Existence Results of Three-Point Boundary Value Problems for Second-order Ordinary Differential Equations

In this paper, we consider three-point boundary value problem

(1.1)

(1.2)

where and .

In the mathematical literature, a number of works have appeared on nonlocal boundary value problems, and one of the first of these was [1]. Il'in and Moiseev initiated the research of multipoint boundary value problems for second-order linear ordinary differential equations, see [2, 3], motivated by the study [4–6] of Bitsadze and Samarskii.

Recently, nonlinear multipoint boundary value problems have been receiving considerable attention, and have been studied extensively by using iteration scheme (e.g., [7]), fixed point theorems in cones (e.g., [8]), and the Leray-Schauder continuation theorem (e.g., [9]). We refer more detailed treatment to more interesting research [10, 11] and the references therein.

The theory of upper and lower solutions is also a powerful tool in studying boundary value problems. For the existence results of two-point boundary value problem, there already are lots of interesting works by applying this essential technique (see [12, 13]). Recently, it is shown that this method plays an important role in proving the existence of solutions for three-point boundary value problems (see [14–16]).

Last but not least, as the singular source term appearing in two-point problems, singular three-point boundary value problems also attract more attention (e.g., [17]).

In this paper, we will discuss the existence of solutions of some general types on three-point boundary value problems by using upper and lower solution method associated with basic degree theory or Schauder's fixed point theorem.

This paper is organized as follows. In Section 2, we give two lemmas which will be extensively used later. In Section 3, when the source term is a Carathéodory function, we consider the Sobolev space defined by

(1.3)

and obtain the existence of -solution in Theorems 3.5 and 3.11. In Section 4, we discuss the singular case, that is, maybe singular at the end points or , or at . We will introduce the -class of functions and another space (see [18, 19]) as follows:

(1.4)

and prove the existence of -solution in Theorems 4.1 and 4.4. Some sufficient conditions for constructing upper and lower solutions are given in each section for applications.

Define by

(2.1)

where and are given as (1.2) and

(2.2)

By direct computations, we get the following results.

 (ii) The function defined by (2.1), is continuous.

1. (iii)
   In the case , we have

   

   (2.4)
    

Lemma 2.2.

If , then the problem

(2.5)

with boundary condition (1.2) has a unique solution such that

(2.6)

where is defined by (2.1).

In this section we first introduce the Carathéodory function as follows.

Definition 3.1.

A function defined on is called a Carathéodory function on if

(i)for almost every is continuous on ;

(ii)for any the function is measurable on ;

(iii)for any , there exists such that for any and for almost every with , we have .

We in this section assume that is a Carathéodory function and discuss the existence of -solution by assuming the existence of upper and lower solutions.

3.1. Existence of -Solutions

We first introduce the definitions of -upper and lower solutions as below.

Definition 3.2.

A function is called a -lower solution of problem (1.1) and (1.2) if it satisfies

(i) , , and

(ii)for any , either , or there exists an open interval containing such that and, for almost every , we have

(3.1)

Definition 3.3.

A function is called a -upper solution of problem (1.1) and (1.2) if it satisfies

(i) , , and

(ii)for any , either , or there exists an open interval containing such that and, for almost every , we have

(3.2)

Before proving our main results, we first consider such a modified problem given as follows:

(3.3)

with boundary condition (1.2), where is defined by

(3.4)

Proposition 3.4.

Let and be respective -lower and upper solutions of problem (1.1) and (1.2) with on . If is a solution of problem (3.3) and (1.2), then , for any .

Proof.

Suppose there exists such that

(3.5)

Case 1.

If , we have , which implies Hence, by Definition 3.2 and the continuity of at , there exist an open interval with , and a neighborhood of contained in such that for almost every ,

(3.6)

Furthermore, it follows from that for , , we have

(3.7)

This implies that the minimum of cannot occur at , a contradiction.

Case 2.

If , by the definition of -lower solution , we then have

(3.8)

And we get a contradiction.

Case 3.

If , it follows from the conclusion of Case 1 that

(3.9)

which is impossible.

Consequently, we obtain on . By the similar arguments as above, we also have

(3.10)

Theorem 3.5.

Let and be -lower and upper solutions of problem (1.1) and (1.2) such that on and let be a Carathéodory function on , where

(3.11)

Then problem (1.1) and (1.2) has at least one solution such that, for all ,

(3.12)

Proof.

