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

1. 

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

Received  January 2015 Revised  March 2015 Published  June 2015

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.
Citation: Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817
References:
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R. P. Agarwal, V. Lakshmikantham and J. J. Nieto, On the concept of solutions for fractional differential equations with uncertainly,, \emph{Nonlinear Anal.}, 72 (2010), 2859. doi: 10.1016/j.na.2009.11.029. Google Scholar

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E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces,, Ph.D thesis, (2001). Google Scholar

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J. Bao and Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients,, \emph{Comput. Math. Appl.}, 59 (2010), 207. doi: 10.1016/j.camwa.2009.08.035. Google Scholar

[4]

J. Bao, Z. Hou and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 2169. doi: 10.1090/S0002-9939-10-10230-5. Google Scholar

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C. Chen and M. Li, On fractional resolvent operator functions,, \emph{Semigroup Forum}, 80 (2010), 121. doi: 10.1007/s00233-009-9184-7. Google Scholar

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C. Chen, M. Li and F. B. Li, On boundary values of fractional resolvent families,, \emph{J. Math. Anal. Appl.}, 384 (2011), 453. doi: 10.1016/j.jmaa.2011.05.074. Google Scholar

[7]

P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 711. doi: 10.1007/s00033-013-0351-z. Google Scholar

[8]

P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces,, \emph{Collect. Math.}, 66 (2015), 63. doi: 10.1007/s13348-014-0106-y. Google Scholar

[9]

J. Cui and L. Yan, Existence result for fractional neutral stochastic integro-differential equations with infinite delay,, \emph{J. Phys. A}, 44 (2011). doi: 10.1088/1751-8113/44/33/335201. Google Scholar

[10]

J. Cui, L. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces,, \emph{J. Korean Stat. Soci.}, 41 (2012), 279. doi: 10.1016/j.jkss.2011.10.001. Google Scholar

[11]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space,, \emph{J. Differential Equations}, 10 (1971), 412. Google Scholar

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[13]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations,, \emph{J. Differential Equations}, 199 (2004), 211. doi: 10.1016/j.jde.2003.12.002. Google Scholar

[14]

M. M. EI-Borai, Some probability densities and fundamental solutions of fractional evolution equations,, \emph{Chaos, 14 (2002), 433. doi: 10.1016/S0960-0779(01)00208-9. Google Scholar

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M. M. EI-Borai, O. L. Mostafa and H. M. Ahmed, Asymptotic stability of some stochastic evolution equations,, \emph{Appl. Math. Comput.}, 144 (2003), 273. doi: 10.1016/S0096-3003(02)00406-X. Google Scholar

[16]

Z. Fan, Characterization of compactness for resolvents and its applications,, \emph{Appl. Math. Comput.}, 232 (2014), 60. doi: 10.1016/j.amc.2014.01.051. Google Scholar

[17]

W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach,, Akademic Verlag, (1995). Google Scholar

[18]

J. Jia, J. Peng and K. Li, Well-posedness of abstract distributed-order fractional diffusion equations,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 605. doi: 10.3934/cpaa.2014.13.605. Google Scholar

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V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations,, \emph{Nonlinear Anal.}, 69 (2008), 1677. doi: 10.1016/j.na.2007.08.042. Google Scholar

[21]

M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families,, \emph{J. Funct. Anal.}, 259 (2010), 2702. doi: 10.1016/j.jfa.2010.07.007. Google Scholar

[22]

K. Li, J. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives,, \emph{J. Funct. Anal.}, 263 (2012), 476. doi: 10.1016/j.jfa.2012.04.011. Google Scholar

[23]

K. Li and J. Peng, Fractional resolvents and fractional evolution equations,, \emph{Appl. Math. Lett.}, 25 (2012), 808. doi: 10.1016/j.aml.2011.10.023. Google Scholar

[24]

K. Li and J. Peng, Fractional abstract Cauchy problems,, \emph{Integr. Equ. Oper. Theory}, 70 (2011), 333. doi: 10.1007/s00020-011-1864-5. Google Scholar

[25]

K. Li and J. Peng, Controllability of fractional neutral stochastic functional differential systems,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 941. doi: 10.1007/s00033-013-0369-2. Google Scholar

[26]

K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications,, Chapman and Hall, (2006). Google Scholar

[27]

J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{J. Math. Anal. Appl.}, 342 (2008), 753. doi: 10.1016/j.jmaa.2007.11.019. Google Scholar

[28]

