Open Access

Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales

Boundary Value Problems20092009:937064

DOI: 10.1155/2009/937064

Received: 21 March 2009

Accepted: 1 July 2009

Published: 16 August 2009

Abstract

We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result.

1. Introduction

In this paper, we are concerned with the following singular three-point boundary value problem (BVP for short) for delay higher-order dynamic equations on time scales:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq7_HTML.gif . The functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq8_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq9_HTML.gif is continuous. Our nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq10_HTML.gif may have singularity at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq11_HTML.gif and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq13_HTML.gif may have singularity at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq14_HTML.gif .

To understand the notations used in (1.1), we recall the following definitions which can be found in [1, 2].

(a) A time scale https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq15_HTML.gif is a nonempty closed subset of the real numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq16_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq17_HTML.gif has the topology that it inherits from the real numbers with the standard topology. It follows that the jump operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq18_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ2_HTML.gif
(1.2)

(supplemented by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq20_HTML.gif ) are well defined. The point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq21_HTML.gif is left-dense, left-scattered, right-dense, right-scattered if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq22_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq25_HTML.gif , respectively. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq26_HTML.gif has a left-scattered maximum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq27_HTML.gif (right-scattered minimum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq28_HTML.gif ), define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq29_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq30_HTML.gif ); otherwise, set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq31_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq32_HTML.gif ). By an interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq33_HTML.gif we always mean the intersection of the real interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq34_HTML.gif with the given time scale, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq35_HTML.gif . Other types of intervals are defined similarly.

(b) For a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq37_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq38_HTML.gif -derivative of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq39_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq40_HTML.gif , denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq41_HTML.gif , is the number (provided it exists) with the property that, given any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq42_HTML.gif , there is a neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq43_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq44_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ3_HTML.gif
(1.3)
(c) For a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq46_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq47_HTML.gif -derivative of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq48_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq49_HTML.gif , denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq50_HTML.gif , is the number (provided it exists) with the property that, given any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq51_HTML.gif , there is a neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq52_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq53_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ4_HTML.gif
(1.4)
(d) If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq54_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq55_HTML.gif then we define the integral
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ5_HTML.gif
(1.5)

Theoretically, dynamic equations on time scales can build bridges between continuous and discrete mathematics. Practically, dynamic equations have been proposed as models in the study of insect population models, neural networks, and many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical of biological problems [2]. Hence, two-point and multipoint boundary value problems for dynamic equations on time scales have attracted many researchers' attention (see, e.g., [119] and references therein). Moreover, singular boundary value problems have also been treated in many papers (see, e.g., [4, 5, 1214, 18] and references therein).

In 2004, J. J. DaCunha et al. [13] considered singular second-order three-point boundary value problems on time scales
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ6_HTML.gif
(1.6)

and obtained the existence of positive solutions by using a fixed point theorem due to Gatica et al. [14], where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq56_HTML.gif is decreasing in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq57_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq58_HTML.gif and may have singularity at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq59_HTML.gif .

In 2006, Boey and Wong [11] were concerned with higher-order differential equation on time scales of the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ7_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq60_HTML.gif are fixed integers satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq62_HTML.gif . They obtained some existence theorems of positive solutions by using Krasnosel'skii fixed point theorem.

Recently, Anderson and Karaca [8] studied higher-order three-point boundary value problems on time scales and obtained criteria for the existence of positive solutions.

The purpose of this paper is to investigate further the singular BVP for delay higher-order dynamic equation (1.1). By the use of the fixed point theorem of cone expansion and compression type, results on the existence of positive solutions to the BVP (1.1) are established.

The paper is organized as follows. In Section 2, we give some lemmas, which will be required in the proof of our main theorem. In Section 3, we prove some theorems on the existence of positive solutions for BVP (1.1). Moreover, we give an example to illustrate our main result.

2. Lemmas

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq63_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq64_HTML.gif be Green's function of the following three-point boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ8_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq66_HTML.gif satisfy the following condition:
  1. (C)
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ9_HTML.gif
    (2.2)
     

Throughout the paper, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq67_HTML.gif .

From [8], we know that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq69_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ10_HTML.gif
(2.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ11_HTML.gif
(2.4)

The following four lemmas can be found in [8].

Lemma 2.1.

Suppose that the condition (C) holds. Then the Green function of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq70_HTML.gif in (2.3) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ12_HTML.gif
(2.5)

Lemma 2.2.

Assume that the condition (C) holds. Then Green's function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq71_HTML.gif in (2.3) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ13_HTML.gif
(2.6)

Remark 2.3.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq72_HTML.gif   If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq73_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq74_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq75_HTML.gif is nonincreasing in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq76_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ14_HTML.gif
(2.7)
Therefore, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ15_HTML.gif
(2.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ16_HTML.gif
(2.9)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq77_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq79_HTML.gif satisfy the other cases, then we get that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq80_HTML.gif is nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq81_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ17_HTML.gif
(2.10)

Lemma 2.4.

