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Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq1_HTML.gif -Point Boundary Value Problems

Boundary Value Problems20092009:191627

DOI: 10.1155/2009/191627

Received: 2 April 2009

Accepted: 23 November 2009

Published: 1 December 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq2_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq3_HTML.gif and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq4_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq5_HTML.gif -point boundary value problems of the following differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ1_HTML.gif
(1.1)
with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq6_HTML.gif are constants, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq8_HTML.gif satisfies the following hypothesis:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq10_HTML.gif is continuous, nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq11_HTML.gif , and nonincreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq12_HTML.gif for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq13_HTML.gif there exists a real number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq14_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq15_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ3_HTML.gif
(1.3)

there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq16_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq17_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq18_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ4_HTML.gif
(1.4)

Remark 1.1.

(i) Inequality (1.3) implies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ5_HTML.gif
(1.5)

Conversely, (1.5) implies (1.3).

(ii) Inequality (1.4) implies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ6_HTML.gif
(1.6)

Conversely, (1.6) implies (1.4).

Remark 1.2.

It follows from (1.3), (1.4) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ7_HTML.gif
(1.7)

When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq19_HTML.gif is increasing with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq20_HTML.gif , singular nonlinear https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq21_HTML.gif -point boundary value problems have been extensively studied in the literature, see [13]. However, when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq22_HTML.gif is increasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq23_HTML.gif , and is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq24_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq25_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq26_HTML.gif and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq27_HTML.gif

When referring to singularity we mean that the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq28_HTML.gif in (1.1) are allowed to be unbounded at the points https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq29_HTML.gif , and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq30_HTML.gif . A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq31_HTML.gif is called a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq32_HTML.gif (positive) solution to (1.1) and (1.2) if it satisfies (1.1) and (1.2) ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq33_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq34_HTML.gif ). A https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq35_HTML.gif (positive) solution to (1.1) and (1.2) is called a smooth (positive) solution if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq37_HTML.gif both exist ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq38_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq39_HTML.gif ). Sometimes, we also call a smooth solution a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq40_HTML.gif solution. It is worth stating here that a nontrivial https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq41_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq42_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq43_HTML.gif -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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq44_HTML.gif is called a lower solution to the problem (1.1), (1.2), if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq45_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ8_HTML.gif
(1.8)

Upper solution is defined by reversing the above inequality signs. If there exist a lower solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq46_HTML.gif and an upper solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq47_HTML.gif to problem (1.1), (1.2) such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq48_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq49_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq50_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq51_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq52_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq53_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq54_HTML.gif

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ9_HTML.gif
(1.9)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ10_HTML.gif
(1.10)

then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq55_HTML.gif

By integrating (1.9) twice and noting (1.10), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ11_HTML.gif
(1.11)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ12_HTML.gif
(1.12)

In view of (1.11) and the definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq56_HTML.gif , we can obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq57_HTML.gif This completes the proof of Lemma 1.3.

Now we state the main results of this paper as follows.

Theorem 1.4.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq58_HTML.gif holds, then a necessary and sufficient condition for the problem (1.1) and (1.2) to have smooth positive solution is that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ13_HTML.gif
(1.13)

Theorem 1.5.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq59_HTML.gif and (1.13) hold, then the smooth positive solution to problem (1.1) and (1.2) is also the unique https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq60_HTML.gif positive solution.

2. Proof of Theorem 1.4

2.1. The Necessary Condition

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq61_HTML.gif 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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ14_HTML.gif
(2.1)
that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq62_HTML.gif is nonincreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq63_HTML.gif Thus, by the Lebesgue theorem, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ15_HTML.gif
(2.2)
It is well known that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq64_HTML.gif can be stated as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ16_HTML.gif
(2.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ17_HTML.gif
(2.4)
By (2.3) and (1.2) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ18_HTML.gif
(2.5)
therefore because of (2.3) and (2.5),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ19_HTML.gif
(2.6)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq65_HTML.gif is a smooth positive solution to (1.1) and (1.2), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ20_HTML.gif
(2.7)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq66_HTML.gif From (2.6), (2.7) it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ21_HTML.gif
(2.8)
Without loss of generality we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq67_HTML.gif This together with the condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq68_HTML.gif implies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ22_HTML.gif
(2.9)

