Journal of Advances in Applied Mathematics
Existence-uniqueness for Stochastic Functional Differential Equations with Non-Lipschitz Coefficients
- Lingying Teng*
College of Computer science and Technology of Southwest University for Nationalities, Chengdu, Sichuan, China
- Xiaohu Wang
Yangtze Center of Mathematics, Sichuan University, Chengdu, Sichuan, China
 L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1972.
 S-E A Mohammed, Stochastic Functional Differential Equations, Pitman Publishing Program, Boston, 1984.
 X. Mao, Stochastic Differential Equations and Applications, Horwood, 1997.
 D. Y. Xu, Y. M. Huang and Z. G. Yang, “Existence theorems for periodic Markov process and stochastic functional differential equations," Discrete Contin. Dyn. Syst. vol.24,no.3, pp. 1005-1023, 2009.
 D. Y. Xu, Z. G. Yang and Y. M. Huang, “Existence-uniqueness and continuation theorems for stochastic functional differential equations," J. Differential Equations, vol.245, pp. 1681-1703, 2008.
 J. Turo, “Successive approximations of solutions to stochastic functional differential equations," Period. Math. Hungar. vol.30, no.1, pp. 87-96, 1995.
 Y. Ren, S. Lu and N. Xia, “Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay," J. Comput. Appl. Math. vol.220, no.1-2, pp. 364-372, 2008.
 D. Y. Xu, B. Li, S.J. Long and L.Y. Teng, “Moment estimate and existence for solutions of stochastic functional differential equations," Nonlinear Anal. vol.108, pp. 128-143, 2014.
 P. Hartman, Ordinary Differential Equations, Birkhauser, Boston 1982.
 I. Hirai and K. Ako, “On generalized Peano’s theorem concerning the Dirichlet problem for semi-linear elliptic differential equations," Proc. Japan Academy, vol.36, pp. 480-485, 1960.
 J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
 J-P. Aubin and H, Frankowska, Set-Valued Analysis, Birkhauser, Basel, 1991.
 O. Kaleva, “The Cauchy problem for fuzzy differential equations," Fuzzy Sets and Systems, vol.35, pp. 389-396, 1990.
 P. E. Kloeden, “Remarks on Peano-like theorems for fuzzy differential equations," Fuzzy Sets and Systems, vol.44, pp. 161-163, 1991.
 Juan J. Nieto, “The Cauchy problem for continuous fuzzy differential equations," Fuzzy Sets and Systems, vol.102, pp. 259-262, 1999.
 V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations," Nonlinear Anal. vol.69, pp. 2677-2682, 2008.
 T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Dover, New York, 2005.
 E. Zeidler, Nonlinear Functional Analysis and its Applications I Fixed-Point Theorems, Springer-Verlag, New York, 1986.
 R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, New York, 1982.
 T. Taniguchi, “On sufficient conditions for nonexplosion of solutions to stochastic differential equations," J. Math. Anal. Appl. vol.153, pp. 549-561, 1990.
 T. Taniguchi, “Successive approximations to solutions of stochastic differential equations," J. Differential Equations, vol.96, pp. 152-169, 1992.
 T. Yamada, “On the successive approximation of solutions of stochastic differential equations," J. Math. Sci. Univ. Kyoto, vol.21, pp. 501-515, 1981.
 G. S. Ladde and S. Seikkala, “Existence, uniqueness and upper estimates for solutions of McShane type stochastic differential systems," Stoch. Anal. Appl. vol.4, pp. 409-429, 1986.
 I. V. Fedorenko, “Existence and uniqueness of a continuous solution of a system of stochastic differential equations of It? type," Russ. Math. Surv. vol.41, pp. 227-228, 1986.
 J. C. Kuang, Applied Inequalities, 3nd edn. Shandong Science and Technology Press, 2004.