Afrika Matematika

, Volume 24, Issue 1, pp 33–53 | Cite as

Bounded solutions to some classes of nonautonomous higher-order differential equations

Article

Abstract

In this paper we study and obtain under some reasonable assumptions the existence of bounded and pseudo-almost automorphic mild solutions to some classes of nonautonomous higher-order abstract differential equations on an infinite dimensional separable Hilbert space. To illustrate our abstract results, we discuss the existence of bounded and pseudo almost automorphic mild solutions to Sine–Gordon type boundary value problems.

Keywords

Exponential dichotomy Acquistapace–Terreni conditions Evolution families Almost automorphic Pseudo-almost automorphic Nonautonomous higher-order differential equation Sine–Gordon equation type Boundary value problems 

Mathematics Subject Classification (2000)

43A60 34B05 34C27 42A75 47D06 35L90 

References

  1. 1.
    Acquistapace P.: Evolution operators and strong solutions of abstract linear parabolic equations. Differ. Integr. Equ. 1, 433–457 (1988)MathSciNetMATHGoogle Scholar
  2. 2.
    Acquistapace P., Terreni B.: A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sem. Mat. Univ. Padova 78, 47–107 (1987)MathSciNetMATHGoogle Scholar
  3. 3.
    Amann H.: Linear and Quasilinear Parabolic Problems. Birkhäuser, Berlin (1995)MATHCrossRefGoogle Scholar
  4. 4.
    Andres J., Bersani A.M., Radová L.: Almost periodic solutions in various metrics of higher-order differential equations with a nonlinear restoring term. Acta Univ. Palacki Olomuc., Fac. rer. nat., Mathematica 45, 7–29 (2006)MATHGoogle Scholar
  5. 5.
    Andres J., Górniewicz L.: Topological Fixed-Points Principles for Boundary Value Problems. Kluwer, Dordrecht (2003)Google Scholar
  6. 6.
    Andres J.: Existence of two almost periodic solutions of pendulum-type equations. Nonlinear Anal. 37(6), 797–804 (1999)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Arendt W., Chill R., Fornaro S., Poupaud C.: L p-Maximal regularity for non-autonomous evolution equations. J. Differ. Equ. 237(1), 1–26 (2007)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Arendt W., Batty C.J.K.: Almost periodic solutions of first- and second-order Cauchy problems. J. Differ. Equ. 137(2), 363–383 (1997)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Baroun M., Boulite S., Diagana T., Maniar L.: Almost periodic solutions to some semilinear non-autonomous thermoelastic plate equations. J. Math. Anal. Appl. 349(1), 74–84 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Baroun M., Boulite S., N’Guérékata G.M., Maniar L.: Almost automorphy of semilinear parabolic equations. Electron. J. Differ. Equ. 2008(60), 1–9 (2008)Google Scholar
  11. 11.
    Bugajewski D., Diagana T.: Almost automorphy of the convolution operator and applications to differential and functional-differential equations. Nonlinear Stud. 13(2), 129–140 (2006)MathSciNetMATHGoogle Scholar
  12. 12.
    Bugajewski D., Diagana T., Mahop C.M.: Asymptotic and pseudo almost periodicity of the convolution operator and applications to differential and integral equations. Z. Anal. Anwend. 25(3), 327–340 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Chicone, C., Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. Amer. Math. Soc. (1999)Google Scholar
  14. 14.
    Cieutat P., Ezzinbi K.: Existence, uniqueness and attractiveness of a pseudo almost automorphic solutions for some dissipative differential equations in Banach spaces. J. Math. Anal. Appl. 354(2), 494–506 (2009)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Diagana T.: Existence of pseudo-almost automorphic solutions to some abstract differential equations with S p-pseudo-almost automorphic coefficients. Nonlinear Anal. 70(11), 3781–3790 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Diagana T.: Pseudo Almost Periodic Functions in Banach Spaces. Nova Science Publishers Inc., New York (2007)MATHGoogle Scholar
  17. 17.
    Diagana, T.: Existence of almost automorphic solutions to some classes of nonautonomous higher-order differential equations. Electron. J. Qual. Theory Differ. Equ. 58, 17 pp (2010)Google Scholar
  18. 18.
