Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 729615, 22 pages
http://dx.doi.org/10.1155/2012/729615
Research Article

Existence of the Mild Solutions for Delay Fractional Integrodifferential Equations with Almost Sectorial Operators

School of Mathematics, Yunnan Normal University, Kunming 650092, China

Received 1 February 2012; Accepted 8 June 2012

Academic Editor: Detlev Buchholz

Copyright © 2012 Fang Li. 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. M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons & Fractals, vol. 149, pp. 823–831, 2004. View at Google Scholar
  2. L. Hu, Y. Ren, and R. Sakthivel, “Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays,” Semigroup Forum, vol. 79, no. 3, pp. 507–514, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. F. Li, “Solvability of nonautonomous fractional integrodifferential equations with infinite delay,” Advances in Difference Equations, vol. 2011, Article ID 806729, 18 pages, 2011. View at Google Scholar · View at Zentralblatt MATH
  4. F. Li, “An existence result for fractional differential equations of neutral type with infinite delay,” Electronic Journal of Qualitative Theory of Differential Equations, no. 52, pp. 1–15, 2011. View at Google Scholar
  5. F. Li, T. J. Xiao, and H. K. Xu, “On nonlinear neutral fractional integrodifferential inclusions with infinite delay,” Journal of Applied Mathematics, vol. 2012, Article ID 916543, 19 pages, 2012. View at Google Scholar
  6. F. Mainardi, P. Paradisi, and R. Goren°o, “Probability distributions generated by fractional diffusion equations,” in Econophysics: An Emerging Science, J. Kertesz and I. Kondor, Eds., Kluwer, Dordrecht, The Netherlands, 2000. View at Google Scholar
  7. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
  8. G. M. Mophou and G. M. N'Guérékata, “Existence of the mild solution for some fractional differential equations with nonlocal conditions,” Semigroup Forum, vol. 79, no. 2, pp. 315–322, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. G. M. Mophou and G. M. N'Guérékata, “Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 61–69, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. G. M. Mophou and G. M. N'Guérékata, “A note on a semilinear fractional differential equation of neutral type with infinite delay,” Advances in Difference Equations, vol. 2010, Article ID 674630, 8 pages, 2010. View at Google Scholar · View at Zentralblatt MATH
  11. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  12. V. Obukhovskii and J.-C. Yao, “Some existence results for fractional functional differential equations,” Fixed Point Theory, vol. 11, no. 1, pp. 85–96, 2010. View at Google Scholar · View at Zentralblatt MATH
  13. G. M. N'Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 5, pp. 1873–1876, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Y. Ren, Y. Qin, and R. Sakthivel, “Existence results for fractional order semilinear integro-differential evolution equations with infinite delay,” Integral Equations and Operator Theory, vol. 67, no. 1, pp. 33–49, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. R.-N. Wang, D.-H. Chen, and T.-J. Xiao, “Abstract fractional Cauchy problems with almost sectorial operators,” Journal of Differential Equations, vol. 252, no. 1, pp. 202–235, 2012. View at Publisher · View at Google Scholar
  16. J. Liang and T. J. Xiao, “Functional-differential equations with infinite delay in Banach spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 497–508, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J. Liang, T.-J. Xiao, and J. van Casteren, “A note on semilinear abstract functional differential and integrodifferential equations with infinite delay,” Applied Mathematics Letters, vol. 17, no. 4, pp. 473–477, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J. Liang and T.-J. Xiao, “Solvability of the Cauchy problem for infinite delay equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no. 3-4, pp. 271–297, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. T.-J. Xiao and J. Liang, The Cauchy Problem for Higher Order Abstract Differential Equations, vol. 1701 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998.
  20. T.-J. Xiao and J. Liang, “Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1442–e1447, 2009. View at Publisher · View at Google Scholar
  21. W. von Wahl, “Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen,” Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse, vol. 11, pp. 231–258, 1972. View at Google Scholar
  22. R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, vol. 55 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1992. View at Zentralblatt MATH
  23. M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, vol. 7 of De Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter & Co., Berlin, Germany, 2001. View at Publisher · View at Google Scholar
  24. D. Henry, Geometric Theory of Semilinear Parabolic Partial Differential Equations, Springer, Berlin, Germany, 1989.
  25. M. Haase, The Functional Calculus for Sectorial Operators, vol. 169 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2006. View at Publisher · View at Google Scholar
  26. F. Periago and B. Straub, “A functional calculus for almost sectorial operators and applications to abstract evolution equations,” Journal of Evolution Equations, vol. 2, no. 1, pp. 41–68, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH