- Research Article
- Open Access
- Published:
Existence Results of Three-Point Boundary Value Problems for Second-order Ordinary Differential Equations
Boundary Value Problems volume 2011, Article number: 901796 (2011)
Abstract
We establish existence results of the following three-point boundary value problems: 
, 
, 
and 
, where 
and 
. The approach applied in this paper is upper and lower solution method associated with basic degree theory or Schauder's fixed point theorem. We deal with this problem with the function 
which is Carathéodory or singular on its domain.
1. Introduction
In this paper, we consider three-point boundary value problem
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
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:
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.
2. Preliminaries
Define 
by
where 
and 
are given as (1.2) and
By direct computations, we get the following results.
Lemma 2.1.
-
(i)
The function
defined by (2.1), is the Green function corresponding for the problem 
(2.3)
defined by (2.1), is the Green function corresponding for the problem 
(ii) The function 
defined by (2.1), is continuous.
-
(iii)
In the case
, we have
(2.4)
, we have
Lemma 2.2.
If 
, then the problem
with boundary condition (1.2) has a unique solution 
such that
where 
is defined by (2.1).
3. Carathéodory Case
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
-SolutionsWe 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
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
Before proving our main results, we first consider such a modified problem given as follows:
with boundary condition (1.2), where 
is defined by
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
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 
,
Furthermore, it follows from 
that for 
, 
, we have
This implies that the minimum of 
cannot occur at 
, a contradiction.
Case 2.
If 
, by the definition of 
-lower solution 
, we then have
And we get a contradiction.
Case 3.
If 
, it follows from the conclusion of Case 1 that
which is impossible.
Consequently, we obtain 
on 
. By the similar arguments as above, we also have
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
Then problem (1.1) and (1.2) has at least one solution 
such that, for all 
,
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
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
Define
where
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:
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
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
We now choose an upper solution 
of the form
To this end, we compute
Clearly, one can choose 
such that
that is,
and choose 
, where 
, which is a positive solution of
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
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
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
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. 
Hence, we have the contradiction since
Case 2.
If 
, by the definition of strict 
-lower solution that 
, we then have
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
Then, problem (1.1) and (1.2) has at least one solution 
such that, for any 
,
Proof.
This is a consequence of Theorem 3.5 and Proposition 3.10 and hence, we omits this proof.
4. Singular Case
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 
,
where
Then problem (1.1) and (1.2) has at least one solution 
such that, for all 
,
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
where
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
where 
is a constant and 
, 
are given as (1.2).
Example 4.3.
Consider the problem (4.7), for 
, 
, 
,
Clearly, 
is a 
-lower solution of (4.7) and
where
From Lemma 2.1, we have 
and define 
. Since, for 
, 
that is, 
, we have, from Lemma 2.2, 
and 
exists. Let
and, by Lemma 2.2 again, choose 
such that
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
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
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 
,
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
for some 
and 
, there is 
such that
where 
is defined as in Lemma 2.1.
for any compact set 
, there is 
such that
Then problem (1.1) and (1.2) with 
has at least one solution
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
Proof.
Step 1.
Construction of lower solutions. Consider 
such that 
and the function
where 
is chosen small enough so that
Next, we choose 
from the Remark 4.5, and let
where 
is small enough so that for some points 
, 
, we have:
Notice that by (4.24) and (4.25), for any 
such that
we have:
Step 2.
Approximation problems. We define for each 
, 
,
and set
We have that, for each index 
, 
is continuous and
where
Hence, the sequence of functions 
converges to 
uniformly on any set 
, where 
is an arbitrary compact subset of 
. Next we define
Each of the functions 
is a continuous function defined on 
, moreover
and the sequence 
converges to 
uniformly on the compact subsets of 
since
Define now a decreasing sequence 
such that
and consider a sequence of the following approximation problems:
where 
.
Step 3.
A lower solution of ( ). It is clear that for any 
,
As the sequence 
is decreasing, we also have
Clearly, 
satisfies
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
From assumption 
, we can find 
and 
such that, for all 
, 
,
Also, one has
where 
is a suitable constant. Hence, we obtain, for such 
and 
,
Let 
be a constant such that
Choose 
such that
that is,
where 
is defined by (2.1). Note that 
is well-defined and 
since 
. It is easy to see that
So by Remark 4.2, there is a solution 
of (4.7) such that
Step 5.
The problem ( ) has at least one solution 
such that
Notice that 
is an upper solution of ( ), since
Step 6.
