The Solvability and Optimal Controls for Some Fractional Impulsive Equation with Finite Delay
DOI:
https://doi.org/10.5890/DNC.2014.06.003Abstract
This paper is concerned with the existence and uniqueness of mild solution of some fractional impulsive equations withfinite delay. Firstly,we introduce the fractional calculus,Gronwall inequality, leray-schauder’sfixed point theorem,Secondly with the help of them, the sufficient condition for the existence and uniqueness of solutions is presented. Finally we give an example to illustrate our main results.References
[1] Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York.
[2] Hu, S. and Papageorgiou, N.S. (1997), Handbook of multivalued Analysis Theory, Kluwer Academic Publishers, Dordrecht Boston, London.
[3] Kilbas, A.A., Hari, M., Srivastava, J., and Trujillo, Juan (2006), Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam.
[4] Lakshmikantham, V., Leela, S. and Vasundhara Devi, J. (2009), Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge.
[5] Diethelm, Kai (2010), The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin & Heidelberg.
[6] Wang, J.R., Zhou, Y. andMedved, Mi. (2012),On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay, Journal of Optimization Theory and Applications, 152, 31-50.
[7] Wang, J.R., Zhou, Y., and Wei, W. (2011), A class of fractional delay nonlinear integrodifferential controlled systems in Banach space, Communications in Nonlinear Science and Numerical Simulation, 16, 4049-4059.
[8] Shu, X.-B., Lai, Y.Z., and Chen, Y.M. (2011) ,The existence ofmild solutions for impulsive fractional partial differential equations, Nonlinear Analysis, 74 ,2003-2011.
[9] Benchohra, M. and Henderson, J. (2008), Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications , 338, 1340-1350.
[10] Balachandran, K. and Zhou, Y. (2012), Relative controllability of fractional dynamical with delays in control, Communications in Nonlinear Science and Numerical Simulation , 17, 3508-3520.
[11] Cuevas, C., Lizama, C. (2008), Almost automorphic solutions to a class of semilinear fractional differential equations, Applied Mathematics Letters, 21, 1315-1319.
[12] Bazhlekova, E. (2001), Fractional Evolution Equations in Banach Spaces, Ph.D Thesis, Eindhoven University of Technology.
[13] Xue, L., Xiong, J.J. (2011), Existence and uniqueness of mild solutions for abstract delay fractional differential equations, Computers & Mathematics with Applications, 62, 1398-1404.
[14] Hernández, E.,O'Regan, D., and Balachandran, K. (2010), On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Analysis, 73, 3462-3471.
[15] Mophou, G.M., N'Guérékata, G.M. (2010), Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Applied Mathematics and Computation, 216, 61-69.
[16] Wang, J.R., Lv, L.L., and Zhou, Y. (2012), New concepts and results in stability of fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 17, 2530-2538.
[17] Mophou, G.M. (2010), Existence and uniqueness of mild solution to impulsive fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72, 1604-1615.
[18] Hale, J., Kato, J. (1978), Phase space for retarded equations with infinte delay, Funkcialaj, 21, 11-41.
[19] Wang, J.R., Fečkan, M., and Zhou, Y. (2011), On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynamics of PDE, 8(4), 345-361.
[20] Wang, J.R., Fečkan, M., and Zhou, Y. (2013), Relaxed controls for nonlinear fractional impulsive fvolution equations, Journal of Optimization Theory and Applications, in press.
[21] Ye, H.P., Gao, J.M., and Ding, Y.S. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 1075-1081.
[22] Zhou,Y. and Jiao, F. (2010), Existence of mild solutions for fractional neutral evolution equations, Computers & Mathematics with Applications, 59, 1063-1077.
[23] Balder, E. (1987), Necessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functional, Nonlinear Analysis: Real World Applications, 11, 1399-1404.
[24] Finner, H. (2007),A generalization of Hölder's inequality and some probability inequalities, the Annals of Probobity, 1893-1901.
[25] Granas, A. and Dugundji, J. (2003), Fixed Point Theory, Springer, New York.
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