Journal of Applied and Pure Mathematics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

STEPANOV-LIKE PSEUDO ALMOST AUTOMORPHIC SOLUTIONS OF CLASS r IN 𝛼-NORM UNDER THE LIGHT OF MEASURE THEORY

  • DJENDODE MBAINADJI (Universite Joseph KI-ZERBO, Unite de Recherche et de Formation en Sciences Exactes et Appliquees, Departement de Mathematiques) ;
  • ISSA ZABSONRE (Universite Joseph KI-ZERBO, Unite de Recherche et de Formation en Sciences Exactes et Appliquees, Departement de Mathematiques)
  • Received : 2022.05.24
  • Accepted : 2023.06.07
  • Published : 2023.07.30
Download PDF
⟨ Previous Next ⟩

Abstract

The aim of this work is to present some interesting results on weighted ergodic functions and prove the existence and uniqueness of Stepanov-like pseudo almost automorphic solutions using the spectral decomposition of the phase space developed by Adimy and co-authors. We also give the next challenge of this work.

Keywords

Acknowledgement

The authors wish to thank the referee for his (her) careful reading and valuable remarks which improve the presentation of the paper.This work is partially support by the Network PDE-MC under grants ISP-Bur01.

References

  1. M. Adimy, A. Elazzouzi and K. Ezzinbi, Reduction principle and dynamic behavoirs for a class of partial functional differential equations, Nonlinear Analysis 71 (2009), 1709-1727.  https://doi.org/10.1016/j.na.2009.01.008
  2. M. Adimy, K. Ezzinbi, M. Laklach, Spectral decomposition for partial neutral functional differential equations, Canadian Applied Mathematics Quarterly 1 (2001), 1-34. 
  3. J. Blot, P. Cieutat and K. Ezzinbi, Measure theory and pseudo almost automorphic functions: New developments and applications, Nonlinear Analysis (2012), 2426-2447. 
  4. S. Bochner, Continuous mappings of almost automorphic and almost automorphic functions, Proc. Natl. Sci. 52 (1964), 907-910.  https://doi.org/10.1073/pnas.52.4.907
  5. S. Bochner, A New Approach to Almost-Periodicity, Proc. Natl. Acad. Sci. 48 (1962), 2039-2043.  https://doi.org/10.1073/pnas.48.12.2039
  6. T. Diagana, K. Ezzinbi and M. Miraoui, Pseudo-almost periodic and pseudo-almost automorphic solutions to some evolution equations involving theoretical measure theory, CUBO A Mathematical Journal 16 (2014), 01-31.  https://doi.org/10.4067/S0719-06462014000200001
  7. T. Diagana, Weighted pseudo-almost periodic solutions to some differential equations, Nonlinear Anal. Theory Methods Appl. 68 (2008), 2250-2260.  https://doi.org/10.1016/j.na.2007.01.054
  8. K. Ezzinbi, H. Toure, Issa Zabsonre Pseudo almost automorphic solutions of class r for some partial functional differential equations, Afrika Matematika 25 (2014), 25-41.  https://doi.org/10.1007/s13370-012-0096-4
  9. K. Ezzinbi, S. Fatajou, N'Guerekata, Pseudo Almost Automorphic Solutions for Dissipative Differential Equations in Banach Spaces, Journal of Mathematical Analysis and Applicatons 351 (2009), 765-772.  https://doi.org/10.1016/j.jmaa.2008.11.017
  10. K. Ezzinbi, S. Fatajou, G.M. N'guerekata, Cn-almost automorphic solutions for partial neutral functional differential equations, Applicable Analysis 86:9 (2007), 1127-1146.  https://doi.org/10.1080/00036810701584591
  11. G.M. N'Guerekata, Spectral theory of bounded functions and applications to evolution equations, Nova Science Publisher, New York, 2017. 
  12. G.M. N'Guerekata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Analysis 68 (2008), 2658-2667.  https://doi.org/10.1016/j.na.2007.02.012
  13. G.M. N'Guerekata, Almost automorphic and almost periodic functions, Kluwer Academic Publishers, New York, 2001. 
  14. A. Pazy, Semigroups of Linear Operators and Application to Partial Differental Equation, Applied Mathematical Sciences, Springer-Verlag, New York, 44, 1983. 
  15. C.C. Travis and G.F. Webb, Existence, stability, and compactness in the α-norm for partial functional differential equations , Transaction of the American Mathematical Society 240 (1978), 129-143. 
  16. T.J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum 76 (2008), 518-524.  https://doi.org/10.1007/s00233-007-9011-y
  17. I. Zabsonre and H. Toure, Pseudo almost periodic and pseudo almost automorphic solutions of class r under the light of measure theory, African Diaspora Journal of Mathematics 19 (2016), 58-86. 
  18. Zhang, Y.K. Chang, G.M. N'Guerekata, New composition theorems of Stepanov-like almost automorphic functions and applications to nonautonomous evolution equations, Nonlinear Anal. RWA 13 (2012), 2866-2879. https://doi.org/10.1016/j.nonrwa.2012.04.016