We are concerned with the higherorder nonlinear threepoint 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 LeraySchauder degree theory. We give two examples to demonstrate our result.
1. Introduction
Higherorder 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 abovementioned references are for twopoint boundary conditions [2–11, 14, 17–22], and threepoint boundary conditions are rarely seen [1, 12, 13, 16, 18]. Furthermore works for nonlinear three point boundary conditions are quite rare in literatures.
The purpose of this article is to study the existence and uniqueness of solutions for higher order nonlinear three point boundary value problem
with nonlinear three point boundary conditions
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 LeraySchauder 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.
2. Preliminary
In this section, we present some definitions and lemmas that are needed to our main results.
Definition 2.1.
are called lower and upper solutions of BVP (1.1), (1.2), respectively, if
Definition 2.2.
Let be a subset of . We say that satisfies the Nagumo condition on if there exists a continuous function such that
Lemma 2.3 (see [10]).
Let be a continuous function satisfying the Nagumo condition on
where are continuous functions such that
Then there exists a constant (depending only on and such that every solution of (1.1) with
satisfies
Lemma 2.4.
Let be a continuous function. Then boundary value problem
has only the trivial solution.
Proof.
Suppose that is a nontrivial solution of BVP (2.6), (2.7). Then there exists such that or . We may assume . There exists such that
Then , . From (2.6) we have
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).
Theorem 3.1.
Assume that
(i)there exist lower and upper solutions of BVP (1.1), (1.2), respectively, such that
(ii) is continuous on , is nonincreasing in on , and is nonincreasing in on and satisfies the Nagumo condition on , where
(iii) is continuous on , and is nonincreasing in and nondecreasing in on ;
(iv) is continuous on , and nonincreasing in and nondecreasing in on
Then BVP (1.1), (1.2) has at least one solution such that for each ,
Proof.
For each define
where , .
For , we consider the auxiliary equation
where is given by the Nagumo condition, with the boundary conditions
Then we can choose a constant such that
In the following, we will complete the proof in four steps.
Step 1.
Show that every solution of BVP (3.5), (3.6) satisfies
independently of .
Suppose that the estimate is not true. Then there exists such that or . We may assume . There exists such that
There are three cases to consider.
Case 1 ().
In this case, and . For , by (3.8), we get the following contradiction:
and for , we have the following contradiction:
Case 2 ().
In this case,
and . For , by (3.6) we have the following contradiction:
For , by (3.9) and condition (iii) we can get the following contradiction:
Case 3 ().
In this case,
and . For , by (3.6) we have the following contradiction:
For , by (3.10) and condition (iv) we can get the following contradiction:
By (3.6), the estimates
are obtained by integration.
Step 2.
Show that there exists such that every solution of BVP (3.5), (3.6) satisfies
independently of .
Let
and define the function as follows:
In the following, we show that satisfies the Nagumo condition on , independently of . In fact, since satisfies the Nagumo condition on , we have
Furthermore, we obtain
Thus, satisfies the Nagumo condition on , independently of . Let
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 .
Step 3.
Show that for , BVP (3.5), (3.6) has at least one solution .
Define the operators as follows:
by
by
with
Since is compact, we have the following compact operator:
defined by
Consider the set
By Steps 1 and 2, the degree is well defined for every and by homotopy invariance, we get
Since the equation has only the trivial solution from Lemma 2.4, by the degree theory we have
Hence, the equation has at least one solution. That is, the boundary value problem
with the boundary conditions
has at least one solution in .
Step 4.
Show that is a solution of BVP (1.1), (1.2).
In fact, the solution of BVP (3.36), (3.37) will be a solution of BVP (1.1), (1.2), if it satisfies
By contradiction, suppose that there exists such that . There exists such that
Now there are three cases to consider.
Case 1 ().
In this case, since on , we have and . By conditions (i) and (ii), we get the following contradiction:
Case 2 ().
In this case, we have
and . By (3.37) and conditions (i) and (iii) we can get the following contradiction:
Case 3 ().
In this case, we have
and . By (3.37) and conditions (i) and (iv) we can get the following contradiction:
Similarly, we can show that on . Hence
Also, by boundary condition (3.37) and condition (i), we have
Therefore by integration we have for each ,
that is,
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.
Theorem 3.2.
Assume that
(i)there exist lower and upper solutions of BVP (1.1), (1.2), respectively, such that
(ii) and its firstorder partial derivatives in are continuous on , on , on and satisfy the Nagumo condition on
(iii) is continuous on and continuously partially differentiable on , and
(iv) is continuous on and continuously partially differentiable on , and
(v)there exists a function such that on and
Then BVP (1.1), (1.2) has a unique solution satisfying (3.3).
Proof.
The existence of a solution for BVP (1.1), (1.2) satisfying (3.3) follows from Theorem 3.1.
Now, we prove the uniqueness of solution for BVP (1.1), (1.2). To do this, we let and are any two solutions of BVP (1.1), (1.2) satisfying (3.3). Let . It is easy to show that is a solution of the following boundary value problem
where for each ,
By conditions (ii), (iii), and (iv), we have that and
Now suppose that there exists such that . Without loss of generality assume , and let
It is easy to see that by condition (v), hence . Let . We have that , on , and there exists a point such that . Furthermore . In fact, if , then . By condition (v) and (3.55) we can easily show that
In particular
Hence
which contradicts to (3.54). Thus . Similarly we can show that . Consequently .
Now, there are two cases to consider, that is
If , then by (3.59) we have
Thus, by (3.53) and condition (v) we have
Consequently, by Taylor's theorem there exists such that
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.
Example 3.3.
Consider the following thirdorder three point BVP:
Let
Choose and . It is easy to check that , and are lower and upper solutions of BVP (3.66), (3.67) respectively, and all the assumptions in Theorem 3.2 are satisfied. Therefore by Theorem 3.2 BVP (3.66), (3.67) has a unique solution satisfying
Example 3.4.
Consider the following fourthorder three point BVP:
Let
Choose and . It is easy to check that , and are lower and upper solutions of BVP (3.70), (3.71), respectively, and all the assumptions in Theorem 3.2 are satisfied. Therefore by Theorem 3.2 BVP (3.70), (3.71) has a unique solution satisfying
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