Abstract and Applied Analysis

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

G. M. N’Guérékata and Marko Kostić

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Abstract

The main aim of this paper is to investigate generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions to a class of abstract (semilinear) multiterm fractional differential inclusions with Caputo derivatives. We illustrate our abstract results with several examples and possible applications.

Article information

Source
Abstr. Appl. Anal., Volume 2018 (2018), Article ID 5947393, 17 pages.

Dates
Received: 13 October 2017
Accepted: 10 December 2017
First available in Project Euclid: 14 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1518577263

Digital Object Identifier
doi:10.1155/2018/5947393

Mathematical Reviews number (MathSciNet)
MR3759805

Zentralblatt MATH identifier
06929590

Citation

N’Guérékata, G. M.; Kostić, Marko. Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions. Abstr. Appl. Anal. 2018 (2018), Article ID 5947393, 17 pages. doi:10.1155/2018/5947393. https://projecteuclid.org/euclid.aaa/1518577263


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References

  • L. Amerio and G. Prouse, Almost-Periodic Functions and Functional Equations, Van Nostrand-Reinhold, New York, NY, USA, 1971.
  • 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.
  • D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, NY, USA, 2009.
  • T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, NY, USA, 2013.
  • G. M. N'Guérékata, Almost Automorphic and Almost Periodicity Functions in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, Netherlands, 2001.
  • 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.
  • M. Kostić, Abstract Volterra Integro-Differential Equations: Almost Periodicity and Asymptotically Almost Periodic Properties of Solutions, Book Manuscript, 2017.
  • M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, London, UK, 1982.
  • T.-J. Xiao and J. Liang, The Cauchy Problem for Higher–Order Abstract Differential Equations, Springer-Verlag, Berlin, Germany, 1998.
  • S. Zaidman, Almost-Periodic Functions in Abstract Spaces, vol. 126, Pitman, Boston, Massachusetts, USA, 1985.
  • E. Bazhlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], Eindhoven University of Technology, Eindhoven, Netherlands, 2001.
  • M. Kostić, Abstract Volterra Integro-Differential Equations, Taylor and Francis Group/CRC Press/Science Publishers, Boca Raton, New York, NY, USA, 2015.
  • 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.
  • M. Sova, “The Laplace transform of analytic vector-valued functions,” Casopis pro Pestovani Matematiky, vol. 104, pp. 267–280, 1979.
  • 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.
  • 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.
  • 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.
  • A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Chapman and Hall/CRC Pure and Applied Mathematics, New York, NY, USA, 1998.
  • M. Kostić, Abstract Degenerate Volterra Integro-Differential Equations: Linear Theory and Applications, Book Manuscript, 2016.
  • K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004, Springer-Verlag, Berlin, Germany, 2010.
  • 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.
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  • J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser-Verlag, Basel, Switzerland, 2012.
  • 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.
  • 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.
  • J. Andres, A. M. Bersani, and R. F. Grande, “Hierarchy of almost-periodic function spaces,” Rendiconti di Matematica, vol. 26, pp. 121–188, 2006.
  • A. S. Besicovitch, Almost Periodic Functions, Dover Publications Inc., New York, NY, USA, 1954.
  • 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.
  • G. M. N'Guérékata, Spectral Theory for Bounded Functions and Applications to Evolution Equations, Nova Science Publishers, New York, NY, USA, 2017.
  • 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.
  • L. Ji and A. Weber, “Dynamics of the heat semigroup on symmetric spaces,” Ergodic Theory Dynam. Systems, vol. 3, pp. 457–468, 2010.
  • 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.
  • 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.
  • M. Kostić, “Degenerate k-regularized (C1,C2)-existence and uniqueness families,” CUBO: A Mathematical Journal, vol. 17, no. 3, pp. 15–41, 2015.
  • 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.
  • A. M. Fink, “Extensions of almost automorphic sequences,” Journal of Mathematical Analysis and Applications, vol. 27, no. 3, pp. 519–523, 1969.
  • S. Abbas, “A note on Weyl pseudo almost automorphic functions and their properties,” Mathematical Sciences, vol. 6, no. 1, p. 29, 2012.
  • 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.
  • M. Kostić, “Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions,” Fractional Differential Calculus, 2017.
  • R. Cross, Multivalued Linear Operators, Marcel Dekker, Inc., New York, NY, USA, 1998.
  • R. W. Carroll and R. W. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, New York, NY, USA, 1976.
  • I. V. Melnikova and A. I. Filinkov, Abstract Cauchy Problems: Three Approaches, Chapman Hall/CRC Press, Boca Raton, New York, NY, USA, 2001.
  • 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.
  • Q. Zheng and M. Li, Regularized Semigroups and Non-Elliptic Differential Operators, Science Press, Beijing, China, 2014.
  • 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. \endinput