Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012, Article ID 131652, 41 pages
http://dx.doi.org/10.1155/2012/131652
Research Article

On a Class of Abstract Time-Fractional Equations on Locally Convex Spaces

1Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125 Novi Sad, Serbia
2Department of Mathematics, Sichuan University, Chengdu 610064, China

Received 18 June 2012; Revised 5 August 2012; Accepted 5 August 2012

Academic Editor: Dumitru Băleanu

Copyright © 2012 Marko Kostić et al. 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. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Complexity, Nonlinearity and Chaos, World Scientific, Singapore, 2012. View at Publisher · View at Google Scholar
  2. J. Klafter, S. C. Lim, and R. Metzler, Eds., Fractional Dynamics in Physics: Recent Advances, World Scientific, Singapore, 2012.
  3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, The Netherlands, 2006.
  4. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar
  5. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, New York, NY, USA, 1999.
  6. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Derivatives and Integrals: Theory and Applications, Gordon and Breach Science, New York, NY, USA, 1993.
  7. T. Abdeljawad, D. Baleanu, F. Jarad, O. G. Mustafa, and J. J. Trujillo, “A Fite type result for sequential fractional differential equations,” Dynamic Systems and Applications, vol. 19, no. 2, pp. 383–394, 2010. View at Google Scholar
  8. T. M. Atanacković, S. Pilipović, and D. Zorica, “A diffusion wave equation with two fractional derivatives of different order,” Journal of Physics A, vol. 40, no. 20, pp. 5319–5333, 2007. View at Publisher · View at Google Scholar
  9. D. Baleanu and J. J. Trujillo, “On exact solutions of a class of fractional Euler-Lagrange equations,” Nonlinear Dynamics, vol. 52, no. 4, pp. 331–335, 2008. View at Publisher · View at Google Scholar
  10. E. Bazhlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], Eindhoven University of Technology, Eindhoven, The Netherlands, 2001.
  11. C. Chen and M. Li, “On fractional resolvent operator functions,” Semigroup Forum, vol. 80, no. 1, pp. 121–142, 2010. View at Google Scholar
  12. A. Karczewska and C. Lizama, “Solutions to stochastic fractional oscillation equations,” Applied Mathematics Letters, vol. 23, no. 11, pp. 1361–1366, 2010. View at Publisher · View at Google Scholar
  13. V. Keyantuo and C. Lizama, “A characterization of periodic solutions for time-fractional differential equations in UMD spaces and applications,” Mathematische Nachrichten, vol. 284, no. 4, pp. 494–506, 2011. View at Publisher · View at Google Scholar
  14. M. Kostić, “(a,k)-regularized C-resolvent families: regularity and local properties,” Abstract and Applied Analysis, Article ID 858242, 27 pages, 2009. View at Publisher · View at Google Scholar
  15. M. Kostić, C.-G. Li, M. Li, and S. Piskarev, “On a class of time-fractional differential equations,” Fractional Calculus Applied Analysis, vol. 15, no. 4, pp. 639–668, 2012. View at Google Scholar
  16. F.-B. Li, M. Li, and Q. Zheng, “Fractional evolution equations governed by coercive differential operators,” Abstract and Applied Analysis, Article ID 438690, 14 pages, 2009. View at Publisher · View at Google Scholar
  17. C. Lizama and H. Prado, “Fractional relaxation equations on Banach spaces,” Applied Mathematics Letters, vol. 23, no. 2, pp. 137–142, 2010. View at Publisher · View at Google Scholar
  18. F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 283–299, 2000. View at Publisher · View at Google Scholar
  19. R. K. Saxena, A. M. Mathai, and H. J. Haubold, “Reaction-diffusion systems and nonlinear waves,” Astrophysics and Space Science, vol. 305, no. 3, pp. 297–303, 2006. View at Google Scholar
  20. M. Kostić, “On a class of (a; k)-regularized C-resolvent families,” Electronic Journal of Qualitative Theory of Differential Equations, preprint.
  21. M. Kostić, “Perturbation theory for abstract Volterra equations,” preprint.
  22. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010.
  23. M. Kostić, “Abstract differential operators generating fractional regularized resolvent families,” preprint.
  24. T.-J. Xiao and J. Liang, “Higher order abstract Cauchy problems: their existence and uniqueness families,” Journal of the London Mathematical Society, vol. 67, no. 1, pp. 149–164, 2003. View at Publisher · View at Google Scholar
  25. V. A. Babalola, “Semigroups of operators on locally convex spaces,” Transactions of the American Mathematical Society, vol. 199, pp. 163–179, 1974. View at Google Scholar
  26. M. Kostić, “Abstract Volterra equations in locally convex spaces,” Science China Math, vol. 55, no. 9, pp. 1797–1825, 2012. View at Google Scholar
  27. M. Kostić, “(a; k)-regularized (C1, C2)-existence and uniqueness families,” preprint.
  28. T.-J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, Springer, Berlin, Germany, 1998.
  29. J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, Switzerland, 1993. View at Publisher · View at Google Scholar
  30. M. Kostić, Generalized Semigroups and Cosine Functions, Matematički Institut, Belgrade, 2011.
  31. M. Kostić, “Generalized well-posedness of hyperbolic Volterra equations of non-scalar type,” preprint.
  32. L. Wu and Y. Zhang, “A new topological approach to the L-uniqueness of operators and the L1-uniqueness of Fokker-Planck equations,” Journal of Functional Analysis, vol. 241, no. 2, pp. 557–610, 2006. View at Publisher · View at Google Scholar
  33. T.-J. Xiao and J. Liang, “Abstract degenerate Cauchy problems in locally convex spaces,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 398–412, 2001. View at Publisher · View at Google Scholar
  34. P. C. Kunstmann, “Banach space valued ultradistributions and applications to abstract Cauchy problems,” preprint.
  35. R. de Laubenfels, Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Mathematics, Springer, New York, NY, USA, 1994.
  36. M. Kostić, “Systems of abstract time-fractional equations,” preprint.
  37. R. de Laubenfels, “Existence and uniqueness families for the abstract Cauchy problem,” Journal of the London Mathematical Society, vol. 44, no. 2, pp. 310–338, 1991. View at Publisher · View at Google Scholar
  38. A. M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York, NY, USA, 1966.