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Abstract and Applied Analysis
Volume 2018, Article ID 5947393, 17 pages
https://doi.org/10.1155/2018/5947393
Research Article

Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

1Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA
2Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia

Correspondence should be addressed to G. M. N’Guérékata; moc.loa@atakereugn

Received 13 October 2017; Accepted 10 December 2017; Published 22 January 2018

Academic Editor: Khalil Ezzinbi

Copyright © 2018 G. M. N’Guérékata and Marko Kostić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. Amerio and G. Prouse, Almost-Periodic Functions and Functional Equations, Van Nostrand-Reinhold, New York, NY, USA, 1971.
  2. W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, vol. 96 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  3. D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, NY, USA, 2009. View at MathSciNet
  4. T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, NY, USA, 2013.
  5. G. M. N'Guérékata, Almost Automorphic and Almost Periodicity Functions in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, Netherlands, 2001.
  6. Y. Hino, T. Naito, N. V. Minh, and J. S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces. Stability and Control: Theory, Methods and Applications, vol. 15, Taylor and Francis Group, London, UK, 2002.
  7. M. Kostić, Abstract Volterra Integro-Differential Equations: Almost Periodicity and Asymptotically Almost Periodic Properties of Solutions, Book Manuscript, 2017. View at MathSciNet
  8. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, London, UK, 1982.
  9. T.-J. Xiao and J. Liang, The Cauchy Problem for Higher–Order Abstract Differential Equations, Springer-Verlag, Berlin, Germany, 1998. View at MathSciNet
  10. S. Zaidman, Almost-Periodic Functions in Abstract Spaces, vol. 126, Pitman, Boston, Massachusetts, USA, 1985.
  11. E. Bazhlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], Eindhoven University of Technology, Eindhoven, Netherlands, 2001.
  12. M. Kostić, Abstract Volterra Integro-Differential Equations, Taylor and Francis Group/CRC Press/Science Publishers, Boca Raton, New York, NY, USA, 2015. View at MathSciNet
  13. M. Kostić, C.-G. Li, and M. Li, “On a class of abstract time-fractional equations on locally convex spaces,” Abstract and Applied Analysis, p. 41, 2012. View at Google Scholar · View at MathSciNet
  14. M. Sova, “The Laplace transform of analytic vector-valued functions,” Casopis pro Pestovani Matematiky, vol. 104, pp. 267–280, 1979. View at Google Scholar
  15. E. Cuesta, “Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations,” Discrete and Continuous Dynamical Systems - Series A, pp. 277–285, 2007. View at Google Scholar · View at MathSciNet
  16. V. Keyantuo, C. Lizama, and M. Warma, “Asymptotic behavior of fractional-order semilinear evolution equations,” Differential and Integral Equations. An International Journal for Theory and Applications, vol. 26, no. 7-8, pp. 757–780, 2013. View at Google Scholar · View at MathSciNet
  17. V. T. Luong, “Decay mild solutions for two-term time fractional differential equations in Banach spaces,” Journal of Fixed Point Theory and Applications, vol. 18, no. 2, pp. 417–432, 2016. View at Publisher · View at Google Scholar · View at Scopus
  18. A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Chapman and Hall/CRC Pure and Applied Mathematics, New York, NY, USA, 1998.
  19. M. Kostić, Abstract Degenerate Volterra Integro-Differential Equations: Linear Theory and Applications, Book Manuscript, 2016.
  20. K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004, Springer-Verlag, Berlin, Germany, 2010. View at Publisher · View at Google Scholar
  21. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Elsevier Science, Amsterdam, Netherlands, 2006.
  22. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  23. J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser-Verlag, Basel, Switzerland, 2012. View at Publisher · View at Google Scholar