J. Luo and T. Taniguchi, Fixed point and stability of stochastic neutral partial differential equations with infinite delays,, \emph{Stoch. Anal. Appl.}, 27 (2009), 1163. doi: 10.1080/07362990903259371 . Google Scholar

[29]

X. Mao, Stochastic Differential Equations and Their Applications,, Horwood Publishing Ltd., (1997). Google Scholar

[30]

M. Niu and B. Xie, Regularity of a fractional partial differential equation driven by space-time white noise,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 1479. doi: 10.1090/S0002-9939-09-10197-1. Google Scholar

[31]

I. Podlubny, Fractional Differential Equations,, Academic Press, (1999). Google Scholar

[32]

T. Poinot and J. C. Trigeassou, Identification of fractional systems using an output-error technique,, \emph{Nonl. Dynamics}, 38 (2004), 133. doi: 10.1007/s11071-004-3751-y. Google Scholar

[33]

J. Prüss, Evolutionary Integral Equations and Applications,, Birkh\, (1993). doi: 10.1007/978-3-0348-8570-6. Google Scholar

[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,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4739406. Google Scholar

[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,, \emph{J. Optim. Theory Appl.}, 149 (2011), 315. doi: 10.1007/s10957-010-9792-0. Google Scholar

[36]

Y. A. Rossikhin and M. V. Shitikova, Application of fractional dericatives to the analysis of damped vibrations of viscoelastic single mass system,, \emph{Acta. Mech.}, 120 (1997), 109. doi: 10.1007/BF01174319. Google Scholar

[37]

J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Development and Applications in Physics and Engineering,, Springer, (2007). doi: 10.1007/978-1-4020-6042-7. Google Scholar

[38]

R. Sakthivel and J. Luo, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays,, \emph{J. Math. Anal. Appl.}, 356 (2009), 1. doi: 10.1016/j.jmaa.2009.02.002. Google Scholar

[39]

R. Sakthivel and Y. Ren, Exponential stability of second-order stochastic evolution equations with Poisson jumps,, \emph{Commu. Nonl. Sci. Nume. Simu.}, 17 (2012), 4517. doi: 10.1016/j.cnsns.2012.04.020. Google Scholar

[40]

R. Sakthivel, P. Revathi and N. I. Mahmudov, Asymptotic stability of fractional stochastic neutral differential equations with infinite delays,, \emph{Abstr. Appl. Anal.}, 2013 (2013). doi: 10.1155/2013/769257. Google Scholar

[41]

R. Sakthivel, P. Revathi and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations,, \emph{Nonlinear Anal.}, 81 (2013), 70. doi: 10.1016/j.na.2012.10.009. Google Scholar

[42]

R. Sakthivel, S. Suganyab and S. M. Anthonib, Approximate controllability of fractional stochastic evolution equations,, \emph{Comput. Math. Appl.}, 63 (2012), 660. doi: 10.1016/j.camwa.2011.11.024. Google Scholar

[43]

K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering,, Kluwer Academic Publishers, (1991). doi: 10.1007/978-94-011-3712-6. Google Scholar

[44]

T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces,, \emph{J. Differential Equations}, 181 (2002), 72. doi: 10.1006/jdeq.2001.4073. Google Scholar

[45]

M. S. Tvazoei, M. Haeri, S. Jafari, S. Bolouki and M. Siami, Some applications of fractional calculus in suppression of chaotic oscillations,, \emph{IEEE Transactions on Industrial Electronics}, 11 (2008), 4094. doi: 10.1109/TIE.2008.925774. Google Scholar

[46]

J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls,, \emph{Nonlinear Anal. Real World Appl.}, 12 (2011), 263. doi: 10.1016/j.nonrwa.2010.06.013. Google Scholar

[47]

S. Westerlund and L. Ekstam, Capacitor theory,, \emph{IEEE Transactions on Dielectrics and Electrical Insulation}, 1 (1994), 826. doi: 10.1109/94.326654. Google Scholar

[48]

Z. Yan and X. Yan, Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay,, \emph{Collec. Math.}, 64 (2013), 235. doi: 10.1007/s13348-012-0063-2. Google Scholar

[49]

Z. Yan and X. Yan, Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces,, \emph{Z. Angew. Math. Phys.}, 64 (2013), 573. doi: 10.1007/s00033-012-0249-1. Google Scholar

[50]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations,, \emph{Comput. Math. Appl.}, 59 (2010), 1063. doi: 10.1016/j.camwa.2009.06.026. Google Scholar

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,, \emph{Nonlinear Anal.}, 72 (2010), 2859. doi: 10.1016/j.na.2009.11.029. Google Scholar

