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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

A Note on Existence of Global Solutions for Impulsive Functional Integrodifferential Systems

Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 397--407 | DOI:10.5890/DNC.2021.09.004

C. Dineshkumar, R. Udhayakumar

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

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Abstract

In our manuscript, we research the existence of global solutions for a class of impulsive abstract functional integrodifferential systems with nonlocal conditions. We proved our outcomes by utilizing the Leray-Schauder's Alternative fixed point theorem. Lastly, a model is presented for illustration of theory.

References

  1. [1]  Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), { Theory of Impulsive Differential Equations}, World Scientific, Singapore.
  2. [2]  Bainov, D.D. and Simeonov, P.S. (1993), { Impulsive Differential Equations: Periodic Solutions and Applications}, Longman Scientific and Technical Group, England.
  3. [3]  Chalishajar, D., Ravichandran, C., Dhanalakshmi, S., and Murugesu, R. (2019), Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces, { Appl. Syst. Innov.}, {\bf 2}(18), 1-17.
  4. [4]  Balachandran, K., Park, D.G., and Kwun, Y.C. (1999), Nonlinear integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces, { Commun. Korean Math. Soc.}, {\bf 14}, 223-231.
  5. [5]  Hern\{a}ndez, E. (2002), Existence results for partial neutral integrodifferential equations with nonlocal conditions, { Dynam. Syst. Appl.}, {\bf 11}(2), 241-252.
  6. [6]  Hern\{a}ndez, E. and Mckibben, M. (2005), Some comments on: ``Existence of solutions of abstarct nonlinear second-order neutral functional integrodifferential equations", { Comput. Math. Appl.}, {\bf 50}, 655-669.
  7. [7]  Kavitha, V., Arjunan, M.M., and Ravichandran, C. (2012), Existence Results for a Second Order Impulsive Neutral Functional Integrodifferential Inclusions in Banach Spaces with Infinite Delay, { J. Nonlinear Sci. Appl}, {\bf 5}, 321-333.
  8. [8]  Kavitha, V. Arjunan, M.M., and Ravichandran, C. (2011), Existence results for impulsive systems with nonlocal conditions in Banach spaces, { J. Nonlinear Sci. Appl}, {\bf 4}(2), 138-151.
  9. [9]  Machado, J.A., Ravichandran, C., Rivero, M., and Trujillo, J.J. (2013), Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, { Fixed Point Theo. Appl}, {\bf 2013}(66), 1-16.
  10. [10]  Mahmudov, N.I., Murugesu, R., Ravichandran, C., and Vijayakumar, V. (2017), Approximate controllability results for fractional semilinear integro-differential inclusions in Hilbert spaces, { Results in Mathematics}, {\bf 71} , 45-61.
  11. [11]  Vijayakumar, V. (2018), Approximate controllability results for impulsive neutral differential inclusions of Sobolev-type with infinite delay, { International Journal of Control}, {\bf 91}(10), 2366-2386.
  12. [12]  Vijayakumar, V., Ravichandran, C., Murugesu, R., and Trujillo, J.J. (2014), Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators, { Applied Math. Comp.}, {\bf 247}, 152-161.
  13. [13]  Vijayakumar, V. (2018) Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces, { IMA J. Math. control Inf.}, {\bf 35}, 297-314.
  14. [14]  Vijayakumar, V., Udhayakumar, R., and Dineshkumar, C. (2020), Approximate controllability of second order nonlocal neutral differential evolution inclusions, { IMA J. Math. control Inf.}, {\bf 00}, 1-19, doi:10.1093/imamici/dnaa001.
  15. [15]  Yan, Z. and Jia, X. (2016), Approximate controllability of impulsive fractional stochastic partial integro-differential inclusions with infinte delay, { IMA J. Math. control Inf.}, {\bf 1-42}, 1590-1639.
  16. [16]  Samoilenko, A.M. and Perestyuk, N.A. (1995), { Impulsive Differential Equations}, World Scientific, Singapore.
  17. [17]  B. Yan, Boundary value problems on the half-line with impulses and infinite delay, (2001), { Journal of Mathematical Analysis and Appli}, {\bf 259}(1), 94-114.
  18. [18]  Benchohra, M., Henderson, J., and Ntouyas, S.K. (2006), { Impulsive Differential Equations and Inclusions, in: Contemporary Mathematics and its Applications}, Vol. 2, Hindawi Publishing Corporation, New York.
