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.
where , is a continuous function, are continuous functions, and are arbitrary given constants. The tools we mainly used are the method of upper and lower solutions and Leray-Schauder degree theory.
Note that for the cases of or in the boundary conditions (1.2), our theorems hold also true. However, for brevity we exclude such cases in this paper.
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
We may now formulate and prove our main results on the existence and uniqueness of solutions for -order three point boundary value problem (1.1), (1.2).
(iii) is continuous on , and is nonincreasing in and nondecreasing in on ;
(iv) is continuous on , and nonincreasing in and nondecreasing in on
where , .
In the following, we will complete the proof in four steps.
independently of .
There are three cases to consider.
Case 1 ( ).
Case 2 ( ).
Case 3 ( ).
are obtained by integration.
independently of .
By Step 1 and Lemma 2.3, there exists such that for . Since and do not depend on , the estimate on is also independent of .
Show that for , BVP (3.5), (3.6) has at least one solution .
Consider the set
has at least one solution in .
Show that is a solution of BVP (1.1), (1.2).
Now there are three cases to consider.
Case 1 ( ).
Case 2 ( ).
Case 3 ( ).
Hence is a solution of BVP (1.1), (1.2) and satisfies (3.3).
Now we give a uniqueness theorem by assuming additionally the differentiability for functions , and , and a kind of estimating condition in Theorem 3.1.
(ii) and its first-order partial derivatives in are continuous on , on , on and satisfy the Nagumo condition on
(v)there exists a function such that on and
Then BVP (1.1), (1.2) has a unique solution satisfying (3.3).
The existence of a solution for BVP (1.1), (1.2) satisfying (3.3) follows from Theorem 3.1.
where for each ,
which contradicts to (3.54). Thus . Similarly we can show that . Consequently .
which is a contradiction.
A similar contradiction can be obtained if . Hence on . By (3.55), we obtain on . This completes the proof of the theorem.
Next we give two examples to demonstrate the application of Theorem 3.2.
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