Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Almost automorphic solutions of evolution equations

Authors: Toka Diagana, Gaston Nguerekata and Nguyen Van Minh
Journal: Proc. Amer. Math. Soc. 132 (2004), 3289-3298
MSC (2000): Primary 34G10; Secondary 43A60
Published electronically: June 18, 2004
MathSciNet review: 2073304
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the existence of almost automorphic mild solutions to equations of the form

\begin{displaymath}\dot u(t)= Au(t)+f(t),\tag*{$(*)$ }\end{displaymath}

where $A$ generates a holomorphic semigroup and $f$ is an almost automorphic function. Since almost automorphic functions may not be uniformly continuous, we introduce the notion of the uniform spectrum of a function. By modifying the method of sums of commuting operators used in previous works for the case of bounded uniformly continuous solutions, we obtain sufficient conditions for the existence of almost automorphic mild solutions to $(*)$ in terms of the imaginary spectrum of $A$and the uniform spectrum of $f$.

References [Enhancements On Off] (What's this?)

  • 1. W. Arendt, F. Räbiger, A. Sourour, Spectral properties of the operator equation AX + XB = Y, Quart. J. Math. Oxford (2), 45 (1994), 133-149. MR 95g:47060
  • 2. W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. MR 2003g:47072
  • 3. B. Basit, Harmonic analysis and asymptotic behavior of solutions to the abstract Cauchy problem, Semigroup Forum 54 (1997), 58-74. MR 98f:47049
  • 4. C.J.K. Batty, W. Hutter, F. Räbiger, Almost periodicity of mild solutions of inhomogeneous periodic Cauchy problems, J. Differential Equations 156 (1999), 309-327. MR 2001c:34116
  • 5. G. Da Prato, P. Grisvard, Sommes d'opérateurs linéaires et équations différentielles opérationelles, J. Math. Pures Appl. 54 (1975), 305-387. MR 56:1129
  • 6. E.B. Davies, ``One-parameter Semigroups", Academic Press, London, 1980. MR 82i:47060
  • 7. . T. Diagana and G. M. N'Guerekata, Some remarks on almost automorphic solutions of some abstract differential equations, Far East J. of Math. Sci. 8(3) (2003), 313-322. MR 2004d:34123
  • 8. K.J. Engel, R. Nagel, One-parameter Semigroups for Linear Evolution Equations. Springer, Berlin, 1999. MR 2000i:47075
  • 9. J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1985. MR 87c:47056
  • 10. Y. Hino, S. Murakami, Almost automorphic solutions for abstract functional differential equations. Preprint.
  • 11. Y. Hino, T. Naito, N.V. Minh, J.S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces. Taylor & Francis, London - New York, 2002. MR 2003i:34115
  • 12. R. Johnson, A linear almost periodic equation with an almost automorphic solution, Proc. Amer. Math. Soc. 82 (1981), no. 2, 199-205. MR 82i:34044a
  • 13. Y. Katznelson, An Introduction to Harmonic Analysis, Dover Publications, New York, 1976. MR 54:10976
  • 14. B.M. Levitan, V.V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow Univ. Publ. House 1978. English translation by Cambridge University Press, 1982. MR 84g:34004
  • 15. S. Murakami, T. Naito, N.V. Minh, Evolution semigroups and sums of commuting operators: a new approach to the admissibility theory of function spaces, J. Differential Equations 164 (2000), 240-285. MR 2001d:47063
  • 16. T. Naito, N.V. Minh, Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations, J. Differential Equations 152 (1999), 358-376. MR 99m:34131
  • 17. T. Naito, N.V. Minh, J. S. Shin, New spectral criteria for almost periodic solutions of evolution equations, Studia Mathematica 145 (2001), 97-111. MR 2002d:34092
  • 18. T. Naito, Nguyen Van Minh, J. Liu, On the bounded solutions of Volterra equations, Applicable Analysis. To appear.
  • 19. J. M. A. M. van Neerven, The Asymptotic Behavior of Semigroups of Linear Operators, Birkhäuser, Basel, 1996. MR 98d:47001
  • 20. G. M. N'Guerekata, Almost automorphic functions and applications to abstract evolution equations, Contemporary Math. 252 (1999), 71-76. MR 2001a:34097
  • 21. G. M. N'Guerekata Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer, Amsterdam, 2001. MR 2003d:43001
  • 22. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Math. Sci. 44, Springer-Verlag, Berlin-New York 1983. MR 85g:47061
  • 23. J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. MR 94h:45010
  • 24. W. M. Ruess, W.H. Summers, Almost periodicity and stability for solutions to functional-differential equations with infinite delay, Differential and Integral Equations, 9 (1996), no. 6, 1225-1252. MR 97m:34159
  • 25. W.M. Ruess, Q.P. Vu, Asymptotically almost periodic solutions of evolution equations in Banach spaces, J. Differential Equations 122 (1995), 282-301. MR 96i:34143
  • 26. W. Shen, Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows, Memoirs of the Amer. Math. Soc. 136 (1998). MR 99d:34088
  • 27. Q.P. Vu, E. Schüler, The operator equation $AX-XB = C$, admissibility, and asymptotic behaviour of differential equations, J. Differential Equations 145 (1998), 394-419. MR 99h:34081
  • 28. S. Zaidman, Topics in abstract differential equations, Pitman Research Notes in Mathematics Series, 304, Longman Scientific & Technical, New York, 1994. MR 95f:34087

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34G10, 43A60

Retrieve articles in all journals with MSC (2000): 34G10, 43A60

Additional Information

Toka Diagana
Affiliation: Department of Mathematics, Howard University, 2441 6th Street N.W., Washington D.C. 20059

Gaston Nguerekata
Affiliation: Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, Maryland 21251

Nguyen Van Minh
Affiliation: Department of Mathematics, Hanoi University of Science, Khoa Toan, Dai Hoc Khoa Hoc Tu Nhien, 334 Nguyen Trai, Hanoi, Vietnam
Address at time of publication: Department of Mathematics, State University of West Georgia, Carrollton, Georgia 30118

Keywords: Analytic semigroup, almost automorphic solution, uniform spectrum, sums of commuting operators
Received by editor(s): July 16, 2003
Published electronically: June 18, 2004
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society