Open Access

Positive Solutions to Nonlinear First-Order Nonlocal BVPs with Parameter on Time Scales

Boundary Value Problems20112011:198598

DOI: 10.1155/2011/198598

Received: 4 May 2010

Accepted: 3 June 2010

Published: 27 June 2011

Abstract

We discuss the existence of solutions for the first-order multipoint BVPs on time scale https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq1_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq5_HTML.gif is a parameter, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq6_HTML.gif is a fixed number, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq8_HTML.gif is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq9_HTML.gif is regressive and rd-continuous, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq13_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq14_HTML.gif . For suitable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq15_HTML.gif , some existence, multiplicity, and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq16_HTML.gif be a time scale (a nonempty closed subset of the real line https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq17_HTML.gif ). We discuss the existence of positive solutions to the first-order multipoint BVPs on time scale https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq18_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq19_HTML.gif is a fixed number, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq21_HTML.gif is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq22_HTML.gif is regressive and rd-continuous, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq23_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq25_HTML.gif is defined in its standard form; see [1, page 59] for details.

The multipoint boundary value problems arise in a variety of different areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross-section and composed of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq26_HTML.gif parts of different densities can be set up as a multipoint boundary value problem [2]; also many problems in the theory of elastic stability can be handled by a multipoint problem [3]. So, the existence of solutions to multipoint boundary value problems have been studied by many authors; see [413] and the reference therein. Especially, in recent years the existence of positive solutions to multipoint boundary value problems on time scales has caught considerable attention; see [1014]. For other background on dynamic equations on time scales, one can see [1, 1518].

Our ideas arise from [13, 16]. In [13], Tian and Ge discussed the existence of positive solutions to nonlinear first-order three-point boundary value problems on time scale https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq27_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq28_HTML.gif is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq29_HTML.gif is regressive and rd-continuous, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq31_HTML.gif . The existence results are based on Krasnoselskii's fixed-point theorem in cones and Leggett-Williams's theorem.

As we can see, if we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq32_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq34_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq35_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq36_HTML.gif , then (1.1) is reduced to (1.2). Because of the influence of the parameter https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq37_HTML.gif , it will be more difficult to solve (1.1) than to solve (1.2).

In 2009, by using the fixed-point index theory, Sun and Li [16] discussed the existence of positive solutions to the first-order PBVPs on time scale https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq38_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ3_HTML.gif
(1.3)

For suitable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq39_HTML.gif , they gave some existence, multiplicity, and nonexistence criteria of positive solutions.

Motivated by the above results, by using the well-known fixed-point index theory [16, 19], we attempt to obtain some existence, multiplicity and nonexistence criteria of positive solutions to (1.1) for suitable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq40_HTML.gif .

The rest of this paper is arranged as follows. Some preliminary results including Green's function are given in Section 2. In Section 3, we obtain some useful lemmas for the proof of the main result. In Section 4, some existence and multiplicity results are established. At last, some nonexistence results are given in Section 5.

2. Preliminaries

Throughout the rest of this paper, we make the following assumptions:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq42_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq43_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq44_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq46_HTML.gif is rd-continuous, which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq47_HTML.gif (where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq48_HTML.gif is defined in [16, 18, 20]).

Moreover, let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ4_HTML.gif
(2.1)

Our main tool is the well-known results from the fixed-point index, which we state here for the convenience of the reader.

Theorem 2.1 (see [19]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq49_HTML.gif be a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq50_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq51_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq52_HTML.gif , we define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq53_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq54_HTML.gif is completely continuous such https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq55_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq56_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq57_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq58_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ5_HTML.gif
(2.2)
(ii)If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq59_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq60_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ6_HTML.gif
(2.3)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq61_HTML.gif be equipped with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq62_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq63_HTML.gif is a Banach space.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq64_HTML.gif , we consider the following linear BVP:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ7_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ8_HTML.gif
(2.5)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq65_HTML.gif , define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ9_HTML.gif
(2.6)

Lemma 2.2.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq66_HTML.gif , the linear BVP (2.4)-(2.5) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq67_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq68_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ10_HTML.gif
(2.7)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ11_HTML.gif
(2.8)

Proof.

By (2.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ12_HTML.gif
(2.9)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ13_HTML.gif
(2.10)
And so,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ14_HTML.gif
(2.11)
Combining this with (2.5), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ15_HTML.gif
(2.12)

Lemma 2.3.

If the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq69_HTML.gif is defined in (2.7), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq70_HTML.gif may be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ16_HTML.gif
(2.13)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ17_HTML.gif
(2.14)

Proof.

