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Bounded Mild Solutions for Semilinear Integro Differential Equations in Banach Spaces

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Abstract

In this paper we study the structure of various classes of spaces of vector-valued functions \({\mathcal{M}(\mathbb{R};X)}\) ranging between periodic functions and bounded continuous functions. Some of these functions are introduced here for the first time. We propose a general operator theoretical approach to study a class of semilinear integro-differential equations. The results obtained are new and they recover, extend or improve variety of recent works.

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References

  1. Amir B., Maniar L.: Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain. Ann. Math. Blaise Pascal 6(1), 1–11 (1999)

    MATH  MathSciNet  Google Scholar 

  2. Andres J., Bersani A.M., Grande R.F.: Hierarchy of almost periodic function spaces. Rend. Mat. Ser. VII Roma 26, 121–188 (2006)

    MATH  MathSciNet  Google Scholar 

  3. Araya D., Lizama C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69(11), 3692–3705 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Blot J., Cieutat P., N’Guérékata G.M., Pennequin D.: Superposition operators between various almost periodic function spaces and applications. Commun. Math. Anal. 6(1), 42–70 (2009)

    MATH  MathSciNet  Google Scholar 

  5. Bochner S.: A new approach to almost periodicity. Proc. Natl. Acad. Sci. USA 48, 2039–2043 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bohr H.: Zur theorie der fast periodischen funktionen. (German) I. Eine verallgemeinerung der theorie der fourierreihen. Acta Math. 45(1), 29–127 (1925)

    Article  MathSciNet  Google Scholar 

  7. Bugajewski D., N’Guérékata G.M.: On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces. Nonlinear Anal. 59, 1333–1345 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Clément Ph., Da Prato G.: Existence and regularity results for an integral equation with infinite delay in a Banach space. Integr. Equ. Oper. Theory 11, 480–500 (1988)

    Article  MATH  Google Scholar 

  9. Coleman B.D., Gurtin M.E.: Equipresence and constitutive equation for rigid heat conductors. Z. Angew. Math. Phys. 18, 199–208 (1967)

    Article  MathSciNet  Google Scholar 

  10. Cuevas C., Lizama C.: Almost automorphic solutions to a class of semilinear fractional differential equations. Appl. Math. Lett. 21, 1315–1319 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dafermos C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dafermos C.M.: An abstract Volterra equation with applications to linear viscoelasticity. J. Differ. Equ. 7, 554–569 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  13. Da Prato G., Lunardi A.: Periodic solutions for linear integrodifferential equations with infinite delay in Banach spaces. Differential Equations in Banach spaces. Lect. Notes Math. 1223, 49–60 (1985)

    Article  MathSciNet  Google Scholar 

  14. Da Prato G., Lunardi A.: Solvability on the real line of a class of linear Volterra integrodifferential equations of parabolic type. Ann. Math. Pura Appl. 4, 67–117 (1988)

    MathSciNet  Google Scholar 

  15. De Bruijn N.G.: The asymptotically periodic behavior of the solutions of some linear functional equations. Am. J. Math. 71, 313–330 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  16. Diagana T.: weighted pseudo almost automorphic functions and applications. C. R. Acad. Sci. Paris Ser. I 343, 643–646 (2006)

    MATH  MathSciNet  Google Scholar 

  17. Diagana T., Hernández E., Dos Santos J.P.C.: Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations. Nonlinear Anal. 71, 248–257 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Diagana T., Henríquez H., Hernández E.: Almost automorphic mild solutions to some partial neutral functional–differential equations and applications. Nonlinear Anal. 69(5-6), 1485–1493 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ding H.-S., Liang J., Xiao T.-J.: Asymptotically almost automorphic solutions for some integro-differential equations with nonlocal conditions. J. Math. Anal. Appl. 338(1), 141–151 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fašangová E., Prüss J.: Asymptotic behaviour of a semilinear viscoelastic beam model. Arch. Math. (Basel) 77, 488–497 (2001)

    MATH  MathSciNet  Google Scholar 

  21. Fink A.M.: Almost automorphic and almost periodic solutions which minimize functionals. Tôhoku Math. J. 20(2), 323–332 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fréchet M.: Les fonctions asymptotiquement presque-periodiques continues (French). C. R. Acad. Sci. Paris 213, 520–522 (1941)

