×

Existence of solutions of nonlinear neutral stochastic differential inclusions in a Hilbert space. (English) Zbl 1081.34083

The authors consider the following nonlinear neutral stochastic differential inclusion in a Hilbert space \[ d(X(t)-f(t,X_t)) \in AX(t)dt+G(t,X_t)dB(t). \] Here, \(X_t\) denotes the solution segment at time \(t\), \(f\) is a sufficiently regular nonlinear function, \(B\) is a Hilbert-space-valued Brownian motion, \(A\) is the infinitesimal generator of a strongly continuous semigroup and \(G\) is a suitable set-valued mapping. The main result is that this equation has – under certain regularity conditions – a mild solution. The main mathematical tool used in the proof is Martelli’s fixed-point theorem. Finally, an example of a neutral stochastic reaction-diffusion inclusion is discussed.

MSC:

34K50 Stochastic functional-differential equations
34K40 Neutral functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K35 Control problems for functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aubin J. P., Differential Inclusions (1984) · Zbl 0538.34007
[2] Deimling K., Nonlinear Functional Analysis (1985) · Zbl 0559.47040
[3] Hu S., Proc. of the Amer. Math. Soc. 123 pp 3043– (1995)
[4] DOI: 10.1016/0022-1236(82)90086-6 · Zbl 0528.60066
[5] Pettersson R., Stochastic and Stochastic Reports 52 pp 107– (1995) · Zbl 0864.60046
[6] Pettersson R., Probability and Mathematical Statistics 17 pp 29– (1997) · Zbl 0880.60062
[7] DOI: 10.1080/07362999408809334 · Zbl 0789.60052
[8] Ahmed N. U., Proceedings of the First World Congress of Nonlinear Analysis pp 1699– (1996)
[9] Balasubramaniam P., Tamkang Journal of Mathematics 33 pp 35– (2002)
[10] Bohnenblust , H. F. , and S. Karlin . 1950 . On the theorem of ville contributions to the theory of games , eds. H.W. Kuhn , and A.W Tucker . Princeton , NJ : Princeton University Press . 155 – 160 . · Zbl 0041.25701
[11] Martelli M., Boll. Un. Mat. Ital. 11 pp 70– (1975)
[12] DOI: 10.1515/9783110874228
[13] Hu S., Handbook of Multivalued Analysis (1997) · Zbl 0887.47001
[14] Banas J., Lect. Not. Pure. Appl. Math. No. 60, in: Measures of Noncompactness in Banach Spaces (1980) · Zbl 0438.47051
[15] Lasota A., Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 pp 781– (1965)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.