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http://dx.doi.org/10.4134/BKMS.b170531

APPROXIMATE CONTROLLABILITY OF SECOND-ORDER NONLOCAL IMPULSIVE FUNCTIONAL INTEGRO-DIFFERENTIAL SYSTEMS IN BANACH SPACES  

Baleanu, Dumitru (Department of Mathematics and Computer Sciences Faculty of Arts and Sciences Cankaya University)
Arjunan, Mani Mallika (Department of Mathematics C. B. M. College)
Nagaraj, Mahalingam (Nadar Saraswathi College of Engineering & Technology)
Suganya, Selvaraj (Department of Mathematics C. B. M. College)
Publication Information
Bulletin of the Korean Mathematical Society / v.55, no.4, 2018 , pp. 1065-1092 More about this Journal
Abstract
This manuscript is involved with a category of second-order impulsive functional integro-differential equations with nonlocal conditions in Banach spaces. Sufficient conditions for existence and approximate controllability of mild solutions are acquired by making use of the theory of cosine family, Banach contraction principle and Leray-Schauder nonlinear alternative fixed point theorem. An illustration is additionally furnished to prove the attained principles.
Keywords
impulsive conditions; nonlocal conditions; integro-differential equations; semigroup theory; cosine family; fixed point;
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1 J. P. Dauer and N. I. Mahmudov, Approximate controllability of semilinear functional equations in Hilbert spaces, J. Math. Anal. Appl. 273 (2002), no. 2, 310-327.   DOI
2 J. Dugundji and A. Granas, Fixed Point Theory. I, Monografie Matematyczne, 61, Panstwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982.
3 H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, NorthHolland Mathematics Studies, 108, North-Holland Publishing Co., Amsterdam, 1985.
4 X. Fu, Approximate controllability for neutral impulsive differential inclusions with nonlocal conditions, J. Dyn. Control Syst. 17 (2011), no. 3, 359-386.   DOI
5 E. Hernandez M and H. R. Henriquez, Impulsive partial neutral differential equations, Appl. Math. Lett. 19 (2006), no. 3, 215-222.   DOI
6 V. Kavitha and M. Mallika Arjunan, Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces, Nonlinear Anal. Hybrid Syst. 4 (2010), no. 3, 441-450.   DOI
7 V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of impulsive differential equations, Series in Modern Applied Mathematics, 6, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
8 M. Li and J. Ma, Approximate controllability of second order impulsive functional differential system with infinite delay in Banach spaces, J. Appl. Anal. Comput. 6 (2016), no. 2, 492-514.
9 N. I. Mahmudov and A. Denker, On controllability of linear stochastic systems, Internat. J. Control 73 (2000), no. 2, 144-151.   DOI
10 N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 (2008), no. 3, 536-546.   DOI
11 N. I. Mahmudov, V. Vijayakumar, and R. Murugesu, Approximate controllability of second-order evolution differential inclusions in Hilbert spaces, Mediterr. J. Math. 13 (2016), no. 5, 3433-3454.   DOI
12 M. Pierri, D. O'Regan, and A. Prokopczyk, On recent developments treating the exact controllability of abstract control problems, Electron. J. Differential Equations 2016, Paper No. 160, 9 pp.
13 R. Sakthivel, N. I. Mahmudov, and J. H. Kim, On controllability of second order nonlinear impulsive differential systems, Nonlinear Anal. 71 (2009), no. 1-2, 45-52.   DOI
14 R. Sakthivel, Y. Ren, and N. I. Mahmudov, On the approximate controllability of semilinear fractional differential systems, Computers & Math. Appl. 62 (2011), 1451-1459.   DOI
15 I. Stamova, Stability Analysis of Impulsive Functional Differential Equations, De Gruyter Expositions in Mathematics, 52, Walter de Gruyter GmbH & Co. KG, Berlin, 2009.
16 C. C. Travis and G. F. Webb, Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston J. Math. 3 (1977), no. 4, 555-567.
17 C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar. 32 (1978), no. 1-2, 75-96.   DOI
18 Z. Yan, Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators, J. Comput. Appl. Math. 235 (2011), no. 8, 2252-2262.   DOI
19 V. Vijayakumar, Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces, IMA Journal of Mathematical Control and Information (2016), 1-18. Available online.
20 V. Vijayakumar, Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert spaces, International Journal of Control (2017), 1-11. Available online.
21 Z. Yan, Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay, Internat. J. Control 85 (2012), no. 8, 1051-1062.   DOI
22 M. M. Arjunan and V. Kavitha, Existence results for impulsive neutral functional differential equations with state-dependent delay, Electron. J. Qual. Theory Differ. Equ. 2009 (2009), no. 26, 13 pp.
23 G. Arthi and K. Balachandran, Controllability of damped second order nonlinear impulsive systems, Nonlinear Functional Anal. Appl. 16 (2011), 227-245.
24 G. Arthi and K. Balachandran, Controllability of second-order impulsive differential and integrodifferential evolution systems with nonlocal conditions, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 19 (2012), no. 6, 693-715.
25 G. Arthi and J. H. Park, On controllability of second-order impulsive neutral integrodifferential systems with infinite delay, IMA J. Math. Control Inform. 32 (2015), no. 3, 639-657.   DOI
26 P. Balasubramaniam, J. Y. Park, and P. Muthukumar, Approximate controllability of neutral stochastic functional differential systems with infinite delay, Stoch. Anal. Appl. 28 (2010), no. 2, 389-400.   DOI
27 Y.-K. Chang, A. Anguraj, and M. Mallika Arjunan, Controllability of impulsive neutral functional differential inclusions with infinite delay in Banach spaces, Chaos Solitons & Fractals 39 (2009), no. 4, 1864-1876.   DOI
28 M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, 2, Hindawi Publishing Corporation, New York, 2006.
29 D. N. Chalishajar, Controllability of second order impulsive neutral functional differential inclusions with infinite delay, J. Optim. Theory Appl. 154 (2012), no. 2, 672-684.   DOI
30 Y.-K. Chang, Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos Solitons & Fractals 33 (2007), 1601-1609.   DOI