Nonlinear Boundary Value Problem of First-Order Impulsive Functional Differential Equations
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Research Article Nonlinear Boundary Value Problem of First-Order Impulsive Functional Differential Equations Kexue Zhang1 and Xinzhi Liu2 1 2
School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1
Correspondence should be addressed to Xinzhi Liu, [email protected] Received 8 December 2009; Accepted 30 January 2010 Academic Editor: Shusen Ding Copyright q 2010 K. Zhang and X. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates the nonlinear boundary value problem for a class of first-order impulsive functional differential equations. By establishing a comparison result and utilizing the method of upper and lower solutions, some criteria on the existence of extremal solutions as well as the unique solution are obtained. Examples are discussed to illustrate the validity of the obtained results.
1. Introduction It is now realized that the theory of impulsive differential equations provides a general framework for mathematical modelling of many real world phenomena. In particular, it serves as an adequate mathematical tool for studying evolution processes that are subjected to abrupt changes in their states. Some typical physical systems that exhibit impulsive behaviour include the action of a pendulum clock, mechanical systems subject to impacts, the maintenance of a species through periodic stocking or harvesting, the thrust impulse maneuver of a spacecraft, and the function of the heart. For an introduction to the theory of impulsive differential equations, refer to 1. It is also known that the method of upper and lower solutions coupled with the monotone iterative technique is a powerful tool for obtaining existence results of nonlinear differential equations 2. There are numerous papers devoted to the applications of this method to nonlinear differential equations in the literature, see 3–9 and references therein. The existence of extremal solutions of impulsive differential equations is considered in papers 3–11. However, only a few papers have implemented the technique in nonlinear boundary value problem of impulsive differential equations 5, 12. In this paper, we will investigate
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Journal of Inequalities and Applications
nonlinear boundary value problem of a class of first-order impulsive functional differential equations. Such equations include the retarded impulsive differential equations as special cases 5, 12–14. The rest of this paper is organized as follows. In Section 2, we establish a new comparison principle and discuss the existence and uniqueness of the solution for first order impulsive functional differential equations with linear boundary condition. We then obtain existence results for extremal solutions and unique solution in Section 3 by using the method of upper and
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