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Jingqi Han, Litan Yan. A TIME FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION DRIVEN BY THE FRACTIONAL BROWNIAN MOTION[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 547-567. doi: 10.11948/2156-907X.20180068
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A TIME FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION DRIVEN BY THE FRACTIONAL BROWNIAN MOTION
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Abstract
Let $B^H$ be a fractional Brownian motion with Hurst index $H>\frac12$. In this paper, we prove the global existence and uniqueness of the equation $ \left\{\begin{array}{ll}{C D_{t}^{\gamma} x(t)=f\left(x_{t}\right)+G\left(x_{t}\right) \frac{d}{d t} B^{H}(t),} & {t \in(0, T]}, \\ {x(t)=\eta(t),} & {t \in[-r, 0]},\end{array}\right. $ where $\max\{H, 2-2H\} < \gamma < 1$, $^CD_t^{\gamma}$ is the Caputo derivative, and $x_t\in \mathcal{C}_r=\mathcal{C}([-r, 0], \mathbb{R})$ with $x_t(u)=x(t+u), u\in[-r, 0]$. We also study the dependence of the solution on the initial condition. -
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References
[1] H. Y. Alfifi, I. B. Saad, S. Turki and et al, Existence and asymptotic behavior of positive solutions for a coupled system of semilinear fractional differential equations, Results Math, 2017, 71(3-4), 705-730. doi: 10.1007/s00025-016-0528-9 [2] C. Bai and J. Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Appl. Math. Comput., 2004, 150(3), 611-621. [3] F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic calculus for fractional Brownian motion and applications, Springer, London, 2008. [4] B. Boufoussi and S. Hajji, Functional differential equations driven by a fractional Brownian motion, Comput. Math. Appl., 2011, 62(2), 746-754. doi: 10.1016/j.camwa.2011.05.055 [5] B. Boufoussi, S. Hajji and E. H. Lakhel, Functional differential equations in Hilbert spaces driven by a fractional Brownian motion, Afrika Mat., 2012, 23(2), 173-194. [6] X. Fernique, Régularité des trajectoires des fonctions aléatoires Gaussiennes, In: Ecole d'été de Probabilités de Saint-Flour. Ⅳ-1974. Lecture Notes in Mathematics, 1974, 480, 1-96. [7] M. Ferrante and C. Rovira, Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H>1/2, Bernoulli, 2006, 12(1), 85-100. [8] M. Ferrante and C. Rovira, Convergence of delay differential equations driven by fractional Brownian motion, J. Evol. Equ., 2010, 10(4), 761-783. doi: 10.1007/s00028-010-0069-8 [9] Y. Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Mem. Amer. Math. Soc., 2005, 175(825). [10] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland, 2006. [11] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motion, fractional noises and applications, SIAM Rev., 1968, 10(4), 422-437. doi: 10.1137/1010093 [12] Y. S. Mishura, Stochastic Calculus for fractional Brownian motion and Related Processes, Springer, Berlin, 2008. [13] I. Nourdin, Selected aspects of fractional Brownian motion, Springer, Milan, 2012. [14] D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 2002, 53(1), 55-82. [15] D. Nualart, Malliavin Calculus and Related Topics, Springer, Berlin, 2006. [16] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, 1993. [17] L. C. Young, An Inequality of Hölder Type Connected with Stieltjes Integration, Acta Math., 1936, 67(1), 251-282. [18] M. Zähle, Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields, 1998, 111(3), 333-374. doi: 10.1007/s004400050171 [19] M. Zähle, Integration with respect to fractal functions and stochastic calculus Ⅱ, Math. Nachr., 2001, 225(1), 145-183. -
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Jingqi Han, Litan Yan. A TIME FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION DRIVEN BY THE FRACTIONAL BROWNIAN MOTION[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 547-567. doi: 10.11948/2156-907X.20180068
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