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On the initial value problem of fractional stochastic evolution equations in Hilbert spaces
1. | Department of Mathematics, Northwest Normal University, Lanzhou 730070, China, China, China |
References:
[1] |
R. P. Agarwal, V. Lakshmikantham and J. J. Nieto, On the concept of solutions for fractional differential equations with uncertainly, Nonlinear Anal., 72 (2010), 2859-2862.
doi: 10.1016/j.na.2009.11.029. |
[2] |
E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph.D thesis, Department of Mathematics, Eindhoven University of Technology, 2001. Available from: http://www.researchgate.net/publication/230675246_Fractional_Evolution_Equations_in_Banach_Spaces. |
[3] |
J. Bao and Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl., 59 (2010), 207-214.
doi: 10.1016/j.camwa.2009.08.035. |
[4] |
J. Bao, Z. Hou and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soc., 138 (2010), 2169-2180.
doi: 10.1090/S0002-9939-10-10230-5. |
[5] |
C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142.
doi: 10.1007/s00233-009-9184-7. |
[6] |
C. Chen, M. Li and F. B. Li, On boundary values of fractional resolvent families, J. Math. Anal. Appl., 384 (2011), 453-467.
doi: 10.1016/j.jmaa.2011.05.074. |
[7] |
P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728.
doi: 10.1007/s00033-013-0351-z. |
[8] |
P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces, Collect. Math., 66 (2015), 63-76 .
doi: 10.1007/s13348-014-0106-y. |
[9] |
J. Cui and L. Yan, Existence result for fractional neutral stochastic integro-differential equations with infinite delay, J. Phys. A, 44 (2011), 335201.
doi: 10.1088/1751-8113/44/33/335201. |
[10] |
J. Cui, L. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces, J. Korean Stat. Soci., 41 (2012), 279-290.
doi: 10.1016/j.jkss.2011.10.001. |
[11] |
R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430. |
[12] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223. |
[13] |
S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.
doi: 10.1016/j.jde.2003.12.002. |
[14] |
M. M. EI-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons and Fractals, 14 (2002), 433-440.
doi: 10.1016/S0960-0779(01)00208-9. |
[15] |
M. M. EI-Borai, O. L. Mostafa and H. M. Ahmed, Asymptotic stability of some stochastic evolution equations, Appl. Math. Comput., 144 (2003), 273-286.
doi: 10.1016/S0096-3003(02)00406-X. |
[16] |
Z. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput., 232 (2014), 60-67.
doi: 10.1016/j.amc.2014.01.051. |
[17] |
W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Akademic Verlag, Berlin, 1995. |
[18] |
J. Jia, J. Peng and K. Li, Well-posedness of abstract distributed-order fractional diffusion equations, Commun. Pure Appl. Anal., 13 (2014), 605-621.
doi: 10.3934/cpaa.2014.13.605. |
[19] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006. |
[20] |
V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 1677-1682.
doi: 10.1016/j.na.2007.08.042. |
[21] |
M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726.
doi: 10.1016/j.jfa.2010.07.007. |
[22] |
K. Li, J. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476-510.
doi: 10.1016/j.jfa.2012.04.011. |
[23] |
K. Li and J. Peng, Fractional resolvents and fractional evolution equations, Appl. Math. Lett., 25 (2012), 808-812.
doi: 10.1016/j.aml.2011.10.023. |
[24] |
K. Li and J. Peng, Fractional abstract Cauchy problems, Integr. Equ. Oper. Theory, 70 (2011), 333-361.
doi: 10.1007/s00020-011-1864-5. |
[25] |
K. Li and J. Peng, Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys., 65 (2014), 941-959.
doi: 10.1007/s00033-013-0369-2. |
[26] |
K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall, London, 2006. |
[27] |
J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760.
doi: 10.1016/j.jmaa.2007.11.019. |
[28] |
J. Luo and T. Taniguchi, Fixed point and stability of stochastic neutral partial differential equations with infinite delays, Stoch. Anal. Appl., 27 (2009), 1163-1173.
doi: 10.1080/07362990903259371 . |
[29] |
X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Ltd., Chichester, 1997. |
[30] |
M. Niu and B. Xie, Regularity of a fractional partial differential equation driven by space-time white noise, Proc. Amer. Math. Soc., 138 (2010), 1479-1489.
doi: 10.1090/S0002-9939-09-10197-1. |
[31] |
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. |
[32] |
T. Poinot and J. C. Trigeassou, Identification of fractional systems using an output-error technique, Nonl. Dynamics, 38 (2004), 133-154.
doi: 10.1007/s11071-004-3751-y. |
[33] |
J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[34] |
Y. Ren and R. Sakthivel, Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps, J. Math. Phys., 53 (2012), 14 pages.
doi: 10.1063/1.4739406. |
[35] |
Y. Ren, Q. Zhou and L. Chen, Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with poisson jumps and infinite delay, J. Optim. Theory Appl., 149 (2011), 315-331.
doi: 10.1007/s10957-010-9792-0. |
[36] |
Y. A. Rossikhin and M. V. Shitikova, Application of fractional dericatives to the analysis of damped vibrations of viscoelastic single mass system, Acta. Mech., 120 (1997), 109-125.
