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On the Stochastic Evolution Equation Driven by Brownian Motion in a Separable Space

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Abstract

This article discusses issues surrounding the concept of a double measure Stepanov-type pseudo-almost periodic (‘pap’) mean-square process with a double measures. Moreover, using stochastic analysis techniques and Banach’s fixed point Theorem, we investigate the uniqueness and existence of the ‘pap’ solution to partial stochastic neutral differential equations with double measure mean-square ’pap’ coefficients of the Stepanov type driven by the Brownian motion in a separable Hilbert space \({\mathcal {K}}\). Therefore, we study its global exponential stability. The concluding segment of our work is exemplified by a practical illustration, affirming the reliability and applicability of our findings.

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References

  1. Belmabrouk, N., Damak, M., Miraoui, M.: Measure pseudo almost periodic solution for a class of nonlinear delayed stochastic evolution equations driven by Brownian motion. Filomat. 35(2), 515–534 (2021)

    Article  MathSciNet  Google Scholar 

  2. Belmabrouk, N., Damak, M., Miraoui, M.: Stochastic Nicholson’s blowflies model with delays. Int. J. Biomath. 16(01), 2250065 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blot, J., Cieutat, P., Ezzinbi, K.: New approch for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications. Appl. Anal. 92(3), 493–526 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cherif, F., Miraoui, M.: New results for a Lasota-Wazewska model. Int. J. Biomath. 12(2), 1950019 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dads, E.A., Ezzinbi, K., Miraoui, M.: \((\mu,\nu )-\)Pseudo almost automorphic solutions for some nonautonomous differential equations. Int. J. Math. 26(11), 1550090 (2015)

    Article  MATH  Google Scholar 

  6. Diaga, T., Ezzinbi, K., Miraoui, M.: Pseudo almost periodic and pseudo almost automorphic solutions to some evolution equations involving theoretical measure theory. CUBO A Math. J. 16, 1061–1093 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Diop, M.A., Ezzinbi, K., Mbaye, M.M.: Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion. Stochastics An Int. J. Prob. Stochastic Processes. 87(6), 1061–1093 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ghanmi, B., Miraoui, M.: Stability of unique pseudo almost periodic solutions with measure. Appl. Math. 64(4), 421–445 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hale, J.K.: Theory of Functional Differential Equations. Springer-Verlag, New York (1977)

    Book  MATH  Google Scholar 

  10. Hernandez, E.: Regularity of solutions of partial neutral functional differential equations with unbounded delay. Proyecciones (Antofagasta). 21(1), 65–95 (2002)

    Article  MathSciNet  Google Scholar 

  11. Hu, Z., Jin, Z.: Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay. Nonlinear Anal. Theory Methods Appl. 71, 5381–5391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kolmanovskii, V., Myshkis, A.: Introduction to the theory and applications of functional differential equations, Springer Science and Business Media. Vol(463), (2013)

  13. Mao, X.: Stochastic differential equations and applications, Elseiver. (2007)

  14. Miraoui, M.: Existence of \(\mu -\)pseudo almost periodic solutions to some evolution equations. Math. Methods in the Appl. Sci. 40(13), 4716–4726 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Miraoui, M.: Measure pseudo almost periodic solutions for differential equations with reflection. Appl. Anal. 101, 938–951 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Miraoui, M.: \(\mu -\)pseudo almost automorphic solutions for some differential equations with reflection of the argument. Numer. Funct. Anal. Optim. 38, 371–394 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Miraoui, M., Missaoui, M.: Existence and exponential stability of the piecewise pseudo almost periodic mild solution for some partial impulsive stochastic neutral evolution equations. Math. Methods in the Appl. Sci. (2023). https://doi.org/10.1002/mma.9465

    Article  MathSciNet  MATH  Google Scholar 

  18. Miraoui, M., Yaakobi, N.: Measure pseudo almost periodic solutions of shunting inhibitory cellular neural networks with mixed delays. Numer. Funct. Anal. Optim. 40(5), 571–585 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Missaoui, M., Rguigui, H., Wannes, S.: Generalized Riccati Wick differential equation and applications. São Paulo J. Math. Sci. 14(4), 1–16 (2020)

    MathSciNet  MATH  Google Scholar 

  20. N’Guérékata, G.M.: Almost automorphic and almost periodic functions in abstract spaces, Springer Science and Business Media. (2001)

  21. Slutsky, E.: Sur les fonctions aléatoires presque périodiques et sur la decomposition des functions aléatoires. Actualités Sceintifiques et industrielles, Actualités scientifiques et industrielles. 738, 33–55 (1938)

    MATH  Google Scholar 

  22. Xia, Z.N.: Weighted pseudo asymptotically periodic mild solutions of evolution equations. Acta Math. Sinica, English Series. 31, 1215–1232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yan, Z., Zhang, H.: Existence of Stepanov-like square-mean pseudo almost periodic solutions to partial stochastic neutral diffential equations. Ann. Funct. Anal. 6, 116–138 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, C.Y.: Pseudo almost periodic solutions of some differential equations I, J. Math. Anal. Appl. 151, 62-76 (1994)

  25. Zhang, C.Y.: Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl., 192, 543-561 (1995)

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Authors and Affiliations

  1. IPEIK, Kairouan University, Kairouan, Tunisia

    Mohsen Miraoui

  2. FSS, Sfax University, Sfax, Tunisia

    Sonia Missaoui

Authors
  1. Mohsen Miraoui
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  2. Sonia Missaoui
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Correspondence to Mohsen Miraoui.

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Miraoui, M., Missaoui, S. On the Stochastic Evolution Equation Driven by Brownian Motion in a Separable Space. Complex Anal. Oper. Theory 17, 132 (2023). https://doi.org/10.1007/s11785-023-01441-w

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