ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Solvability of a Stochastic Differential Equation with Nonlocal and Integral Conditions

Author(s):

A.M.A. El-Sayed

Faculty of Science
Alexandria University
Alexandria, Egypt

amasayed@alexu.edu.eg

M. E. I. El-Gendy

Faculty of Science
Damanhour University
Behera, Egypt

maysa_elgendy@yahoo.com

Abstract:

In this paper we are concerned with a stochastic differential equation with nonlocal condition. We study the existence of a unique mean square continuous solution. The continuous dependencies of the solution with respect to the random initial value and deterministic coefficients of the nonlocal condition are shown. A stochastic differential equation with an integral condition is considered as well.

Keywords

References:

  1. P. Balasubramaniam, J. Y. Parkand and A. V. A. Kumar, Existence of solutions for semilinear neutral stochastic functional differential equations with nonlocal conditions, Nonlinear Anal., 71(2009), 1049-1058
  2. A. T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bulletin of the American Mathematical Society, 82, 5(1976)
  3. A. Boucherif and Radu Precup, On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 4, 2(2003), 205-212
  4. L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Applicable analysis, 40(1991), 11-19
  5. P. Chen and Y. Li, Existence and uniqueness of strong solutions for nonlocal evolution equations, Electronic Journal of Differential Equations, 2014(2014), 1-9
  6. A. M. A. El-Sayed, R. O. Abd El-Rahman and M. El-Gendy, Uniformly stable solution of a nonlocal problem of coupled system of differential equations, Differ. Equ. Appl., 5, 3(2013), 355-365
  7. A. M. A. El-Sayed, R. O. Abd El-Rahman and M. El-Gendy, Existence of solution of a coupled system of differential equation with nonlocal conditions, Malaya Journal Of Mathematics, 2, 4(2014), 345-351
  8. A. M. A. EL-Sayed and E. O. Bin-Tahir, An arbitraty fractional order differential equation with internal nonlocal and integral conditions, advances in Pure Mathematics, 1, 3(2011), 59-62
  9. D. Gordeziani and G. Avalishvili, Investigation of the nonlocal initial boundary value problems for some hyperbolic equations, Hiroshima Math. J., 31(2001), 345-366
  10. D. Isaacson, Stochastic integrals and derivatives, The Annals of Mathematical Statistics, 40, 5(1969), 1610-1616
  11. S. Itoh, Random fixed point theorems with an application to random differential equations in Banach spaces, Journal Of Mathematical Analysis And Applications, 67(1979), 261-273
  12. B. Oksendal, Stochastic differential equations (An introduction with applications), Springer-Verlag Berlin Heidelberg New York, (2013)
  13. M. Rockner, R. Zhu and X. Zhu, Existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions, Nonlinear Analysis: Theory, Methods and Applications, 125(2015), 358-397
  14. R. Sakthivel, P. Revathi and Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlilear Anal., 81(2013), 70-86
  15. T. T. Soong, Random differential equations in science and engineering, Mathematics in Science and Engineering, 103, (1973)
  16. D. W. Stroock, Topics in stochastic differential equations, Tata Institute of Fundamental Research Bombay, (1982)
  17. S. Watanabe and T. Yamada, On uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., (1971), 155-167 and 553-563

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