© J.-P. Sun and F.-X. Jin. 2010
Received: 5 November 2010
Accepted: 14 December 2010
Published: 19 December 2010
In this paper, we are concerned with the following nonlinear third-order -point boundary value problem: , , , , . Some existence criteria of solution and positive solution are established by using the Schauder fixed point theorem. An example is also included to illustrate the importance of the results obtained.
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 .
The main tool used was Mawhin's continuation theorem.
Throughout, we always assume that and . The purpose of this paper is to consider the local properties of on some bounded sets and establish some existence criteria of solution and positive solution for the BVP (1.3) by using the Schauder fixed point theorem. An example is also included to illustrate the importance of the results obtained.
2. Main Results
The following theorem guarantees the existence of solution for the BVP (1.3).
On the basis of Theorem 2.2, we now give some existence results of nonnegative solution and positive solution for the BVP (1.3).
Assume that all the conditions of Theorem 2.3 are fulfilled. Then the BVP (1.3) has one positive solution if one of the following conditions is satisfied:
A simple calculation shows that and . Thus, if we choose and , then all the conditions of Theorem 2.3 and (i) of Corollary 2.4 are fulfilled. It follows from Corollary 2.4 that the BVP (2.35) has a positive solution.
This paper was supported by the National Natural Science Foundation of China (10801068).
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