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On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay

  • * Corresponding author: V. Vijayakumar

    * Corresponding author: V. Vijayakumar 
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  • In our manuscript, we organize a group of sufficient conditions of neutral integro-differential inclusions of Sobolev-type with infinite delay via resolvent operators. By applying Bohnenblust-Karlin's fixed point theorem for multivalued maps, we proved our results. Lastly, we present an application to support the validity of the study.

    Mathematics Subject Classification: Primary: 34G10, 34G25; Secondary: 93B05.


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  • [1] H. M. Ahmed, Controllability for Sobolev type fractional integro-differential systems in a Banach space, Advances in Difference Equations, 167 (2012), 1-10.  doi: 10.1186/1687-1847-2012-167.
    [2] K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional integrodifferential equations of Sobolev type, Computers and Mathematics with Applications, 64 (2012), 3406-3413.  doi: 10.1016/j.camwa.2011.12.051.
    [3] H. F. Bohnenblust and  S. KarlinOn a Theorem of Ville. Contributions to the Theory of Games, Vol. I, Princeton University Press, Princeton, NJ, 1950. 
    [4] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, Journal of Mathematical Analysis and Applications, 162 (1991), 494-505.  doi: 10.1016/0022-247X(91)90164-U.
    [5] L. Byszewski and H. Akca, On a mild solution of a semilinear functional–differential evolution nonlocal problem, Journal of Applied Mathematics and Stochastic Analysis, 10 (1997), 265-271.  doi: 10.1155/S1048953397000336.
    [6] Y.-K. Chang and W.-T. Li, Controllability of Sobolev type semilinear functional differential and integrodifferential inclusions with an unbounded delay, Georgian Mathematical Journal, 13 (2006), 11-24.  doi: 10.1515/GMJ.2006.11.
    [7] Y. K. Chang, Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos Solitons and Fractals, 33 (2007), 1601-1609.  doi: 10.1016/j.chaos.2006.03.006.
    [8] K. Deimling, Multivalued Differential Equations, Walter De Gruyter & Co., Berlin, 1992. doi: 10.1515/9783110874228.
    [9] J. P. C. dos Santos, C. Cuevas and B. de Andrade, Existence results for a fractional equation with state–dependent delay, Advances in Difference Equations, (2011), Art. ID 642013, 1–15. doi: 10.1155/2011/642013.
    [10] J. P. C. dos SantosV. 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, 14 (2013), 59-71. 
    [11] R. Grimmer and A. J. Pritchard, Analytic resolvent operators for integral equations in Banach space, Journal of Differential Equations, 50 (1983), 234-259.  doi: 10.1016/0022-0396(83)90076-1.
    [12] R. Grimmer and J. Prüss, On linear Volterra equations in Banach spaces, Computers and Mathematics with Applications, 11 (1985), 189-205.  doi: 10.1016/0898-1221(85)90146-4.
    [13] J. K. Hale and S. M. V. Lunel, Introduction to Functional–Differential Equations. Applied Mathematical Sciences, Vol. 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
    [14] J. K. Hale, Partial neutral functional–differential equations, Revue Roumaine de Mathematiques Pures et Appliquees, 39 (1994), 339-344. 
    [15] A. Harrat and A. Debbouche, Sobolev type fractional delay impulsive equations with alpha–Sobolev resolvent families and integral conditions, Nonlinear Studies, 20 (2013), 549-558. 
    [16] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis (Theory), Kluwer Academic Publishers, Dordrecht, 1997.
    [17] V. Kavitha and M. Mallika Arjunan, Controllability of non–densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces, Nonlinear Analysis: Hybrid Systems, 4 (2010), 441-450.  doi: 10.1016/j.nahs.2009.11.002.
    [18] K. D. Kucche and M. B. Dhakne, Sobolev–type Volterra–Fredholm functional integrodifferential equations in Banach spaces, Bulletin of Parana's Mathematical Society, 32 (2014), 237-253.  doi: 10.5269/bspm.v32i1.19901.
    [19] A. Lasota and Z. Opial, An application of the Kakutani–Ky Fan theorem in the theory of ordinary differential equations or noncompact acyclic–valued map, Bulletin L'Academie Polonaise des Science, Serie des Sciences Mathematiques, Astronomiques et Physiques, 13 (1965), 781-786. 
    [20] J. H. Lightbourne III and S. Rankin, A partial functional-differential equation of Sobolev type, Journal of Mathematical Analysis and Applications, 93 (1983), 328-337.  doi: 10.1016/0022-247X(83)90178-6.
    [21] J. A. MachadoC. RavichandranM. Rivero and J. J. Trujillo, Controllability results for impulsive mixed–type functional integro–differential evolution equations with nonlocal conditions, Fixed Point Theory and Applications, 66 (2013), 1-16.  doi: 10.1186/1687-1812-2013-66.
