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)
Multiplicity of solutions to two-point boundary value problems for second-order impulsive differential equations. (English) Zbl 1165.34329
The authors consider the impulsive boundary value problem $$\aligned &-u'' = f(t,u,u'), \quad t \in (0,1), t \ne t_k,\\ &\triangle u(t_k) = I_k(u(t_k)), -\triangle u'(t_k) = N_k(u(t_k)), \quad k = 1,\ldots,m,\\ &au(0) - bu'(0) = 0, \quad cu(1) + du'(1) = 0, \endaligned $$ where $0 < t_1 < \ldots < t_m < 1$, the functions $f$, $I_k$, $N_k$ are continuous. Sufficient conditions for the existence of at least three solutions are obtained. Main results are proved by using lower and upper solutions and Leray-Schauder degree theory.

MSC:
34B37Boundary value problems for ODE with impulses
47N20Applications of operator theory to differential and integral equations
34B15Nonlinear boundary value problems for ODE
WorldCat.org
Full Text: DOI
References:
[1] Bainov, D. D.; Simeonov, P. S.: Systems with impulse effect. (1989) · Zbl 0671.34052
[2] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[3] Guo, D.: Multiple solutions for nth order impulsive integro-differential equations in Banach spaces. Nonlinear. anal. 60, 955-976 (2005) · Zbl 1069.45010
[4] Liu, X.; Jiang, D.: Multiple positive solutions of dirichet boundary value problems for second order impulsive differential equations. J. math. Anal. appl. 321, 501-514 (2006) · Zbl 1103.34015
[5] He, Z.; He, X.: Monotone iterative technique for impulsive integro-differential equations with periodic boundary value problems. Comput. math. Appl. 48, 73-84 (2004) · Zbl 1070.65136
[6] Ding, W.; Han, M.; Mi, J.: Periodic boundary value problems for second-order impulsive functional equations. Comput. math. Appl. 50, 491-507 (2005) · Zbl 1095.34042
[7] Chen, L.; Sun, J.: Boundary value problem of second order impulsive functional differential equations. J. math. Anal. appl. 323, 708-720 (2006) · Zbl 1111.34047
[8] Nieto, J. J.; Rodríguez-López, R.: Remarks on periodic boundary value problem for functional differential equations. J. comput. Appl. math. 158, 339-353 (2003) · Zbl 1036.65058
[9] Liu, Z.; Li, F.: Multiple positive solutions of nonlinear two-point boundary value problems. J. math. Anal. appl. 203, 610-625 (1996) · Zbl 0878.34016
[10] Hu, L.; Liu, L.; Wu, Y.: Positive solutions of nonlinear singular two-point boundary value problem for second-order impulsive differential equations. Appl. math. Comput. 196, 550-562 (2008) · Zbl 1144.34017
[11] Henderson, J.; Thompson, H. B.: Existence of multiple solutions for second-order boundary value problems. J. differ. Equations 166, 443-454 (2000) · Zbl 1013.34017
[12] Du, Z.; Xue, C.; Ge, W.: Multiple solutions for three-point boundary value problems nonlinear terms depending on the first order derivate. Arch. math. 84, 341-349 (2005) · Zbl 1074.34011
[13] Khan, R. A.; Webb, J. R. L.: Existence of at least three solutions of a second-order three-point boundary value problem. Nonlinear anal. 64, 1356-1366 (2006) · Zbl 1101.34005
[14] Du, Z.; Liu, W.; Lin, X.: Multiple solutions to a three-point boundary value problem for higher order ordinary differential equations. J. math. Anal. appl. 335, 1207-1218 (2007) · Zbl 1133.34011
[15] Lee, Y.; Liu, X.: Study of singular for second order impulsive differential equations. J. math. Anal. appl. 331, 159-176 (2007) · Zbl 1120.34018
[16] Guo, D.: Existence of solutions of boundary value problem for second order impulsive differential equations in Banach space. J. math. Anal. appl. 181, 407-421 (1994) · Zbl 0807.34076