[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Approximate Controllability Results for Impulsive Partial Functional Nonlocal Integro-differential Evolution Systems through Resolvent Operators

Discontinuity, Nonlinearity, and Complexity 7(3) (2018) 305--325 | DOI:10.5890/DNC.2018.09.008

Mahalingam Nagaraj$^{1}$, Selvaraj Suganya$^{2}$, Dumitru Baleanu$^{3}$, Mani Mallika Arjunan$^{2}$

$^{1}$ Department of Mathematics, Nadar Saraswathi College of Engineering & Technology, Theni-625531, Tamil Nadu, India

$^{2}$ Department of Mathematics, C. B. M. College, Kovaipudur, Coimbatore - 641042, Tamil Nadu, India

$^{3}$ Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey, and Institute of Space Sciences, Magurele-Bucharest, Romania

Download Full Text PDF

 

Abstract

This paper investigates the existence and approximate controllability results for a class of impulsive functional integro-differential evolution systems with nonlocal conditions via resolvent operators in Banach spaces. By making utilization of Banach contraction principle and Schaefer’s fixed point theorem along with resolvent operators and semigroup theory, we build up the desired results. As an application, we also consider an impulsive partial functional integro-differential equations.

References

  1. [1]  Chang, Y.K. (2007), Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos Solitons & Fractals, 33, 1601-1609.
  2. [2]  Chang, Y.K., Anguraj, A., and Arjunan, M.M. (2009), Controllability of impulsive neutral functional differential inclusions with infinite delay in Banach spaces, Chaos Solitons & Fractals, 39(4), 1864-1876.
  3. [3]  Kavitha, V. and Mallika Arjunan, M. (2010), Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces, Nonlinear Analysis: Hybrid Systems, 4, 441-450.
  4. [4]  Mahmudov, N.I., Vijayakumar, V., and Murugesu, R. (2017), Approximate controllability of second-order evolution differential inclusions in Hilbert spaces, Mediterranean Journal of Mathematics, 71(1), 45-61.
  5. [5]  Mahmudov, N.I. and Denker, A. (2000), On controllability of linear stochastic systems, International Journal of Control, 73, 144-151.
  6. [6]  Pierri, M., O’Regan, D., and Prokopczyk, A. (2016), On recent developments treating the exact controllability of abstract control problems, Electronic Journal of Differential Equations, 2016(160), 1-9.
  7. [7]  Vijayakumar, V. (2016), Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces, IMA Journal of Mathematical Control and Information, 1-18, DOI: 10.1093/imamci/dnw049.
  8. [8]  Vijayakumar, V. (2017), Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert spaces, International Journal of Control, (2017), 1-11, DOI:10.1080/00207179.2016.1276633.
  9. [9]  Pavlidis, T. and Jury, E.I. (1965), Analysis of a new class of pulse-frequency modulated feedback systems, IEEE Transactions on Automatic Control, 10, 35-43.
  10. [10]  Schmaedeke, W.W. (1965), Optimal control theory for nonlinear vector differential equations containing measures, SIAM Journal on Control, 3(2), 231-280.
  11. [11]  Pavlidis, T. (1967), Tability of systems described by differential equations containing impulses, IEEE Transactions on Automatic Control, 12, 43-45.
  12. [12]  Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of Impulsive Differential Equations, World Scientific, Singapore, Teaneck, London.
  13. [13]  Agarwal, R.P., Hristova, S., and O’Regan, D. (2017), P-Moment exponential stability of Caputo fractional differential equations with random impulses, Discontinuity, Nonlinearity, and Complexity, 6(1), 49-63.
  14. [14]  Anguraj, A., Kanjanadevi, S., and Trujillo, J.J. (2017), Existence of mild solutions of abstract fractional differential equations with fractional non-instantaneous impulsive conditions, Discontinuity, Nonlinearity, and Complexity, 6(2), 173-183.
  15. [15]  Bonotto, E.M., Bortolan, M.C., Carvalho, A.N., and Czaja, R. (2015), Global attractors for impulsive dynamical systems - a precompact approach, Journal of Differential Equations, 259(7), 2602-2625.
  16. [16]  Fen, F.T. and Karaca, I.Y. (2015), Nonlinear four-point impulsive fractional differential equations with p-Laplacian operator, Discontinuity, Nonlinearity, and Complexity, 4(4), 467-486.
  17. [17]  Li, Y., Yang, L., and Wu, W. (2915), Anti-periodic solution for impulsive BAM neural networks with time-varying leakage delays on time scales, Neurocomputing, 149B(3), 536-545.
  18. [18]  Liu, S., Wang, J., and Wei, W. (2015), A study on iterative learning control for impulsive differential equations, Communications in Nonlinear Science and Numerical Simulation, 24(1-3), 4-10.
  19. [19]  Qian, D., Chen, L., and Sun, X. (2015), Periodic solutions of superlinear impulsive differential equations: A geometric approach, Journal of Differential Equations, 258(9), 3088-3106.
  20. [20]  Shen, B., Zhai, J., Gao, J., Feng, E., and Xiu, X. (2016), Nonlinear state-dependent impulsive system and its parameter identification in microbial fed-batch culture, Applied Mathematical Modelling, 40(2), 1126-1136.
  21. [21]  Wang, X. and Jia, J. (2015), Dynamic of a delayed predator - prey model with birth pulse and impulsive harvesting in a polluted environment, Physics A: Statistical Mechanics and its Applications, 422, 1-15.