  24. W. M. Ruess and W. H. Summers, “Asymptotic almost periodicity and motions of semigroups of operators,” Linear Algebra and its Applications, vol. 84, no. C, pp. 335–351, 1986. View at Publisher · View at Google Scholar · View at Scopus
  25. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, New York, NY, USA, 1993.
  26. J. Andres, A. M. Bersani, and R. F. Grande, “Hierarchy of almost-periodic function spaces,” Rendiconti di Matematica, vol. 26, pp. 121–188, 2006. View at Google Scholar
  27. A. S. Besicovitch, Almost Periodic Functions, Dover Publications Inc., New York, NY, USA, 1954.
  28. G. M. N’Guérékata and A. Pankov, “Stepanov-like almost automorphic functions and monotone evolution equations,” Nonlinear Analysis. Theory, Methods and Applications. An International Multidisciplinary Journal, vol. 68, no. 9, pp. 2658–2667, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  29. G. M. N’Guérékata, Spectral Theory for Bounded Functions and Applications to Evolution Equations, Nova Science Publishers, New York, NY, USA, 2017.
  30. H. R. Henríquez, “On Stepanov-almost periodic semigroups and cosine functions of operators,” Journal of Mathematical Analysis and Applications, vol. 146, no. 2, pp. 420–433, 1990. View at Publisher · View at Google Scholar · View at Scopus
  31. L. Ji and A. Weber, “Dynamics of the heat semigroup on symmetric spaces,” Ergodic Theory Dynam. Systems, vol. 3, pp. 457–468, 2010. View at Google Scholar
  32. M. Kostić, “Some contributions to the theory of abstract degenerate volterra integro-differential equations,” Journal of Mathematics and Statistics, vol. 12, no. 2, pp. 65–76, 2016. View at Publisher · View at Google Scholar · View at Scopus
  33. M. Kostić, “Existence of generalized almost periodic and asymptotic almost periodic solutions to abstract Volterra integro-differential equations,” Electronic Journal of Differential Equations, vol. 2017, no. 239, pp. 1–30, 2017. View at Google Scholar · View at MathSciNet
  34. M. Kostić, “Degenerate k-regularized (C1,C2)-existence and uniqueness families,” CUBO: A Mathematical Journal, vol. 17, no. 3, pp. 15–41, 2015. View at Google Scholar · View at MathSciNet
  35. M. Kostić, “Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations,” Journal of Mathematical Analysis and Applications, Paper No. 239, 30 pages, 2017. View at Google Scholar · View at MathSciNet
  36. A. M. Fink, “Extensions of almost automorphic sequences,” Journal of Mathematical Analysis and Applications, vol. 27, no. 3, pp. 519–523, 1969. View at Publisher · View at Google Scholar · View at Scopus
  37. S. Abbas, “A note on Weyl pseudo almost automorphic functions and their properties,” Mathematical Sciences, vol. 6, no. 1, p. 29, 2012. View at Publisher · View at Google Scholar
  38. F. Bedouhene, N. Challali, O. Mellah, P. Raynaud de Fitte, and M. Smaali, “Almost automorphy and various extensions for stochastic processes,” Journal of Mathematical Analysis and Applications, vol. 429, no. 2, pp. 1113–1152, 2015. View at Publisher · View at Google Scholar · View at Scopus
  39. M. Kostić, “Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions,” Fractional Differential Calculus, 2017. View at Google Scholar
  40. R. Cross, Multivalued Linear Operators, Marcel Dekker, Inc., New York, NY, USA, 1998. View at MathSciNet
  41. R. W. Carroll and R. W. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, New York, NY, USA, 1976.
  42. I. V. Melnikova and A. I. Filinkov, Abstract Cauchy Problems: Three Approaches, Chapman Hall/CRC Press, Boca Raton, New York, NY, USA, 2001.
  43. G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, Inverse and Ill-Posed Problems (Book 42), VSP, Utrecht, Boston, USA, 2003.
  44. Q. Zheng and M. Li, Regularized Semigroups and Non-Elliptic Differential Operators, Science Press, Beijing, China, 2014.
  45. W. Von Wahl, “Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen,” Nachrichten von der Gesellschaft der wissenschaften zu Gottingen, Mathematisch-Physikalische Klasse, vol. 11, pp. 231–258, 1972. View at Google Scholar