- Research Article
- Open Access
© 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
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).
Now we state the main results of this paper as follows.
2.1. The Necessary Condition
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
2.3. The Sufficient Condition
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
It follows from (3.1) that
By (1.2), it is easy to check that there exist the following two possible cases:
From this it follows that
Moreover, when the positive solution exists, it is unique.
By analogous methods, we have the following results.
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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