  19. [19]  Sivasankaran, S., Mallika Arjunan, M., and Vijayakumar, V. (2011), Existence of global solutions for second order impulsive abstract partial differential equations, { Nonlinear Anal. TMA}, {\bf 74}(17), 6747-6757.
  20. [20]  Vijayakumar, V. and Henr\{i}quez, H.R. (2018), Existence of global solutions for a class of abstract second order nonlocal Cauchy problem with impulsive conditions in Banach spaces, { Numerical Functional Analysis and Optimization}, {\bf 39}(6), 704-736.
  21. [21]  Chang, Y.K. (2007), Controllability of impulsive functional differential systems with infinite delay in Banach spaces, { Chaos Solitons $\&$ Fractals}, {\bf 33}, 1601-1609.
  22. [22]  Chang, Y.K., Anguraj, A., and Mallika Arjunan, M. (2009), Controllability of impulsive neutral functional differential inclusions with infinite delay in Banach spaces, { Chaos Solitons $\&$ Fractals}, {\bf 39}(4), 1864-1876.
  23. [23]  Byszewski, L. (1991), Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, { J. Math. Anal. Appl.}, { \bf 162}(2), 494-505.
  24. [24]  Byszewski, L. and Lakshmikantham, V. (1990), Theorem about existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, { Appl. Anal.}, {\bf 40}, 11-19.
  25. [25]  Cuevas, C., Hern\{a}ndez, E., and Rabelo, M. (2009), The existence of solutions for impulsive neutral functional differential equations, { Comput. Math. Appl.}, {\bf 58}, 744-757.
  26. [26]  Hern\{a}ndez, E. and Tanaka Aki, S.M. (2009), Global solutions for abstract functional differential equations with nonlocal conditions, { Elect. J. Quali. Theo. Diff. Equ.}, {\bf 50}, 1-8.
  27. [27]  Hern\{a}ndez, E., Tanaka Aki, S.M., and Henr\{\i}quez, H.R. (2008), Global solutions for abstract impulsive partial differential equations, { Comput. Math. Appl.}, {\bf 56}, 1206-1215.
  28. [28]  Hern\{a}ndez, E. and Henr\{i}quez, H.R. (2004), Global solutions for a functional second order abstract Cauchy problem with nonlocal conditions, { Annales Polonici Mathematici.}, {\bf 83}, 149-170.
  29. [29]  Sivasankaran, S., Mallika Arjunan, M., and Vijayakumar, V. (2011), Existence of global solutions for impulsive functional differential equations with nonlocal conditions, { J. Nonlinear Sci. Appl.}, {\bf 4}(2), 102-114.
  30. [30]  Sivasankaran, S., Vijayakumar, V., and Mallika Arjunan, M. (2011), Existence of global solutions for impulsive abstract partial neutral functional differential equations, { Int. J. Nonlinear Sci.} {\bf 11}(4), 412-426.
  31. [31]  Vijayakumar, V., Sivasankaran, S., and Mallika Arjunan, M. (2011), Existence of global solutions for second order impulsive abstract functional integrodifferential equations, { Dyn. Contin. Discrete Impuls. Syst.}, {\bf 18}, 747-766.
  32. [32]  Pazy, A. (1983), { Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences}, {\bf44}, Springer-Verlag, New York-Berlin.
  33. [33]  Hino, Y., Murakami, S., and Naito, T. (1991), { In Functional-differential equations with infinite delay, Lecture notes in Mathematics}, {\bf 1473}, Springer-Verlog, Berlin.
  34. [34]  Granas, A. and Dugundji, J. (2003), { Fixed Point Theory}, Springer-Verlag, New York.
  35. [35]  Martin, R.H. (1987), { Nonlinear Operators and Differential Equations in Banach Spaces}, Robert E. Krieger Publ. Co., Florida.
  36. [36]  Rogovchenko, Y.V. (1997), Impulsive evolution systems: Main results and new trends, { Dynam. Contin. Discrete Impuls. Syst.}, {\bf3}(1), 57-88.
  37. [37]  Rogovchenko,Y.V. (1997), Nonlinear impulsive evolution systems and application to population models,\ { J. Math. Anal. Appl.}, {\bf207}(2), 300-315.
  38. [38]  Balachandran, K., Park, J.Y., and Chandrasekaran, M. (2002), Nonlocal Cauchy problem for delay integrodifferential equations of Sobolve type in Banach spaces, { Appl. Math. Lett.}, {\bf 15}(7), 845-854.
  39. [39]  Ezzinbi, K., Fu, X., and Hilal, K. (2007), Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions, { Nonlinear Anal.}, 67, 1613-1622.
  40. [40]  Fu, X. and Ezzinbi, K. (2003), Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, { Nonlinear Anal.}, {\bf 4}, 215-227.
  41. [41]  Fu, X. (2004), On solutions of neutral nonlocal evolution equations with nondense domain, { J. Math. Anal. Appl.}, {\bf 299}, 392-410.

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