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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 914592, 9 pages
The Solvability and Optimal Controls for Some Fractional Impulsive Equation
1School of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, China
2College of Sciences, Guangxi University for Nationalities Nanning, Guangxi 530006, China
3Faculty of Mathematics and Computer Science, Jagiellonian University, ul. S. Lojasiewicza 6, 30-348 Krakow, Poland
Received 21 April 2013; Revised 18 June 2013; Accepted 27 July 2013
Academic Editor: Naseer Shahzad
Copyright © 2013 Xianghu Liu 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.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- 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 B.V., Amsterdam, The Netherlands, 2006.
- V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
- K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004, Springer, Berlin, Germany, 2010.
- J. Wang, Y. Zhou, and M. Medved, “On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay,” Journal of Optimization Theory and Applications, vol. 152, no. 1, pp. 31–50, 2012.
- J. Wang, Y. Zhou, and W. Wei, “A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 4049–4059, 2011.
- X. B. Shu, Y. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Analysis, vol. 74, no. 5, pp. 2003–2011, 2011.
- M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1340–1350, 2008.
- K. Balachandran, Y. Zhou, and J. Kokila, “Relative controllability of fractional dynamical systems with delays in control,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 9, pp. 3508–3520, 2012.
- C. Cuevas and C. Lizama, “Almost automorphic solutions to a class of semilinear fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 12, pp. 1315–1319, 2008.
- E. Bazhlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], Eindhoven University of Technology, 2001.
- L. Xue and J. Xiong, “Existence and uniqueness of mild solutions for abstract delay fractional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1398–1404, 2011.
- H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007.
- J. Wang, M. Fečkan, and Y. Zhou, “Relaxed controls for nonlinear fractional impulsive evolution equations,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 13–32, 2013.
- S. Hu and N. S. Papageorgiou, Handbook of multivalued Analysis, Kluwer Academic Publishers, London, UK, 1997.
- J. Wang, M. Fečkan, and Y. Zhou, “On the new concept of solutions and existence results for impulsive fractional evolution equations,” Dynamics of Partial Differential Equations, vol. 8, no. 4, pp. 345–361, 2011.
- Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1063–1077, 2010.
- E. J. Balder, “Necessary and sufficient conditions for -strong-weak lower semicontinuity of integral functionals,” Nonlinear Analysis, vol. 11, no. 12, pp. 1399–1404, 1987.