Weighted Stepanov-Like Pseudo Almost Automorphic Solutions of Class r for Some Partial Differential Equations

  • Hamidou ToureEmail author
  • Issa Zabsonre
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 275)


The aim of this work is to study weighted Stepanov-like pseudo almost automorphic functions using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions. We also study the existence and uniqueness of \((\mu ,\nu )\) -Weighted Stepanov-like pseudo almost automorphic solutions of class r for some neutral partial functional differential equations in a Banach space when the delay is distributed using the spectral decomposition of the phase space developed by Adimy and co-authors. Here we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille-Yosida condition, the delayed part are assumed to be pseudo almost automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument.


Automorphic solutions Pseudo periodic functions Lipschitz condition 



The authors would like to thank the referees for their careful reading of this article. Their valuable suggestions and critical remarks made numerous improvements throughout this article.


  1. 1.
    Adimy, M., Elazzouzi, A., Ezzimbi, K.: Bohr-Neugebauer type theorem for some partial neutral functional differential equations. Nonlinear Anal. Theory Methods Appl. 66(5), 1145–1160 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adimy, M., Ezzinbi, K., Laklach, M.: Spectral decomposition for partial neutral functional differential equations. Can. Appl. Math. Q. 1, 1–34 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Blot, J., Cieutat, P., Ezzinbi, K.: New approach for weighted pseudo almost periodic functions under the light of measure theory, basic results and applications. Appl. Anal. 92(3), 493–526 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bochner, S.: Continuous mappings of almost automorphic and almost automorphic functions. Proc. Natl. Sci. USA 52, 907–910 (1964)CrossRefGoogle Scholar
  5. 5.
    Bochner, S.: A New Approach to Almost-Periodicity. Proc. Natl. Acad. Sci. USA 48, 2039–2043 (1962)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Diagana, T., Ezzinbi, K., Miraoui, M.: Pseudo-almost periodic and pseudo-almost automorphic solutions to some evolution equations involving theoretical measure theory. CUBO A Math. J. 16(02), 01–31 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ezzinbi, K., Fatajou, S., N’Guérékata, G.M.: Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces. J. Math. Anal. Appl. 351(2), 765–772 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ezzinbi, K., Fatajou, S., N’Guérékata, G.M.: \(C^n\)-almost automorphic solutions for partial neutral functional differential equations. Appl. Anal. 86(9), 1127–1146 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ezzinbi, K., N’Guérékata, G.M.: Almost automorphic solutions for some partial functional differential equations. Appl. Math. Lett. 328, 344–358 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    N’Guérékata, G.M.: Spectral Theory of Bounded Functions and Applications to Evolution Equations. Nova Science Publisher, New York (2017)Google Scholar
  11. 11.
    N’Guérékata, G.M., Pankov, A.: Stepanov-like almost automorphic functions and monotone evolution equations. Nonlinear Anal. 68, 2658–2667 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    N’Guérékata, G.M.: Topics in Almost Automorphy. Springer, New York (2005)Google Scholar
  13. 13.
    N’Guérékata, G.M.: Almost Automorphic and Almost Periodic Functions. Kluwer Academic Publishers, New York (2001)Google Scholar
  14. 14.
    Xiao, T.J., Liang, J., Zhang, J.: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76, 518–524 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zabsonré, I., Touré, H.: Pseudo almost periodic and pseudo almost automorphic solutions of class \(r\) under the light of measure theory. Afr. Diaspora J. Math. 19(1), 58–86 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Zhang, R., Chang, Y.K., N’Guérékata, G.M.: New composition theorems of Stepanov-like almost automorphic functions and applications to nonautonomous evolution equations. Nonlinear Anal. RWA 13, 2866–2879 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Unité de Recherche et de Formation en Sciences Exactes et Appliquées, Département de MathématiquesUniversité de Ouaga 1 Pr Joseph Ki-ZerboOuagadougou 03Burkina Faso

Personalised recommendations