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Pseudo-almost periodic solutions of a class of semilinear fractional differential equations

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Abstract

We study existence and uniqueness of a pseudo-almost periodic (of class infinity) mild solution to the semilinear fractional equation \(\partial_{t}^{\alpha}u=Au+\partial_{t}^{\alpha-1}f(\cdot,u),1<\alpha<2\), where A is a linear operator of sectorial negative type.

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References

  1. Agarwal, R.P., Diagana, T., Hernández, E.: Weighted pseudo almost periodic solutions to some partial neutral functional differential equations. J. Nonlinear Convex Anal. 8(3), 397–415 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivate. Adv. Differ. Equ. 2009, 74 (2009). Article ID 981728

    MathSciNet  Google Scholar 

  3. Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109(3), 973–1033 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Agarwal, R.P., de Andrade, B., Cuevas, C.: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Anal., Real World Appl. 11, 3532–3554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Agarwal, R.P., de Andrade, B., Cuevas, C.: On type of periodicity and ergodicity to a class of integral equations with infinite delay. J. Nonlinear Convex Anal. 11(2), 309–335 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Agarwal, R.P., de Andrade, B., Cuevas, C.: On type of periodicity and ergodicity to a class of fractional order differential equations. Adv. Differ. Equ. 2010, 1–25 (2010). Article ID 179750

    Google Scholar 

  7. Agarwal, R.P., Lakshmikantham, V., Nieto, J.J.: On the concept of solution for fractional differential equations with uncertainly. Nonlinear Anal. 72(6), 2859–2862 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Agarwal, R.P., Cuevas, C., Soto, H., El-Gebeily, M.: Asymptotic periodicity for some evolution equations in Banach spaces. Nonlinear Anal. (to appear)

  9. Ahmad, B., Nieto, J.J.: Existence results for nonlinear boundary value problems of fractional integro-differential equations with integral boundary conditions. Bound. Value Probl. 2009, 11 (2009). Article ID 708576

    MathSciNet  Google Scholar 

  10. Ahn, W., Mcvinish, R.: Fractional differential equations driven by Levy noise. J. Appl. Math. Stoch. Anal. 16(2), 97–119 (2003)

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Belmekki, M., Nieto, J.J., Rodriguez-Lopez, R.: Existence of periodic solution for a nonlinear fractional differential equation. Bound. Value Probl. 2009, 18 (2009). Article ID 324561

    Article  MathSciNet  Google Scholar 

  13. Caicedo, A., Cuevas, C., Mophou, G.M., N’Guérékata, G.M.: Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces. J. Franklin Inst. (2010, to appear)

  14. Cuesta, E.: Asymptotic behavior of the solutions of fractional integro-differential equations and some time discretizations. Discrete Contin. Dyn. Syst. 277–285 (2007)

  15. Cuesta, E., Palencia, C.: A numerical method for an integro-differential equation with memory in Banach spaces: Qualitative properties. SIAM J. Numer. Anal. 41, 1232–1241 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cuesta, E., Lubich, Ch., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673–696 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cuevas, C., de Souza, J.C.: S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations. Appl. Math. Lett. 22, 865–870 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cuevas, C., de Souza, J.C.: Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Anal. 72(3–4), 1683–1689 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Cuevas, C., Hernández, E.: Pseudo-almost periodic solutions for abstract partial functional differential equation. Appl. Math. Lett. 22, 534–538 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cuevas, C., Pinto, M.: Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain. Nonlinear Anal. 45(1), 73–83 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cuevas, C., Rabelo, M., Soto, H.: Pseudo almost automorphic solutions to a class of semilinear fractional differential equations. Commun. Appl. Nonlinear Anal. 17, 33–48 (2010)

    MathSciNet  Google Scholar 

  22. Diagana, T.: Pseudo almost periodic solutions to some differential equations. Nonlinear Anal. 60(7), 1277–1286 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Diagana, T.: Weighted pseudo almost periodic functions and applications. C. R. Acad. Sci. Paris Ser. I 343(10), 643–646 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Diagana, T.: Existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equation. Electron. J. Qual. Theory Differ. Equ. (3), 1–12 (2007)

  25. Diagana, T.: Pseudo Almost Periodic Functions in Banach Spaces. Nova Science Publishers, New York (2007)

    MATH  Google Scholar 

  26. Diagana, T.: Weighted pseudo almost periodic solutions to some differential equations. Nonlinear Anal. 68(8), 2250–2260 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Diagana, T.: Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations. J. Math. Anal. Appl. 350, 18–28 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Diagana, T., Hernández, E.: Existence and uniqueness of pseudo-almost periodic solutions to some abstract partial neutral functional-differential equations and applications. J. Math. Anal. Appl. 327(1), 776–791 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Diagana, T., Mahop, C.M.: Pseudo almost periodic solutions to a neutral delay integral equation. Cubo 9(1), 47–55 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Diagana, T., Mahop, C.M., N’Guérékata, G.M.: Pseudo almost periodic solutions to some semilinear differential equations. Math. Comput. Model. 43(1–2), 89–96 (2006)

