Abstract
The nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions for first-order semilinear stochastic functional differential equations in Hilbert spaces.
Open Access
On nondensely defined semilinear stochastic functional differential equations with nonlocal conditions
Received09 Oct 2004
Revised05 Mar 2005
Accepted07 Mar 2005
Published05 Mar 2006
References
W. Arendt, “Vector-valued Laplace transforms and Cauchy problems,” Israel Journal of Mathematics, vol. 59, no. 3, pp. 327–352, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNetP. Balasubramaniam and S. K. Ntouyas, “Global existence for semilinear stochastic delay evolution equations with nonlocal conditions,” Soochow Journal of Mathematics, vol. 27, no. 3, pp. 331–342, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNetA. T. Bharucha-Reid, Random Integral Equations, Academic Press, New York, 1972.
View at: Zentralblatt MATH | MathSciNetL. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 162, no. 2, pp. 494–505, 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNetG. Da Prato and E. Sinestrari, “Differential operators with nondense domain,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, vol. 14, no. 2, pp. 285–344, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNetG. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1992.
View at: Zentralblatt MATH | MathSciNetJ. Dugundji and A. Granas, Fixed Point Theory. I, vol. 61 of Monografie Matematyczne, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982.
View at: Zentralblatt MATH | MathSciNetT. Hida, Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes, no. 13, Carleton University, Ontario, 1975.
View at: MathSciNetT. Hida, H.-H. Kuo, J. Potthoff, and L. Streit, White Noise. An Infinite Dimensional Calculus, vol. 253 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1993.
View at: Zentralblatt MATH | MathSciNetH. Kellerman and M. Hieber, “Integrated semigroups,” Journal of Functional Analysis, vol. 84, no. 1, pp. 160–180, 1989.
View at: Google Scholar | Zentralblatt MATH | MathSciNetK. Liu, “Carathéodory approximate solutions for a class of semilinear stochastic evolution equations with time delays,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 349–364, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNetM. A. McKibben, “Second-order damped functional stochastic evolution equations in Hilbert space,” Dynamic Systems and Applications, vol. 12, no. 3-4, pp. 467–487, 2003.
View at: Google Scholar | MathSciNetM. A. McKibben, “Second-order neutral stochastic evolution equations with heredity,” Journal of Applied Mathematics and Stochastic Analysis. JAMSA, vol. 2004, no. 2, pp. 177–192, 2004.
View at: Google Scholar | MathSciNetK. Sobczyk, Stochastic Differential Equations. With Applications to Physics and Engineering, vol. 40 of Mathematics and Its Applications (East European Series), Kluwer Academic, Dordrecht, 1991.
View at: Zentralblatt MATH | MathSciNetT. Taniguchi, “Successive approximations to solutions of stochastic differential equations,” Journal of Differential Equations, vol. 96, no. 1, pp. 152–169, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNetC. P. Tsokos and W. J. Padgett, Random Integral Equations with Applications to Life Sciences and Engineering, Academic Press, New York, 1974.
View at: Zentralblatt MATH | MathSciNet
Copyright
Copyright © 2006 M. Benchohra et al. This is an open access article distributed under the
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