Journal of Integral Equations and Applications

Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions

Pengyu Chen, Ahmed Abdelmonem, and Yongxiang Li

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The aim of this paper is to discuss the global existence, uniqueness and asymptotic stability of mild solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. A sufficient condition is given for judging the relative compactness of a class of abstract continuous family of functions on infinite intervals. With the aid of this criteria the compactness of the solution operator for the problem studied on the half line is obtained. The theorems proved in this paper improve and extend some related results in this direction. Discussions are based on stochastic analysis theory, analytic semigroup theory, relevant fixed point theory and the well known Gronwall-Bellman type inequality. An example to illustrate the feasibility of our main results is also given.

Article information

Source
J. Integral Equations Applications, Volume 29, Number 2 (2017), 325-348.

Dates
First available in Project Euclid: 17 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1497664831

Digital Object Identifier
doi:10.1216/JIE-2017-29-2-325

Mathematical Reviews number (MathSciNet)
MR3663526

Zentralblatt MATH identifier
1371.34086

Subjects
Primary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03] 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25] 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
Stochastic evolution equation nonlocal initial condition analytic semigroups existence asymptotic stability

Citation

Chen, Pengyu; Abdelmonem, Ahmed; Li, Yongxiang. Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions. J. Integral Equations Applications 29 (2017), no. 2, 325--348. doi:10.1216/JIE-2017-29-2-325. https://projecteuclid.org/euclid.jiea/1497664831


Export citation

References

  • J. Bao, Z. Hou and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential equations, Proc. Amer. Math. Soc. 138 (2010), 2169–2180.
  • L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Appl. Anal. 162 (1991), 494–505.
  • ––––, Existence and uniqueness of a classical solutions to a functional-differential abstract nonlocal Cauchy problem, J. Math. Appl. Stoch. Anal. 12 (1999), 91–97.
  • P. Chen and Y. Li, Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces, Collect. Math. 66 (2015), 63–76.
  • P. Chen, Y. Li and X. Zhang, On the initial value problem of fractional stochastic evolution equations in Hilbert spaces, Comm. Pure Appl. Anal. 14 (2015), 1817–1840.
  • P. Chen, X. Zhang and Y. Li, Nonlocal problem for fractional stochastic evolution equations with solution operators, Fract. Calc. Appl. Anal. 19 (2016), 1507–1526.
  • C. Corduneanu, Principles of differential and integral equations, Allyn and Bacon, Boston, 1971.
  • J. Cui, L. Yan and X. Wu, Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces, J. Korean Stat. Soc. 41 (2012), 279–290.
  • G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992.
  • K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York, 1985.
  • K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl. 179 (1993), 630–637.
  • M.M. El-Borai, O.L. Mostafa and H.M. Ahmed, Asymptotic stability of some stochastic evolution equations, Appl. Math. Comp. 144 (2003), 273–286.
  • K. Ezzinbi, X. Fu and K. Hilal, Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, Nonlin. Anal. 67 (2007), 1613–1622.
  • W. Grecksch and C. Tudor, Stochastic evolution equations: A Hilbert space approach, Akademic Verlag, Berlin, 1995.
  • D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes Math. 840, Springer-Verlag, New York, 1981.
  • J. Liang, J.H. Liu and T.J. Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlin. Anal. 57 (2004), 183–189.
  • K. Liu, Stability of infinite dimensional stochastic differential equations with applications, Chapman and Hall, London, 2006.
  • J. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. Math. Anal. Appl. 342 (2008), 753–760.
  • M. McKibben, Discoving evolution equations with applications, I, Deterministic models, Chapman and Hall/CRC, New York, 2011.
  • A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, Berlin, 1983.
  • Y. Ren, Q. Zhou and L. Chen, Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay, J. Oper. Th. Appl. 149 (2011), 315–331.
  • R. Sakthivel, Yong Ren, Amar Debbouche and N.I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Appl. Anal. 95 (2016), 2361–2382.
  • K. Sobczyk, Stochastic differential equations with applications to physics and engineering, Kluwer Academic Publishers, London, 1991.
  • T. Taniguchi, K. Liu and A. Truman, Existence, uniqueness and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. Diff. Equat. 181 (2002), 72–91.
  • I.I. Vrabie, Existence in the large for nonlinear delay evolution inclutions with nonlocal initial conditions, J. Funct. Anal. 262 (2012), 1363–1391.
  • T.J. Xiao and J. Liang, Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlin. Anal. 63 (2005), 225–232.
  • X. Xue, Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces, Nonlin. Anal. 70 (2009), 2593–2601.