On the other hand, notice that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq69_HTML.gif is a smooth positive solution to (1.1), (1.2), we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ23_HTML.gif
(2.10)
therefore, there exists a positive number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq70_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq71_HTML.gif Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq73_HTML.gif It follows from (1.7) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ24_HTML.gif
(2.11)
Consequently https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq74_HTML.gif which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ25_HTML.gif
(2.12)
From (2.9) and (2.12) it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ26_HTML.gif
(2.13)

which is the required inequality.

2.2. The Existence of Lower and Upper Solutions

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq75_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq76_HTML.gif thus
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ27_HTML.gif
(2.14)
Otherwise, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq77_HTML.gif then there exists a real number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq78_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq79_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq80_HTML.gif this contradicts with the condition that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq81_HTML.gif is integrable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq82_HTML.gif By condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq83_HTML.gif and (2.14) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ28_HTML.gif
(2.15)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ29_HTML.gif
(2.16)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq84_HTML.gif

Suppose that (1.13) holds. Let

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ30_HTML.gif
(2.17)

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ31_HTML.gif
(2.18)
and there exists a positive number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq85_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ32_HTML.gif
(2.19)
By (2.14) and (2.16) we see, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq86_HTML.gif is sufficiently small, then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ33_HTML.gif
(2.20)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ34_HTML.gif
(2.21)
Then from (2.19) and (2.21) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ35_HTML.gif
(2.22)

Consequently, with the aid of (2.20), (2.22) and the condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq87_HTML.gif we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ36_HTML.gif
(2.23)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ37_HTML.gif
(2.24)
From (2.17), (2.21) it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ38_HTML.gif
(2.25)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ39_HTML.gif
(2.26)

therefore, (2.23)–(2.26) imply that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq88_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq89_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq90_HTML.gif if and only if

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ40_HTML.gif
(2.27)

Then, we will define an auxiliary function. For all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq91_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ41_HTML.gif
(2.28)

By the assumption of Theorem 1.4, we have that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq92_HTML.gif is continuous.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq93_HTML.gif be a sequence satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq95_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq96_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq97_HTML.gif be a sequence satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ42_HTML.gif
(2.29)

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq98_HTML.gif let us consider the following nonsingular problem:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ43_HTML.gif
(2.30)

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ44_HTML.gif
(2.31)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq99_HTML.gif is defined in the proof of Lemma 1.3. It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq100_HTML.gif is a completely continuous operator and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq101_HTML.gif is a bounded set. Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq102_HTML.gif is a solution to (2.30) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq103_HTML.gif Using the Schauder's fixed point theorem, we assert that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq104_HTML.gif has at least one fixed point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq105_HTML.gif

We claim that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ45_HTML.gif
(2.32)

From this it follows that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ46_HTML.gif
(2.33)

Suppose by contradiction that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq106_HTML.gif is not satisfied on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq107_HTML.gif . Let

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ47_HTML.gif
(2.34)
therefore
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ48_HTML.gif
(2.35)

Since by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq108_HTML.gif and (2.30) we obviously have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq109_HTML.gif

Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ49_HTML.gif
(2.36)

So,when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq110_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq111_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ50_HTML.gif
(2.37)

Therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq112_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq113_HTML.gif is an upper convex function in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq114_HTML.gif .

By (2.30) and (2.36), for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq115_HTML.gif we have the following two cases:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq116_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq117_HTML.gif

For case (i): it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq118_HTML.gif this is a contradiction.

For case (ii): in this case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq119_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq120_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq121_HTML.gif , thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq122_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq123_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq124_HTML.gif From https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq125_HTML.gif we see https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq126_HTML.gif which is in contradiction with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq127_HTML.gif

From this it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq128_HTML.gif

Similarly, we can verify that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq129_HTML.gif Consequently (2.32) holds.

