Discontinuity, Nonlinearity, and Complexity
A Note on Existence of Global Solutions for Impulsive Functional Integrodifferential Systems
Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 397407  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
Download Full Text PDF
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 LeraySchauder's Alternative fixed point theorem. Lastly, a model is presented for illustration of theory.
References

[1]  Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), { Theory of Impulsive Differential Equations}, World Scientific, Singapore.


[2]  Bainov, D.D. and Simeonov, P.S. (1993), { Impulsive Differential Equations: Periodic Solutions and Applications}, Longman Scientific and Technical Group, England.


[3]  Chalishajar, D., Ravichandran, C., Dhanalakshmi, S., and Murugesu, R. (2019), Existence of Fractional Impulsive Functional IntegroDifferential Equations in Banach Spaces, { Appl. Syst. Innov.}, {\bf 2}(18), 117.


[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}, 223231.


[5]  Hern\{a}ndez, E. (2002), Existence results for partial neutral integrodifferential equations with nonlocal conditions, { Dynam. Syst. Appl.}, {\bf 11}(2), 241252.


[6]  Hern\{a}ndez, E. and Mckibben, M. (2005), Some comments on: ``Existence of solutions of abstarct nonlinear secondorder neutral functional integrodifferential equations", { Comput. Math. Appl.}, {\bf 50}, 655669.


[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}, 321333.


[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), 138151.


[9]  Machado, J.A., Ravichandran, C., Rivero, M., and Trujillo, J.J. (2013), Controllability results for impulsive mixedtype functional integrodifferential evolution equations with nonlocal conditions, { Fixed Point Theo. Appl}, {\bf 2013}(66), 116.


[10]  Mahmudov, N.I., Murugesu, R., Ravichandran, C., and Vijayakumar, V. (2017), Approximate controllability results for fractional semilinear integrodifferential inclusions in Hilbert spaces, { Results in Mathematics}, {\bf 71} , 4561.


[11]  Vijayakumar, V. (2018), Approximate controllability results for impulsive neutral differential inclusions of Sobolevtype with infinite delay, { International Journal of Control}, {\bf 91}(10), 23662386.


[12]  Vijayakumar, V., Ravichandran, C., Murugesu, R., and Trujillo, J.J. (2014), Controllability results for a class of fractional semilinear integrodifferential inclusions via resolvent operators, { Applied Math. Comp.}, {\bf 247}, 152161.


[13]  Vijayakumar, V. (2018) Approximate controllability results for abstract neutral integrodifferential inclusions with infinite delay in Hilbert spaces, { IMA J. Math. control Inf.}, {\bf 35}, 297314.


[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}, 119, doi:10.1093/imamici/dnaa001.


[15]  Yan, Z. and Jia, X. (2016), Approximate controllability
of impulsive fractional stochastic partial integrodifferential
inclusions with infinte delay, { IMA J. Math. control Inf.},
{\bf 142}, 15901639.


[16]  Samoilenko, A.M. and Perestyuk, N.A. (1995), { Impulsive Differential Equations}, World Scientific, Singapore.


[17]  B. Yan, Boundary value problems on the halfline with impulses and infinite delay, (2001), { Journal of Mathematical Analysis and Appli}, {\bf 259}(1), 94114.


[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]  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), 67476757.


[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), 704736.


[21]  Chang, Y.K. (2007), Controllability of impulsive functional differential systems with infinite delay in Banach spaces, { Chaos Solitons $\&$ Fractals}, {\bf 33}, 16011609.


[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), 18641876.


[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), 494505.


[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}, 1119.


[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}, 744757.


[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}, 18.


[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}, 12061215.


[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}, 149170.


[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), 102114.


[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), 412426.


[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}, 747766.


[32]  Pazy, A. (1983), { Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences}, {\bf44}, SpringerVerlag, New YorkBerlin.


[33]  Hino, Y., Murakami, S., and Naito, T. (1991), { In Functionaldifferential equations with infinite delay, Lecture notes in Mathematics}, {\bf 1473}, SpringerVerlog, Berlin.


[34]  Granas, A. and Dugundji, J. (2003), { Fixed Point Theory}, SpringerVerlag, New York.


[35]  Martin, R.H. (1987), { Nonlinear Operators and Differential Equations in Banach Spaces}, Robert E. Krieger Publ. Co., Florida.


[36]  Rogovchenko, Y.V. (1997), Impulsive evolution systems: Main results and new trends, { Dynam. Contin. Discrete Impuls. Syst.}, {\bf3}(1), 5788.


[37]  Rogovchenko,Y.V. (1997), Nonlinear impulsive evolution systems and application to population models,\ { J. Math. Anal. Appl.}, {\bf207}(2), 300315.


[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), 845854.


[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, 16131622.


[40]  Fu, X. and Ezzinbi, K. (2003), Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, { Nonlinear Anal.}, {\bf 4}, 215227.


[41]  Fu, X. (2004), On solutions of neutral nonlocal evolution equations with nondense domain, { J. Math. Anal. Appl.}, {\bf 299}, 392410.
