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June  2021, 10(2): 271-296. doi: 10.3934/eect.2020066

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

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore - 632 014, Tamil Nadu, India

* Corresponding author: V. Vijayakumar

Received  November 2019 Revised  March 2020 Published  June 2020

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.

Citation: V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066
References:
[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.  Google Scholar

[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.  Google Scholar

[3] H. F. Bohnenblust and S. Karlin, On a Theorem of Ville. Contributions to the Theory of Games, Vol. I, Princeton University Press, Princeton, NJ, 1950.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

K. Deimling, Multivalued Differential Equations, Walter De Gruyter & Co., Berlin, 1992. doi: 10.1515/9783110874228.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. K. Hale, Partial neutral functional–differential equations, Revue Roumaine de Mathematiques Pures et Appliquees, 39 (1994), 339-344.   Google Scholar

[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.   Google Scholar

[16]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis (Theory), Kluwer Academic Publishers, Dordrecht, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3] H. F. Bohnenblust and S. Karlin, On a Theorem of Ville. Contributions to the Theory of Games, Vol. I, Princeton University Press, Princeton, NJ, 1950.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

K. Deimling, Multivalued Differential Equations, Walter De Gruyter & Co., Berlin, 1992. doi: 10.1515/9783110874228.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. K. Hale, Partial neutral functional–differential equations, Revue Roumaine de Mathematiques Pures et Appliquees, 39 (1994), 339-344.   Google Scholar

[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.   Google Scholar

[16]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis (Theory), Kluwer Academic Publishers, Dordrecht, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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