    MathSciNet  Google Scholar 

  23. Gao H., Wang K., Wei F., Ding X.: Massera-type theorem and asymptotically periodic logistic equations. Nonlinear Anal. Real World Appl. 7, 1268–1283 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gripenberg G., Londen S.-O., Staffans O.: Volterra Integral And Functional Equations. In: Encyclopedia of Mathematics and its Applications, vol. 34. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  25. Henríquez H.R., Lizama C.: Compact almost periodic solutions to integral equations with infinite delay. Nonlinear Anal. 71, 6029–6037 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Henríquez H.R., Pierri M., Táboas P.: Existence of S-asymptotically ω-periodic solutions for abstract neutral equations. Bull. Aust. Math. Soc. 78, 365–382 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Henríquez H.R., Pierri M., Táboas P.: On S-asymptotically ω-periodic functions on Banach spaces and applications. J. Math. Anal. Appl. 343(2), 1119–1130 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Levitan, B.M., Zhikov, V.V.: Almost Periodic Functions and Differential Equations. Moscow University House, Moscow (1978) (English Translation by Cambridge University Press, 1982)

  29. Li H., Huang F., Li J.: Composition of pseudo almost periodic functions and semilinear differential equations. J. Math. Anal. Appl. 255, 436–446 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Liang J., Zhang J., Xiao T.J.: Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J. Math. Anal. Appl. 340(2), 1493–1499 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lizama C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  32. Mophou, G.M., N’Guérékata, G.M.: On some classes of almost automorphic functions and applications to fractional differential equations (2010, to appear)

  33. N’Guérékata G.M.: Topics in Almost Automorphy. Springer, New York (2005)

    MATH  Google Scholar 

  34. N’Guérékata G.M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer, New York (2001)

    MATH  Google Scholar 

  35. N’Guérékata G.M.: Quelques remarques sur les fonctions asymptotiquement presque automorphes (French). Ann. Sci. Math. Quebec 7(2), 185–191 (1983)

    MATH  MathSciNet  Google Scholar 

  36. N’Guérékata G.M.: Comments on almost automorphic and almost periodic functions in Banach spaces. Far East J. Math. Sci. (FJMS) 17(3), 337–344 (2005)

    MATH  MathSciNet  Google Scholar 

  37. N’Guérékata G.M.: Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations. Semigroup Forum 69, 80–86 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  38. Nguyen V.M., Naito T., N’Guérékata G.M.: A spectral countability condition for almost automorphy of solutions of differential equations. Proc. Am. Math. Soc. 134, 3257–3266 (2006)

    Article  MATH  Google Scholar 

  39. Nunziato J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–304 (1971)

    MATH  MathSciNet  Google Scholar 

  40. Prüss, J.: Evolutionary integral equations and applications. In: Monographs in Mathematics, vol. 87. Birkhäuser, Boston (1993)

  41. Renardy, M., Hrusa, W.J., Nohel, J.A.: Mathematical problems in viscoelasticity. In: Pitman Monographs Pure Applied Mathematics, vol. 35. Longman Sci. Tech., Harlow, Essex (1988)

  42. Sforza D.: Existence in the large for a semilinear integrodifferential equation with infinite delay. J. Differ. Equ. 120, 289–303 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  43. 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)

    Article  MATH  MathSciNet  Google Scholar 

  44. Yuan R.: Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument. Nonlinear Anal. 41(7–8), 871–890 (2000)

    Article  MathSciNet  Google Scholar 

  45. Zhang C.Y.: Integration of vector-valued pseudo-almost periodic functions. Proc. Am. Math. Soc. 121(1), 167–174 (1994)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago, Chile

    Carlos Lizama

  2. Department of Mathematics, Morgan State University, Baltimore, MD, 21251, USA

    Gaston M. N’Guérékata

Authors
  1. Carlos Lizama
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  2. Gaston M. N’Guérékata
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Corresponding author

Correspondence to Gaston M. N’Guérékata.

Additional information

C. Lizama was partially supported by Project FONDECYT 1100485.

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Lizama, C., N’Guérékata, G.M. Bounded Mild Solutions for Semilinear Integro Differential Equations in Banach Spaces. Integr. Equ. Oper. Theory 68, 207–227 (2010). https://doi.org/10.1007/s00020-010-1799-2

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  • DOI: https://doi.org/10.1007/s00020-010-1799-2

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