[2]

E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces,, Ph.D thesis, (2001). Google Scholar

[3]

J. Bao and Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients,, \emph{Comput. Math. Appl.}, 59 (2010), 207. doi: 10.1016/j.camwa.2009.08.035. Google Scholar

[4]

J. Bao, Z. Hou and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 2169. doi: 10.1090/S0002-9939-10-10230-5. Google Scholar

[5]

C. Chen and M. Li, On fractional resolvent operator functions,, \emph{Semigroup Forum}, 80 (2010), 121. doi: 10.1007/s00233-009-9184-7. Google Scholar

[6]

C. Chen, M. Li and F. B. Li, On boundary values of fractional resolvent families,, \emph{J. Math. Anal. Appl.}, 384 (2011), 453. doi: 10.1016/j.jmaa.2011.05.074. Google Scholar

[7]

P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 711. doi: 10.1007/s00033-013-0351-z. Google Scholar

[8]

P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces,, \emph{Collect. Math.}, 66 (2015), 63. doi: 10.1007/s13348-014-0106-y. Google Scholar

[9]

J. Cui and L. Yan, Existence result for fractional neutral stochastic integro-differential equations with infinite delay,, \emph{J. Phys. A}, 44 (2011). doi: 10.1088/1751-8113/44/33/335201. Google Scholar

[10]

J. Cui, L. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces,, \emph{J. Korean Stat. Soci.}, 41 (2012), 279. doi: 10.1016/j.jkss.2011.10.001. Google Scholar

[11]

R. F. Curtain and P. L. Falb, Stochastic differential equations in Hilbert space,, \emph{J. Differential Equations}, 10 (1971), 412. Google Scholar

[12]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar

[13]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations,, \emph{J. Differential Equations}, 199 (2004), 211. doi: 10.1016/j.jde.2003.12.002. Google Scholar

[14]

M. M. EI-Borai, Some probability densities and fundamental solutions of fractional evolution equations,, \emph{Chaos, 14 (2002), 433. doi: 10.1016/S0960-0779(01)00208-9. Google Scholar

[15]

M. M. EI-Borai, O. L. Mostafa and H. M. Ahmed, Asymptotic stability of some stochastic evolution equations,, \emph{Appl. Math. Comput.}, 144 (2003), 273. doi: 10.1016/S0096-3003(02)00406-X. Google Scholar

[16]

Z. Fan, Characterization of compactness for resolvents and its applications,, \emph{Appl. Math. Comput.}, 232 (2014), 60. doi: 10.1016/j.amc.2014.01.051. Google Scholar

[17]

W. Grecksch and C. Tudor, Stochastic Evolution Equations: A Hilbert Space Approach,, Akademic Verlag, (1995). Google Scholar

[18]

J. Jia, J. Peng and K. Li, Well-posedness of abstract distributed-order fractional diffusion equations,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 605. doi: 10.3934/cpaa.2014.13.605. Google Scholar

[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, (2006). Google Scholar

[20]

V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations,, \emph{Nonlinear Anal.}, 69 (2008), 1677. doi: 10.1016/j.na.2007.08.042. Google Scholar

[21]

M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families,, \emph{J. Funct. Anal.}, 259 (2010), 2702. doi: 10.1016/j.jfa.2010.07.007. Google Scholar

[22]

K. Li, J. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives,, \emph{J. Funct. Anal.}, 263 (2012), 476. doi: 10.1016/j.jfa.2012.04.011. Google Scholar

[23]

K. Li and J. Peng, Fractional resolvents and fractional evolution equations,, \emph{Appl. Math. Lett.}, 25 (2012), 808. doi: 10.1016/j.aml.2011.10.023. Google Scholar

[24]

K. Li and J. Peng, Fractional abstract Cauchy problems,, \emph{Integr. Equ. Oper. Theory}, 70 (2011), 333. doi: 10.1007/s00020-011-1864-5. Google Scholar

[25]

K. Li and J. Peng, Controllability of fractional neutral stochastic functional differential systems,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 941. doi: 10.1007/s00033-013-0369-2. Google Scholar

[26]

K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications,, Chapman and Hall, (2006). Google Scholar

[27]

J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays,, \emph{J. Math. Anal. Appl.}, 342 (2008), 753. doi: 10.1016/j.jmaa.2007.11.019. Google Scholar

[28]

J. Luo and T. Taniguchi, Fixed point and stability of stochastic neutral partial differential equations with infinite delays,, \emph{Stoch. Anal. Appl.}, 27 (2009), 1163. doi: 10.1080/07362990903259371 . Google Scholar

[29]

X. Mao, Stochastic Differential Equations and Their Applications,, Horwood Publishing Ltd., (1997). Google Scholar

[30]

M. Niu and B. Xie, Regularity of a fractional partial differential equation driven by space-time white noise,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 1479. doi: 10.1090/S0002-9939-09-10197-1. Google Scholar

[31]

I. Podlubny, Fractional Differential Equations,, Academic Press, (1999). Google Scholar

[32]

T. Poinot and J. C. Trigeassou, Identification of fractional systems using an output-error technique,, \emph{Nonl. Dynamics}, 38 (2004), 133. doi: 10.1007/s11071-004-3751-y. Google Scholar

[33]

J. Prüss, Evolutionary Integral Equations and Applications,, Birkh\, (1993). doi: 10.1007/978-3-0348-8570-6. Google Scholar

[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,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.4739406. Google Scholar

[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,, \emph{J. Optim. Theory Appl.}, 149 (2011), 315. doi: 10.1007/s10957-010-9792-0. Google Scholar

[36]

Y. A. Rossikhin and M. V. Shitikova, Application of fractional dericatives to the analysis of damped vibrations of viscoelastic single mass system,, \emph{Acta. Mech.}, 120 (1997), 109. doi: 10.1007/BF01174319. Google Scholar

[37]

J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Development and Applications in Physics and Engineering,, Springer, (2007). doi: 10.1007/978-1-4020-6042-7. Google Scholar

[38]

R. Sakthivel and J. Luo, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays,, \emph{J. Math. Anal. Appl.}, 356 (2009), 1. doi: 10.1016/j.jmaa.2009.02.002. Google Scholar

[39]

R. Sakthivel and Y. Ren, Exponential stability of second-order stochastic evolution equations with Poisson jumps,, \emph{Commu. Nonl. Sci. Nume. Simu.}, 17 (2012), 4517. doi: 10.1016/j.cnsns.2012.04.020. Google Scholar

[40]

R. Sakthivel, P. Revathi and N. I. Mahmudov, Asymptotic stability of fractional stochastic neutral differential equations with infinite delays,, \emph{Abstr. Appl. Anal.}, 2013 (2013). doi: 10.1155/2013/769257. Google Scholar

[41]

R. Sakthivel, P. Revathi and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations,, \emph{Nonlinear Anal.}, 81 (2013), 70. doi: 10.1016/j.na.2012.10.009. Google Scholar

[42]

R. Sakthivel, S. Suganyab and S. M. Anthonib, Approximate controllability of fractional stochastic evolution equations,, \emph{Comput. Math. Appl.}, 63 (2012), 660. doi: 10.1016/j.camwa.2011.11.024. Google Scholar

[43]

K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering,, Kluwer Academic Publishers, (1991). doi: 10.1007/978-94-011-3712-6. Google Scholar

[44]

T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces,, \emph{J. Differential Equations}, 181 (2002), 72. doi: 10.1006/jdeq.2001.4073. Google Scholar

[45]

M. S. Tvazoei, M. Haeri, S. Jafari, S. Bolouki and M. Siami, Some applications of fractional calculus in suppression of chaotic oscillations,, \emph{IEEE Transactions on Industrial Electronics}, 11 (2008), 4094. doi: 10.1109/TIE.2008.925774. Google Scholar

[46]

J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls,, \emph{Nonlinear Anal. Real World Appl.}, 12 (2011), 263. doi: 10.1016/j.nonrwa.2010.06.013. Google Scholar

[47]

S. Westerlund and L. Ekstam, Capacitor theory,, \emph{IEEE Transactions on Dielectrics and Electrical Insulation}, 1 (1994), 826. doi: 10.1109/94.326654. Google Scholar

[48]

Z. Yan and X. Yan, Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay,, \emph{Collec. Math.}, 64 (2013), 235. doi: 10.1007/s13348-012-0063-2. Google Scholar

[49]

Z. Yan and X. Yan, Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces,, \emph{Z. Angew. Math. Phys.}, 64 (2013), 573. doi: 10.1007/s00033-012-0249-1. Google Scholar

[50]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations,, \emph{Comput. Math. Appl.}, 59 (2010), 1063. doi: 10.1016/j.camwa.2009.06.026. Google Scholar

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Kai Yan, Zhaoyang Yin. On the initial value problem for higher dimensional Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1327-1358. doi: 10.3934/dcds.2015.35.1327

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