zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. (English) Zbl 1173.34048

Summary: We study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive differential equation with nonlocal conditions

u ' (t)=Au(t)+f(t,u(t)),0tK,tt i ,u(0)+g(u)=u 0 ,Δu(t i )=I i (u(t i )),i=1,2,,p,0<t 1 <t 2 <<t p <K·

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 Δu(t i + )-u(t i - ) constitutes an impulsive condition. New results are obtained.

MSC:
34K30Functional-differential equations in abstract spaces
34K45Functional-differential equations with impulses
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 · doi:10.1016/j.aml.2004.01.010
[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 · doi:10.1016/0022-247X(91)90164-U
[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, Contemporary mathematics and its applications 2 (2006) · 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 · doi:10.1080/00036819008839989
[7]Boucherif, A.; Precup, R.: On the nonlocal initial value problem for first order differential equations, Fixed point theory 4, No. 2, 205-212 (2003) · Zbl 1050.34001
[8]Ezzinbi, K.; Fu, X.; Hilal, K.: Existence and regularity in the α-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear anal. 67, No. 5, 1613-1622 (2007) · Zbl 1119.35105 · doi:10.1016/j.na.2006.08.003
[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 · doi:10.1006/jmaa.1993.1276
[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 · doi:10.1006/jmaa.1993.1022
[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 · doi:10.1016/S0362-546X(01)00743-X
[14]Lin, Y.; Liu, J. H.: Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear anal. 26, 1023-1033 (1996) · Zbl 0916.45014 · doi:10.1016/0362-546X(94)00141-0
[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 · doi:10.1016/j.na.2004.02.007
[16]Liang, J.; Xiao, T. J.: Semilinear integrodifferential equations with nonlocal initial conditions, Comput. math. Appl. 47, 863-875 (2004) · Zbl 1068.45014 · doi:10.1016/S0898-1221(04)90071-5
[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)
[19]N’guérékata, Gaston M.: Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions, , 843-849 (2006) · 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