# STABILITY IN THE α-NORM FOR SOME STOCHASTIC PARTIAL FUNCTIONAL INTEGRODIFFERENTIAL EQUATIONS

• Accepted : 2018.05.30
• Published : 2019.01.01
• 237 4

#### Abstract

In this work, we study the existence, uniqueness and stability in the ${\alpha}$-norm of solutions for some stochastic partial functional integrodifferential equations. We suppose that the linear part has an analytic resolvent operator in the sense given in Grimmer [8] and the nonlinear part satisfies a $H{\ddot{o}}lder$ type condition with respect to the ${\alpha}$-norm associated to the linear part. Firstly, we study the existence of the mild solutions. Secondly, we study the exponential stability in pth moment (p > 2). Our results are illustrated by an example. This work extends many previous results on stochastic partial functional differential equations.

#### Keywords

analytic resolvent operators;fractional power;stochastic partial functional integrodifferential equations;Wiener process;Picard iteration;mild solution;exponential stability

#### References

1. G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.
2. M. A. Diop, K. Ezzinbi, and M. Lo, Existence and uniqueness of mild solutions to some neutral stochastic partial functional integrodifferential equations with non-Lipschitz coefficients, Int. J. Math. Math. Sci. 2012 (2012), Art. ID 748590, 12 pp.
3. K. Ezzinbi, S. Ghnimi, and M. A. Taoudi, Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay, Nonlinear Anal. Hybrid Syst. 4 (2010), no. 1, 54-64. https://doi.org/10.1016/j.nahs.2009.07.006
4. K. Ezzinbi, H. Megdiche, and A. Rebey, Existence of solutions in the ${\alpha}$norm for partial differential equations of neutral type with finite delay, Electron. J. Differential Equations (2010), no. 157, 11 pp.
5. K. Ezzinbi, H. Toure, and I. Zabsonre, Local existence and regularity of solutions for some partial functional integrodifferential equations with infinite delay in Banach spaces, Nonlinear Anal. 70 (2009), no. 9, 3378-3389. https://doi.org/10.1016/j.na.2008.05.006
6. T. E. Govindan, Stability properties in the ${\alpha}$norm of solutions of stochastic partial functional differential equations, Differential Integral Equations 23 (2010), no. 5-6, 401-418.
7. R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc. 273 (1982), no. 1, 333-349. https://doi.org/10.1090/S0002-9947-1982-0664046-4
8. R. C. Grimmer and A. J. Pritchard, Analytic resolvent operators for integral equations in Banach space, J. Differential Equations 50 (1983), no. 2, 234-259. https://doi.org/10.1016/0022-0396(83)90076-1
9. V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and its Applications, 463, Kluwer Academic Publishers, Dordrecht, 1999.
10. K. Liu, Stability of infinite dimensional stochastic differential equations with applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135, Chapman & Hall/CRC, Boca Raton, FL, 2006.
11. R. Ravi Kumar, Nonlocal Cauchy problem for analytic resolvent integrodifferential equations in Banach spaces, Appl. Math. Comput. 204 (2008), no. 1, 352-362.
12. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
13. T. Taniguchi, On the estimate of solutions of perturbed linear differential equations, J. Math. Anal. Appl. 153 (1990), no. 1, 288-300. https://doi.org/10.1016/0022-247X(90)90279-O
14. 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), no. 1, 72-91. https://doi.org/10.1006/jdeq.2001.4073
15. C. C. Travis and G. F. Webb, Existence, stability, and compactness in the ${\alpha}$norm for partial functional differential equations, Trans. Amer. Math. Soc. 240 (1978), 129-143.
16. E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3 (1979), no. 2, 127-167.
17. J. P. C. dos Santos, S. M. Guzzo, and M. N. Rabelo, Asymptotically almost periodic solutions for abstract partial neutral integro-differential equation, Adv. Difference Equ. 2010 (2010), Art. ID 310951, 26 pp.
18. J. Wu, Theory and applications of partial functional-differential equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996.