    [22] N. I. Mahmudov and A. Denker, On controllability of linear stochastic systems, International Journal of Control, 73 (2000), 144-151.  doi: 10.1080/002071700219849.
    [23] N. I. Mahmudov, Approximate controllability of fractional Sobolev–type evolution equations in Banach spaces, Abstract and Applied Analysis, 2013, Art. ID 502839, 1–9. doi: 10.1155/2013/502839.
    [24] N. I. Mahmudov, V. Vijayakumar and R. Murugesu, Approximate controllability of second–order evolution differential inclusions in Hilbert spaces, Mediterranean Journal of Mathematics, 13 (2016), 3433—3454. doi: 10.1007/s00009-016-0695-7.
    [25] N. I. MahmudovR. MurugesuC. Ravichandran and and V. Vijayakumar, Approximate controllability results for fractional semilinear integro–differential inclusions in Hilbert spaces, Results in Mathematics, 71 (2017), 45-61.  doi: 10.1007/s00025-016-0621-0.
    [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.
    [27] B. Radhakrishnan, A. Mohanraj and V. Vinoba, Existence of solutions for nonlinear impulsive neutral integro–differential equations of Sobolev type with nonlocal conditions in Banach spaces, Electronic Journal of Differential Equations, (2013), 1–13.
    [28] R. Ravi Kumar, Nonlocal Cauchy problem for analytic resolvent operator integrodifferential equations in Banach spaces, Applied Mathematics and Computation, 204 (2008), 352-362.  doi: 10.1016/j.amc.2008.06.050.
    [29] P. RevathiR. Sakthivel and Y. Ren, Stochastic functional differential equations of Sobolev–type with infinite delay, Statistics & Probability Letters, 109 (2016), 68-77.  doi: 10.1016/j.spl.2015.10.019.
    [30] R. SakthivelR. Ganesh and S. M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Applied Mathematics and Computation, 225 (2013), 708-717.  doi: 10.1016/j.amc.2013.09.068.
    [31] R. Sakthivel, E. R. Anandhi and N. I. Mahmudov, Approximate controllability of second-order systems with state-dependent delay, Numerical Functional Analysis and Optimization, 29, (2008), 1347–1362. doi: 10.1080/01630560802580901.
    [32] R. SakthivelY. RenA. Debbouche and N. I. Mahmudov, Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Applicable Analysis, 95 (2016), 2361-2382.  doi: 10.1080/00036811.2015.1090562.
    [33] N. ValliammalC. Ravichandran and J. H. Park, On the controllability of fractional neutral integrodifferential delay equations with nonlocal conditions, Mathematical Methods in the Applied Sciences, 40 (2017), 5044-5055.  doi: 10.1002/mma.4369.
    [34] V. Vijayakumar and R. Murugesu, Controllability for a class of second order evolution differential inclusions without compactness, Applicable Analysis, 98 (2019), 1367-1385.  doi: 10.1080/00036811.2017.1422727.
    [35] V. Vijayakumar, Approximate controllability results for non–densely defined fractional neutral differential inclusions with Hille–Yosida operators, International Journal of Control, 92 (2019), 2210-2222.  doi: 10.1080/00207179.2018.1433331.
    [36] V. VijayakumarR. MurugesuR. Poongodi and S. Dhanalakshmi, Controllability of second order impulsive nonlocal Cauchy problem via measure of noncompactness, Mediterranean Journal of Mathematics, 14 (2017), 29-51.  doi: 10.1007/s00009-016-0813-6.
    [37] V. Vijayakumar, Approximate controllability results for analytic resolvent integro–differential inclusions in Hilbert spaces, International Journal of Control, 91 (2018), 204-214.  doi: 10.1080/00207179.2016.1276633.
    [38] V. Vijayakumar, Approximate controllability results for impulsive neutral differential inclusions of Sobolev–type with infinite delay, International Journal of Control, 91 (2018), 2366-2386.  doi: 10.1080/00207179.2017.1346300.
    [39] V. Vijayakumar, S. Sivasankaran and M. Mallika Arjunan, Existence of solutions for second-order impulsive neutral functional integro-differential equations with infinite delay, Nonlinear Studies, 19 (2012), 327-343.
    [40] V. VijayakumarS. Sivasankaran and M. Mallika Arjunan, Existence of global solutions for second order impulsive abstract functional integrodifferential equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 18 (2011), 747-766. 
    [41] J. WangM. Feckan and Y. Zhou, Controllability of Sobolev type fractional evolution systems, Dynamics of Partial Differential Equations, 11 (2014), 71-87.  doi: 10.4310/DPDE.2014.v11.n1.a4.
    [42] B. Yan, Boundary value problems on the half–line with impulses and infinite delay, Journal of Mathematical Analysis and Applications, 259 (2001), 94-114.  doi: 10.1006/jmaa.2000.7392.
    [43] Y. ZhouV. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evolution Equations and Control Theory, 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507.
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