We consider the modified problem (3.3) and (1.2) with respect to the given and . Consider the Banach space with supremum and the operator by

(3.13)

for , where is defined as (2.1). Since is a Carathéodory function on , for almost all and for all , there exists a function , we have

(3.14)

Define

(3.15)

where

(3.16)

(3.17)

It is clear that is a closed, bounded and convex set in and one can show that is a completely continuous mapping by Arzelà-Ascoli theorem and Lebesgue dominated convergence theorem. By applying Schauder's fixed point theorem, we obtain that has a fixed point in which is a solution of problem (3.3) and (1.2). From Proposition 3.4, this fixed point is also a solution of problem (1.1) and (1.2). Hence, we complete the proof.

We further illustrate the use of Theorem 3.11 in the following second-order differential equation:

(3.18)

with the boundary condition (1.2).

Corollary 3.6.

Assume that is a Carathéodory function satisfying is essentially bounded for , where is a constant large enough. Assume further that and there exists a constant such that

(3.19)

Then, problem (3.18) and (1.2) has at least one solution.

Proof.

By hypothesis, for any given small enough such that and for almost all , for any large enough, we have

(3.20)

We now choose an upper solution of the form

(3.21)

To this end, we compute

(3.22)

Clearly, one can choose such that

(3.23)

that is,

(3.24)

and choose , where , which is a positive solution of

(3.25)

Hence, if is large enough, we can show that and , where , which implies that is a positive -upper solution. In the same way we construct a -lower solution on .

3.2. Nontangency Solution

In this subsection, we afford another stronger lower and upper solutions to get a strict inequality of the solution between them.

Definition 3.7.

A function is a strict -lower solution of problem (1.1) and (1.2), if it is not a solution of problem (1.1) and (1.2), , and for any , one of the following is satisfied:

(i) ;

(ii)there exist an interval and such that int , and for almost every , for all we have

(3.26)

Definition 3.8.

A function is a strict -upper solution of problem (1.1) and (1.2), if it is not a solution of problem (1.1) and (1.2), , and for any , one of the following is satisfied:

(i) ,

(ii)there exist an interval and such that int , and for almost every , for all we have

(3.27)

Remark 3.9.

Every strict -lower(upper) solution of problem (1.1) and (1.2) is a -lower(upper) solution.

Now we are going to show that the solution curve of problem (1.1) and (1.2) cannot be tangent to upper or lower solutions from below or above.

Proposition 3.10.

Let and be respective strict -lower and upper solutions of problem (1.1) and (1.2) with on . If is a solution of problem (1.1) and (1.2) with on , then , for any .

Proof.

As is not a solution, is not identical to . Assume, the conclusion does not hold, then

(3.28)

exists. Hence, has minimum at , that is, .

Case 1.

Set . Since has minimum at , we have . According to the Definition 3.7, there exist , and with such that, for every , , and for a.e.

(3.29)

Hence, we have the contradiction since

(3.30)

Case 2.

If , by the definition of strict -lower solution that , we then have

(3.31)

And we get a contradiction.

Case 3.

If , repeat the same arguments in Case 3 of the proof of Proposition 3.4. Therefore, we obtain on . The inequality on can be proved by the similar arguments as above.

Theorem 3.11.

Let and be strict -lower and upper solutions of problem (1.1) and (1.2) such that on and let be a Carathéodory function, where

(3.32)

Then, problem (1.1) and (1.2) has at least one solution such that, for any ,

(3.33)

Proof.

This is a consequence of Theorem 3.5 and Proposition 3.10 and hence, we omits this proof.

In this section we give a more general existence result than Theorem 3.11 by assuming the existence of -lower and upper solutions. This makes us to deal with problem (1.1) and (1.2), where the function is singular at the end point and .

Theorem 4.1.

Let and be -lower and upper solutions of problem (1.1) and (1.2) such that on and let satisfy the following conditions:

(i)for almost every is continuous on ;

(ii)for any the function is measurable on ;

(iii)there exists a function such that, for all ,

(4.1)

where

(4.2)

Then problem (1.1) and (1.2) has at least one solution such that, for all ,

(4.3)

Proof.

Consider the modified problem (3.3) and (1.2) with respect to the given and and define by (3.13). Note that by Lemma 2.2, is well defined. Define

(4.4)

where

(4.5)

and is defined by (3.17). The rest arguments are similar to the proof of Theorem 3.5.

Remark 4.2.

