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Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$
Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity
1. | School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China |
2. | Department of Mathematics, Shanghai Normal University, Shanghai 200234, China |
3. | College of Mathematics, Sichuan University, Chengdu, China |
In the setting of mean-square exponential dichotomies, we study the existence and uniqueness of mean-square almost automorphic solutions of non-autonomous linear and nonlinear stochastic differential equations.
References:
[1] |
L. Arnold,
Stochastic Differential Equations: Theory and Applications, New York, 1974. |
[2] |
S. Bochner,
A new approach to almost periodicity, J. Differential Equations, 256 (2014), 1350-1367.
doi: 10.1073/pnas.48.12.2039. |
[3] |
J. Campos and M. Tarallo,
Almost automorphic linear dynamics by Favard theory, J. Differential Equations, 256 (2014), 1350-1367.
doi: 10.1016/j.jde.2013.10.018. |
[4] |
T. Caraballo and D. Cheban,
Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅰ, J. Differential Equations, 246 (2009), 108-128.
doi: 10.1016/j.jde.2008.04.001. |
[5] |
T. Caraballo and D. Cheban,
Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅱ, J. Differential Equations, 246 (2009), 1164-1186.
doi: 10.1016/j.jde.2008.07.025. |
[6] |
K. Chang, Z. Zhao and G. M. N'Guérékata,
Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Comput. Math. Appl., 61 (2011), 384-391.
doi: 10.1016/j.camwa.2010.11.014. |
[7] |
Z. Chen and W. Lin,
Square-mean pseudo almost automorphic process and its application to stochastic evolution equations, J. Funct. Anal., 261 (2011), 69-89.
doi: 10.1016/j.jfa.2011.03.005. |
[8] |
Z. Chen and W. Lin,
Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations, J. Math. Pures Appl., 100 (2013), 476-504.
doi: 10.1016/j.matpur.2013.01.010. |
[9] |
W. A. Coppel,
Dichotomy in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, New York/Berlin, 1978.
doi: 10.1007/BFb0067780. |
[10] |
T. Diagana,
Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, 2013.
doi: 10.1007/978-3-319-00849-3. |
[11] |
H. Ding, W. Long and G. M. N'Guérékata,
Almost automorphic solutions of nonautonomous evolution equations, Nonlinear Anal., 70 (2009), 4158-4164.
doi: 10.1016/j.na.2008.09.005. |
[12] |
J. D. Dollard and C. N. Friedman,
Product Integration with Applications to Differential Equations, Addison-Wesley Publishing Company, Reading, Massachusetts, 1979.
doi: 10.1017/CBO9781107340701.005. |
[13] |
L. C. Evans,
An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, RI, 2013.
doi: 10.1090/mbk/082. |
[14] |
M. Fu and Z. Liu,
Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc., 138 (2010), 3689-3701.
doi: 10.1090/S0002-9939-10-10377-3. |
[15] |
R. D. Gill and S. Johansen,
A survey of product integration with a view toward application in survival analysis, Ann. Stat., 18 (1990), 1501-1555.
doi: 10.1214/aos/1176347865. |
[16] |
D. J. Higham,
Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numerical Anal., 38 (2000), 753-769.
doi: 10.1137/S003614299834736X. |
[17] |
R. A. Johnson,
A linear, almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc., 82 (1981), 199-205.
doi: 10.1090/S0002-9939-1981-0609651-0. |
[18] |
P. E. Kloeden and T. Lorenz,
Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.
doi: 10.1016/j.jde.2012.05.016. |
[19] |
A. G. Ladde and G. S. Ladde,
An Introduction to Differential Equations: Stochastic Modeling, Methods, and Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
doi: 10.1142/8384. |
[20] |
Z. Liu and K. Sun,
Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 226 (2014), 1115-1149.
doi: 10.1016/j.jfa.2013.11.011. |
[21] |
C. Lizama and J. G. Mesquita,
Almost automorphic solutions of non-autonomous difference equations, J. Math. Anal. Appl., 407 (2013), 339-349.
doi: 10.1016/j.jmaa.2013.05.032. |
[22] |
X. Mao,
Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
[23] |
P. R. Masani,
Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc., 61 (1947), 147-192.
doi: 10.1090/S0002-9947-1947-0018719-6. |
[24] |
J. Massera and J. Schäffer,
Linear Differential Equations and Function Spaces, in: Pure and Applied Mathematics, vol. 21, Academic Press, 1966. |
[25] |
G. M. N'Guérékata,
Topics in Almost Automorphy, Springer, New York, Boston, Dordrecht, London, Moscow, 2005. |
[26] |
O. Perron,
Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
[27] |
L. Schlesinger,
Neue Grundlagen für einen infinitesimalkalkul der Matrizen, Math. Zeit., 33 (1931), 33-61.
doi: 10.1007/BF01174342. |
[28] |
W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows
Mem. Amer. Math. Soc. , 136 (1998), x+93 pp.
doi: 10.1090/memo/0647. |
[29] |
A. Slavík,
Product Integration, its History and Applications, Matfyzpress, Prague, 2007. |
[30] |
O. M. Stanzhyts'kyi,
Investigation of exponential dichotomy of Itô stochastic systems by using quadratic forms, Ukr. Mat. Zh., 53 (2001), 1545-1555.
|
[31] |
D. Stoica,
Uniform exponential dichotomy of stochastic cocycles, Stochastic Process. Appl., 120 (2010), 1920-1928.
doi: 10.1016/j.spa.2010.05.016. |
[32] |
W. A. Veech,
On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.
doi: 10.2307/1970363. |
[33] |
V. Volterra,
Sulle equazioni differenziali lineari, Rendiconti Accademia dei Lincei, 4 (1887), 393-396.
