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)
Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators. (English) Zbl 1200.34095
The authors consider a class of first order impulsive neutral differential equations as well as a class of first order impulsive neutral integrodifferential equations, both governed by nonlinear perturbations of a densely defined and closed linear operator generating an analytic compact semigroup, and subjected to nonlocal initial conditions. By imposing some technical conditions on the data and using Sadovskii’s fixed point theorem, they prove some existence results which cover the case when the nonlinear part is defined on a fractional power of the linear part. An illustrative example is included.

MSC:
34K45Functional-differential equations with impulses
34K40Neutral functional-differential equations
34K30Functional-differential equations in abstract spaces
47D06One-parameter semigroups and linear evolution equations
47N20Applications of operator theory to differential and integral equations
WorldCat.org
Full Text: DOI
References:
[1] Benchohra, M.; Henderson, J.; Ntouyas, S. K.: Impulsive differential equations and inclusions, (2006) · Zbl 1130.34003
[2] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[3] Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations, (1995) · Zbl 0837.34003
[4] Hernández, E.; Henriquez, H. R.: Impulsive partial neutral differential equations, Appl. math. Lett. 19, No. 3, 215-222 (2006) · Zbl 1103.34068 · doi:10.1016/j.aml.2005.04.005
[5] Hernández, E.; Henriquez, H. R.; Marco, R.: Existence of solutions for a class of impulsive partial neutral functional differential equations, J. math. Anal. appl. 331, No. 2, 1135-1158 (2007) · Zbl 1123.34062 · doi:10.1016/j.jmaa.2006.09.043
[6] Liu, J. H.: Nonlinear impulsive evolution equations, Dynam. contin. Discrete impuls. Syst. 6, No. 1, 77-85 (1999) · Zbl 0932.34067
[7] Rogovchenko, Y. V.: Impulsive evolution systems: Main results and new trends, Dynam. contin. Discrete impuls. Syst. 3, No. 1, 57-88 (1997) · Zbl 0879.34014
[8] Rogovchenko, Y. V.: Nonlinear impulse evolution systems and applications to population models, J. math. Anal. appl. 207, No. 2, 300-315 (1997) · Zbl 0876.34011 · doi:10.1006/jmaa.1997.5245
[9] Chang, J. C.; Liu, H.: Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the a-norm, Nonlinear anal. 71, 3759-3768 (2009) · Zbl 1185.34112 · doi:10.1016/j.na.2009.02.035
[10] Chang, Y. K.; Anguraj, A.; Karthikeyan, K.: Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Nonlinear anal. (2009) · Zbl 1178.34071
[11] Chang, Y. K.; Nieto, J. J.: Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Numer. funct. Anal. optim. 30, 227-244 (2009) · Zbl 1176.34096 · doi:10.1080/01630560902841146 · http://www.informaworld.com/smpp/./content~db=all~content=a910367252
[12] 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, 1613-1622 (2007) · Zbl 1119.35105 · doi:10.1016/j.na.2006.08.003
[13] Fu, X.; Ezzinbi, K.: Existence of solutions for neutral functional differential evolutions equations with nonlocal conditions, Nonlinear anal. 54, 215-227 (2003) · Zbl 1034.34096 · doi:10.1016/S0362-546X(03)00047-6
[14] Fu, X.; Cao, Y.: Existence for neutral impulsive differential inclusions with nonlocal conditions, Nonlinear anal. 68, 3707-3718 (2008) · Zbl 1156.34063 · doi:10.1016/j.na.2007.04.013
[15] Byszewski, L.: Theorems about the existence and uniqueness of a solution of a semilinear evolution nonlocal Cauchy problem, J. math. Anal. appl. 162, 496-505 (1991) · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U
[16] Deng, K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. math. Anal. appl. 179, 630-637 (1993) · Zbl 0798.35076 · doi:10.1006/jmaa.1993.1373
[17] Byszewski, L.; Lakshmikantham, V.: Theorems about existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. anal. 40, 11-19 (1990) · Zbl 0694.34001 · doi:10.1080/00036819008839989
[18] Akca, H.; Boucherif, A.; Covachev, V.: Impulsive functional differential equations with nonlocal conditions, Int. J. Math. math. Sci. 29, No. 5, 251-256 (2002) · Zbl 1005.34072 · doi:10.1155/S0161171202012887 · http://www.hindawi.com/journals/ijmms/volume-29/S0161171202012887.html
[19] Anguraj, A.; Karthikeyan, K.: Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear anal. 70, No. 7, 2717-2721 (2009) · Zbl 1165.34416 · doi:10.1016/j.na.2008.03.059
[20] Chang, Y. -K.; Anguraj, A.; Arjunan, M. Mallika: Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions, J. appl. Math. comput. 28, No. 1, 79-91 (2008) · Zbl 1160.34072 · doi:10.1007/s12190-008-0078-8
[21] Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983) · Zbl 0516.47023
[22] Sadovskii, B. N.: On a fixed point principle, Funct. anal. Appl. 1, No. 2, 74-76 (1967) · Zbl 0165.49102
[23] Travis, C. C.; Webb, G. F.: Existence, stability and compactness with ${\alpha}$-norm for partial functional differential equations, Trans. amer. Math. soc. 240, 129-143 (1978) · Zbl 0414.34080 · doi:10.2307/1998809
[24] Yosida, K.: Functional analysis, (1971) · Zbl 0217.16001