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FRACTIONAL NONLOCAL INTEGRODIFFERENTIAL EQUATIONS AND ITS OPTIMAL CONTROL IN BANACH SPACES

  • Wang, Jinrong (COLLEGE OF SCIENCE, GUIZHOU UNIVERSITY) ;
  • Wei, W. (COLLEGE OF SCIENCE, GUIZHOU UNIVERSITY) ;
  • Yang, Y. (COLLEGE OF TECHNOLOGY, GUIZHOU UNIVERSITY)
  • Received : 2009.12.16
  • Accepted : 2010.06.02
  • Published : 2010.06.25

Abstract

In this paper, a class of fractional integrodifferential equations of mixed type with nonlocal conditions is considered. First, using contraction mapping principle and Krasnoselskii's fixed point theorem via Gronwall's inequailty, the existence and uniqueness of mild solution are given. Second, the existence of optimal pairs of systems governed by fractional integrodifferential equations of mixed type with nonlocal conditions is also presented.

Keywords

References

  1. L. Byszewski; Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162(1991), 494-505. https://doi.org/10.1016/0022-247X(91)90164-U
  2. L. Byszewski; Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems, Dynam. Systems Appl., 5(1996), 595-605.
  3. L. Byszewski and H. Akca; Existence of solutions of a semilinear functional differential evolution nonlocal problem, Nonlinear Anal., 34(1998), 65-72. https://doi.org/10.1016/S0362-546X(97)00693-7
  4. L. Byszewski and H. Akca; On a mild solution of a semilinear functional-differential evolution nonlocal problem, J. Appl. Math. Stochastic Anal., 10(1997), 265-271. https://doi.org/10.1155/S1048953397000336
  5. K. Balachandran and M. Chandrasekaran; The nonlocal Cauchy problem for semilinear integrodifferential equation with devating argument, Proceedings of the Edinburgh Mathematical Society, 44(2001), 63-70. https://doi.org/10.1017/S0013091598001060
  6. K. Balachandran and R. R. Kumar; Existence of solutions of integrodifferential evoluition equations with time varying delays, Applied Mathematics E-Notes, 7(2007), 1-8.
  7. K. Balachandran, J. Y. Park; Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal. 71(2009), 4471-4475. https://doi.org/10.1016/j.na.2009.03.005
  8. M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab; Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338(2008), 1340-1350. https://doi.org/10.1016/j.jmaa.2007.06.021
  9. M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab; Existence results for fractional functional differential inclusions with infinite delay and application to control theory, Fract. Calc. Appl. Anal., 11(2008), 35-56.
  10. Yong-Kui Chang, V. Kavitha, M. Mallika Arjunan; Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order, Nonlinear Anal., 71(2009), 5551-5559. https://doi.org/10.1016/j.na.2009.04.058
  11. Diethelm, A. D. Freed; On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.), Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, 1999, pp. 217-224.
  12. M. M. El-Borai; Semigroup and some nonlinear fractional differential equations, Applied Mathematics and Computation, 149(2004), 823-831. https://doi.org/10.1016/S0096-3003(03)00188-7
  13. L. Gaul, P. Klein, S. Kempfle; Damping description involving fractional operators, Mech. Syst. Signal Process., 5(1991), 81-88. https://doi.org/10.1016/0888-3270(91)90016-X
  14. W. G. Glockle, T. F. Nonnenmacher; A fractional calculus approach of self-similar protein dynamics, Biophys. J., 68(1995), 46-53. https://doi.org/10.1016/S0006-3495(95)80157-8
  15. R. Hilfer; Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  16. S. Hu and N. S. Papageorgiou, Handbook of multivalued Analysis (Theory), Kluwer Academic Publishers, Dordrecht Boston, London, 1997.
  17. Lanying Hu, Yong Ren and R. Sakthivel, Existence and uniqueness of mild solutions for semilinear integrodifferential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum, (in press).
  18. A. A. Kilbas, Hari M. Srivastava, J. Juan Trujillo; Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V, Amsterdam, 2006.
  19. X. Li and J. Yong, Optimal control theory for infinite dimensional systems, Birkhauser Boston, 1995.
  20. V. Lakshmikantham, S. Leela and J. Vasundhara Devi; Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.
  21. V. Lakshmikantham; Theory of fractional differential equations, Nonlinear Anal., 60(2008), 3337-3343.
  22. V. Lakshmikantham, A. S. Vatsala; Basic theory of fractional differential equations, Nonlinear Anal., 69(2008), 2677-2682. https://doi.org/10.1016/j.na.2007.08.042
  23. K. S. Miller, B. Ross; An Introduction to the Fractional Calculus and Differential Equations, JohnWiley, New York, 1993.
  24. F. Mainardi, Fractional calculus; Some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997, pp. 291-348.
  25. F. Metzler, W. Schick, H. G. Kilian, T. F. Nonnenmache; Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys., 103(1995), 7180-7186. https://doi.org/10.1063/1.470346
  26. G. M. Mophou, G. M. N'Guerekata; Mild solutions for semilinear fractional differential equations, Electron. J. Differ. Equ., 21(2009), 1-9. https://doi.org/10.1007/s10884-008-9127-0
  27. G. M. Mophou, G. M. N'Gu¶er¶ekata; Existence of mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum, 79(2009), 315-322. https://doi.org/10.1007/s00233-008-9117-x
  28. I. Podlubny; Fractional Differential Equations, Academic Press, San Diego, 1999.
  29. JinRong Wang, X. Xiang and W. Wei; A class of nonlinear integrodifferential impulsive periodic systems of mixed type and optimal controls on Banach spacs, Journal of Applied Mathematics and Computing, (in press).

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