Uniqueness and Parameter Dependence of Positive Solution of Fourth-Order Nonhomogeneous BVPs

Boundary value problems (BVPs for short) consisting of fourth-order differential equation and four-point homogeneous boundary conditions have received much attention due to their striking applications. For example, Chen et al. [1] studied the fourth-order nonlinear differential equation

(1.1)

with the four-point homogeneous boundary conditions

(1.2)

(1.3)

where . By means of the upper and lower solution method and Schauder fixed point theorem, some criteria on the existence of positive solutions to the BVP (1.1)–(1.3) were established. Bai et al. [2] obtained the existence of solutions for the BVP (1.1)–(1.3) by using a nonlinear alternative of Leray-Schauder type. For other related results, one can refer to [3–5] and the references therein.

Recently, nonhomogeneous BVPs have attracted many authors' attention. For instance, Ma [6, 7] and L. Kong and Q. Kong [8–10] studied some second-order multipoint nonhomogeneous BVPs. In particular, L. Kong and Q. Kong [10] considered the following second-order BVP with multipoint nonhomogeneous boundary conditions

(1.4)

where and are nonnegative parameters. They derived some conditions for the above BVP to have a unique solution and then studied the dependence of this solution on the parameters and . Sun [11] discussed the existence and nonexistence of positive solutions to a class of third-order three-point nonhomogeneous BVP. The authors in [12] studied the multiplicity of positive solutions for some fourth-order two-point nonhomogeneous BVP by using a fixed point theorem of cone expansion/compression type. For more recent results on higher-order BVPs with nonhomogeneous boundary conditions, one can see [13–16].

Inspired greatly by the above-mentioned excellent works, in this paper we are concerned with the following Sturm-Liouville BVP consisting of the fourth-order differential equation:

(1.5)

and the four-point nonhomogeneous boundary conditions

(1.6)

(1.7)

where and are nonnegative parameters. Under the following assumptions:

(A1) and are nonnegative constants with , , , , and

(A2) is continuous and monotone increasing in for every ;

(A3) there exists such that

(1.8)

we prove the uniqueness of positive solution for the BVP (1.5)–(1.7) and study the dependence of this solution on the parameters .

First, we recall some fundamental definitions.

Definition 2.1.

Let be a Banach space with norm . Then

(1) a nonempty closed convex set is said to be a cone if for all and , where is the zero element of

(2) every cone in defines a partial ordering in by

(3) a cone is said to be normal if there exists such that implies that

(4) a cone is said to be solid if the interior of is nonempty.

Definition 2.2.

Let be a solid cone in a real Banach space an operator, and Then is called a -concave operator if

(2.1)

Next, we state a fixed point theorem, which is our main tool.

Lemma 2.3 (see [17]).

Assume that is a normal solid cone in a real Banach space and is a -concave increasing operator. Then has a unique fixed point in

The following two lemmas are crucial to our main results.

Lemma 2.4.

Assume that and are defined as in (A1) and . Then for any the BVP consisting of the equation

(2.2)

and the boundary conditions (1.6) and (1.7) has a unique solution

(2.3)

where

(2.4)

Proof.

Let

(2.5)

Then

(2.6)

By (2.5) and (1.6), we know that

(2.7)

On the other hand, in view of (2.5) and (1.7), we have

(2.8)

So, it follows from (2.6) and (2.8) that

(2.9)

which together with (2.7) implies that

(2.10)

Lemma 2.5.

Assume that (A1) holds. Then

(1) for

(2) for

(3) for

For convenience, we denote and . In the remainder of this paper, the following notations will be used:

(1) if at least one of approaches ;

(2) if for ;

(3) if for and at least one of them is strict.

Let . Then is a Banach space, where is defined as usual by the sup norm.

Our main result is the following theorem.

Theorem 3.1.

Assume that (A1)–(A3) hold. Then the BVP (1.5)–(1.7) has a unique positive solution for any , where . Furthermore, such a solution satisfies the following properties:

(P1)

(P2) is strictly increasing in , that is,

(3.1)

(P3) is continuous in , that is, for any given

(3.2)

Proof.

Let . Then is a normal solid cone in with For any , if we define an operator as follows:

(3.3)

then it is not difficult to verify that is a positive solution of the BVP (1.5)–(1.7) if and only if is a fixed point of .

Now, we will prove that has a unique fixed point by using Lemma 2.3.

First, in view of Lemma 2.5, we know that

Next, we claim that is a -concave operator.

In fact, for any and it follows from (3.3) and (A3) that

(3.4)

which shows that is -concave.

Finally, we assert that is an increasing operator.

Suppose that and By (3.3) and (A2), we have

(3.5)

which indicates that is increasing.

Therefore, it follows from Lemma 2.3 that has a unique fixed point which is the unique positive solution of the BVP (1.5)–(1.7). The first part of the theorem is proved.

In the rest of the proof, we will prove that such a positive solution satisfies properties (P1), (P2), and (P3).

First,

(3.6)

which together with for implies (P1).

Next, we show (P2). Assume that Let

(3.7)

Then for We assert that Suppose on the contrary that Since is a -concave increasing operator and for given , is strictly increasing in , we have

(3.8)

which contradicts the definition of Thus, we get for And so,

(3.9)

which indicates that is strictly increasing in .

Finally, we prove (P3). For any given we first suppose that with From (P2), we know that

(3.10)

Let

(3.11)

Then and for If we define

(3.12)

then and

(3.13)

which together with the definition of implies that

(3.14)

So,

(3.15)

Therefore,

(3.16)

In view of (3.10) and (3.16), we obtain that

(3.17)

which together with the fact that as shows that

(3.18)

Similarly, we can also prove that

(3.19)

Hence, (P3) holds.