- Research Article
- Open Access
Periodic Boundary Value Problems for Second-Order Functional Differential Equations
© X. Fu and W. Wang. 2010
- Received: 11 June 2009
- Accepted: 28 January 2010
- Published: 8 March 2010
This note considers a periodic boundary value problem for a second-order functional differential equation. We extend the concept of lower and upper solutions and obtain the existence of extreme solutions by using upper and lower solution method.
- Boundary Condition
- Differential Equation
- Banach Space
- Integral Equation
- Unique Solution
Upper and lower solution method plays an important role in studying boundary value problems for nonlinear differential equations; see  and the references therein. Recently, many authors are devoted to extend its applications to boundary value problems of functional differential equations [2–5]. Suppose is one upper solution or lower solution of periodic boundary value problems for second-order differential equation; the condition is required. A neutral problem is that whether we can define upper and lower solution without assuming . The aim of the present paper is to discuss the following second order functional differential equation
In this paper, we extended the concept of lower and upper solutions for (1.1). By using the method of upper and lower solutions and monotone iterative technique, we obtained the existence of extreme solutions for the boundary value problem (1.1).
We now present the main results of this section.
The proof of Lemma 2.3 is similar to that of Lemma 2.2, here we omit it.
In this section, we consider the boundary value problem
Finally, we show that (3.1) has a solution by several steps.
It is easy to check that (3.6) is equivalent to the integral equation
Let the following conditions hold.
We shall show that
This work is supported by Scientific Research Fund of Hunan Provincial Education Department (09B033) and the NNSF of China (10871062).
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