# EXISTENCE AND CONTROLLABILITY OF FRACTIONAL NEUTRAL INTEGRO-DIFFERENTIAL SYSTEMS WITH STATE-DEPENDENT DELAY IN BANACH SPACES

• Accepted : 2016.03.07
• Published : 2016.03.25

#### Abstract

In view of ideas for semigroups, fractional calculus, resolvent operator and Banach contraction principle, this manuscript is generally included with existence and controllability (EaC) results for fractional neutral integro-differential systems (FNIDS) with state-dependent delay (SDD) in Banach spaces. Finally, an examples are also provided to illustrate the theoretical results.

#### References

1. D. Baleanu, J.A.T. Machado, and A.C.J. Luo, Fractional Dynamics and Control, Springer, New York, USA, (2012).
2. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amesterdam, (2006).
3. G. Bonanno, R. Rodriguez-Lopez, and S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations, Fractional Calculus and Applied Analysis, 17(3) (2014), 717-744. https://doi.org/10.2478/s13540-014-0196-y
4. R. Rodriguez-Lopez and S. Tersian, Multiple solutions to boundary value problem for impulsive fractional differential equations, Fractional Calculus and Applied Analysis, 17(4) (2014), 1016-1038.
5. R.P. Agarwal, V. Lupulescu, D. O'Regan, and G. Rahman, Fractional calculus and fractional differential equations in nonreflexive Banach spaces, Communications in Nonlinear Science and Numerical Simulation, 20(1) (2015), 59-73. https://doi.org/10.1016/j.cnsns.2013.10.010
6. E. Keshavarz, Y. Ordokhani, and M. Razzaghi, Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Applied Mathematical Modelling, 38 (2014), 6038-6051. https://doi.org/10.1016/j.apm.2014.04.064
7. Z. Lv and B. Chen, Existence and uniqueness of positive solutions for a fractional switched system, Abstract and Applied Analysis, 2014 (2014), Article ID 828721, 7 pages.
8. Y. Wang, L. Liu, and Y. Wu, Positive solutions for a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters, Advances in Difference Equations, 2014(268), 1-24.
9. B. Ahmad, S.K. Ntouyas, and A. Alsaed, Existence of solutions for fractional q-integro-difference inclusions with fractional q-integral boundary conditions, Advances in Difference Equations, 2014(257), 1-18.
10. R.P. Agarwal and B.D. Andrade, On fractional integro-differential equations with statedependent delay, Comp. Math. App., 62 (2011), 1143-1149. https://doi.org/10.1016/j.camwa.2011.02.033
11. M. Benchohra and F. Berhoun, Impulsive fractional differential equations with statedependent delay, Commun. Appl. Anal., 14(2) (2010), 213-224.
12. K. Aissani and M. Benchohra, Fractional integro-differential equations with statedependent delay, Advances in Dynamical Systems and Applications, 9(1) (2014), 17-30.
13. J. Dabas and G.R. Gautam, Impulsive neutral fractional integro-differential equation with state-dependent delay and integral boundary condition, Electronic Journal of Differential Equations, 2013(273) (2013), 1-13.
14. S. Suganya, M. Mallika Arjunan, and J.J. Trujillo, Existence results for an impulsive fractional integro-differential equation with state-dependent delay, Applied Mathematics and Computation, 266 (2015), 54-69. https://doi.org/10.1016/j.amc.2015.05.031
15. J.P.C. Dos Santos, C. Cuevas, and B. de Andrade, Existence results for a fractional equations with state-dependent delay, Advances in Difference Equations, (2011), 1-15.
16. J.P.C. Dos Santos, M. Mallika Arjunan, and C. Cuevas, Existence results for fractional neutral integrodifferential equations with state-dependent delay, Comput. Math. Appl., 62 (2011), 1275-1283. https://doi.org/10.1016/j.camwa.2011.03.048
17. Z. Yan, Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA Journal of Mathematical Control and Information, 30 (2013), 443-462. https://doi.org/10.1093/imamci/dns033
18. Z. Yan and X. Jia, Approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay, Collect. Math., 66 (2015), 93-124. https://doi.org/10.1007/s13348-014-0109-8
19. V. Vijayakumar, C. Ravichandran, and R. Murugesu, Approximate controllability for a class of fractional neutral integro-differential inclusions with state-dependent delay, Nonlinear Studies, 20(4) (2013), 513-532.
20. R.P. Agarwal, J.P.C. Dos Santos, and C. Cuevas, Analytic resolvent operator and existence results for fractional integro-differential equations, J. Abstr. Differ. Equ. Appl., 2(2) (2012), 26-47.
21. V. Vijayakumar, A. Selvakumar, and R. Murugesu, Controllability for a class of fractional neutral integro-differential equations with unbounded delay, Applied Mathematics and Computation, 232 (2014), 303-312. https://doi.org/10.1016/j.amc.2014.01.029
22. C. Rajivganthi, P. Muthukumar, and B. Ganesh Priya, Approximate controllability of fractional stochastic integro-differential equations with infinite delay of order 1 < ${\alpha}$ < 2, IMA Journal of Mathematical Control and Information, (2015), 1-15, Available Online.
23. B.D. Andrade and J.P.C. Dos Santos, Existence of solutions for a fractional neutral integro-differential equation with unbounded delay, Electronic Journal of Differential Equations, 2012(90) (2012), 1-13.
24. J.P.C. Dos Santos, V. Vijayakumar, and R. Murugesu, Existence of mild solutions for nonlocal Cauchy problem for fractional neutral integro-differential equation with unbounded delay, Communications in Mathematical Analysis, X (2011), 1-13.
25. X. Shu and Q.Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1 < ${\alpha}$ < 2, Comput. Math. Appl., 64 (2012), 2100-2110. https://doi.org/10.1016/j.camwa.2012.04.006
26. Z. Yan and X. Jia, Impulsive problems for fractional partial neutral functional integrodifferentialinclusions with infinite delay and analytic resolvent operators, Mediterr.Math., 11 (2014), 393-428. https://doi.org/10.1007/s00009-013-0349-y
27. Z. Yan and F. Lu, On approximate controllability of fractional stochastic neutral integrodifferentialinclusions with infinite delay, Applicable Analysis, (2014), 1235-1258.
28. L. Kexue, P. Jigen, and G. Jinghuai, Controllability of nonlocal fractional differentialsystems of order ${\alpha}$ ${\in}$ (1, 2] in Banach spaces, Rep. Math. Phys., 71 (2013), 33-43. https://doi.org/10.1016/S0034-4877(13)60020-8
29. R. Sakthivel, R. Ganesh, Y. Ren, and S. Marshal Anthoni, Approximate controllabilityof nonlinear fractional dynamical systems, Commun. Nonlinear Sci. Numer. Simulat., 18(2013), 3498-3508. https://doi.org/10.1016/j.cnsns.2013.05.015
30. J. Dabas, A. Chauhan, and M. Kumar, Existence of the mild solutions for impulsive fractionalequations with infinite delay, International Journal of Differential Equations, 2011,Article ID 793023, 20 pages.
31. J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial.Ekvac., 21 (1978), 11-41.
32. Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with UnboundedDelay, Springer-Verlag, Berlin, (1991).
33. X. Fu and R. Huang, Existence of solutions for neutral integro-differential equations withstate-dependent delay, Appl. Math. Comp., 224 (2013), 743-759. https://doi.org/10.1016/j.amc.2013.09.010
34. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, (1983).
35. N.I. Mahmudov and A. Denker, On controllability of linear stochastic systems, Int. J.Control, 73 (2000), 144-151. https://doi.org/10.1080/002071700219849

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