On initial value problem of random fractional differential equation with impulses

Abstract

In this paper, we prove the existence and uniqueness of solution for random fractional differential equation with impulses via Banach fixed point theorem and Schauder fixed point theorem. Moreover, the continuous dependence of the solution on the initial data is investigated.

Keywords

• Random fractional differential equation with impulses,second-order stochastic processes,mean square continuous solution

References

1. [1] B. Bayour and D. Torres, Existence of solution to a local fractional nonlinear differential equation, J. Comput. Appl. Math., 312, 127–133, 2017.
2. [2] A. Bharucha-Reid, Random integral equations, Academic Press, New York, 1972.
3. [3] A. El-Sayed, The mean square riemann-liouville stochastic fractional derivative and stochastic fractional order differential equation, Math. Sci. Res. J., 9, 142–150, 2005.
4. [4] A. El-Sayed, On the stochastic fractional calculus operators, J. Frac. Calc. Appl., 6, 101–109, 2015.
5. [5] F. Hafiz, The fractional calculus for some stochastic processes, Stoch. Anal. Appl., 22, 507–523, 2004.
6. [6] F. Hafiz, A. El-Sayed and M. El-Tawil, On a stochastic fractional calculus, Frac. Calc. Appl. Anal., 4, 81–90, 2001.
7. [7] A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, Volume 204, North-Holland Mathematics Studies, Elsevier Science Inc., 2006.
8. [8] G. Ladde and V. Lakshmikantham, Random differential inequalities, Academic Press, New York, 1980.

9. [9] V. Lakshmikantham, S. Leela and J. V. Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, 2009.
10. [10] V. Lupulescu and S. Ntouyas, Random fractional differential equations, Int. Elec. J. Pure Appl. Math., 4, 119–136, 2012.
11. [11] V. Lupulescu, D. O’Regan and G. ur Rahman, Existence results for random fractional differential equations, Opuscula Mathematica, 34, 813–825, 2014.
12. [12] Z.-D. Mei, J.-G. Peng and J.-H. Gao, Existence and uniqueness of solutions for nonlinear general fractional differential equations in banach spaces, Indagat. Math., 26, 669–678, 2015.
13. [13] K. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley-Interscience, 1993.
14. [14] Z. Shuorui and S. Jitao, On existence and uniqueness of random impulsive differential equations, J. Syst. Sci. Complex., 29, 300–314, 2016.
15. [15] T. Soong, Random differential equations in science and engineering, Academic Press.
16. [16] N. Tobias and R. Florian, Random differential equations in scientific computing, De Gruyter Open, Berlin, 2013.
17. [17] H. Vu, Random fractional functional differential equations, Int. J. Nonlin. Anal. Appl., 7, 253–267, 2016.
18. [18] H. Vu, N. Phung and N. Phuong, On fractional random differential equations with delay, Opuscula Mathematica, 36, 541–556, 2016.
19. [19] D. Yang and J. Wang, Non-instantaneous impulsive fractional-order implicit differential equations with random effects, Stoch. Anal. Appl., 35, 719–741, 2017.
20. [20] Y. Zou and G. He, On the uniqueness of solutions for a class of fractional differential equations, Appl. Math. Lett., 74, 68–73, 2017.