This paper mainly considers the existence of the compact almost automorphic mild solution for a class of stochastic nonlinear differential equations. More specifically, based on $ C_{0} $-semigroup theory, Hölder inequality, Burkholder–Davis–Gundy inequality and Lebesgue dominated convergence theorem, we obtain that the $ K $-mild solution is uniformly continuous and is relatively compact, etc. Combined with the subvariant functional method, we give some sufficient conditions to make sure that there exists at least one minimal $ K $-mild solution; further, if the minimal $ K $-mild solution is unique, then it is compact and almost automorphic. Moreover, we provide an example to illustrate the main presented results.
Citation: Ping Zhu. Dynamics of the compact almost automorphic solution for a class of stochastic nonlinear differential equations[J]. AIMS Mathematics, 2025, 10(7): 15893-15911. doi: 10.3934/math.2025712
This paper mainly considers the existence of the compact almost automorphic mild solution for a class of stochastic nonlinear differential equations. More specifically, based on $ C_{0} $-semigroup theory, Hölder inequality, Burkholder–Davis–Gundy inequality and Lebesgue dominated convergence theorem, we obtain that the $ K $-mild solution is uniformly continuous and is relatively compact, etc. Combined with the subvariant functional method, we give some sufficient conditions to make sure that there exists at least one minimal $ K $-mild solution; further, if the minimal $ K $-mild solution is unique, then it is compact and almost automorphic. Moreover, we provide an example to illustrate the main presented results.
| [1] |
H. Bohr, Zur theorie der fastperiodischen funktionen, Acta Math., 46 (1925), 101–214. https://doi.org/10.1007/BF02543859 doi: 10.1007/BF02543859
|
| [2] | S. Bochner, Curvature and Betti numbers in real and complex vector bundles, Univ. Torino. Rend. Sem. Mat., 15 (1955), 225–253. |
| [3] | T. Diagana, Almost automorphic type and almost periodic type functions in abstract spaces, Switzerland: Springer International Publishing, 2013. https://doi.org/10.1007/978-3-319-00849-3 |
| [4] |
T. Zhang, Y. Li, J. Zhou, Almost automorphic strong oscillation in time-fractional parabolic equations, Fractal Fract., 7 (2023), 88. https://doi.org/10.3390/fractalfract7010088 doi: 10.3390/fractalfract7010088
|
| [5] | W. Shen, Y. Wang, D. Zhou, Almost automorphically and almost periodically forced circle flows of almost periodic parabolic equations on $S^{1}$, J. Dyn. Diff. Equat., 32, (2019), 1687–1729. https://doi.org/10.1007/s10884-019-09786-7 |
| [6] | G. N'Guérékata, Almost automorphic and almost periodic functions in abstract spaces, New York: Kluwer Academic Publishers, 2001. |
| [7] |
J. Liu, X. Song, Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations, J. Funct. Anal., 258 (2010), 196–207. https://doi.org/10.1016/j.jfa.2009.06.007 doi: 10.1016/j.jfa.2009.06.007
|
| [8] |
M. Yang, (Weighted pseudo) almost automorphic solutions in distribution for fractional stochastic differential equations driven by Levy noise, Filomat, 35 (2021), 2403–2424. https://doi.org/10.2298/FIL2107403Y doi: 10.2298/FIL2107403Y
|
| [9] |
D. Cheban, Z. X. Liu, Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations, J. Differ. Equations, 269 (2020), 3652–3685. https://doi.org/10.1016/j.jde.2020.03.014 doi: 10.1016/j.jde.2020.03.014
|
| [10] |
Y. Li, S. Shen, Compact almost automorphic function on time scales and its application, Qual. Theory Dyn. Syst., 20 (2021), 86. https://doi.org/10.1007/s12346-021-00522-5 doi: 10.1007/s12346-021-00522-5
|
| [11] |
Y. Cao, B. Li, Existence and global exponential stability of compact almost automorphic solutions for Clifford-valued high-order Hopfield neutral neural networks with $D$ operator, AIMS Math., 7 (2022), 6182–6203. https://doi.org/10.3934/math.2022344 doi: 10.3934/math.2022344
|
| [12] |
H. R. Henríquez, C. Lizama, Compact almost automorphic solutions to integral equations with infinite delay, Nonlinear Anal., 71 (2009), 6029–6037. https://doi.org/10.1016/j.na.2009.05.042 doi: 10.1016/j.na.2009.05.042
|
| [13] | A. M. Fink, Almost automorphic and almost periodic solutions which minimize functional, Tôhoku Math. J., 20 (1968), 323–332. https://doi.org/10.2748/tmj/1178243139 |
| [14] |
P. Cieutat, S. Fatajou, G. N'Guérékata, Bounded and almost automorphic solutions of some nonlinear differential equations in Banach spaces, Nonlinear Anal., 71 (2009), 674–684. https://doi.org/10.1016/j.na.2008.10.100 doi: 10.1016/j.na.2008.10.100
|
| [15] |
P. Cieutat, K. Ezzinbi, Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities, J. Funct. Anal., 260 (2011), 2598–2634. https://doi.org/10.1016/j.jfa.2011.01.002 doi: 10.1016/j.jfa.2011.01.002
|
Ping Zhu. Dynamics of the compact almost automorphic solution for a class of stochastic nonlinear differential equations[J]. AIMS Mathematics, 2025, 10(7): 15893-15911. doi: 10.3934/math.2025712
DownLoad: