Browse > Article
http://dx.doi.org/10.12941/jksiam.2010.14.2.079

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)
Publication Information
Journal of the Korean Society for Industrial and Applied Mathematics / v.14, no.2, 2010 , pp. 79-91 More about this Journal
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
Fractional integrodifferential equations of mixed type; nonlocal conditions; Krasnoselskii's fixed point theorem; existence; optimal control;
Citations & Related Records
연도 인용수 순위
  • Reference
1 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.   DOI   ScienceOn
2 K. Balachandran, J. Y. Park; Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal. 71(2009), 4471-4475.   DOI   ScienceOn
3 L. Gaul, P. Klein, S. Kempfle; Damping description involving fractional operators, Mech. Syst. Signal Process., 5(1991), 81-88.   DOI   ScienceOn
4 G. M. Mophou, G. M. N'Guerekata; Mild solutions for semilinear fractional differential equations, Electron. J. Differ. Equ., 21(2009), 1-9.   DOI   ScienceOn
5 X. Li and J. Yong, Optimal control theory for infinite dimensional systems, Birkhauser Boston, 1995.
6 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.   DOI   ScienceOn
7 I. Podlubny; Fractional Differential Equations, Academic Press, San Diego, 1999.
8 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).
9 V. Lakshmikantham, S. Leela and J. Vasundhara Devi; Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.
10 V. Lakshmikantham; Theory of fractional differential equations, Nonlinear Anal., 60(2008), 3337-3343.
11 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.   DOI   ScienceOn
12 V. Lakshmikantham, A. S. Vatsala; Basic theory of fractional differential equations, Nonlinear Anal., 69(2008), 2677-2682.   DOI   ScienceOn
13 K. S. Miller, B. Ross; An Introduction to the Fractional Calculus and Differential Equations, JohnWiley, New York, 1993.
14 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.
15 R. Hilfer; Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
16 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).
17 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.
18 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.
19 M. M. El-Borai; Semigroup and some nonlinear fractional differential equations, Applied Mathematics and Computation, 149(2004), 823-831.   DOI   ScienceOn
20 W. G. Glockle, T. F. Nonnenmacher; A fractional calculus approach of self-similar protein dynamics, Biophys. J., 68(1995), 46-53.   DOI   ScienceOn
21 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.   DOI   ScienceOn
22 S. Hu and N. S. Papageorgiou, Handbook of multivalued Analysis (Theory), Kluwer Academic Publishers, Dordrecht Boston, London, 1997.
23 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.   DOI   ScienceOn
24 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.
25 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.
26 L. Byszewski; Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems, Dynam. Systems Appl., 5(1996), 595-605.
27 L. Byszewski and H. Akca; Existence of solutions of a semilinear functional differential evolution nonlocal problem, Nonlinear Anal., 34(1998), 65-72.   DOI   ScienceOn
28 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.   DOI   ScienceOn
29 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.   DOI