Existence of mild solutions of second-order impulsive differential equations in Banach spaces

Adel Jawahdou

doi:10.26637/mjm1102/001

Abstract

We discuss the existence of solutions for second-order impulsive differential equation with nonlocal conditions in Banach spaces. Our approach is based on the generalization of Schauder fixed point
principle that is Darbo fixed point theorem. An example is also presented to for illustration.

Keywords:

Second order differential equations, mild solution, impulse, nonlocal condition, Kuratowski measures of noncompactness.

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Adel Jawahdou Department of Mathematics, Bizerte Preparatory Engineering Institute, Tunisia. https://orcid.org/0000-0002-3551-1236

Pages: 117-126
Date Published: 01-04-2023
Vol. 11 No. 02 (2023): Malaya Journal of Matematik (MJM)

J. Bana` s, K. Goebel, Measures of Noncompactness in Banach Spaces, In Lecture Notes in Pure and pplied Mathematics, Volume 60, Marcel Dekker, New York, 1980.

D. D. Bainov, P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, in Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 66, Longman Scientific & Technical, Harlow, 1993, copublished in the United States with John Wiley & Sons, Inc., New York.

M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclutions, Contemp. Math. Appl., vol. 2, Hindawi Publ. Corp., 2006, https://doi.org/10.1155/9789775945501. DOI: https://doi.org/10.1155/9789775945501

Y. Peng, and X. Xiang, Necessary conditions of optimality for second-order nonlinear impulsive integro-

differential equations on Banach spaces, Nonlinear Anal. Real World Appl., 11(2010), 3121–3130,

https://doi.org/10.1016/j.nonrwa.2009.11.007. DOI: https://doi.org/10.1016/j.nonrwa.2009.11.007

P. Chen, and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63(2013), 731–744, https://doi.org/10.1007/s00025-012-0230-5. DOI: https://doi.org/10.1007/s00025-012-0230-5

E.M. Hernandez, H.R. Henriquez, and M.A. McKibben, Existence results for abstract impulsive second order neutral functional differential equations, Nonlinear Anal., 70(2009), 2736–2751, https://doi.org/10.1016/j.na.2008.03.062. DOI: https://doi.org/10.1016/j.na.2008.03.062

V. Colao, L. Mugliam, and H. Xu, Existence of solutions for a second-order differential equation with non-instantaneous impulses and delay, Annali di Matematica, 195(2016), 697–716, https://doi.org/10.1007/s10231-015-0484-0. DOI: https://doi.org/10.1007/s10231-015-0484-0

K. Deimling; Nonlinear Functional Analysis, Springer-Verlag, New York, 1985, https://doi.org/10.1007/978-3-662- 00547-7. DOI: https://doi.org/10.1007/978-3-662-00547-7

Z. Fan, and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct.Anal., 258(2010), 1709–1727, https://doi.org/10.1016/j.jfa.2009.10.023. DOI: https://doi.org/10.1016/j.jfa.2009.10.023

H. Akca, V. Covachev, and Z. Covacheva, Existence theorem for a second-order impulsive functional differential equation with a nonlocal condition, J. Nonlinear Convex Anal., 17(2016), 1129–1136.

D. Guo, Existence of positive solutions for n th-order nonlinear impulsive singular integro-differential equations in Banach spaces, Nonlinear Anal., 68(2008), 2727–2740, https://doi.org/10.1016/j.na.2007.02.019. DOI: https://doi.org/10.1016/j.na.2007.02.019

D. Guo, Existence of solutions of Boundary value problems for nonlinear Second order impulsive Differential Equations in Banach spaces, J. Math. Anal. Appl., 181(1994), 407–421, https://doi.org/10.1006/jmaa.1994.1031. DOI: https://doi.org/10.1006/jmaa.1994.1031

H.P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7(1983), 1351–1371, https://doi.org/10.1016/0362-546X(83)90006-8. DOI: https://doi.org/10.1016/0362-546X(83)90006-8

E. Hernández, and D. O’Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc.,141(2013), 1641–1649, https://doi.org/10.1090/S0002-9939-2012-11613-2. DOI: https://doi.org/10.1090/S0002-9939-2012-11613-2

V. Lakshmikantham, D. D. Bainov, P. S. Simeonov; Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989, https://doi.org/10.1142/0906. DOI: https://doi.org/10.1142/0906

Y. Li, Existence of solutions of initial value problems for abstract semilinear evolution equations, Acta Math. Sin., 48(2005), 1089–1094 (in Chinese).

Y. Li, and Z. Liu, Monotone iterative technique for addressing impulsive integro-differential equtions in Banach spaces, Nonlinear Anal., 66(2007), 83–92, https://doi.org/10.1016/j.na.2005.11.013. DOI: https://doi.org/10.1016/j.na.2005.11.013

J. Liang, J. H. Liu, and T. J. Xiao, Nonlocal impulsive problems for integrodifferential equations, Math. Comput. Modelling, 49(2009), 798–804, https://doi.org/10.1016/j.mcm.2008.05.046. DOI: https://doi.org/10.1016/j.mcm.2008.05.046

J.Liang, J.H.Liu, andT.J.Xiao, Periodicsolutionsofdelayimpulsivedifferentialequations, NonlinearAnal., 74(2011), 6835–6842, https://doi.org/10.1016/j.na.2011.07.008. DOI: https://doi.org/10.1016/j.na.2011.07.008

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-verlag, Berlin, 1983, https://doi.org/10.1007/978-1-4612-5561-1. DOI: https://doi.org/10.1007/978-1-4612-5561-1

M. Pierri, D. O’Regan, and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 219(2013), 6743–6749, https://doi.org/10.1016/j.amc.2012.12.084. DOI: https://doi.org/10.1016/j.amc.2012.12.084

A. M. Samoilenko, and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995, https://doi.org/10.1142/2892. DOI: https://doi.org/10.1142/2892

J. Wang, and X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput., 46(2014), 321–334, https://doi.org/10.1007/s12190-013-0751-4. DOI: https://doi.org/10.1007/s12190-013-0751-4

J. Wang, Y. Zhou, and Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242(2014), 649–657, https://doi.org/10.1016/j.amc.2014.06.002. DOI: https://doi.org/10.1016/j.amc.2014.06.002

C. C. Travis, and G. F. Webb, An abstract second order semilinear Volterra-integro-differential equation, SIAM J. Math. Anal., 10(1979), 412–424, https://doi.org/10.1137/0510038. DOI: https://doi.org/10.1137/0510038