    Diagana T., N’Guérékata G.M.: Pseudo almost periodic mild solutions to hyperbolic evolution equations in abstract intermediate Banach spaces. Applicable Anal. 85(6–7), 769–780 (2006)MATHCrossRefGoogle Scholar
  19. 19.
    Engel K.J., Nagel R.: One parameter semigroups for linear evolution equations Graduate Texts in Mathematics. Springer, Berlin (1999)Google Scholar
  20. 20.
    Ezzinbi K., Fatajou S., N’Guérékata G.M.: Pseudo almost automorphic solutions to some neutral partial functional differential equations in Banach space. Nonlinear Anal. 70(4), 1641–1647 (2009)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    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)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Leiva H.: Existence of bounded solutions of a second-order system with dissipation. J. Math. Anal. Appl. 237, 288–302 (1999)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Leiva H., Sivoli Z.: Existence, stability and smoothness of a bounded solution for nonlinear time-varying theormoelastic plate equations. J. Math. Anal. Appl. 285, 191–211 (1999)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lions J.-L., Peetre J.: Sur une classe d’espaces d’interpolation. Inst. Hautes tudes Sci. Publ. Math. 19, 5–68 (1964)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems. PNLDE vol. 16. Birkhäuser Verlag, Basel (1995)CrossRefGoogle Scholar
  26. 26.
    Liang J., Zhang J., Xiao T.-J.: Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J. Math. Anal. Appl. 340, 1493–1499 (2008)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Liang J., N’Guérékata G.M., Xiao T.-J., Zhang J.: Some properties of pseudo almost automorphic functions and applications to abstract differential equations. Nonlinear Anal. 70(7), 2731–2735 (2009)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Mawhin J.: Bounded solutions of second-order semicoercive evolution equations in a Hilbert space and nonlinear telegraph equations. Rend. Sem. Mat. Univ. Pol. Torino. 58(3), 361–374 (2000)MathSciNetMATHGoogle Scholar
  29. 29.
    N’Guérékata G.M.: Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces. Kluwer Academic/Plenum Publishers, New York-London-Moscow (2001)MATHGoogle Scholar
  30. 30.
    Pazy A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)CrossRefGoogle Scholar
  31. 31.
    Prüss J.: Evolutionary Integral Equations and Applications. Birkhäuser, Boston (1993)MATHCrossRefGoogle Scholar
  32. 32.
    Schnaubelt R.: Sufficient conditions for exponential stability and dichotomy of evolution equations. Forum Math. 11(5), 543–566 (1999)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Sinestrari E.: On the abstract Cauchy problem of parabolic type in spaces of continuous functions. J. Math. Anal. Appl. 107(1), 16–66 (1985)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Xiao T.J., Liang J., Zhang J.: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76(3), 518–524 (2008)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Xiao T.J., Zhu X.-X., Liang J.: Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications. Nonlinear Anal. 70(11), 4079–4085 (2009)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Xiao T.J., Liang J.: Second-order linear differential equations with almost periodic solutions. Acta Math. Sinica (N.S.) 7, 354–359 (1991)MathSciNetMATHGoogle Scholar
  37. 37.
    Xiao T.J., Liang J.: Complete second-order linear differential equations with almost periodic solutions. J. Math. Anal. Appl. 163, 136–146 (1992)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Xiao, T.J., Liang, J.: The Cauchy problem for higher-order abstract differential equations. In: Lecture Notes in Mathematics, vol. 1701. Springer, Berlin (1998)Google Scholar
  39. 39.
    Yagi A.: Parabolic equations in which the coefficients are generators of infinitely differentiable semigroups II. Funkcial. Ekvac. 33, 139–150 (1990)MathSciNetMATHGoogle Scholar
  40. 40.
    Yagi A.: Abstract quasilinear evolution equations of parabolic type in Banach spaces. Boll. Un. Mat. Ital. B 5(7), 341–368 (1991)MathSciNetMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsHoward UniversityWashingtonUSA

Personalised recommendations