On the initial value problem of fractional stochastic evolution equations in Hilbert spaces

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September 2015, 14(5): 1817-1840. doi: 10.3934/cpaa.2015.14.1817

On the initial value problem of fractional stochastic evolution equations in Hilbert spaces

Pengyu Chen ^1, , Yongxiang Li ^1, and Xuping Zhang ^1,

1.	Department of Mathematics, Northwest Normal University, Lanzhou 730070, China, China, China

Received January 2015 Revised March 2015 Published June 2015

Abstract
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In this article, we are concerned with the initial value problem of fractional stochastic evolution equations in real separable Hilbert spaces. The existence of saturated mild solutions and global mild solutions is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions by using $\alpha$-order fractional resolvent operator theory, the Schauder fixed point theorem and piecewise extension method. Furthermore, the continuous dependence of mild solutions on initial values and orders as well as the asymptotical stability in $p$-th moment of mild solutions for the studied problem have also been discussed. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract results.

Keywords: Wiener process., continuous dependence, asymptotical stability, initial value problem, Fractional stochastic evolution equations, $\alpha$-order fractional resolvent operator.

Mathematics Subject Classification: 34A12, 35R11, 47J35, 60H1.

Citation: Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817

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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