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A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients. (English) Zbl 1217.60054
Summary: We study the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations under some Carathéodory-type conditions on the coefficients by means of the successive approximation. In particular, we generalize and improve the results of T.E. Govindan [Stochastics 77, No. 2, 139–154 (2005; Zbl 1115.60064)] and J. Bao and Z. Hou [Comput. Math. Appl. 59, No. 1, 207–214 (2010; Zbl 1189.60122)].
MSC:
60H15Stochastic partial differential equations
35R10Partial functional-differential equations
35R60PDEs with randomness, stochastic PDE
References:
[1]Liu, K.: Stability of infinite dimensional stochastic differential equations with applications, (2006)
[2]Govindan, T. E.: Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77, 139-154 (2005) · Zbl 1115.60064 · doi:10.1080/10451120512331335181
[3]Bao, J.; Hou, Z.: Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Comput. math. Appl. 59, 207-214 (2010) · Zbl 1189.60122 · doi:10.1016/j.camwa.2009.08.035
[4]Taniguchi, T.: Successive approximations to solutions of stochastic differential equations, J. differential equations 96, 152-169 (1992) · Zbl 0744.34052 · doi:10.1016/0022-0396(92)90148-G
[5]Turo, J.: Successive approximations of solutions to stochastic functional differential equations, Period. math. Hungar. 30, 87-96 (1995) · Zbl 0816.60055 · doi:10.1007/BF01876930
[6]Cao, G.; He, K.; Zhang, X.: Successive approximations of infinite dimensional SDES with jump, Stoch. syst. 5, 609-619 (2005) · Zbl 1082.60048 · doi:10.1142/S0219493705001584
[7]Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1992)