Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

8 Issues per year


IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

Online
ISSN
2191-0294
See all formats and pricing
More options …
Volume 19, Issue 5

Issues

Pseudo Almost Periodicity and Its Applications to Impulsive Nonautonomous Partial Functional Stochastic Evolution Equations

Zuomao Yan / Xiumei Jia
Published Online: 2018-06-08 | DOI: https://doi.org/10.1515/ijnsns-2017-0086

Abstract

In this paper, we establish a new composition theorem for pseudo almost periodic functions under non-Lipschitz conditions. We apply this new composition theorem together with a fixed-point theorem for condensing maps to investigate the existence of p-mean piecewise pseudo almost periodic mild solutions for a class of impulsive nonautonomous partial functional stochastic evolution equations in Hilbert spaces, and then, the exponential stability of p-mean piecewise pseudo almost periodic mild solutions is studied. Finally, an example is given to illustrate our results.

Keywords: impulsive nonautonomous partial functionalstochastic evolution equations; p-mean piecewise pseudo almostperiodic functions; composition theorem; evolution family; fixed point

MSC 2010: 34A37; 60H10; 35B15; 34F05

References

  • [1]

    C. Y. Zhang, Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl. 181 (1994), 62–76.CrossrefWeb of ScienceGoogle Scholar

  • [2]

    B. Amir, L. Maniar, Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain, Ann. Math. Blaise Pascal. 6 (1999), 1–11.Crossref

  • [3]

    H. Li, F. Huang, J. Li, Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 255 (2001), 436–446.CrossrefGoogle Scholar

  • [4]

    T. Diagana, C. M. Mahop, G. M. N’Guérékata, B. Toni, Existence and uniqueness of pseudo almost periodic solutions to some classes of semilinear differential equations and applications, Nonlinear Anal. 64 (2006), 2442–2453.Crossref

  • [5]

    E. M. Hernández, H. R. Henr&’ıquez, Pseudo almost periodic solutions for non-autonomous neutral differential equations with unbounded delay, Nonlinear Anal. RWA 9 (2008), 430–437.CrossrefGoogle Scholar

  • [6]

    T. Diagana, Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations. Nonlinear Anal. 69 (2008), 4277–4285.Web of ScienceCrossrefGoogle Scholar

  • [7]

    Z. Hu, Z. Jin, Stepanov-like pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations, Nonlinear Anal. 75 (2012), 244–252.Web of ScienceCrossref

  • [8]

    E. Alvarez, C. Lizama, Weighted pseudo almost periodic solutions to a class of semilinear integro-differential equations in Banach spaces, Adv. Difference Equ. 2015 (2015), 1–18.Web of ScienceGoogle Scholar

  • [9]

    D. Prato, C. Tudor, Periodic and almost periodic solutions for semilinear stochastic equations, Stoch. Anal. Appl. 13 (1995), 13–33.CrossrefGoogle Scholar

  • [10]

    A. Ya. Dorogovtsev, O. A. Ortega, On the existence of periodic solutions of a stochastic equation in a Hilbert space, Visnik Kiiv. Univ. Ser. Mat. Mekh. 115 (1988), 21–30.

  • [11]

    P. H. Bezandry, T. Diagana, Existence of almost periodic solutions to some stochastic differential equations, Appl. Anal. 86 (2007), 819–827.Web of ScienceCrossrefGoogle Scholar

  • [12]

    P. Crewe, Almost periodic solutions to stochastic evolution equations on Banach spaces, Stoch. Dyn. 13 (2013), 1250027, 1–23.Google Scholar

  • [13]

    X.-L. Li, Square-mean almost periodic solutions to some stochastic evolution equations, Acta Math. Sin. (Engl. Ser.) 30 (2014), 881–898.CrossrefGoogle Scholar

  • [14]

    J. Cao, Q. Yang, Z. Huang, Q. Liu, Asymptotically almost periodic solutions of stochastic functional differential equations, Appl. Math. Comput. 218 (2011), 1499–1511.Web of ScienceGoogle Scholar

  • [15]

    J. Cao, Q. Yang, Z. Huang, On almost periodic mild solutions for stochastic functional differential equations, Nonlinear Anal. RWA 13 (2012), 275–286.Crossref

  • [16]

    C.A. Tudor, M. Tudor, Pseudo almost periodic solutions of some stochastic differential equations, Math. Rep. (Bucur.) 1 (1999), 305–314.Google Scholar

  • [17]

    Z. Yan, H. Zhang, Existence of Stepanov-like square-mean pseudo almost periodic solutions to partial stochastic neutral differential equations, Ann. Funct. Anal. 6 (2015), 116–138.CrossrefWeb of ScienceGoogle Scholar

  • [18]

    P. H. Bezandry, T. Diagana, Square-mean almost periodic solutions nonautonomous stochastic differential equations, Electron. J. Differ. Equ. 2007 (2007), 1–10.Google Scholar

  • [19]

    P. H. Bezandry, T. Diagana, Existence of square-mean almost periodic mild solutions to some nonautonomous stochastic second-order differential equations, Electron. J. Differ. Equ. 2010 (2010), 1–25.

