In this paper, sufficient conditions are given for the existence of solutions for a class of second order stochastic differential inclusions in Hilbert space with the help of Leray-Schauder Nonlinear Alternative.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1090,
author = {P. Balasubramaniam and S.K. Ntouyas},
title = {Existence of solutions for second order stochastic differential inclusions in Hilbert spaces},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {27},
year = {2007},
pages = {365-384},
zbl = {1155.34034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1090}
}
P. Balasubramaniam; S.K. Ntouyas. Existence of solutions for second order stochastic differential inclusions in Hilbert spaces. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 27 (2007) pp. 365-384. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1090/
[000] [1] N.U. Ahmed, Nonlinear stochastic differential inclusions on Banach space, Stochastic Anal. Appl. 12 (1994), 1-10. | Zbl 0789.60052
[001] [2] J.P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. | Zbl 0538.34007
[002] [3] P. Balasubramaniam, Existence of solutions of functional stochastic differential inclusions, Tamkang J. Math. 33 (2002), 35-43. | Zbl 1012.35085
[003] [4] P. Balasubramaniam, S.K. Ntouyas and D. Vinayagam, Existence of solutions of nonlinear stochastic differential inclusions in a Hilbert space, Comm. Appl. Nonlinear Anal. 12 (2005), 1-15. | Zbl 1086.34055
[004] [5] P. Balasubramaniam and J.Y. Park, Nonlocal Cauchy problem for second order stochastic evolution equations in Hilbert spaces, Dynamic Syst. Appl. (in press). | Zbl 1151.34046
[005] [6] J. Ball, Initial boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61-90. | Zbl 0254.73042
[006] [7] J. Bochenek, An abstract nonlinear second order differential equation, Ann. Polon. Math. 2 (1991), 155-166. | Zbl 0724.34069
[007] [8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. | Zbl 0761.60052
[008] [9] W.E. Fitzgibbon, Global existence and boundedness of solutions to the extensible beam equation, SIAM J. Math. Anal. 13 (1982), 739-745. | Zbl 0506.73057
[009] [10] K. Deimling, Multivalued Differential Equations, de Gruyter, New York, 1992.
[010] [11] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. | Zbl 1025.47002
[011] [12] S. Hu and N.S. Papageorgiou, On the existence of periodic solutions for non-convex valued differential inclusions in ℝⁿ, Proc. Amer. Math. Soc. 123 (1995), 3043-3050. | Zbl 0851.34014
[012] [13] S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Vol. I. Theory, Kluwer Academic Publishers, Dordrecht, Boston, London, 1997. | Zbl 0887.47001
[013] [14] D.N. Keck and M.A. McKibben, Functional integro-differential stochastic evolution equations in Hilbert space, J. Appl. Math. Stochastic Anal. 16 (2003), 127-147. | Zbl 1031.60061
[014] [15] P. Kree, Diffusion equation for multivalued stochastic differential equations, J. Funct. Anal. 49 (1982), 73-90. | Zbl 0528.60066
[015] [16] A. Lasota and Z. Opial, An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786. | Zbl 0151.10703
[016] [17] N.I. Mahmudov and M.A. McKibben, Abstract second-order damped McKean-Vlasov stochastic evolution equations, Stochastic Anal. Appl. 24 (2006), 303-328. | Zbl 1102.35044
[017] [18] M.A. McKibben, Second-order neutral stochastic evolution equations with heredity, J. Appl. Math. Stochastic Anal. 2 (2004), 177-192. | Zbl 1077.34062
[018] [19] M. Michta and J. Motyl, Second order stochastic inclusion, Stochastic Anal. Appl. 22 (2004), 701-720. | Zbl 1061.60043
[019] [20] M. Martelli, A Rothe's type theorem for non-compact acyclic-valued map, Boll. Unione Mat. Ital. 4 (11) (1975), 70-76. | Zbl 0314.47035
[020] [21] S.K. Ntouyas, Global existence results for certain second order delay integrodifferential equations with nonlocal conditions, Dynam. Systems Appl. 7 (1998), 415-425. | Zbl 0914.35148
[021] [22] S.K. Ntouyas and P.Ch. Tsamatos, Global existence for second order functional semilinear equations, Period. Math. Hungar. 31 (1995), 223-228.
[022] [23] N. Papageorgiou, Boundary value problems for evolution inclusions, Comment. Math. Univ. Carol. 29 (1988), 355-362. | Zbl 0696.35074
[023] [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
[024] [25] R. Pettersson, Yosida approximations for multivalued stochastic differential equations, Stochastics and Stochastics Reports 52 (1995), 107-120. | Zbl 0864.60046
[025] [26] R. Pettersson, Existence theorem and Wong-Zakai approximations for multivalued stochastic differential equations, Probability and Mathematical Statistics 17 (1997), 29-45. | Zbl 0880.60062
[026] [27] T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Differential Equations 181 (2002), 72-91. | Zbl 1009.34074
[027] [28] C.C. Travis and G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Mathematica Academiae Scientiarum Hungaricae 32 (1978), 75-96. | Zbl 0388.34039
[028] [29] C.C. Travis and G.F. Webb, An abstract second order semilinear Volterra integrodifferential equation, SIAM J. Math. Anal. 10 (1979), 412-424. | Zbl 0406.45014