International Journal of Mathematics Trends and Technology
Volume 65 | Issue 11 | Year 2019 | Article Id. IJMTT-V65I11P501 | DOI : https://doi.org/10.14445/22315373/IJMTT-V65I11P501
This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii's fixed point theorem, we get the controllability of nonlocal impulsive differential equation under compactness conditions, Lipschitz conditions and mixed-type conditions respectively.
[1] L.Byszewski; Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494-505.
[2] W.Fitzgibbon; Semilinear integrodifferential equations in Banach spaces, Nonlinear Anal.: Theory, Methods Appl. 4 (1980), 745-760.
[3] V.Lakshmikantham, D.D. Bainov, P.S.D. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[4] M.Benchohra, J.Henderson, S.K.Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing, New York, 2006.
[5] D.D.Bainov, P.S.Simeonov Impulsive differential equations: Periodic solutions and applications, Longman scientific and technical group, England, 1993.
[6] J.Banas. K.Goebel, Measure of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., Marcel Dekker, New York, vol. 60, 1980.
[7] M.Guo, X.Xue,R.Li, Controllability of impulsive evolution inclusions with nonlocal conditions, J. Optim. Theory Appl., 120, 355-374, 2004.
[8] E.Hernandez, M.Rabelo, H.R.Henrquez, Existence of solutions for impulsive partial neutral functional differential equations, J. Math. Anal. Appl., 331, 1135-1158, 2007.
[9] Bashir Ahamed, K.Malar, K.Karthikeyan, A study of nonlocal problems of impulsive integro differential equations with measure of noncompactness, Advances in differential equations, 2013.
[10] Shaochun Ji, Gang Li, A unified approach to nonlocal impulsice differential equations with measure of noncompactness, Advances in differential equations, 2012.
[11] Y.K.Chang, Controllability of impulsive functional differential systems with infinite delay in Banach spaces. Chaos, Solitons and Fractals, 33, 1601-1609, 2007.
[12] B.Liu, Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Analysis, 60, 1533-1552, 2005.
[13] S.Li, G.Li, M.Wang, Controllability of impulsive differential systems with nonlocal conditions, Applied Mathematics and Computation, 217, 6981-6989, 2011.
[14] D.N.Chalishajar, Controllability of impulsive partial neutral functional differential equations with infinite delay, Int. Journal of Math. Analysis, Vol.5, 8, 369 - 380, 2011.
[15] D.N.Chalishajar, F.S.Acharya, Controllability of neutral impulsive differential inclusions with non-local conditions, Applied Mathematics, 2, 1486 - 1496, 2011.
[16] S.Selvi, M.Mallika Arjunan, Controllability results for impulsive differential systems with infinite delay, Applied Mathematics, 42, 6187 - 6192, 2012.
[17] S.Selvi, M.Mallika Arjunan, Controllability results for impulsive differential systems with finite delay, J. Nonlinear Sci. Appl., 5, 206 - 219, 2012.
[18] Lizhen Chen, Gang Li, Approximate Controllability of impulsive differential equations with nonlocal conditions, Int. Journal of Nonlinear Science, Vol.10, 4, 438 - 446, 2010.
[19] S.Ji. S.Wen, Nonlocal Cauchy problem for impulsive differential equations in Banach spaces, Int. J. Nonlinear Sci., 10(1), 88-95, 2010.
[20] S.Ji, G.Li, Existence results for impulsive differential inclusions with nonlocal conditions, Comput. Math. Appl., 62, 1908-1915, 2011.
[21] J.Li, J.J.Nieto, J.Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal.Appl., 325, 226-236, 2007.
[22] M.Benchohra, J.Henderson, S.K.Ntouyas, An existence result for first-order impulsive func-tional differential equations in Banach spaces, Comput. Math. Appl., 42. 1303-1310, 2001.
[23] N.Abada, M.Benchohra, H.Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differ. Equ., 246, 3834-3863, 2009.
[24] X.Xue, Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces, Nonlinear Anal., 70, 2593-2601, 2009.
[25] Q.Dong, G.Li, Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces, Electron. J. Qual. Theory Differ. Equ., 47, 2009.
[26] R.P.Agarwal, M.Benchohra, D.Seba, On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math., 55, 221-230, 2009.
[27] J.Liang, J.H.Liu, T.J.Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math.Comput. Model., 49, 798-804, 2009.
[28] L.Zhu, Q.Dong,G.Li, Impulsive differential equations with nonlocal conditions in general Banach spaces, Adv. Differ.Equ., 10, 2012.
[29] J.Sun, X.Zhang, The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations, Acta Math. Sinica (Chin. Ser.), 48, 439-446, 2005.
[30] V.Obukhovski, P.Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Anal., 70, 3424-3436, 2009. (1, 2.3, 2.4)
[31] L.S.Liu, F.Guo, C.X.Wu, Y.H.Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl., 309, 638-649, 2005.
K. Karthikeyan, D. Rameshkumar, "Controllability of Nonlocal Impulsive Differential Equations With Measure of Noncompactness," International Journal of Mathematics Trends and Technology (IJMTT), vol. 65, no. 11, pp. 1-16, 2019. Crossref, https://doi.org/10.14445/22315373/IJMTT-V65I11P501
PDF 
Abstract 
Keywords 
References 
Citation