  • [20]

    P. H. Bezandry, T. Diagana, Almost Periodic Stochastic Processes, Springer-Verlag New York Inc., 2011.Web of Science

  • [21]

    P. H. Bezandry, T. Diagana, P-th mean pseudo almost automorphic mild solutions to some nonautonomous stochastic differential equations, Afr. Diaspora J. Math. 12 (2011), 60–79.

  • [22]

    M. A. Diop, K. Ezzinbi, M. M. Mbaye, Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion, Stochastics 87 (2015), 1061–1093.Web of ScienceCrossref

  • [23]

    A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.

  • [24]

    H. R. Henr&’ıquez, B. De Andrade, M. Rabelo, Existence of almost periodic solutions for a class of abstract impulsive differential equations. ISRN Math. Anal. 2011 (2011), Article ID 632687, 1–21.

  • [25]

    J. Liu, C. Zhang, Existence and stability of almost periodic solutions for impulsive differential equations. Adv. Differ. Equ. 2012 (2012), 1–14.Web of ScienceGoogle Scholar

  • [26]

    G. T. Stamov, Almost Periodic Solutions of Impulsive Differential Equations, Springer, Berlin, 2012.Google Scholar

  • [27]

    G. T. Stamov, J. O. Alzabut, Almost periodic solutions for abstract impulsive differential equations. Nonlinear Anal. 72 (2010), 2457–2464.Google Scholar

  • [28]

    G. T. Stamov, I. M. Stamova, Almost periodic solutions for impulsive fractional differential equations. Dyn. Syst. 29 (2014), 119–132.CrossrefGoogle Scholar

  • [29]

    J. Liu, C. Zhang, Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations, Adv. Differ. Equ. 2013 (2013), 1–21.Google Scholar

  • [30]

    F. Chérif, Pseudo almost periodic solutions of impulsive differential equations with delay, Differ. Equ. Dyn. Syst. 22 (2014), 73–91.CrossrefGoogle Scholar

  • [31]

    D. D. Bainov, P. S. Simeonov, Impulsive Differential Equations, Asymptotic properties of the solutions, World Scientific, Singapore, 1995.Google Scholar

  • [32]

    R. Sakthivel, J. Luo, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl. 356 (2009), 1–6.CrossrefWeb of ScienceGoogle Scholar

  • [33]

    L. Hu, Y. Ren, Existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays, Acta Appl. Math. 111 (2010), 303–317.CrossrefWeb of ScienceGoogle Scholar

  • [34]

    Z. Yan, X. Yan, Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay, Collect. Math. 64 (2013), 235–250.CrossrefWeb of ScienceGoogle Scholar

  • [35]

    R. J. Zhang, N. Ding, L. S. Wang, Mean square almost periodic solutions for impulsive stochastic differential equations with delays, J. Appl. Math. 2012 (2012), Article ID 414320, 1–14.Web of ScienceGoogle Scholar

  • [36]

    J. Liu, C. Zhang, Existence and stability of almost periodic solutions to impulsive stochastic differential equations. Cubo 15 (2013), 77–96.CrossrefGoogle Scholar

  • [37]

    K. J. Engel, R. Nagel, One parameter semigroups for linear evolution equations, in: Graduate texts in Mathematics, Springer-Verlag, 2000.Google Scholar

  • [38]

    C. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press, Beijing, 2003.Google Scholar

  • [39]

    B. N. Sadovskii, On a fixed-point principle. Funct. Anal. Appl. 1 (1967), 74–76.Google Scholar

  • [40]

    P. Acquistapace, F. Flandoli, B. Terreni, Initial boundary value problems and optimal control for nonautonomous parabolic systems, SIAM J. Control Optim. 29 (1991), 89–118.CrossrefGoogle Scholar

  • [41]

    L. Maniar, S. Roland, Almost periodicity of inhomogeneous parabolic evolution equations. In: Lecture Notes in Pure and Applied Mathematics, vol. 234, pp.299–318. Dekker, New York, 2003.Google Scholar

  • [42]

    A. Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl. 90 (1982), 12–44.CrossrefGoogle Scholar

About the article

Received: 2017-04-16

Accepted: 2018-05-20

Published Online: 2018-06-08

Published in Print: 2018-07-26


This work is supported by the National Natural Science Foundation of China (11461019), the President Fund of Scientific Research Innovation and Application of Hexi University (xz2013-10).


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 5, Pages 511–529, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0086.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in