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 (
).

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 (
).

and

. For

, by (3.6) we have the following contradiction:

For

, by (3.10) and condition (iv) we can get the following contradiction:

are obtained by integration.

Step 2.

Show that there exists

such that every solution

of BVP (3.5), (3.6) satisfies

independently of
.

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

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:

Since

is compact, we have the following compact operator:

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 (
).

and

. By (3.37) and conditions (i) and (iii) we can get the following contradiction:

Case 3 (
).

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

,

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 first-order 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

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 third-order three point BVP:

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 fourth-order three point BVP:

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