The main aim of this paper is to prove the existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay, which the initial data belong to the phase space ℬ((−∞,0];ℝd). The vital work of this paper is to extend the initial function space of the paper (Wei and Wang, J. Math. Anal. Appl. 331:516–531, 2007) and give some examples to show that the phase space ℬ((−∞,0];ℝd) exists. In addition, this paper builds a Banach space ℳ2((−∞,T],ℝd) with a new norm in order to discuss the existence and uniqueness of the solution for such equations with infinite delay.
Phase space Neutral stochastic functional differential equations Infinite delay Existence Uniqueness
Mathematics Subject Classification (2000)
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Atkinson, F., Haddock, J.: On determining phase space for functional differential equations. Funkcial. Ekvac. 31, 331–347 (1988)
Brayton, R.: Nonlinear oscillations in a distributed network. Q. Appl. Math. 4, 289–301 (1976)
Coleman, B., Gurtin, M.: Equipresence and constitutive equations for rigid heat conductors. Z. Angew. Math. Phys. 18, 199–208 (1967)
Desch, W., Grimmer, R., Schappacher, W.: Wellposedness and wave propagation for a class of integrodifferential equations in Banach space. J. Differ. Equ. 74, 39–411 (1988)
Wei, F., Wang, K.: The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. J. Math. Anal. Appl. 331, 516–531 (2007)
Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, Berlin (1996)