Representation and Control of Infinite Dimensional Systems, vol. 1. Birkhuser, Boston (1992)

    MATH  Google Scholar 

  4. Boruga Toma, P., Borlea Pǎtraşcu, D.I., Toth, D.M.-M.: On uniform stability with growth rates in Banach spaces. IEEE 15th International Symposium on Applied Computational Intelligence and Informatics (SACI). 000393-000396 (2021). https://doi.org/10.1109/SACI51354.2021.9465607

  5. Chen, P., Zhang, X., Li, Y.: Non-autonomous evolution equations of parabolic type with non-instantaneous impulses. Mediterr. J. Math. 118, 1–14 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Choi, S.K., Koo, N.J., Im, D.M.: \(h\)-stability for linear dynamic equations on time scales. J. Math. Anal. Appl. 324, 707–720 (2006)

    Article  MathSciNet  Google Scholar 

  7. Damak, H., Hammami, M.A., Kalitine, B.: On the global uniform asymptotic stability of time-varying systems. Differ. Equ. Dyn. Syst. 22, 113–124 (2014)

    Article  MathSciNet  Google Scholar 

  8. Damak, H., Hammami, M.A.: Stabilization and practical asymptotic stability of abstract differential equations. Numer. Func. Anal. Opt. 37, 1235–1247 (2016)

    Article  MathSciNet  Google Scholar 

  9. Damak, H., Hammami, M.A.: Asymptotic stability of a perturbed abstract differential equations in Banach spaces. Operators Matrices 14(1), 129–138 (2020)

    Article  MathSciNet  Google Scholar 

  10. Damak, H., Kicha, A., Hammami, M.A.: A converse theorem on practical \(h\)-stability of nonlinear systems. Mediterr. J. Math. 17(88), 1–18 (2020)

    MathSciNet  MATH  Google Scholar 

  11. Damak, H., Kicha, A., Hammami, M.A.: A converse theorem for practical \(h\)-stability of time-varying nonlinear systems. New Zealand J. Math. 50, 109–123 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Gil, M.: Integrally small perturbation of semigroups and stability of partial differential equations. Int. J. Partial Differ. Equ. 2013, 1–5 (2013)

    Article  Google Scholar 

  13. Goo, Y.H., Ry, D.H.: \(h\)-stability for perturbed integro-differential systems. J. Chungcheong. Math. Soc. 21, 511–517 (2008)

    Google Scholar 

  14. Hai, P.V.: On the polynomial stability of evolution families. Appl. Anal. 95, 1239–1255 (2015)

    Article  MathSciNet  Google Scholar 

  15. Hammi, M., Hammami, M.A.: Gronwall-Bellman type integral inequalities and applications to global uniform asymtotic stability. Cubo. A Math. J. 17, 53–70 (2015)

    Article  Google Scholar 

  16. Ion, A.V.: Lyapunov stability of the zero solution of a perturbed abstract parabolic non-autonomous equation. ROMAI J. 4, 105–114 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Lakshmikantham, V., Leela, S., Martynuk, A.A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1998)

    Google Scholar 

  18. Lupa, N., Megan, M., Popa, I.L.: On weak exponential stability of evolution operators in Banach spaces. Nonlinear Anal. 73, 2445–2450 (2010)

    Article  MathSciNet  Google Scholar 

  19. Megan, M.: On \(h\)-stability of evolution operators. In: Corduneanu, C. (ed.) Qualitative Problems for Differential Equations and Control Theory, pp. 33–40. World Science Publishing, Singapore (1995)

    Google Scholar 

  20. Megan, M., Sasu, B., Sasu, A.L.: On uniform exponential stability of evolution families. Riv. Mat. Univ. Parma 4, 27–43 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Megan, M., Ceausu, T., Ramneantu, M.L.: Polynomial stability of evolution operators in Banach spaces. Opusc. Mathematica. 31, 279–288 (2011)

    Article  MathSciNet  Google Scholar 

  22. Mihit, C.L.: On uniform \(h\)-stability of evolution operators in Banach spaces. Theory Appl. Math. Comput. Sci. 1, 19–27 (2016)

    MathSciNet  MATH  Google Scholar 

  23. Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  Google Scholar 

  24. Pinto, M.: Perturbations of asymptotically stable differential systems. Analysis 4, 161–175 (1984)

    Article  MathSciNet  Google Scholar 

  25. Pinto, M.: Stability of nonlinear differential systems. Appl. Anal. 43, 1–20 (1992)

    Article  MathSciNet  Google Scholar 

  26. Pinto, M.: Asymptotic integration of a system resulting from the perturbation of an \(h\)-system. J. Math. Anal. Appl. 131, 194–216 (1988)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank the editor and the reviewers for a number of valuable comments and constructive suggestions that have improved the quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences of Sax, Road of Soukra, km 3,5, BP:1171-3000, Sfax, Tunisia

    Hanen Damak

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  1. Hanen Damak
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Correspondence to Hanen Damak.

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Communicated by V. Ravichandran.

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Damak, H. On Uniform h-stability of Non-autonomous Evolution Equations in Banach Spaces. Bull. Malays. Math. Sci. Soc. 44, 4367–4381 (2021). https://doi.org/10.1007/s40840-021-01173-0

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  • DOI: https://doi.org/10.1007/s40840-021-01173-0

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