  22. [22]  Yu, X. andWang, J. (2015), Periodic boundary value problems for nonlinear impulsive evolution equations on Banach spaces, Communications in Nonlinear Science and Numerical Simulation, 22(1-3), 980-989.
  23. [23]  Dugundji, J. and Granas, A. (1982), Fixed Point Theory, MonografieMat. PWN, Warsaw.
  24. [24]  Grimmer, R. (1982), Resolvent operators for integral equations in Banach space, Transactions of the American Mathematical Society, 48, 333-349.
  25. [25]  Chen, G. and Grimmer, R. (1982), Integral equations as evolution equations, Journal of Differential Equations, 45, 53-74.
  26. [26]  Hannsgen, K.B. (1976), The resolvent kernel of an integrodifferential equation in Hilbert space, SIAM Journal on Mathematical Analysis, 7, 481-490.
  27. [27]  Miller, R.K. (1975), Volterra integral equations in a Banachspace, Funkcialaj Ekvacioj, 18, 163-193.
  28. [28]  Miller, R.K. (1978), An integrodifferential equation for rigid heat conductors with memory, Journal of Mathematical Analysis and Applications, 66, 313-332.
  29. [29]  Byszewski, L. (1991), Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, Journal of Mathematical Analysis and Applications, 162, 494-505.
  30. [30]  Byszewski, L. and Lakshmikantham, V. (1990), Theorem about existence and uniqueness of a solutions of a nonlocal Cauchy problem in a Banach space, Applicable Analysis, 40, 11-19.
  31. [31]  Liang, J., Liu, J.H., and Xiao, T.J. (2004), Nonlocal Cauchy problems governed by compact operator families, Nonlinear Analysis: Theory, Methods & Applications, 57, 183-189.
  32. [32]  Aizicovici, S. and Mckibben, M. (2000), Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear Analysis: Theory, Methods & Applications, 39, 649-668.
  33. [33]  Liang, J., Liu, J.H., and Xiao, T.J. (2006), Nonlocal Cauchy problems for nonautonomous evolution equations, Communications on Pure and Applied Analysis, 5, 529-535.
  34. [34]  Balachandran, K. and Divya, S. (2017), Controllability of nonlinear neutral fractional integrodifferential systems with infinite delay, Journal of Applied Nonlinear Dynamics, 6(3), 333-344.
  35. [35]  Cernea, A. (2017), On the solutions of some boundary value problems for integro-differential inclusions of fractional order, Journal of Applied Nonlinear Dynamics, 6(2), 173-179.
  36. [36]  Chang, Y.K., Kavitha, V., and Arjunan, M.M. (2010), Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators, Nonlinear Analysis: Hybrid Systems, 4, 32-43.
  37. [37]  Chang, J.C. and Liu, H. (2009), Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the α-norm, Nonlinear Analysis: Theory, Methods & Applications, 71, 3759-3768.
  38. [38]  Ezzinbi, K, Fu, X., and Hilal, K. (2007), Existence and regularity in the α-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Analysis:Theory, Methods & Applications, 67, 1613-1622.
  39. [39]  Ezzinbi, K. and Fu, X. (2004), Existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions, Nonlinear Analysis:Theory, Methods & Applications, 57, 1029-1041.
  40. [40]  Fu, X. and Ezzinbi, K. (2003), Existence of solutions for neutral functional differential evolutions equations with nonlocal conditions, Nonlinear Analysis:Theory, Methods & Applications, 54, 215-227.
  41. [41]  Joice Nirmala, R. and Balachandran, K. (2016), Controllability of fractional nonlinear systems in Banach spaces, Journal of Applied Nonlinear Dynamics, 5(4), 485-494.
  42. [42]  Liang, J., Liu, J.H., and Xiao, T.J. (2009), Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Mathematical and Computer Modelling, 49, 798-804.
  43. [43]  Veretennikova, M. and Kolokoltsov, V. (2017), The fractional Hamilton-Jacobi-Bellman equation, Journal of Applied Nonlinear Dynamics, 6(1), 45-56.
  44. [44]  Yan, Z. (2009), Nonlinear functional integro-differential evolution equations with nonlocal conditions in Banach spaces, Mathematical Communications, 14, 35-45.
  45. [45]  Yan, Z. (2009), On solutions of semilinear evolution integro-differential equations with nonlocal conditions, Tamkang Journal of Mathematics, 40(3), 257-269.
  46. [46]  Yan, Z. (2011), Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators, Journal of Computational and Applied Mathematics, 235, 2252-2262.
  47. [47]  Yan, Z. and Pu, Z. (2009), On solution of a nonlinear functional integrodifferential equations with nonlocal conditions in Banach spaces, Results in Mathematics, 55, 493-505.
  48. [48]  Yan, Z. andWei, P. (2010), Existence of solutions for nonlinear functional integro-differential evolution equations with nonlocal conditions, Aequationes Mathematicae, 79, 213-228.
  49. [49]  Yan, Z. (2010), Nonlocal problems for delay integrodifferential equations in Banach spaces, Differential Equations & Applications, 2(1), 15-25.
  50. [50]  Grimmer, R. and Liu, J.H. (1994), Integrated semigroups and integrodifferential equations, Semigroup Forum, 48, 79-95.
  51. [51]  Pruss, J. (1983), On resolvent operators for linear integrodifferential equations of Volterra type, Journal of Integral Equations, 5, 211-236.
  52. [52]  Liang, J., Liu, J.H., and Xiao, T.J. (2008), Nonlocal impulsive problems for integrodifferential equations, In: 6th International Conference on Differential Equations and Dynamical Systems, May 22-26, 2008, Baltimore, Maryland, USA.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates