Fractional neutral stochastic differential equations driven by \(\alpha\)-stable process

Volume 10, Issue 9, pp 4713--4723 http://dx.doi.org/10.22436/jnsa.010.09.14

Publication Date: September 09, 2017 Submission Date: March 01, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1954 Downloads
• 3336 Views

Authors

Zhi Li - School of Information and Mathematics, Yangtze University, Jingzhou 434023, China.

Abstract

In this paper, we are concerned with a class of fractional neutral stochastic partial differential equations driven by \(\alpha\)-stable process. By the stochastic analysis technique, the properties of operator semigroup and combining the Banach fixed-point theorem, we prove the existence and uniqueness of the mild solutions to this kind of equations driven by \(\alpha\)-stable process. In the end, an example is given to demonstrate the theory of our work.

Share and Cite

Keywords

• Fractional neutral SDEs
• \(\alpha\)-stable process
• existence and uniqueness.

MSC

•  60H15
•  60G15
•  60H05

References

• [1] D. Applebaum, Lévy processes and stochastic calculus, Second edition, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2009)
  • View Article
  • MathSciNet
  • Google Scholar


• [2] D. Baleanu, H. Jafari, H. Khan, S. J. Johnston , Results for mild solution of fractional coupled hybrid boundary value problems, Open Math., 13 (2015), 601–608.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] J.-H. Bao, C.-G. Yuan, Numerical analysis for neutral SPDEs driven by \(\alpha\)-stable processes, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014), 17 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] J. Cui, L.-T. Yan , Existence result for fractional neutral stochastic integro-differential equations with infinite delay, J. Phys. A, 44 (2011), 16 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (1992)
  • Google Scholar


• [6] Z. Dong, L.-H. Xu, X.-C. Zhang, Invariant measures of stochastic 2D Navier-Stokes equation driven by \(\alpha\)-stable processes , Electron. Commun. Probab., 16 (2011), 678–688.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] U. Haagerup , The best constants in the Khintchine inequality, Studia Math., 70 (1982), 231–283.
  • Google Scholar


• [8] R. Jahanipur , Nonlinear functional differential equations of monotone-type in Hilbert spaces, Nonlinear Anal., 72 (2010), 1393–1408.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] F. Jiao, Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62 (2011), 1181–1199.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] P. Kalamani, D. Baleanu, S. Selvarasu, M. M. Arjunan, On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions, Adv. Difference Equ., 163 (2016), 36 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] M. Kerboua, A. Debbouche, D. Baleanu , Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 16 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] K.-X. Li, Stochastic delay fractional evolution equations driven by fractional Brownian motion, Math. Methods Appl. Sci., 38 (2015), 1582–1591.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] K.-X. Li, J.-G. Peng , Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys., 65 (2014), 941–959.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] K.-X. Li, J.-G. Peng, J.-H. Gao, Existence results for semilinear fractional differential equations via Kuratowski measure of noncompactness, Fract. Calc. Appl. Anal., 15 (2012), 591–610.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] K. Liu , Stability of infinite dimensional stochastic differential equations with applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL (2006)
  • MathSciNet
  • Google Scholar


• [16] C. Lizama, G. M. N'Guérékata, Mild solutions for abstract fractional differential equations, Appl. Anal., 92 (2013), 1731–1754.
  • View Article
  • MathSciNet
  • Google Scholar


• [17] J.-W. Luo, K. Liu, Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stochastic Process. Appl., 118 (2008), 864–895.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] F. Mainardi, Fractional calculus and waves in linear viscoelasticity, An introduction to mathematical models, Imperial College Press, London (2010)
  • View Article
  • MathSciNet
  • Google Scholar


• [19] A. Pazy , Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York (1992)
  • View Article
  • MathSciNet
  • Google Scholar


• [20] E. Priola, J. Zabczyk , Structural properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theory Related Fields, 149 (2011), 97–137.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] R. Sakthivel, P. Revathi, Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal., 81 (2013), 70–86.
  • View Article
  • MathSciNet
  • Google Scholar


• [22] R. Sakthivel, S. Suganya, S. M. Anthoni , Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl., 63 (2012), 660–668.
  • View Article
  • Google Scholar


• [23] G. Samorodnitsky, M. S. Taqqu , Stable non-Gaussian random processes, Stochastic models with infinite variance, Stochastic Modeling, Chapman & Hall, New York (1994)
  • MathSciNet
  • Google Scholar


• [24] K.-I. Sato, Lévy processes and infinitely divisible distributions , Translated from the 1990 Japanese original, Revised by the author, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1999)
  • Google Scholar


• [25] L.-H. Xu , Ergodicity of the stochastic real Ginzburg-Landau equation driven by \(\alpha\)-stable noises, Stochastic Process. Appl., 123 (2013), 3710–3736.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] Y.-C. Zang, J.-P. Li, Stability in distribution of neutral stochastic partial differential delay equations driven by \(\alpha\)-stable process, Adv. Difference Equ., 2014 (2014), 16 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] X.-C. Zhang , Derivative formulas and gradient estimates for SDEs driven by \(\alpha\)-stable processes, Stochastic Process. Appl., 123 (2013), 1213–1228.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063–1077.
  • View Article
  • Google Scholar


• [29] Y. Zhou, F. Jiao , Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. Real World Appl., 11 (2010), 4465–4475.
  • View Article
  • Google Scholar