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq71_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ18_HTML.gif
(2.15)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq73_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ19_HTML.gif
(2.16)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq75_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ20_HTML.gif
(2.17)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq77_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ21_HTML.gif
(2.18)

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq78_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq79_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ22_HTML.gif
(2.19)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq81_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ23_HTML.gif
(2.20)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq83_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ24_HTML.gif
(2.21)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq85_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ25_HTML.gif
(2.22)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq87_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ26_HTML.gif
(2.23)

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq88_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ27_HTML.gif
(2.24)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq90_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ28_HTML.gif
(2.25)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq92_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ29_HTML.gif
(2.26)
()For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq94_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ30_HTML.gif
(2.27)

Lemma 2.4.

Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq95_HTML.gif has the following properties.

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq96_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq97_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq98_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq99_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq100_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq101_HTML.gif

Proof.

This proof is similar to [13, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq102_HTML.gif ], so we omit it.

Now, we define a cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq103_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq104_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ31_HTML.gif
(2.28)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq105_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq106_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq108_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq109_HTML.gif , define an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq110_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ32_HTML.gif
(2.29)

Similar to the proof of [13, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq111_HTML.gif ], we can see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq112_HTML.gif is completely continuous. By the above discussions, its not difficult to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq113_HTML.gif being a solution of BVP (1.1) equals the solution that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq114_HTML.gif is a fixed point of the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq115_HTML.gif .

3. Some Lemmas

Lemma 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq116_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq118_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq119_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ33_HTML.gif
(3.1)

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq120_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq121_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq122_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ34_HTML.gif
(3.2)

Lemma 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq123_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq124_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq126_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ35_HTML.gif
(3.3)

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq127_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq128_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq129_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ36_HTML.gif
(3.4)

Lemma 3.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq130_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq131_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ37_HTML.gif
(3.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq132_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq133_HTML.gif .

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq134_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq135_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq136_HTML.gif . So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ38_HTML.gif
(3.6)

4. Some Existence and Multiplicity Results

Theorem 4.1.

Assume that (H1) and (H2) hold and that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq137_HTML.gif . Then the BVP (1.1) has at least two positive solutions for
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ39_HTML.gif
(4.1)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq138_HTML.gif . Then it follows from (4.1) and Lemma 3.3 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ40_HTML.gif
(4.2)
In view of Theorem 2.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ41_HTML.gif
(4.3)
Now, combined with the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq139_HTML.gif , we may choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq140_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq141_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq142_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq143_HTML.gif uniformly, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq144_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ42_HTML.gif
(4.4)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ43_HTML.gif
(4.5)
In view of (4.1), (4.4), (4.5), and Lemma 3.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ44_HTML.gif
(4.6)
It follows from Theorem 2.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ45_HTML.gif
(4.7)
By (4.3) and (4.7), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ46_HTML.gif
(4.8)

This shows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq145_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq146_HTML.gif , which is a positive solution of the BVP (1.1).

Now, by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq147_HTML.gif , there exits an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq148_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq149_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq151_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq152_HTML.gif is chosen so that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ47_HTML.gif
(4.9)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq153_HTML.gif . Then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq154_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq155_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq156_HTML.gif . So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ48_HTML.gif
(4.10)
In view of (4.1), (4.9), and Lemma 3.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ49_HTML.gif
(4.11)
It follows from Theorem 2.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ50_HTML.gif
(4.12)
By (4.3) and (4.12), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ51_HTML.gif
(4.13)

This shows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq157_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq158_HTML.gif , which is another positive solution of the BVP (1.1).

Similar to the proof of Theorem 4.1, we have the following results.

Theorem 4.2.

Suppose that (H1) and (H2) hold and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ52_HTML.gif
(4.14)

Then,

(i)equation (1.1) has at least one positive solution if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq159_HTML.gif ,

(ii)equation (1.1) has at least one positive solution if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq160_HTML.gif ,

(iii)equation (1.1) has at least two positive solutions if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq161_HTML.gif .

Theorem 4.3.

Assume that (H1) and (H2) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq162_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq163_HTML.gif , then the BVP (1.1) has at least two positive solutions for
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ53_HTML.gif
(4.15)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq164_HTML.gif . Then it follows from (4.15) and Lemma 3.3 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ54_HTML.gif
(4.16)
In view of Theorem 2.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ55_HTML.gif
(4.17)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq165_HTML.gif , we may choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq166_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq167_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq169_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq170_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq171_HTML.gif So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ56_HTML.gif
(4.18)
In view of (4.15), (4.18), and Lemma 3.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ57_HTML.gif
(4.19)
It follows from Theorem 2.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ58_HTML.gif
(4.20)
By (4.17) and (4.20), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ59_HTML.gif
(4.21)

This shows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq172_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq173_HTML.gif , which is a positive solution of the BVP (1.1).

Now, by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq174_HTML.gif , there exists an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq175_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq176_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq178_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq179_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ60_HTML.gif
(4.22)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq180_HTML.gif . Then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq181_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq182_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq183_HTML.gif . So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ61_HTML.gif
(4.23)
Combined with (4.22) and Lemma 3.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ62_HTML.gif
(4.24)
It follows from Theorem 2.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ63_HTML.gif
(4.25)
By (4.17) and (4.25), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ64_HTML.gif
(4.26)

This shows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq184_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq185_HTML.gif , which is another positive solution of the BVP (1.1).