|
show all references
References:
[1] |
L. Arnold,
Stochastic Differential Equations: Theory and Applications, New York, 1974. |
[2] |
S. Bochner,
A new approach to almost periodicity, J. Differential Equations, 256 (2014), 1350-1367.
doi: 10.1073/pnas.48.12.2039. |
[3] |
J. Campos and M. Tarallo,
Almost automorphic linear dynamics by Favard theory, J. Differential Equations, 256 (2014), 1350-1367.
doi: 10.1016/j.jde.2013.10.018. |
[4] |
T. Caraballo and D. Cheban,
Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅰ, J. Differential Equations, 246 (2009), 108-128.
doi: 10.1016/j.jde.2008.04.001. |
[5] |
T. Caraballo and D. Cheban,
Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition Ⅱ, J. Differential Equations, 246 (2009), 1164-1186.
doi: 10.1016/j.jde.2008.07.025. |
[6] |
K. Chang, Z. Zhao and G. M. N'Guérékata,
Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces, Comput. Math. Appl., 61 (2011), 384-391.
doi: 10.1016/j.camwa.2010.11.014. |
[7] |
Z. Chen and W. Lin,
Square-mean pseudo almost automorphic process and its application to stochastic evolution equations, J. Funct. Anal., 261 (2011), 69-89.
doi: 10.1016/j.jfa.2011.03.005. |
[8] |
Z. Chen and W. Lin,
Square-mean weighted pseudo almost automorphic solutions for non-autonomous stochastic evolution equations, J. Math. Pures Appl., 100 (2013), 476-504.
doi: 10.1016/j.matpur.2013.01.010. |
[9] |
W. A. Coppel,
Dichotomy in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, New York/Berlin, 1978.
doi: 10.1007/BFb0067780. |
[10] |
T. Diagana,
Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, 2013.
doi: 10.1007/978-3-319-00849-3. |
[11] |
H. Ding, W. Long and G. M. N'Guérékata,
Almost automorphic solutions of nonautonomous evolution equations, Nonlinear Anal., 70 (2009), 4158-4164.
doi: 10.1016/j.na.2008.09.005. |
[12] |
J. D. Dollard and C. N. Friedman,
Product Integration with Applications to Differential Equations, Addison-Wesley Publishing Company, Reading, Massachusetts, 1979.
doi: 10.1017/CBO9781107340701.005. |
[13] |
L. C. Evans,
An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, RI, 2013.
doi: 10.1090/mbk/082. |
[14] |
M. Fu and Z. Liu,
Square-mean almost automorphic solutions for some stochastic differential equations, Proc. Amer. Math. Soc., 138 (2010), 3689-3701.
doi: 10.1090/S0002-9939-10-10377-3. |
[15] |
R. D. Gill and S. Johansen,
A survey of product integration with a view toward application in survival analysis, Ann. Stat., 18 (1990), 1501-1555.
doi: 10.1214/aos/1176347865. |
[16] |
D. J. Higham,
Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numerical Anal., 38 (2000), 753-769.
doi: 10.1137/S003614299834736X. |
[17] |
R. A. Johnson,
A linear, almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc., 82 (1981), 199-205.
doi: 10.1090/S0002-9939-1981-0609651-0. |
[18] |
P. E. Kloeden and T. Lorenz,
Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.
doi: 10.1016/j.jde.2012.05.016. |
[19] |
A. G. Ladde and G. S. Ladde,
An Introduction to Differential Equations: Stochastic Modeling, Methods, and Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
doi: 10.1142/8384. |
[20] |
Z. Liu and K. Sun,
Almost automorphic solutions for stochastic differential equations driven by Lévy noise, J. Funct. Anal., 226 (2014), 1115-1149.
doi: 10.1016/j.jfa.2013.11.011. |
[21] |
C. Lizama and J. G. Mesquita,
Almost automorphic solutions of non-autonomous difference equations, J. Math. Anal. Appl., 407 (2013), 339-349.
doi: 10.1016/j.jmaa.2013.05.032. |
[22] |
X. Mao,
Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
[23] |
P. R. Masani,
Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc., 61 (1947), 147-192.
doi: 10.1090/S0002-9947-1947-0018719-6. |
[24] |
J. Massera and J. Schäffer,
Linear Differential Equations and Function Spaces, in: Pure and Applied Mathematics, vol. 21, Academic Press, 1966. |
[25] |
G. M. N'Guérékata,
Topics in Almost Automorphy, Springer, New York, Boston, Dordrecht, London, Moscow, 2005. |
[26] |
O. Perron,
Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.
doi: 10.1007/BF01194662. |
[27] |
L. Schlesinger,
Neue Grundlagen für einen infinitesimalkalkul der Matrizen, Math. Zeit., 33 (1931), 33-61.
doi: 10.1007/BF01174342. |
[28] |
W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows
Mem. Amer. Math. Soc. , 136 (1998), x+93 pp.
doi: 10.1090/memo/0647. |
[29] |
A. Slavík,
Product Integration, its History and Applications, Matfyzpress, Prague, 2007. |
[30] |
O. M. Stanzhyts'kyi,
Investigation of exponential dichotomy of Itô stochastic systems by using quadratic forms, Ukr. Mat. Zh., 53 (2001), 1545-1555.
|
[31] |
D. Stoica,
Uniform exponential dichotomy of stochastic cocycles, Stochastic Process. Appl., 120 (2010), 1920-1928.
doi: 10.1016/j.spa.2010.05.016. |
[32] |
W. A. Veech,
On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.
doi: 10.2307/1970363. |
[33] |
V. Volterra,
Sulle equazioni differenziali lineari, Rendiconti Accademia dei Lincei, 4 (1887), 393-396.
|
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