Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular -Point Boundary Value Problems

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 .

In this paper,we will consider the existence and uniqueness of positive solutions to a class of second-order singular -point boundary value problems of the following differential equation:

with

where are constants, and satisfies the following hypothesis:

is continuous, nondecreasing on , and nonincreasing on for each fixed there exists a real number such that for any ,

there exists a function , and is integrable on such that

Remark 1.1.

(i) Inequality (1.3) implies

Conversely, (1.5) implies (1.3).

(ii) Inequality (1.4) implies

Conversely, (1.6) implies (1.4).

Remark 1.2.

It follows from (1.3), (1.4) that

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.

A function is called a lower solution to the problem (1.1), (1.2), if and satisfies

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

Proof.

Let

then

By integrating (1.9) twice and noting (1.10), we have

where

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.

Theorem 1.4.

Suppose that holds, then a necessary and sufficient condition for the problem (1.1) and (1.2) to have smooth positive solution is that

Theorem 1.5.

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.

It follows from

that is nonincreasing on Thus, by the Lebesgue theorem, we have

It is well known that can be stated as

where

By (2.3) and (1.2) we have

therefore because of (2.3) and (2.5),

Since is a smooth positive solution to (1.1) and (1.2), we have

Let From (2.6), (2.7) it follows that

Without loss of generality we may assume that This together with the condition implies

On the other hand, notice that is a smooth positive solution to (1.1), (1.2), we have

therefore, there exists a positive number such that Obviously, and It follows from (1.7) that

Consequently which implies that

From (2.9) and (2.12) it follows that

which is the required inequality.

2.2. The Existence of Lower and Upper Solutions

Since is integrable on thus

Otherwise, if then there exists a real number such that when this contradicts with the condition that is integrable on By condition and (2.14) we have

where

Suppose that (1.13) holds. Let

Since by (1.13), (2.17) we obviously have

and there exists a positive number such that

By (2.14) and (2.16) we see, if is sufficiently small, then

Let

Then from (2.19) and (2.21) we have

Consequently, with the aid of (2.20), (2.22) and the condition we have

From (2.17), (2.21) it follows that

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.

Let be a sequence satisfying and as and let be a sequence satisfying

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

We claim that

From this it follows that

Suppose by contradiction that is not satisfied on . Let

therefore

Since by the definition of and (2.30) we obviously have

Let

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:

(i)

(ii)

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.

Using the method of [4] and [5, Theorem ], we can obtain that there is a positive solution to (1.1), (1.2) such that and a subsequence of converges to on any compact subintervals of

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:

(1)

(2)

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

consequently

Similarly

From (3.1), (3.4), and (3.5) we have

On the other hand, (3.2), (1.7), and condition yield

that is,

therefore

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.

Remark 4.1.

The typical function satisfying is where

Remark 4.2.

Condition includes e-concave function (see [6]) as special case. For example, Liu and Yu [7] consider the existence and uniqueness of positive solution to a class of singular boundary value problem under the following condition:

where and is nondecreasing on , nonincreasing on Clearly, condition is weaker than the above condition (4.1).

In fact, for any from (4.1) it follows that

On the other hand, for any from (4.1) it follows that

that is,

In what follows, by using the results obtained in this paper, we study the boundary value problem

where We have the following theorem.

Theorem 4.3.

A necessary and sufficient condition for problem (4.4) to have smooth positive solution is that

Moreover, when the positive solution exists, it is unique.

Remark 4.4.

Consider (1.1) and the following singular -point boundary value conditions:

By analogous methods, we have the following results.

Assume that is a positive solution to (1.1) and (4.6), then can be stated

where is defined in (2.4).

Theorem 4.5.

Suppose that holds, then a necessary and sufficient condition for the problem (1.1) and (4.6) to have smooth positive solution is that

Theorem 4.6.

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).

1. Du, X, Zhao, Z: A necessary and sufficient condition of the existence of positive solutions to singular sublinear three-point boundary value problems. Applied Mathematics and Computation. 186(1), 404–413 (2007). Publisher Full Text

2. Du, X, Zhao, Z: On existence theorems of positive solutions to nonlinear singular differential equations. Applied Mathematics and Computation. 190(1), 542–552 (2007). Publisher Full Text

3. Wei, Z: A necessary and sufficient condition for the existence of positive solutions of singular super-linear -point boundary value problems. Applied Mathematics and Computation. 179(1), 67–78 (2006). Publisher Full Text

4. Zhang, Y: Positive solutions of singular sublinear Emden-Fowler boundary value problems. Journal of Mathematical Analysis and Applications. 185(1), 215–222 (1994). Publisher Full Text

5. Hartman, P: Ordinary Differential Equations, Brikhäuser, Boston, Mass, USA (1982)

6. Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering,p. viii+275. Academic Press, Boston, Mass, USA (1988)

7. Liu, Y, Yu, H: Existence and uniqueness of positive solution for singular boundary value problem. Computers & Mathematics with Applications. 50(1-2), 133–143 (2005). PubMed Abstract | Publisher Full Text