Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems
© M. Pei and S. K. Chang. 2009
Received: 5 February 2009
Accepted: 14 July 2009
Published: 19 August 2009
We are concerned with the higher-order nonlinear three-point boundary value problems: with the three point boundary conditions ; where is continuous, are continuous, and are arbitrary given constants. The existence and uniqueness results are obtained by using the method of upper and lower solutions together with Leray-Schauder degree theory. We give two examples to demonstrate our result.
Higher-order boundary value problems were discussed in many papers in recent years; for instance, see [1–22] and references therein. However, most of all the boundary conditions in the above-mentioned references are for two-point boundary conditions [2–11, 14, 17–22], and three-point boundary conditions are rarely seen [1, 12, 13, 16, 18]. Furthermore works for nonlinear three point boundary conditions are quite rare in literatures.
In this section, we present some definitions and lemmas that are needed to our main results.
Lemma 2.3 (see ).
has only the trivial solution.
which is a contradiction. Hence BVP (2.6), (2.7) has only the trivial solution.
3. Main Results
In the following, we will complete the proof in four steps.
There are three cases to consider.
are obtained by integration.
Now there are three cases to consider.
The existence of a solution for BVP (1.1), (1.2) satisfying (3.3) follows from Theorem 3.1.
which is a contradiction.
Next we give two examples to demonstrate the application of Theorem 3.2.
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