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Existence Results of Neutral Stochastic Partial Dynamic Equations with Stepanov Terms on Time Scales

  • NEUTRAL STOCHASTIC PARTIAL DYNAMIC EQUATIONS
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Abstract

The purpose of this paper is to investigate the existence and uniqueness of the square mean weighted pseudo almost periodic solution for a neutral stochastic partial dynamic equations with weighted Stepanov-like pseudo almost periodic terms on time scales. Firstly, we introduce a time scale version of the weighted Stepanov-like pseudo-almost periodic processes. Finally, an example is provided to illustrate our abstract results.

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REFERENCES

  1. Ch. Yi Zhang, “Integration of vector-valued pseudo almost periodic functions,” Proc. Am. Math. Soc. 121, 167–174 (1994). https://doi.org/10.1090/S0002-9939-1994-1186140-8

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Y. Zhang, “Pseudo almost periodic solutions of some differential equations,” J. Math. Anal. Appl. 181, 62–76 (1994). https://doi.org/10.1006/jmaa.1994.1005

    Article  MathSciNet  MATH  Google Scholar 

  3. T. Diagana, “Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations,” Nonlinear Anal.: Theory, Methods Appl. 69, 4277–4285 (2008). https://doi.org/10.1016/j.na.2007.10.051

    Article  MATH  Google Scholar 

  4. T. Diagana, G. M. Mophou, and G. M. N’Guérékata, “Existence of weighted pseudo-almost periodic solutions to some classes of differential equations with S p-weighted pseudo almost periodic coefficients,” Nonlinear Anal.: Theory, Methods Appl. 72, 430–438 (2010). https://doi.org/10.1016/j.na.2009.06.077

    Article  MATH  Google Scholar 

  5. T. Diagana and M. Zitane, “Stepanov-like pseudo-almost periodic functions in Lebesgue spaces with variable exponents,” in New Frontiers of Multidisciplinary Research in STEAM-H (Science, Technology, Engineering, Agriculture, Mathematics, and Health), Ed. by B. Toni, Springer Proceedings in Mathematics and Statistics, Vol. 90 (Springer, Cham, 2014), pp. 295–314. https://doi.org/10.1007/978-3-319-07755-0_13

  6. M. Es-saiydy and M. Zitane, “Weighted Stepanov-like pseudo almost periodicity on time scales and applications,” Differ. Equations Dyn. Syst. (2020). https://doi.org/10.1007/s12591-020-00543-7

    Book  MATH  Google Scholar 

  7. R. Agarwal, M. Bohner, D. O’Regan, and A. Peterson, “Dynamic equations on time scales: A survey,” J. Comput. Appl. Math. 141 (1–2), 1–26 (2002). https://doi.org/10.1016/S0377-0427(01)00432-0

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Bohner and A. Peterson, “A survey of exponential functions on time scales,” Cubo Mat. Educ. 3, 285–301 (2001).

    MathSciNet  MATH  Google Scholar 

  9. M. Es-saiydy, I. Oumadane, and M. Zitane, “Stepanov stability for delayed Lotka–Volterra recurrent neural networks on time scales,” Appl. Anal. 102, 921–937 (2023). https://doi.org/10.1080/00036811.2021.1967330

  10. P. H. Bezandry and T. Diagana, “Existence of square-mean almost periodic mild solutions to some nonautonomous stochastic second-order differential equations,” Electron. J. Differ. Equations 124, 1–25 (2010).

    MathSciNet  MATH  Google Scholar 

  11. P. H. Bezandry and T. Diagana, “Existence of S 2-almost periodic solutions to a class of nonautonomous stochastic evolution equations,” Electron. J. Qual. Theory Differ. Equations 35, 1–19 (2008).

    Article  MATH  Google Scholar 

  12. G. M. N’Guérékata, “Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations,” Semigroup Forum 69, 80–86 (2004). https://doi.org/10.1007/s00233-003-0021-0

    Article  MathSciNet  MATH  Google Scholar 

  13. Z. Yan and H. W. Zhang, “Existence of Stepanov-like square-mean pseudo almost periodic solutions to partial stochastic neutral differential equations,” Ann. Funct. Anal. 6, 116–138 (2015). https://doi.org/10.15352/afa/06-1-10

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Es-saiydy, M. Zarhouni, and M. Zitane, “Weighted pseudo almost periodic solution to partial stochastic neutral evolution equations,” Asia Pac. J. Math. 9, 28924 (2022). https://doi.org/10.28924/APJM/9-1

    Article  Google Scholar 

  15. E. Hernández, “Existence results for partial neutral functional integrodifferential equations with unbounded delay,” J. Math. Anal. Appl. 292, 194–210 (2004). https://doi.org/10.1016/j.jmaa.2003.11.052

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Hernández and H. R. Henríquez, “Existence of periodic solutions of partial neutral functional differential equations with unbounded delay,” J. Math. Anal. Appl. 221, 499–522 (1998). https://doi.org/10.1006/jmaa.1997.5899

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Wu and H. Xia, “Rotating waves in neutral partial functional-differential equations,” J. Dyn. Differ. Equations 11, 209–238 (1999). https://doi.org/10.1023/A:1021973228398

    Article  MathSciNet  MATH  Google Scholar 

  18. Yo. Li and Ch. Wang, “Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales,” Adv. Differ. Equations 2012, 77 (2012). https://doi.org/10.1186/1687-1847-2012-77

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Bohner and S. Sanyal, “The stochastic dynamic exponential and geometric Brownian motion on isolated time scales,” Commun. Math. Anal. 8, 120–135 (2010).

    MathSciNet  MATH  Google Scholar 

  20. S. Zaidman, Almost-Periodic Functions in Abstract Spaces, Research Notes in Mathematics, Vol. 126 (Pitman, Boston, Mass., 1985).

  21. Yo. Li and L. Zhao, “Weighted pseudo-almost periodic functions on time scales with applications to cellular neural networks with discrete delays,” Math. Methods Appl. Sci. 40, 1905–1921 (2016). https://doi.org/10.1002/mma.4107

    Article  MathSciNet  MATH  Google Scholar 

  22. Ch.-H. Tang and H.-X. Li, “Bochner-like transform and Stepanov almost periodicity on time scales with applications,” Symmetry 10, 566 (2018). https://doi.org/10.3390/sym10110566

    Article  MATH  Google Scholar 

  23. C. Y. Zhang, “Pseudo almost periodic solutions of some differential equations, II,” J. Math. Anal. Appl. 192, 543–561 (1995). https://doi.org/10.1006/jmaa.1995.1189

    Article  MathSciNet  MATH  Google Scholar 

  24. Zh. Hu and Zh. Jin, “Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay,” Nonlinear Anal.: Theory, Methods Appl. 71, 5381–5391 (2009). https://doi.org/10.1016/j.na.2009.04.032

    Article  MATH  Google Scholar 

  25. A. Ichikawa, “Stability of semilinear stochastic evolution equations,” J. Math. Anal. Appl. 90, 12–44 (1982). https://doi.org/10.1016/0022-247X(82)90041-5

    Article  MathSciNet  MATH  Google Scholar 

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Es-saiydy, M., Zarhouni, M. & Zitane, M. Existence Results of Neutral Stochastic Partial Dynamic Equations with Stepanov Terms on Time Scales. Russ Math. 67, 73–89 (2023). https://doi.org/10.3103/S1066369X23020020

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  • DOI: https://doi.org/10.3103/S1066369X23020020

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