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:

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.

then

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

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.