# Journal of Advances in Applied Mathematics

### Existence-uniqueness for Stochastic Functional Differential Equations with Non-Lipschitz Coefficients

Download PDF (245.5 KB) PP. 127 - 142 Pub. Date: July 31, 2017

### Author(s)

**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

### Abstract

### Keywords

### References

[1] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1972.

[2] S-E A Mohammed, Stochastic Functional Differential Equations, Pitman Publishing Program, Boston, 1984.

[3] X. Mao, Stochastic Differential Equations and Applications, Horwood, 1997.

[4] 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.

[5] 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.

[6] J. Turo, “Successive approximations of solutions to stochastic functional differential equations," Period. Math. Hungar. vol.30, no.1, pp. 87-96, 1995.

[7] 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.

[8] 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.

[9] P. Hartman, Ordinary Differential Equations, Birkhauser, Boston 1982.

[10] 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.

[11] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.

[12] J-P. Aubin and H, Frankowska, Set-Valued Analysis, Birkhauser, Basel, 1991.

[13] O. Kaleva, “The Cauchy problem for fuzzy differential equations," Fuzzy Sets and Systems, vol.35, pp. 389-396, 1990.

[14] P. E. Kloeden, “Remarks on Peano-like theorems for fuzzy differential equations," Fuzzy Sets and Systems, vol.44, pp. 161-163, 1991.

[15] Juan J. Nieto, “The Cauchy problem for continuous fuzzy differential equations," Fuzzy Sets and Systems, vol.102, pp. 259-262, 1999.

[16] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations," Nonlinear Anal. vol.69, pp. 2677-2682, 2008.

[17] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Dover, New York, 2005.

[18] E. Zeidler, Nonlinear Functional Analysis and its Applications I Fixed-Point Theorems, Springer-Verlag, New York, 1986.

[19] R. K. Miller and A. N. Michel, Ordinary Differential Equations, Academic Press, New York, 1982.

[20] T. Taniguchi, “On sufficient conditions for nonexplosion of solutions to stochastic differential equations," J. Math. Anal. Appl. vol.153, pp. 549-561, 1990.

[21] T. Taniguchi, “Successive approximations to solutions of stochastic differential equations," J. Differential Equations, vol.96, pp. 152-169, 1992.

[22] T. Yamada, “On the successive approximation of solutions of stochastic differential equations," J. Math. Sci. Univ. Kyoto, vol.21, pp. 501-515, 1981.

[23] 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.

[24] 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.

[25] J. C. Kuang, Applied Inequalities, 3nd edn. Shandong Science and Technology Press, 2004.