doi: 10.1007/BF01174319. |
[37] |
J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Development and Applications in Physics and Engineering, Springer, The Netherlands, 2007.
doi: 10.1007/978-1-4020-6042-7. |
[38] |
R. Sakthivel and J. Luo, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl., 356 (2009), 1-6.
doi: 10.1016/j.jmaa.2009.02.002. |
[39] |
R. Sakthivel and Y. Ren, Exponential stability of second-order stochastic evolution equations with Poisson jumps, Commu. Nonl. Sci. Nume. Simu., 17 (2012), 4517-4523.
doi: 10.1016/j.cnsns.2012.04.020. |
[40] |
R. Sakthivel, P. Revathi and N. I. Mahmudov, Asymptotic stability of fractional stochastic neutral differential equations with infinite delays, Abstr. Appl. Anal., 2013 (2013), Article ID 769257, 9 pages.
doi: 10.1155/2013/769257. |
[41] |
R. Sakthivel, P. Revathi and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal., 81 (2013), 70-86.
doi: 10.1016/j.na.2012.10.009. |
[42] |
R. Sakthivel, S. Suganyab and S. M. Anthonib, Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl., 63 (2012), 660-668.
doi: 10.1016/j.camwa.2011.11.024. |
[43] |
K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, London, 1991.
doi: 10.1007/978-94-011-3712-6. |
[44] |
T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations, 181 (2002), 72-91.
doi: 10.1006/jdeq.2001.4073. |
[45] |
M. S. Tvazoei, M. Haeri, S. Jafari, S. Bolouki and M. Siami, Some applications of fractional calculus in suppression of chaotic oscillations, IEEE Transactions on Industrial Electronics, 11 (2008), 4094-4101.
doi: 10.1109/TIE.2008.925774. |
[46] |
J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 12 (2011), 263-272.
doi: 10.1016/j.nonrwa.2010.06.013. |
[47] |
S. Westerlund and L. Ekstam, Capacitor theory, IEEE Transactions on Dielectrics and Electrical Insulation, 1 (1994), 826-839.
doi: 10.1109/94.326654. |
[48] |
Z. Yan and X. Yan, Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay, Collec. Math., 64 (2013), 235-250.
doi: 10.1007/s13348-012-0063-2. |
[49] |
Z. Yan and X. Yan, Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces, Z. Angew. Math. Phys., 64 (2013), 573-590.
doi: 10.1007/s00033-012-0249-1. |
[50] |
Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.
doi: 10.1016/j.camwa.2009.06.026. |
show all references
References:
[1] |
R. P. Agarwal, V. Lakshmikantham and J. J. Nieto, On the concept of solutions for fractional differential equations with uncertainly, Nonlinear Anal., 72 (2010), 2859-2862.
doi: 10.1016/j.na.2009.11.029. |
[2] |
E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph.D thesis, Department of Mathematics, Eindhoven University of Technology, 2001. Available from: http://www.researchgate.net/publication/230675246_Fractional_Evolution_Equations_in_Banach_Spaces. |
[3] |
J. Bao and Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. Math. Appl., 59 (2010), 207-214.
doi: 10.1016/j.camwa.2009.08.035. |
[4] |
J. Bao, Z. Hou and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soc., 138 (2010), 2169-2180.
doi: 10.1090/S0002-9939-10-10230-5. |
[5] |
C. Chen and M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121-142.
doi: 10.1007/s00233-009-9184-7. |
[6] |
C. Chen, M. Li and F. B. Li, On boundary values of fractional resolvent families, J. Math. Anal. Appl., 384 (2011), 453-467.
doi: 10.1016/j.jmaa.2011.05.074. |
[7] |
P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728.
doi: 10.1007/s00033-013-0351-z. |
[8] |
P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces, Collect. Math., 66 (2015), 63-76 .
doi: 10.1007/s13348-014-0106-y. |
[9] |
J. Cui and L. Yan, Existence result for fractional neutral stochastic integro-differential equations with infinite delay, J. Phys. A, 44 (2011), 335201.
doi: 10.1088/1751-8113/44/33/335201. |
[10] |
J. Cui, L. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces, J. Korean Stat. Soci., 41 (2012), 279-290.
doi: 10.1016/j.jkss.2011.10.001. |
[11] |
R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space, J. Differential Equations, 10 (1971), 412-430. |
[12] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9780511666223. |
[13] |
S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.
doi: 10.1016/j.jde.2003.12.002. |
[14] |
M. M. EI-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons and Fractals, 14 (2002), 433-440.
doi: 10.1016/S0960-0779(01)00208-9. |
[15] |
M. M. EI-Borai, O. L. Mostafa and H. M. Ahmed, Asymptotic stability of some stochastic evolution equations, Appl. Math. Comput., 144 (2003), 273-286.
doi: 10.1016/S0096-3003(02)00406-X. |
[16] |
Z. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput., 232 (2014), 60-67.
doi: 10.1016/j.amc.2014.01.051. |
[17] |
W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach, Akademic Verlag, Berlin, 1995. |
[18] |
J. Jia, J. Peng and K. Li, Well-posedness of abstract distributed-order fractional diffusion equations, Commun. Pure Appl. Anal., 13 (2014), 605-621.
