Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. (English) Zbl 1173.34048

Liang, Jin; Liu, James H.; Xiao, Ti-Jun

Math. Comput. Modelling 49, No. 3-4, 798-804 (2009).

Summary: We study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive differential equation with nonlocal conditions
\[ \begin{cases} u'(t)=Au(t)+f(t,u(t)),\quad 0\leq t\leq K,\;t\neq t_i,\\ u(0)+g(u)=u_0,\\ \Delta u(t_i)=I_i(u(t_i)),\quad i=1,2,\dots,p,\;0<t_1<t_2<\cdots < t_p<K.\end{cases} \]
by combining and extending some earlier work on equations with nonlocal conditions and equations with impulsive conditions. Here, \(A\) is the generator of a strongly continuous semigroup in a Banach space, \(g\) constitutes a nonlocal condition, and \(\Delta u(t^+_i)-u(t^-_i)\) constitutes an impulsive condition. New results are obtained.

MSC:

34K30	Functional-differential equations in abstract spaces
34K45	Functional-differential equations with impulses

Keywords:

nonlocal and impulsive conditions

References:

[1]	Ahmed, N. U., Optimal feedback control for impulsive systems on the space of finitely additive measures, Publ. Math. Debrecen, 70, 371-393 (2007) · Zbl 1164.34026
[2]	Aizicovici, S.; Lee, Haewon, Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. Math. Lett., 18, 401-407 (2005) · Zbl 1084.34002
[3]	Byszewski, L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162, 494-505 (1991) · Zbl 0748.34040
[4]	Byszewski, L., Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems, Dynam. Systems Appl., 5, 595-605 (1996) · Zbl 0869.47034
[5]	Benchohra, M.; Henderson, J.; Ntouyas, S., (Impulsive Differential Equations and Inclusions. Impulsive Differential Equations and Inclusions, Contemporary Mathematics and its Applications, vol. 2 (2006), Hindawi Publ. Corp.) · Zbl 1130.34003
[6]	Byszewski, L.; Lakshmikantham, V., Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Applicable Anal., 40, 11-19 (1990) · Zbl 0694.34001
[7]	Boucherif, A.; Precup, R., On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 4, 2, 205-212 (2003) · Zbl 1050.34001
[8]	Ezzinbi, K.; Fu, X.; Hilal, K., Existence and regularity in the \(\alpha \)-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal., 67, 5, 1613-1622 (2007) · Zbl 1119.35105
[9]	Guo, D.; Liu, X., Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces, J. Math. Anal. Appl., 177, 538-552 (1993) · Zbl 0787.45008
[10]	Henriquez, H. R.; Hernandez, E.; Akca, H., Global solutions for an abstract Cauchy problem with nonlocal conditions, Internat. J. Math. Manuscripts, 1 (2007)
[11]	Jackson, D., Existence and uniqueness of solutions of a semilinear nonlocal parabolic equations, J. Math. Anal. Appl., 172, 256-265 (1993) · Zbl 0814.35060
[12]	Liu, J. H., Nonlinear impulsive evolution equations, Dynam. Contin. Discrete Impuls. Sys., 6, 77-85 (1999) · Zbl 0932.34067
[13]	Liang, J.; van Casteren, J.; Xiao, T. J., Nonlocal Cauchy problems for semilinear evolution equations, Nonlinear Anal., 50, 173-189 (2002) · Zbl 1009.34052
[14]	Lin, Y.; Liu, J. H., Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal., 26, 1023-1033 (1996) · Zbl 0916.45014
[15]	Liang, J.; Liu, J. H.; Xiao, T. J., Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal., 57, 183-189 (2004) · Zbl 1083.34045
[16]	Liang, J.; Xiao, T. J., Semilinear integrodifferential equations with nonlocal initial conditions, Comput. Math. Appl., 47, 863-875 (2004) · Zbl 1068.45014
[17]	Liu, X.; Willms, A., Stability analysis and applications to large scale impulsive systems: A new approach, Canad. Appl. Math. Quart., 3, 419-444 (1995) · Zbl 0849.34044
[18]	Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems (1995), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0816.35001
[19]	N’Guérékata, Gaston M., Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions, (Differential & Difference Equations and Applications (2006), Hindawi Publ. Corp.: Hindawi Publ. Corp. New York), 843-849 · Zbl 1147.35329
[20]	Rogovchenko, Y., Impulsive evolution systems: Main results and new trends, Dynam. Contin. Discrete Impuls. Sys., 3, 57-88 (1997) · Zbl 0879.34014
[21]	Zavalishchin, A., Impulse dynamic systems and applications to mathematical economics, Dynam. Systems Appl., 3, 443-449 (1994) · Zbl 0805.34009