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:

(1.1)

with

(1.2)

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 ,

(1.3)

there exists a function , and is integrable on such that

(1.4)

Remark 1.1.

(i) Inequality (1.3) implies

(1.5)

Conversely, (1.5) implies (1.3).

(ii) Inequality (1.4) implies

(1.6)

Conversely, (1.6) implies (1.4).

Remark 1.2.

It follows from (1.3), (1.4) that

(1.7)

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

(1.8)

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

(1.9)

(1.10)

then

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

(1.11)

where

(1.12)

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

(1.13)

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

(2.1)

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

(2.2)

It is well known that can be stated as

(2.3)

where

(2.4)

By (2.3) and (1.2) we have

(2.5)

therefore because of (2.3) and (2.5),

(2.6)

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

(2.7)

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

(2.8)

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

(2.9)

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

(2.10)

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

(2.11)

Consequently which implies that

(2.12)

From (2.9) and (2.12) it follows that

(2.13)

which is the required inequality.

2.2. The Existence of Lower and Upper Solutions

Since is integrable on thus

(2.14)

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

(2.15)

(2.16)

where

Suppose that (1.13) holds. Let

(2.17)

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

(2.18)

and there exists a positive number such that

(2.19)

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

(2.20)

Let

(2.21)

Then from (2.19) and (2.21) we have

(2.22)

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

(2.23)

(2.24)

From (2.17), (2.21) it follows that

(2.25)

(2.26)

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

(2.27)

Then, we will define an auxiliary function. For all

(2.28)

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

(2.29)

For each let us consider the following nonsingular problem:

(2.30)

Obviously, it follows from the proof of Lemma 1.3 that problem (2.30) is equivalent to the integral equation

(2.31)

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

(2.32)

From this it follows that

(2.33)

Suppose by contradiction that is not satisfied on . Let

(2.34)

therefore

(2.35)

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

Let

(2.36)

So,when , we have and

(2.37)

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

(3.1)

It follows from (3.1) that

(3.2)

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

(3.3)

consequently

(3.4)

Similarly

(3.5)

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

(3.6)

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

(3.7)

that is,

(3.8)

therefore

(3.9)

From this it follows that

(3.10)

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

(3.11)

Noticing (3.11) and we have

(3.12)

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:

(4.1)

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

(4.2)

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

(4.3)

that is,

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

(4.4)

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

(4.5)

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

Remark 4.4.

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

(4.6)

By analogous methods, we have the following results.

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

(4.7)

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

(4.8)

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).

Affiliations

1. School of Mathematics Sciences, Qufu Normal University, Qufu, Shandong, 273165, China

   • Xinsheng Du
   •  & Zengqin Zhao

Corresponding author

Correspondence to Xinsheng Du.

Reprints and Permissions