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Abstract and Applied Analysis
Volume 2014, Article ID 430418, 9 pages
http://dx.doi.org/10.1155/2014/430418
Research Article

Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 31 October 2013; Accepted 11 December 2013; Published 8 January 2014

Academic Editor: Yong Zhou

Copyright © 2014 Ya-Ning Li and Hong-Rui Sun. 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. Mijatović, S. Pilipović, and F. Vajzović, “α-times integrated semigroups (α+),” Journal of Mathematical Analysis and Applications, vol. 210, no. 2, pp. 790–803, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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. T. E. Duncan, Y. Hu, and B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion. I. Theory,” SIAM Journal on Control and Optimization, vol. 38, no. 2, pp. 582–612, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  5. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. View at MathSciNet
  6. V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  7. J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 262–272, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Wang, Y. Zhou, and W. Wei, “Fractional sewage treatment models with impulses at variable times,” Applicable Analysis, vol. 92, pp. 1959–1979, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. C. Chen and M. Li, “On fractional resolvent operator functions,” Semigroup Forum, vol. 80, no. 1, pp. 121–142, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Kostić, “(a,k)-regularized C-resolvent families: regularity and local properties,” Abstract and Applied Analysis, vol. 2009, Article ID 858242, 27 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  11. M. Li, C. Chen, and F.-B. Li, “On fractional powers of generators of fractional resolvent families,” Journal of Functional Analysis, vol. 259, no. 10, pp. 2702–2726, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Kexue and P. Jigen, “Fractional abstract Cauchy problems,” Integral Equations and Operator Theory, vol. 70, no. 3, pp. 333–361, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. K. Li, J. Peng, and J. Jia, “Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives,” Journal of Functional Analysis, vol. 263, no. 2, pp. 476–510, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. C. Lizama, “An operator theoretical approach to a class of fractional order differential equations,” Applied Mathematics Letters, vol. 24, no. 2, pp. 184–190, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. C. Lizama, “Regularized solutions for abstract Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 243, no. 2, pp. 278–292, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. G. da Prato and M. Iannelli, “Linear integro-differential equations in Banach spaces,” Rendiconti del Seminario Matematico dell'Università di Padova, vol. 62, pp. 207–219, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. E. Bajlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], Eindhoven University of Technology, 2001.
  18. J. Peng and K. Li, “A novel characteristic of solution operator for the fractional abstract Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 786–796, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, Switzerland, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  20. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  21. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet