- Research Article
- Open Access
Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular -Point Boundary Value Problems
© X. Du and Z. Zhao. 2009
- Received: 2 April 2009
- Accepted: 23 November 2009
- Published: 1 December 2009
This paper investigates the existence and uniqueness of smooth positive solutions to a class of singular m-point boundary value problems of second-order ordinary differential equations. A necessary and sufficient condition for the existence and uniqueness of smooth positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Our nonlinearity may be singular at and/or .
- Differential Equation
- Real Number
- Integral Equation
- Partial Differential Equation
- Ordinary Differential Equation
where are constants, and satisfies the following hypothesis:
there exists a function , and is integrable on such that
Conversely, (1.5) implies (1.3).
Conversely, (1.6) implies (1.4).
When is increasing with respect to , singular nonlinear -point boundary value problems have been extensively studied in the literature, see [1–3]. However, when is increasing on , and is decreasing on , the study on it has proceeded very slowly. The purpose of this paper is to fill this gap. In addition, it is valuable to point out that the nonlinearity may be singular at and/or
When referring to singularity we mean that the functions in (1.1) are allowed to be unbounded at the points , and/or . A function is called a (positive) solution to (1.1) and (1.2) if it satisfies (1.1) and (1.2) ( for ). A (positive) solution to (1.1) and (1.2) is called a smooth (positive) solution if and both exist ( for ). Sometimes, we also call a smooth solution a solution. It is worth stating here that a nontrivial nonnegative solution to the problem (1.1), (1.2) must be a positive solution. In fact, it is a nontrivial concave function satisfying (1.2) which, of course, cannot be equal to zero at any point
To seek necessary and sufficient conditions for the existence of solutions to the above problems is important and interesting, but difficult. Thus, researches in this respect are rare up to now. In this paper, we will study the existence and uniqueness of smooth positive solutions to the second-order singular -point boundary value problem (1.1) and (1.2). A necessary and sufficient condition for the existence of smooth positive solutions is given by constructing lower and upper solutions and with the maximal principle. Also, the uniqueness of the smooth positive solutions is studied.
Upper solution is defined by reversing the above inequality signs. If there exist a lower solution and an upper solution to problem (1.1), (1.2) such that , then is called a couple of upper and lower solution to problem (1.1), (1.2).
To prove the main result, we need the following maximal principle.
Lemma 1.3 (maximal principle).
Suppose that , and . If such that then
In view of (1.11) and the definition of , we can obtain This completes the proof of Lemma 1.3.
Now we state the main results of this paper as follows.
Suppose that and (1.13) hold, then the smooth positive solution to problem (1.1) and (1.2) is also the unique positive solution.
2.1. The Necessary Condition
Suppose that is a smooth positive solution to the boundary value problem (1.1) and (1.2). We will show that (1.13) holds.
On the other hand, notice that is a smooth positive solution to (1.1), (1.2), we have
which is the required inequality.
2.2. The Existence of Lower and Upper Solutions
Suppose that (1.13) holds. Let
Since by (1.13), (2.17) we obviously have
Consequently, with the aid of (2.20), (2.22) and the condition we have
therefore, (2.23)–(2.26) imply that are lower and upper solutions to the problem (1.1) and (1.2), respectively.
2.3. The Sufficient Condition
First of all, we define a partial ordering in by if and only if
Then, we will define an auxiliary function. For all
By the assumption of Theorem 1.4, we have that is continuous.
For each let us consider the following nonsingular problem:
Obviously, it follows from the proof of Lemma 1.3 that problem (2.30) is equivalent to the integral equation
where is defined in the proof of Lemma 1.3. It is easy to verify that is a completely continuous operator and is a bounded set. Moreover, is a solution to (2.30) if and only if Using the Schauder's fixed point theorem, we assert that has at least one fixed point
From this it follows that
Suppose by contradiction that is not satisfied on . Let
Since by the definition of and (2.30) we obviously have
So,when , we have and
Therefore that is, is an upper convex function in .
By (2.30) and (2.36), for we have the following two cases:
For case (i): it is clear that this is a contradiction.
For case (ii): in this case Since is decreasing on , thus, that is, is decreasing on From we see which is in contradiction with
From this it follows that
Similarly, we can verify that Consequently (2.32) holds.
Suppose that and are positive solutions to (1.1) and (1.2), and at least one of them is a smooth positive solution. If for any without loss of generality, we may assume that for some Let
It follows from (3.1) that
By (1.2), it is easy to check that there exist the following two possible cases:
Assume that case holds. By on it is easy to see that exist (finite or ), moreover, one of them must be finite. The same conclusion is also valid for It follows from (3.2) that
On the other hand, (3.2), (1.7), and condition yield
From this it follows that
If on then, by (3.6) we have and then which imply that there exists a positive number such that on It follows from (3.2) that therefore Substituting into (1.1) and using condition , we have
Noticing (3.11) and we have
which contradicts with the condition that Therefore, and on Thus, , which contradicts with (3.6). So case is impossible.
By analogous methods, we can obtain a contradiction for case . So for any which implies that the result of Theorem 1.5 holds.
The typical function satisfying is where
where and is nondecreasing on , nonincreasing on Clearly, condition is weaker than the above condition (4.1).
where We have the following theorem.
Moreover, when the positive solution exists, it is unique.
By analogous methods, we have the following results.
where is defined in (2.4).
Suppose and (4.8) hold, then the smooth positive solution to problem (1.1) and (4.6) is also unique positive solution.
Research supported by the National Natural Science Foundation of China (10871116), the Natural Science Foundation of Shandong Province (Q2008A03) and the Doctoral Program Foundation of Education Ministry of China (200804460001).
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