doi: 10.3934/cpaa.2014.13.605. |
[19] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006. |
[20] |
V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 1677-1682.
doi: 10.1016/j.na.2007.08.042. |
[21] |
M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726.
doi: 10.1016/j.jfa.2010.07.007. |
[22] |
K. Li, J. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476-510.
doi: 10.1016/j.jfa.2012.04.011. |
[23] |
K. Li and J. Peng, Fractional resolvents and fractional evolution equations, Appl. Math. Lett., 25 (2012), 808-812.
doi: 10.1016/j.aml.2011.10.023. |
[24] |
K. Li and J. Peng, Fractional abstract Cauchy problems, Integr. Equ. Oper. Theory, 70 (2011), 333-361.
doi: 10.1007/s00020-011-1864-5. |
[25] |
K. Li and J. Peng, Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys., 65 (2014), 941-959.
doi: 10.1007/s00033-013-0369-2. |
[26] |
K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall, London, 2006. |
[27] |
J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl., 342 (2008), 753-760.
doi: 10.1016/j.jmaa.2007.11.019. |
[28] |
J. Luo and T. Taniguchi, Fixed point and stability of stochastic neutral partial differential equations with infinite delays, Stoch. Anal. Appl., 27 (2009), 1163-1173.
doi: 10.1080/07362990903259371 . |
[29] |
X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Ltd., Chichester, 1997. |
[30] |
M. Niu and B. Xie, Regularity of a fractional partial differential equation driven by space-time white noise, Proc. Amer. Math. Soc., 138 (2010), 1479-1489.
doi: 10.1090/S0002-9939-09-10197-1. |
[31] |
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. |
[32] |
T. Poinot and J. C. Trigeassou, Identification of fractional systems using an output-error technique, Nonl. Dynamics, 38 (2004), 133-154.
doi: 10.1007/s11071-004-3751-y. |
[33] |
J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[34] |
Y. Ren and R. Sakthivel, Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps, J. Math. Phys., 53 (2012), 14 pages.
doi: 10.1063/1.4739406. |
[35] |
Y. Ren, Q. Zhou and L. Chen, Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with poisson jumps and infinite delay, J. Optim. Theory Appl., 149 (2011), 315-331.
doi: 10.1007/s10957-010-9792-0. |
[36] |
Y. A. Rossikhin and M. V. Shitikova, Application of fractional dericatives to the analysis of damped vibrations of viscoelastic single mass system, Acta. Mech., 120 (1997), 109-125.
doi: 10.1007/BF01174319. |
[37] |
J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Development and Applications in Physics and Engineering, Springer, The Netherlands, 2007.
doi: 10.1007/978-1-4020-6042-7. |
[38] |
R. Sakthivel and J. Luo, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl., 356 (2009), 1-6.
doi: 10.1016/j.jmaa.2009.02.002. |
[39] |
R. Sakthivel and Y. Ren, Exponential stability of second-order stochastic evolution equations with Poisson jumps, Commu. Nonl. Sci. Nume. Simu., 17 (2012), 4517-4523.
doi: 10.1016/j.cnsns.2012.04.020. |
[40] |
R. Sakthivel, P. Revathi and N. I. Mahmudov, Asymptotic stability of fractional stochastic neutral differential equations with infinite delays, Abstr. Appl. Anal., 2013 (2013), Article ID 769257, 9 pages.
doi: 10.1155/2013/769257. |
[41] |
R. Sakthivel, P. Revathi and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal., 81 (2013), 70-86.
doi: 10.1016/j.na.2012.10.009. |
[42] |
R. Sakthivel, S. Suganyab and S. M. Anthonib, Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl., 63 (2012), 660-668.
doi: 10.1016/j.camwa.2011.11.024. |
[43] |
K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic Publishers, London, 1991.
doi: 10.1007/978-94-011-3712-6. |
[44] |
T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations, 181 (2002), 72-91.
doi: 10.1006/jdeq.2001.4073. |
[45] |
M. S. Tvazoei, M. Haeri, S. Jafari, S. Bolouki and M. Siami, Some applications of fractional calculus in suppression of chaotic oscillations, IEEE Transactions on Industrial Electronics, 11 (2008), 4094-4101.
doi: 10.1109/TIE.2008.925774. |
[46] |
J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 12 (2011), 263-272.
doi: 10.1016/j.nonrwa.2010.06.013. |
[47] |
S. Westerlund and L. Ekstam, Capacitor theory, IEEE Transactions on Dielectrics and Electrical Insulation, 1 (1994), 826-839.
doi: 10.1109/94.326654. |
[48] |
Z. Yan and X. Yan, Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay, Collec. Math., 64 (2013), 235-250.
doi: 10.1007/s13348-012-0063-2. |
[49] |
Z. Yan and X. Yan, Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces, Z. Angew. Math. Phys., 64 (2013), 573-590.
doi: 10.1007/s00033-012-0249-1. |
[50] |
Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.
doi: 10.1016/j.camwa.2009.06.026. |
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