Using the method of [4] and [5, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq130_HTML.gif ], we can obtain that there is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq131_HTML.gif positive solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq132_HTML.gif to (1.1), (1.2) such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq133_HTML.gif and a subsequence of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq134_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq135_HTML.gif on any compact subintervals of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq136_HTML.gif

3. Proof of Theorem 1.5

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq137_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq138_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq139_HTML.gif positive solutions to (1.1) and (1.2), and at least one of them is a smooth positive solution. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq140_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq141_HTML.gif without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq142_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq143_HTML.gif Let

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ51_HTML.gif
(3.1)

It follows from (3.1) that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ52_HTML.gif
(3.2)

By (1.2), it is easy to check that there exist the following two possible cases:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq144_HTML.gif

(2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq145_HTML.gif

Assume that case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq146_HTML.gif holds. By https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq147_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq148_HTML.gif it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq149_HTML.gif exist (finite or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq150_HTML.gif ), moreover, one of them must be finite. The same conclusion is also valid for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq151_HTML.gif It follows from (3.2) that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ53_HTML.gif
(3.3)

consequently

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ54_HTML.gif
(3.4)

Similarly

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ55_HTML.gif
(3.5)
From (3.1), (3.4), and (3.5) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ56_HTML.gif
(3.6)

On the other hand, (3.2), (1.7), and condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq152_HTML.gif yield

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ57_HTML.gif
(3.7)

that is,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ58_HTML.gif
(3.8)

therefore

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ59_HTML.gif
(3.9)

From this it follows that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ60_HTML.gif
(3.10)

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq153_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq154_HTML.gif then, by (3.6) we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq155_HTML.gif and then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq156_HTML.gif which imply that there exists a positive number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq157_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq158_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq159_HTML.gif It follows from (3.2) that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq160_HTML.gif therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq161_HTML.gif Substituting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq162_HTML.gif into (1.1) and using condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq163_HTML.gif , we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ61_HTML.gif
(3.11)

Noticing (3.11) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq164_HTML.gif we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ62_HTML.gif
(3.12)

which contradicts with the condition that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq165_HTML.gif Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq166_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq167_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq168_HTML.gif Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq169_HTML.gif , which contradicts with (3.6). So case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq170_HTML.gif is impossible.

By analogous methods, we can obtain a contradiction for case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq171_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq172_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq173_HTML.gif which implies that the result of Theorem 1.5 holds.

4. Concerned Remarks and Applications

Remark 4.1.

The typical function satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq174_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq175_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq176_HTML.gif

Remark 4.2.

Condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq177_HTML.gif 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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ63_HTML.gif
(4.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq179_HTML.gif is nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq180_HTML.gif , nonincreasing on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq181_HTML.gif Clearly, condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq182_HTML.gif is weaker than the above condition (4.1).

In fact, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq183_HTML.gif from (4.1) it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ64_HTML.gif
(4.2)
On the other hand, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq184_HTML.gif from (4.1) it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ65_HTML.gif
(4.3)

that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq185_HTML.gif

In what follows, by using the results obtained in this paper, we study the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ66_HTML.gif
(4.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq186_HTML.gif We have the following theorem.

Theorem 4.3.

A necessary and sufficient condition for problem (4.4) to have smooth positive solution is that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ67_HTML.gif
(4.5)

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

Remark 4.4.

Consider (1.1) and the following singular https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq187_HTML.gif -point boundary value conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ68_HTML.gif
(4.6)

By analogous methods, we have the following results.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq188_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq189_HTML.gif positive solution to (1.1) and (4.6), then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq190_HTML.gif can be stated
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ69_HTML.gif
(4.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq191_HTML.gif is defined in (2.4).

Theorem 4.5.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq192_HTML.gif holds, then a necessary and sufficient condition for the problem (1.1) and (4.6) to have smooth positive solution is that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ70_HTML.gif
(4.8)

Theorem 4.6.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq193_HTML.gif and (4.8) hold, then the smooth positive solution to problem (1.1) and (4.6) is also unique https://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq194_HTML.gif positive solution.

Declarations

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).

Authors’ Affiliations

(1)
School of Mathematics Sciences, Qufu Normal University

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Copyright

© X. Du and Z. Zhao. 2009

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