Monotone Positive Solution of Nonlinear Third-Order BVP with Integral Boundary Conditions
© The Author(s) Jian-Ping Sun and Hai-Bao Li. 2010
Received: 7 September 2010
Accepted: 31 October 2010
Published: 2 November 2010
This paper is concerned with the following third-order boundary value problem with integral boundary conditions , where and . By using the Guo-Krasnoselskii fixed-point theorem, some sufficient conditions are obtained for the existence and nonexistence of monotone positive solution to the above problem.
Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on .
Throughout this paper, we always assume that and . Some sufficient conditions are established for the existence and nonexistence of monotone positive solution to the BVP (1.3). Here, a solution of the BVP (1.3) is said to be monotone and positive if , and for . Our main tool is the following Guo-Krasnoselskii fixed-point theorem .
Lemma 2.2 (see ).
Lemma 2.3 (see ).
which implies that is equicontinuous. Again, by Arzela-Ascoli theorem, we know that has a convergent subsequence in . Therefore, has a convergent subsequence in . Thus, we have shown that is a compact operator.
3. Main Results
The proof is similar to that of Theorem 3.1 and is therefore omitted.
This is a contradiction. Therefore, the BVP (1.3) has no monotone positive solution.
Similarly, we can prove the following theorem.
So, it follows from Theorem 3.1 that the BVP (3.14) has at least one monotone positive solution.
This work was supported by the National Natural Science Foundation of China (10801068).
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