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Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose

Mathematical Citation Quotient (MCQ) 2017: 0.71

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Eberlein weak almost periodic solutions for a class of integro-differential equations with infinite delay

Khalil Ezzinbi
  • Corresponding author
  • Départment de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakesh, Morocco
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Samir Fatajou
  • Laboratoire L. M. C.; Départment de Mathématiques et Informatique, Faculté Polydisciplinaire-Sa, Université Cadi Ayyad, Sidi Bouzid, Morocco
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/ Fatima Zohra Elamrani
Published Online: 2018-12-05 | DOI: https://doi.org/10.1515/msds-2018-0010


In thiswork,we provide sufficient conditions ensuring the existence and uniqueness of an Eberlein weakly almost periodic solutions for some semilinear integro-differential equations with infinite delay in Banach spaces. For illustration, we provide an example arising in viscoelasticity theory.

Keywords: Integro-differential equation; Eberlein weak almost periodic solution; immediately norm continuous C0-semigroup


  • [1] E. Ait Dads E, K. Ezzinbi and S. Fatajou, Weakly almost periodic solutions for some differential equations in a Banach space, Nonlinear Studies, 4 2 (1997) 157 − 170.Google Scholar

  • [2] E. Ait Dads E, K. Ezzinbi and S. Fatajou, Weakly almost periodic solutions for the inhomogeneous linear equations and periodic processes in a Banach space, Dynamic Systems and Applications, 6 (1997), 507 − 516.Google Scholar

  • [3] E. Ait Dads E, K. Ezzinbi and S. Fatajou, Asymptotic behaviour of solutions for some differential equations in Banach spaces, Afr. Diapora J. Math., 12 1 (2011), 1 − 18.Google Scholar

  • [4] J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups, Wiley Inerscience, 1989.Google Scholar

  • [5] S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. USA 48 (1962), 2039 − 2043.CrossrefGoogle Scholar

  • [6] H. Bohr, Zur theorie der fastperiodisch functionen : Eine Versall gemeinerung det theorie der Fourierreihen. Acta. Math. Bd. 45 (1925), 29 − 127.Google Scholar

  • [7] J. Chen, T. Xiao, and J. Liang, Uniform exponential stability of solutions to abstract Volterra equations, J. Evol. Equ. 49 (2009), 661 − 674.Web of ScienceGoogle Scholar

  • [8] W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, TAMS. 67 (1949), 217 − 240.Google Scholar

  • [9] T. Diagana, G. M. Mophou and G. M. N’Guérékata, On the existence of mild solutions to some semilinear fractional integro-differential equations, Electron. J. Qual. Theory Differ. Equ. 58 (2010), 1 − 17.Google Scholar

  • [10] T. Diagana and G.M. N’Guérékata, Almost automorphic solutions to some classes of partial evolution equations, 866 (2007), 462 − 466.Web of ScienceGoogle Scholar

  • [11] T. Diagana, Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations, Nonlinear Analysis 69(2008), 4277 − 4285.Google Scholar

  • [12] T. Diagana, H. R. Henriquez and E. M. Hernandez, Almost automorphic mild solutions to some partial neutral functional-differential equations and applications, Nonlinear Analysis : Theory, Methods and Applications, 69 5 (2008), 1485−1493.Google Scholar

  • [13] T. Diagana, Pseudo almost periodic functions in Banach spaces, Nova Science Publishers, Inc., New York, 2007.Google Scholar

  • [14] K. De Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math, 105 (1961), 63 − 97.Google Scholar

  • [15] K. De Leeuw and I. Glicksberg, Almost periodic functions on semigroups, Acta Math, 105 (1961), 99 − 140.Google Scholar

  • [16] N. Dunford and J. T. Schwartz, Linear Operators, John Wiley and Sons, Vol 1. (1959).Google Scholar

  • [17] C. F. Dunkl and D. E. Ramirez, Topics in Harmonic Analysis, Appleton-Centry-Crofts, New York, 1971.Google Scholar

  • [18] A. Grothendiek, Critères de compacité dans les espaces fonctionels généraux, Amer. J. Math. 74 (1952), 168 − 186.Google Scholar

  • [19] U. Krengel, Ergodic Theorems, De Gruyter Studies in Math. 1985.Google Scholar

  • [20] C. Lizama and V. Poblete, Maximal regularity for perturbed integral equations on periodic Lebesgue spaces, J. Math. Anal. 348 2 (2008), 775 − 786.Web of ScienceGoogle Scholar

  • [21] C. Lizama and R. Ponce, Bounded solutions to a class of semilinear integro-differential equations in Banach spaces, Nonlinear Analysis., 74 (2011), 3397 − 3406.Google Scholar

  • [22] C. Lizama and R. Ponce, Almost automorphic solutions to abstract Volterra equations on the line, Nonlinear Analysis., 74 (2011), 3805 − 3814.Google Scholar

  • [23] M. A. Meyers and K. K. Chawla, Mechanical Behavior of Materials (Second Edition), Cambridge University, Wiley, New York (2009).Google Scholar

  • [24] G. M. N’Guérékata, Existence and uniqueness of almost automorphic mild solutions of some semilinear abstract differential equations, Semigroup Forum, 69 (2004), 80 − 86.Google Scholar

  • [25] G. M. N’Guérékata, Topics in Almost Automorphy, Springer Verlag, New York, (2005).Google Scholar

  • [26] G. M. N’Guérékata, Almost automorphic solutions to second-order semilinear evolution equations, Nonlinear Anal., 71 (2009), e432 − e435.Google Scholar

  • [27] V. Poblete, Solutions of second-order integro-differential equations on periodic Besov spaces, Proc. Edinb. Math. Soc., 502 (2007), 477 − 492.Web of ScienceGoogle Scholar

  • [28] G. Da Prato, A. Lunardi, Periodic solutions for linear integro-differential with infinite delay in Banach space, Differential Equations in Banach spaces, Lecture notes in Math. 1223 (1985), 49 − 60.Google Scholar

  • [29] J. Pruss, Evolutionary Integral Equations and Applications, Monographs Math., 87 Birkhauser Verlag, (1993).Google Scholar

  • [30] W.Rudin, Weak almost periodic functions and Fourier Stieljes transforms, Duke Math. J. 26 (1959), 215 − 220.Google Scholar

  • [31] W. M. Ruess, W. H. Summers, Integration of asymptotically almost periodic functions and weak asymptotic almost periodicity, Dissertationes Math., 279 (1989).Google Scholar

  • [32] W. M. Ruess and W. H. Summers, Weak almost periodicity and the strongly ergodiclLimit Theorem for periodic evolution systems, J.Funct. Anal., 94 (1990), 177 − 195.CrossrefGoogle Scholar

  • [33] W. M. Ruess and W. H. Summers, Weak almost periodic semigroups of operators, Pacific J. Math., 43 (1990), 175 − 193.Google Scholar

  • [34] W. M. Ruess and W. H. Summers, Ergodic theorems for semigroups of operators, Proc. Amer. Math. Soc. 114 (1992), 423 − 432.Google Scholar

About the article

Received: 2017-04-14

Accepted: 2018-10-04

Published Online: 2018-12-05

Published in Print: 2018-11-01

Citation Information: Nonautonomous Dynamical Systems, Volume 5, Issue 1, Pages 127–137, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2018-0010.

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© by Khalil Ezzinbi, et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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