Similar to the proof of Theorem 4.3, we have the following results.

Theorem 4.4.

Suppose that (H1) and (H2) hold and that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ65_HTML.gif
(4.27)

Then,

(i)equation (1.1) has at least one positive solution if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq186_HTML.gif ,

(ii)equation (1.1) has at least one positive solution if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq187_HTML.gif ,
  1. (iii)

    equation (1.1) has at least two positive solutions if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq188_HTML.gif .

     

Theorem 4.5.

Suppose that (H1) and (H2) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq189_HTML.gif , then the BVP (1.1) has at least one positive solution for
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ66_HTML.gif
(4.28)

Proof.

We only deal with the case that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq190_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq191_HTML.gif . The other three cases can be discussed similarly.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq192_HTML.gif satisfy (4.28) and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq193_HTML.gif be chosen such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ67_HTML.gif
(4.29)
From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq194_HTML.gif , we know that there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq195_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq196_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq197_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq198_HTML.gif . So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ68_HTML.gif
(4.30)
This combines with (4.29) and Lemma 3.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ69_HTML.gif
(4.31)
It follows from Theorem 2.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ70_HTML.gif
(4.32)
On the other hand, from the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq199_HTML.gif , there exists an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq200_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq201_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq202_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq203_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq204_HTML.gif . Then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq205_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq206_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq207_HTML.gif . So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ71_HTML.gif
(4.33)
Combined with (4.29) and Lemma 3.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ72_HTML.gif
(4.34)
It follows from Theorem 2.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ73_HTML.gif
(4.35)
By (4.32) and (4.35), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ74_HTML.gif
(4.36)

which implies that the BVP (1.1) has at least one positive solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq208_HTML.gif .

Remark 4.6.

By making some minor modifications to the proof of Theorem 4.5, we can obtain the existence of at least one positive solution, if one of the following conditions is satisfied:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq209_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq210_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq211_HTML.gif .

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq212_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq213_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq214_HTML.gif .

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq215_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq217_HTML.gif .

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq218_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq219_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq220_HTML.gif .

Remark 4.7.

From Conditions (ii) and (iv) of Remark 4.6, we know that the conclusion in Theorem 4.5 holds for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq221_HTML.gif in these two cases. By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq222_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq223_HTML.gif , there exist two positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq224_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq225_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ75_HTML.gif
(4.37)
This is the condition of Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq226_HTML.gif of [13]. By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq227_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq228_HTML.gif , there exist two positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq229_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq230_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ76_HTML.gif
(4.38)

This is the condition of Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq231_HTML.gif of [13]. So, our conclusions extend and improve the results of [13].

5. Some Nonexistence Results

Theorem 5.1.

Assume that (H1) and (H2) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq232_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq233_HTML.gif , then the BVP (1.1) has no positive solutions for sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq234_HTML.gif .

Proof.

In view of the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq235_HTML.gif , there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq236_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq237_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq238_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq239_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq240_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ77_HTML.gif
(5.1)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ78_HTML.gif
(5.2)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq241_HTML.gif and we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ79_HTML.gif
(5.3)

We assert that the BVP (1.1) has no positive solutions for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq242_HTML.gif .

Suppose on the contrary that the BVP (1.1) has a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq243_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq244_HTML.gif . Then from (5.3) and Lemma 3.2, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ80_HTML.gif
(5.4)

which is a contradiction.

Theorem 5.2.

Assume that (H1) and (H2) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq245_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq246_HTML.gif , then the BVP (1.1) has no positive solutions for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq247_HTML.gif .

Proof.

By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq248_HTML.gif , there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq249_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq250_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq251_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq252_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq253_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq254_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq255_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ81_HTML.gif
(5.5)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ82_HTML.gif
(5.6)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq256_HTML.gif and we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ83_HTML.gif
(5.7)

We assert that the BVP (1.1) has no positive solutions for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq257_HTML.gif .

Suppose on the contrary that the BVP (1.1) has a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq258_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq259_HTML.gif . Then from (5.7) and Lemma 3.1 we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_Equ84_HTML.gif
(5.8)

which is a contradiction.

Corollary 5.3.

Assume that (H1) and (H2) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq260_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq261_HTML.gif , then the BVP (1.1) has no positive solutions for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2011%2F198598/MediaObjects/13661_2010_Article_29_IEq262_HTML.gif .

Declarations

Acknowledgments

This work was supported by the NSFC Young Item (no. 70901016), HSSF of Ministry of Education of China (no. 09YJA790028), Program for Innovative Research Team of Liaoning Educational Committee (no. 2008T054), the NSF of Liaoning Province (no. L09DJY065), and NWNU-LKQN-09-3

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University
(2)
School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics

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© C. Gao and H. Luo. 2011

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