We have similar results of Theorems 3.5–4.1, respectively, for (1.1) equipped with

(4.6)

where is a constant and , are given as (1.2).

Example 4.3.

Consider the problem (4.7), for , , ,

(4.7)

Clearly, is a -lower solution of (4.7) and

(4.8)

where

(4.9)

From Lemma 2.1, we have and define . Since, for ,

(4.10)

that is, , we have, from Lemma 2.2, and exists. Let

(4.11)

and, by Lemma 2.2 again, choose such that

(4.12)

Note that according to the direct computation, we see that is well-defined and is bounded by . Next, let . By Young's inequality, it follows that

(4.13)

Hence, such is a -upper solution of (4.7) and on . Clearly, satisfies (i), (ii) of Theorem 4.1. By using Young's inequality again, for , we have

(4.14)

and . Therefore, satisfies the assumption (iii) of Theorem 4.1. Consequently, we conclude that this problem has at least one solution such that, for all ,

(4.15)

Notice that in Theorem 4.1, one can only deal with the case that is singular at end points , . However, when is singular at , there is no hope to obtain the solutions directly from Theorem 4.1. We will establish the following theorem to deal with this case by constructing upper and lower solutions to solve this problem.

Theorem 4.4.

Assume

the function is continuous;

there exists and for any compact set , there is such that

(4.16)

for some and , there is such that

(4.17)

where is defined as in Lemma 2.1.

for any compact set , there is such that

(4.18)

Then problem (1.1) and (1.2) with has at least one solution

(4.19)

Remark 4.5 (see [12, Remark ]).

Assumption is equivalent to the assumption that there exists and a function such that:

(i) for all ,

(ii) , for all , ,

(iii) , for all ,

where

(4.20)

Proof.

Step 1.

Construction of lower solutions. Consider such that and the function

(4.21)

where is chosen small enough so that

(4.22)

Next, we choose from the Remark 4.5, and let

(4.23)

where is small enough so that for some points , , we have:

(4.24)

(4.25)

Notice that by (4.24) and (4.25), for any such that

(4.26)

we have:

(4.27)

(4.28)

Step 2.

Approximation problems. We define for each , ,

(4.29)

and set

(4.30)

We have that, for each index , is continuous and

(4.31)

where

(4.32)

Hence, the sequence of functions converges to uniformly on any set , where is an arbitrary compact subset of . Next we define

(4.33)

Each of the functions is a continuous function defined on , moreover

(4.34)

and the sequence converges to uniformly on the compact subsets of since

(4.35)

Define now a decreasing sequence such that

(4.36)

and consider a sequence of the following approximation problems:

(Pn)

where .

Step 3.

A lower solution of ( ). It is clear that for any ,

(4.37)

As the sequence is decreasing, we also have

(4.38)

Clearly, satisfies

(4.39)

It follows from (4.25) and (4.27) that is a lower solution of ( ).

Step 4.

Existence of a solution of (4.7) such that

(4.40)

From assumption , we can find and such that, for all , ,

(4.41)

Also, one has

(4.42)

where is a suitable constant. Hence, we obtain, for such and ,

(4.43)

Let be a constant such that

(4.44)

Choose such that

(4.45)

that is,

(4.46)

where is defined by (2.1). Note that is well-defined and since . It is easy to see that

(4.47)

So by Remark 4.2, there is a solution of (4.7) such that

(4.48)

Step 5.

The problem ( ) has at least one solution such that

(4.49)

Notice that is an upper solution of ( ), since

(4.50)

Step 6.

Existence of a solution. Consider the pointwise limit

(4.51)

It is clear that, for any ,

(4.52)

and therefore on . Let be a compact interval. There is an index such that for all and therefore for these ,

(4.53)

Moreover, we have

(4.54)

By Arzelá-Ascoli theorem it is standard to conclude that is a solution of problem (1.1) and (1.2) on the interval . Since is arbitrary, we find that and for all ,

(4.55)

Since

(4.56)

it remains only to check the continuity of at . This can be deduced from the continuity of and the fact that as .

Example 4.6.

Consider the following problem , for , , ,

(4.57)

Let , where . Obviously, satisfies and . Moreover, for any given and for any compact set , for small enough, we have

(4.58)

Hence, holds. Furthermore, for large enough, , we have, from Young's inequality by choosing and ,

(4.59)

where . Hence, holds. By Theorem 4.4, has at least one solution

(4.60)