×

Nonlinear boundary value problem of first-order impulsive functional differential equations. (English) Zbl 1203.34101

The authors investigate a nonlinear boundary value problem for a class of first-order impulsive functional differential equations. They establish a new comparison principle and discuss the existence and uniqueness of the solution for first order impulsive functional differential equations with linear boundary conditions. In the next section there are obtained existence results for extremal solutions and unique solution using the method of upper and lower solution and the monotone iterative technique. In the last section there are given two examples to illustrate the abstract results.

MSC:

34K10 Boundary value problems for functional-differential equations
34K45 Functional-differential equations with impulses
34A45 Theoretical approximation of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Lakshmikanthan V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
[2] Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275. · Zbl 0661.47045
[3] Franco D, Liz E, Nieto JJ, Rogovchenko YV: A contribution to the study of functional differential equations with impulses. Mathematische Nachrichten 2000, 218: 49-60. 10.1002/1522-2616(200010)218:1<49::AID-MANA49>3.0.CO;2-6 · Zbl 0966.34073
[4] Nieto JJ, Rodríguez-López R: Boundary value problems for a class of impulsive functional equations. Computers & Mathematics with Applications 2008, 55(12):2715-2731. 10.1016/j.camwa.2007.10.019 · Zbl 1142.34362
[5] Chen L, Sun J: Nonlinear boundary value problem of first order impulsive functional differential equations. Journal of Mathematical Analysis and Applications 2006, 318(2):726-741. 10.1016/j.jmaa.2005.08.012 · Zbl 1102.34052
[6] Li L, Shen J: Periodic boundary value problems for functional differential equations with impulses. Mathematica Scientia 2005, 25A: 237-244. · Zbl 1081.34528
[7] Liu X, Guo D: Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces. Journal of Mathematical Analysis and Applications 1997, 216(1):284-302. 10.1006/jmaa.1997.5688 · Zbl 0889.45016
[8] Guo D, Liu X: Periodic boundary value problems for impulsive integro-differential equations in Banach spaces. Nonlinear World 1996, 3(3):427-441. · Zbl 0902.34018
[9] Liu X, Guo D: Initial value problems for first order impulsive integro-differential equations in Banach spaces. Communications on Applied Nonlinear Analysis 1995, 2(1):65-83. · Zbl 0858.34068
[10] Guo D, Liu X: First order impulsive integro-differential equations on unbounded domain in a Banach space. Dynamics of Continuous, Discrete and Impulsive Systems 1996, 2(3):381-394. · Zbl 0873.45008
[11] Franco D, Nieto JJ: First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2000, 42(2):163-173. 10.1016/S0362-546X(98)00337-X · Zbl 0966.34025
[12] Yang X, Shen J: Nonlinear boundary value problems for first order impulsive functional differential equations. Applied Mathematics and Computation 2007, 189(2):1943-1952. 10.1016/j.amc.2006.12.085 · Zbl 1125.65074
[13] Ding W, Mi J, Han M: Periodic boundary value problems for the first order impulsive functional differential equations. Applied Mathematics and Computation 2005, 165(2):433-446. 10.1016/j.amc.2004.06.022 · Zbl 1081.34081
[14] Luo Z, Jing Z: Periodic boundary value problem for first-order impulsive functional differential equations. Computers & Mathematics with Applications 2008, 55(9):2094-2107. · Zbl 1144.34044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.