Abstract
In this paper we prove the existence of mild solutions for a second-order impulsive semilinear stochastic differential inclusion with an infinite-dimensional standard cylindrical Wiener process and Poisson jumps. We consider non convex-valued cases.
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References
Ahmed, N.U.: Nonlinear stochastic differential inclusions on Banach space. Stoch. Anal. Appl. 12, 1–10 (1994)
Balasubramaniam, P., Ntouyas, S.K., Vinayagam, D.: Existence of solutions of semilinear stochastic delay evolution inclusions in a Hilbert space. J. Math. Anal. Appl. 305, 438–451 (2005)
Balasubramaniam, P.: Existence of solutions of functional stochastic differential inclusions. Tamkang J. Math. 33(1), 35–43 (2002)
Dugundji, J., Granas, A.: Fixed Point Theory. Springer, New York (2003)
Henriquez, H.R.: Existence of solutions of nonautonomous second order functional differential equations with infinite delay. Nonlinear Anal. Theory Methods 74, 3333–3352 (2011)
Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York (2006)
Deimling, K.: Multi-valued Differential Equations. De Gruyter, Berlin (1992)
Lasota, A., Opial, Z.: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 13, 781–786 (1965)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Gard, T.C.: Introduction to Stochastic Differential Equations. Marcel Dekker, New York (1988)
Gikhman, I.I., Skorokhod, A.: Stochastic Differential Equations. Springer, Berlin (1972)
Sobczyk, H.: Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer Academic Publishers, London (1991)
Tsokos, C.P., Padgett, W.J.: Random Integral Equations with Applications to Life Sciences and Engineering. Academic Press, New York (1974)
Bharucha-Reid, A.T.: Random Integral Equations. Academic Press, New York (1972)
Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)
Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 4th edn. Springer, Berlin (1995)
Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Spaces, vol. 293. Cambridge University Press, Cambridge (2002)
Situ, R.: Theory of Stochastic Differential Equations with Jumps and Applications. Springer, Berlin (2005)
Sato, K.I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Xu, Y., Pei, B., Guo, G.: Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise. Appl. Math. Comput. 263, 398–409 (2015)
Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1998)
Mueller, C.: The heat equation with Lévy noise. Stoch. Process. Appl. 74, 67–82 (1998)
Hausenblas, E.: Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure. Electron. J. Probab. 10, 1496–1546 (2005) (electronic)
Kallianpur, G., Xiong, J.: A nuclear space-valued stochastic differential equation driven by Poisson random measures. In: Stochastic Partial Differential Equations and Their Applications. Lecture Notes in Control and Information Sciences, vol. 176, pp. 135–143. Springer, Berlin (1987)
Albeverio, S., Wu, J., Zhang, T.: Parabolic SPDEs driven by Poisson white noise. Stoch. Process. Appl. 74, 21–36 (1998)
Mytnik, L.: Stochastic partial differential equation driven by stable noise. Probab. Theory Relat. Fields. 123(2), 157–201 (2002)
Blouhi, T., Nieto, J., Ouahab, A.: Existence and uniqueness results for systems of impulsive stochastic differential equations. Ukr. Math. J. (2016) (to appear)
Blouhi, T., Caraballo, T., Ouahab, A.: Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion. Stoch. Anal. Appl. 34(5), 792–834 (2016)
Taniguchi, T.: The existence and asymptotic behaviour of solutions to non-Lipschitz stochastic functional evolution equations driven by Poisson jumps. Stochastics 82, 339–363 (2010)
Deimling, K.: Multivalued Differential Equations. Walter De Gruyter, Berlin (1992)
Ouahab, A.: Some Perov’s and Karsnosel’skii type fixed point results and application. Commun. Appl. Anal. 19, 623–642 (2015)
Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces, vol. 108. Elsevier, Amsterdam (2011)
Heikkila, S., Lakshmikantham, V.: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations. Marcel Dekker, New York (1994)
Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. In: Lecture Notes in Mathematics, vol. 580, Springer (2006)
Górniewicz, L.: Topological fixed point theory of multi-valued mappings. In: Mathematics and Its Applications, vol. 495. Kluwer Academic Publishers, Dordrecht (1999)
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The authors would like to thank very much the anonymous referees for their careful reading and valuable comments on this work.
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Blouhi, T., Ferhat, M. Coupled System of Second-Order Stochastic Neutral Differential Inclusions Driven by Wiener Process and Poisson Jumps. Differ Equ Dyn Syst 30, 1011–1025 (2022). https://doi.org/10.1007/s12591-018-00450-y
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DOI: https://doi.org/10.1007/s12591-018-00450-y
Keywords
- Non-autonomous stochastic inclusions
- Second-order system
- Poisson jumps
- Impulses
- Matrix convergent to zero
- Generalized Banach space
- Fixed point
- Set-valued analysis