Existence of a solution. Consider the pointwise limit
It is clear that, for any 
,
and therefore 
on 
. Let 
be a compact interval. There is an index 
such that 
for all 
and therefore for these 
,
Moreover, we have
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 
,
Since
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 
,
, 
,
Let 
, where 
. Obviously, 
satisfies 
and 
. Moreover, for any given 
and for any compact set 
, for 
small enough, we have
Hence, 
holds. Furthermore, for 
large enough, 
, we have, from Young's inequality by choosing 
and 
,
where 
. Hence, 
holds. By Theorem 4.4, 
has at least one solution
References
- 1.
Bitsadze AV, Samarskii AA: On some of the simplest generalizations of linear elliptic boundary-value problems. Doklady Akademii Nauk SSSR 1969, 185: 739–740.
- 2.
Il'in VA, Moiseev EI: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations 1987, 23(7):803–810.
- 3.
Il'in VA, Moiseev EI: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differential Equations 1987, 23(7):979–987.
- 4.
Bitsadze AV: On the theory of nonlocal boundary value problems. Soviet Mathematics—Doklady 1984, 30(1):8–10.
- 5.
Bitsadze AV: On a class of conditionally solvable nonlocal boundary value problems for harmonic functions. Soviet Mathematics—Doklady 1985, 31(1):91–94.
- 6.
Bitsadze AV, Samarskii AA: On some simple generalizations of linear elliptic boundary problems. Soviet Mathematics—Doklady 1969, 10(2):398–400.
- 7.
Yao Q: Successive iteration and positive solution for nonlinear second-order three-point boundary value problems. Computers & Mathematics with Applications 2005, 50(3–4):433–444. 10.1016/j.camwa.2005.03.006
- 8.
Yao Q: On the positive solutions of a second-order three-point boundary value problem with Caratheodory function. Southeast Asian Bulletin of Mathematics 2004, 28(3):577–585.
- 9.
Gupta CP, Trofimchuk SI: A sharper condition for the solvability of a three-point second order boundary value problem. Journal of Mathematical Analysis and Applications 1997, 205(2):586–597. 10.1006/jmaa.1997.5252
- 10.
Ma R: Positive solutions of a nonlinear m -point boundary value problem. Computers & Mathematics with Applications 2001, 42(6–7):755–765. 10.1016/S0898-1221(01)00195-X
- 11.
Thompson HB, Tisdell C: Three-point boundary value problems for second-order, ordinary, differential equations. Mathematical and Computer Modelling 2001, 34(3–4):311–318. 10.1016/S0895-7177(01)00063-2
- 12.
De Coster C, Habets P: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. In Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations, CISM Courses and Lectures. Volume 371. Edited by: Zanolin F. Springer, Vienna, Austria; 1993:1–78.
- 13.
Lü H, O'Regan D, Agarwal RP: Upper and lower solutions for the singular p -Laplacian with sign changing nonlinearities and nonlinear boundary data. Journal of Computational and Applied Mathematics 2005, 181(2):442–466. 10.1016/j.cam.2004.11.037
- 14.
Du Z, Xue C, Ge W: Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative. Archiv der Mathematik 2005, 84(4):341–349. 10.1007/s00013-004-1196-7
- 15.
Khan RA, Webb JRL: Existence of at least three solutions of a second-order three-point boundary value problem. Nonlinear Analysis. Theory, Methods & Applications 2006, 64(6):1356–1366. 10.1016/j.na.2005.06.040
- 16.
Minghe P, Chang SK: The generalized quasilinearization method for second-order threepoint boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 2008, 68(9):2779–2790. 10.1016/j.na.2007.02.025
- 17.
Qu WB, Zhang ZX, Wu JD: Positive solutions to a singular second order three-point boundary value problem. Applied Mathematics and Mechanics 2002, 23(7):854–866. 10.1007/BF02456982
- 18.
De Coster C, Habets P: Two-Point Boundary Value Problems: Lower and Upper Solutions. Springer, Berlin, Germany; 1984.
- 19.
Habets P, Zanolin F: Upper and lower solutions for a generalized Emden-Fowler equation. Journal of Mathematical Analysis and Applications 1994, 181(3):684–700. 10.1006/jmaa.1994.1052
Author information
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wang, S., Tsai, L. Existence Results of Three-Point Boundary Value Problems for Second-order Ordinary Differential Equations. Bound Value Probl 2011, 901796 (2011). https://doi.org/10.1155/2011/901796
Received:
Revised:
Accepted:
Published:
Keywords
- Fixed Point Theorem
- Open Interval
- Lower Solution
- Nonlocal Boundary
- Linear Ordinary Differential Equation