    Article  MATH  Google Scholar 

  31. Diagana, T., Mahop, C.M., N’Guérékata, G.M., Toni, B.: Existence and uniqueness of pseudo almost periodic solutions to some classes of semilinear differential equations and applications. Nonlinear Anal. 64(11), 2442–2453 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. dos Santos, J.P.C., Cuevas, C.: Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations. Appl. Math. Lett. 23, 960–965 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Eidelman, S.D., Kochubei, A.N.: Cauchy problem for fractional diffusion equations. J. Differ. Equ. 199, 211–255 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gorenflo, R., Mainardi, F.: Fractional Calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Vienna (1997)

    Google Scholar 

  35. Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speed. Arch. Ration. Mech. Anal. 31, 113–126 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hernández, E., Henríquez, H.: Pseudo-almost periodic solutions for non-autonomous neutral differential equations with unbounded delay. Nonlinear Anal., Real World Appl. 9, 430–437 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hino, Y., Murakami, S., Naito, T.: Functional-Differential Equations with Infinite Delay. Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (2002)

    Google Scholar 

  38. Lakshmikantham, V.: Theory of fractional differential equations. Nonlinear Anal. 60(10), 3337–3343 (2008)

    MathSciNet  Google Scholar 

  39. Lakshmikantham, V., Devi, J.V.: Theory of fractional differential equations in Banach spaces. Eur. J. Pure Appl. Math. 1, 38–45 (2008)

    MathSciNet  MATH  Google Scholar 

  40. Lakshmikantham, V., Vatsala, A.: Theory of fractional differential inequalities and applications. Commun. Appl. Anal. 11(3–4), 395–402 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Lakshmikantham, V., Vatsala, A.: Basic theory of fractional differential equations. Nonlinear Anal. 69(3), 2677–2682 (2008)

    MathSciNet  MATH  Google Scholar 

  42. Mophou, G.M., N’Guérékata, G.M.: Mild solutions for semilinear fractional differential equations. Electron. J. Differ. Equ. 2009(21), 1–9 (2009)

    Google Scholar 

  43. Mophou, G.M., N’Guérékata, G.M.: Existence of mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 79(2), 315–322 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mophou, G.M., N’Guérékata, G.M.: On integral solutions of some nonlocal fractional differential equations with nondense domain. Nonlinear Anal. (2009). doi:101016/j.na.2009.03.029

    Google Scholar 

  45. Mophou, G.M., N’Guérékata, G.M., Valmorin, V.: Pseudo almost automorphic solutions of a neutral functional fractional differential equations. Int. J. Evol. Equ. 4(2), 249–259 (2010)

    MathSciNet  MATH  Google Scholar 

  46. Mophou, G.M., N’Guérékata, G.M., Valmorin, V.: Corrigendum to Pseudo almost automorphic solutions of a neutral functional fractional differential equations. Int. J. Evol. Equ. 5(1), 1–3 (2010)

    MathSciNet  Google Scholar 

  47. N’Guérékata, G.M.: Cauchy problem for some fractional abstract differential equation with nonlocal conditions. Nonlinear Anal. 70(5), 1873–1876 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Nunziato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–204 (1971)

    MathSciNet  MATH  Google Scholar 

  49. Zhang, C.: Pseudo almost periodic functions and their applications. Ph.D. Thesis, The University of Western Ontario (1992)

  50. Zhang, C.: Pseudo almost periodic solutions of some differential equations. J. Math. Anal. Appl. 151, 62–76 (1994)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  52. Zhang, C.: Pseudo almost periodic solutions of some differential equations II. J. Math. Anal. Appl. 192, 543–561 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhang, C.: Almost Periodic Type and Ergodicity. Kluwer Academic, Norwell (2003)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL, 32901, USA

    Ravi P. Agarwal

  2. Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE, 50540-740, Brazil

    Claudio Cuevas

  3. Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54-D, Temuco, Chile

    Herme Soto

Authors
  1. Ravi P. Agarwal
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  2. Claudio Cuevas
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  3. Herme Soto
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Corresponding author

Correspondence to Ravi P. Agarwal.

Additional information

The second author is partially supported by CNPQ/Brazil under Grant 300365/2008-0.

The third author is partially supported by DIUFRO 120231.

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Agarwal, R.P., Cuevas, C. & Soto, H. Pseudo-almost periodic solutions of a class of semilinear fractional differential equations. J. Appl. Math. Comput. 37, 625–634 (2011). https://doi.org/10.1007/s12190-010-0455-y

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  • DOI: https://doi.org/10.1007/s12190-010-0455-y

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