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Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular
-Point Boundary Value Problems
Boundary Value Problems volume 2009, Article number: 191627 (2009)
Abstract
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
.
1. Introduction and the Main Results
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ1_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ2_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ3_HTML.gif)
there exists a function , and
is integrable on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ4_HTML.gif)
Remark 1.1.
(i) Inequality (1.3) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ5_HTML.gif)
Conversely, (1.5) implies (1.3).
(ii) Inequality (1.4) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ6_HTML.gif)
Conversely, (1.6) implies (1.4).
Remark 1.2.
It follows from (1.3), (1.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ9_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ10_HTML.gif)
then
By integrating (1.9) twice and noting (1.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ11_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ13_HTML.gif)
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. Proof of Theorem 1.4
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ14_HTML.gif)
that is nonincreasing on
Thus, by the Lebesgue theorem, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ15_HTML.gif)
It is well known that can be stated as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ16_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ17_HTML.gif)
By (2.3) and (1.2) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ18_HTML.gif)
therefore because of (2.3) and (2.5),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ19_HTML.gif)
Since is a smooth positive solution to (1.1) and (1.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ20_HTML.gif)
Let From (2.6), (2.7) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ21_HTML.gif)
Without loss of generality we may assume that This together with the condition
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ22_HTML.gif)
On the other hand, notice that is a smooth positive solution to (1.1), (1.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ23_HTML.gif)
therefore, there exists a positive number such that
Obviously,
and
It follows from (1.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ24_HTML.gif)
Consequently which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ25_HTML.gif)
From (2.9) and (2.12) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ26_HTML.gif)
which is the required inequality.
2.2. The Existence of Lower and Upper Solutions
Since is integrable on
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ29_HTML.gif)
where
Suppose that (1.13) holds. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ30_HTML.gif)
Since by (1.13), (2.17) we obviously have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ31_HTML.gif)
and there exists a positive number such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ32_HTML.gif)
By (2.14) and (2.16) we see, if is sufficiently small, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ33_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ34_HTML.gif)
Then from (2.19) and (2.21) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ35_HTML.gif)
Consequently, with the aid of (2.20), (2.22) and the condition we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ37_HTML.gif)
From (2.17), (2.21) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ38_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ40_HTML.gif)
Then, we will define an auxiliary function. For all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ41_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ42_HTML.gif)
For each let us consider the following nonsingular problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ43_HTML.gif)
Obviously, it follows from the proof of Lemma 1.3 that problem (2.30) is equivalent to the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ45_HTML.gif)
From this it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ46_HTML.gif)
Suppose by contradiction that is not satisfied on
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ47_HTML.gif)
therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ48_HTML.gif)
Since by the definition of and (2.30) we obviously have
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ49_HTML.gif)
So,when , we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ50_HTML.gif)
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
3. Proof of Theorem 1.5
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ51_HTML.gif)
It follows from (3.1) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ52_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ53_HTML.gif)
consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ54_HTML.gif)
Similarly
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ55_HTML.gif)
From (3.1), (3.4), and (3.5) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ56_HTML.gif)
On the other hand, (3.2), (1.7), and condition yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ57_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ58_HTML.gif)
therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ59_HTML.gif)
From this it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ60_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ61_HTML.gif)
Noticing (3.11) and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ62_HTML.gif)
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.
4. Concerned Remarks and Applications
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ63_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ64_HTML.gif)
On the other hand, for any from (4.1) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ65_HTML.gif)
that is,
In what follows, by using the results obtained in this paper, we study the boundary value problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ66_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ67_HTML.gif)
Moreover, when the positive solution exists, it is unique.
Remark 4.4.
Consider (1.1) and the following singular -point boundary value conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ68_HTML.gif)
By analogous methods, we have the following results.
Assume that is a
positive solution to (1.1) and (4.6), then
can be stated
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ69_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ70_HTML.gif)
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.
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Acknowledgment
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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Du, X., Zhao, Z. Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular -Point Boundary Value Problems.
Bound Value Probl 2009, 191627 (2009). https://doi.org/10.1155/2009/191627
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DOI: https://doi.org/10.1155/2009/191627