Majid Amintorabi, Ruhollah Jahanipur

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In this paper, we study semilinear stochastic evolution equations with semimonotone nonlinearity and multiplicative noise in L p spaces for 2 ≤ p < ∞. We do not impose any coercivity or Lipschitz condition on the nonlinear part of equations. We prove the existence, uniqueness and measurability of the mild solutions.The proofs of the existence and uniqueness are based on a version of the Itˆ o type inequality which is stronger than analogues inequalities.


Semilinear stochastic evolution equations; semimonotone nonlinearity; multiplicative noise; Lipschitz condition.

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F. Albiac and N. J. Kalton: Topics in Banach Space Theory. Graduate Texts in Mathematics, Springer-Verlag, 2006.

F. E. Browder: Nonlinear equations of evolution. Ann. of Math. 80 (1964), 485–523.

Z. Brze´ zniak: Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal. 4 (1995), no.1, 1–45.

Z. Brze´ zniak: On stochastic convolution in Banach spaces and applications. Stoch. Stoch. Rep. 61 (1997), no. 34, 245–295.

Z. Brze´ zniak and S. Peszat: Maximal inequalities and exponential estimates for stochastic convolutions in Banach spaces. in Stochastic processes, physics and geometry: new interplays, I–Leipzig 1999, CMS Conf. Proc., vol. 28, Amer. Math.

Soc., Providence, RI, 2000, 55–64.

Z. Brze´ zniak and J. M. A. M. van Neerven and M. C. Veraar and L. Weis: Itô formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differential Equations 245 (2008), 30–58.

R. F. Curtain and A. J. Pritchard: Infinite dimensional linear system theory. LN in: Control and Information Sciences vol. 8, Springer-Verlag, New York,

Berlin, 1978.

G. Da Prato and J. Zabczyk: Equations In Infinite Dimensions. 2nd ed., Cambridge University Press, 2014.

D. Freeman and E. Odell and Th. Schlumprecht and A. Zsák: Unconditional structures of translates for L p (R d ). Israel J. Math.203 (2014), 189–209.

H. D. Hamedani and B. Z. Zangeneh:The existence, uniqueness and measurability of stopped semilinear integral equations. Stochastic Anal. Appl. 25 (2007), 493–518.

A. Ichikawa: Some inequalities for martingales and stochastic convolutions. Stochastic Anal. Appl. 4 (1986), no. 3, 329–339.

S. Itoh: Random fixed point theorems with an application to random differential equations in Banach spaces. J. Math. Anal. Appl. 67 (1979), 261–273.

R. Jahanipur and B. Z. Zangeneh: Stability of semilinear stochastic evolution equations with monotone nonlinearity. Math. Ineq. Appl.3 (2000), 593–614.

R. Jahanipur: Stability of stochastic delay evolution equations with monotone nonlinearity. Stoch. Anal. Appl. 21 (2003), 161–181.

R. Jahanipur: Nonlinear functional differential equations of monotone-type in Hilbert spaces. Nonlinear Anal.72 (2010), 1393–1408.

R. Jahanipur: Stochastic functional evolution equations with monotone nonlinearity: Existence and stability of the mild solutions. J. Differential Equations 248 (2010), 1230–1255.

R. Jahanipur: Boundedness and continuity of the mild solutions of semilinear stochastic functional evolution equations. Math. Ineq. Appl. 15 (2012), no. 2, 423–436.

T. Kato: Nonlinear Evolution Equations in Banach Spaces. Proc. Symp. App. Math. 17 (1964), 50–67.

P. Kotelenze: A submartingale type inequality with applications to stochastic evolution equations. Stochastics. 8 (1982), 139–151.

N. V. Krylov and B. L. Rozovskii:Stochastic Evolution Equations. J. of Soviet Mathematics. 16 (1981), 1233-1277.

J. M. A. M. van Neerven and M. C. Veraar: On the stochastic Fubini theorem

in infinite dimensions. Stochastic Partial Differential Equations and Applications VII - Levico Terme 2004. Lecture Notes Pure Appl. Math., vol. 245, CRC Press: pp. 323–336, 2005.

J. M. A. M. van Neerven and M. C. Veraar and L. Weis: Stochastic integration in UMD spaces. Ann. Probab. 35 (2007), no. 4, 1438–1478.

J. M. A. M. van Neerven and M. C. Veraar and L. Weis: Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal.255 (2008), no.4, 940–993.

J. M. A. M. van Neerven: γ-Radonifying operators − a survey. In: The AM-SIANU Workshop on Spectral Theory and Harmonic Analysis. Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 44, pages 1–61. Austral. Nat. Univ., Canberra, 2010.

J. M. A. M. van Neerven and J. Zhu: A maximal inequality for stochastic convolutions in 2-smooth Banach spaces. Electron. Commun. Probab. 6 (2011), 689–705.

E. Pardoux: Equations aux d´ eriv´ es partielles stochastiques non lineaires monotones: Etude de solutions fortes de type Itô. Ph. D. Thesis, Paris-Sud University,

Paris, 1975.

C. Prévôt and M. Röckner: A concise course on stochastic partial differential equations. Springer, 2007.

E. Salavati and B. Z. Zangeneh: Semilinear stochastic evolution equations of monotone type with Lvy noise. Proceedings of Dyn. Sys. and Appl. 6 (2012), 380–387.

E. Salavati and B. Z. Zangeneh: A maximal inequality for pth power of stochastic convolution integrals. J. Ineq. Appl. 2016 (2016), no. 155, 1–16.

E. Salavati and B. Z. Zangeneh: Stochastic evolution equations with multi-

plicative Poisson noise and monotone nonlinearity. Bulletin of the Iranian Mathematical Society. 43 (2017), no. 5, 1287–1299.

L. Tubaro: An estimate of Burkholder type for stochastic processes defined by the stochastic integral. Stochastic Anal. Appl. 2 (1984), 187–192.

M. Verrar and S. Cox: Vector-valued decoupling and the Burkholder-Davis-Gundy inequality. Illinois J. Math. 55 (2011), no. 1, 343–375.

B. Z. Zangeneh: Semilinear stochastic evolution equations. Ph. D. thesis, University of British Columbia, Vancouver, B.C. Canada, 1990.

B. Z. Zangeneh: Measurability of the solution of a semilinear evolution equation. Progr. Probab. 24 (1991), 335–351.

B. Z. Zangeneh: Semilinear stochastic evolution equations with monotone nonlinearities. Stoch. Stoch. Rep. 53 (1995), 129–174.

B. Z. Zangeneh: An energy-type inequality. Math. Ineq. Appl. 1 (1998), no. 3, 453–461.



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