Assume that (C) holds. Then Green's function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq82_HTML.gif in (2.3) verifies the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ18_HTML.gif
(2.11)

Remark 2.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq83_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq84_HTML.gif , then we find
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ19_HTML.gif
(2.12)
So there exists a misprint on [8, Page 2431, line 23]. From (2.3), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ20_HTML.gif
(2.13)
Consequently, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ21_HTML.gif
(2.14)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq85_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq86_HTML.gif , then, from (2.8), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ22_HTML.gif
(2.15)

Remark 2.6.

If we set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq87_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ23_HTML.gif
(2.16)
Denote
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ24_HTML.gif
(2.17)
Thus we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ25_HTML.gif
(2.18)

Lemma 2.7.

Assume that condition (C) is satisfied. For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq88_HTML.gif as in (2.3), put https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq89_HTML.gif and recursively define
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ26_HTML.gif
(2.19)
for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq90_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq91_HTML.gif is Green's function for the homogeneous problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ27_HTML.gif
(2.20)

Lemma 2.8.

Assume that (C) holds. Denote
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ28_HTML.gif
(2.21)
then Green's function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq92_HTML.gif in Lemma 2.7 satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ29_HTML.gif
(2.22)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ30_HTML.gif
(2.23)

Proof.

We proceed by induction on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq93_HTML.gif . We denote the statement by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq94_HTML.gif . From Lemma 2.7, it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ31_HTML.gif
(2.24)
and from (2.18), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ32_HTML.gif
(2.25)

So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq95_HTML.gif is true.

We now assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq96_HTML.gif is true for some positive integer https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq97_HTML.gif . From Lemma 2.7, it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ33_HTML.gif
(2.26)

So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq98_HTML.gif holds. Thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq99_HTML.gif is true by induction.

Lemma 2.9 (see [20]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq100_HTML.gif be a real Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq101_HTML.gif a cone. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq102_HTML.gif is completely continuous operator such that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq103_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq105_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq106_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq107_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq109_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq110_HTML.gif .

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq111_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq112_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq113_HTML.gif .

3. Main Results

We assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq115_HTML.gif are strictly decreasing and strictly increasing sequences, respectively, with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq116_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq118_HTML.gif . A Banach space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq119_HTML.gif is the set of real-valued continuous (in the topology of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq120_HTML.gif ) functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq121_HTML.gif defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq122_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ34_HTML.gif
(3.1)
Define a cone by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ35_HTML.gif
(3.2)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ36_HTML.gif
(3.3)

Assume that

(C1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq123_HTML.gif is continuous;

(C2) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ37_HTML.gif
(3.4)

for constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq124_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq125_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq126_HTML.gif ;

(C3) the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq127_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq128_HTML.gif is continuous satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ38_HTML.gif
(3.5)

We seek positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq129_HTML.gif , satisfying (1.1). For this end, we transform (1.1) into an integral equation involving the appropriate Green function and seek fixed points of the following integral operator.

Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq130_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ39_HTML.gif
(3.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq131_HTML.gif .

Proposition 3.1.

Let (C1), (C2), and (C3) hold, and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq132_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq133_HTML.gif be fixed constants with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq134_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq135_HTML.gif is completely continuous.

Proof.

We separate the proof into four steps.

Step 1.

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq136_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq137_HTML.gif is bounded.

By condition (C3), there exists some positive integer https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq138_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ40_HTML.gif
(3.7)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ41_HTML.gif
(3.8)
here, we used the fact that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq139_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq140_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ42_HTML.gif
(3.9)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ43_HTML.gif
(3.10)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ44_HTML.gif
(3.11)
Then we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ45_HTML.gif
(3.12)

Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq141_HTML.gif is bounded and well defined.

Step 2.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq142_HTML.gif . For every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq143_HTML.gif , we get from (2.22)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ46_HTML.gif
(3.13)
Then by the above inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ47_HTML.gif
(3.14)

This leads to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq144_HTML.gif .

Step 3.

We will show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq145_HTML.gif is continuous. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq146_HTML.gif be any sequence in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq147_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq148_HTML.gif . Notice also that as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq149_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ48_HTML.gif
(3.15)
Now these together with (C2) and the Lebesgue dominated convergence theorem [10] yield that as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq150_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ49_HTML.gif
(3.16)

Step 4.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq151_HTML.gif is compact.

Define
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ50_HTML.gif
(3.17)
and an operator sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq152_HTML.gif for a fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq153_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ51_HTML.gif
(3.18)
Clearly, the operator sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq154_HTML.gif is compact by using the Arzela-Ascoli theorem [3], for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq155_HTML.gif . We will prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq156_HTML.gif converges uniformly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq157_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq158_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq159_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ52_HTML.gif
(3.19)

From (C1), (C2), and the Lebesgue dominated convergence theorem [10], we see that the right-hand side (3.19) can be sufficiently small for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq160_HTML.gif beingbig enough. Hence the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq161_HTML.gif of compact operators converges uniformly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq162_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq163_HTML.gif so that operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq164_HTML.gif is compact. Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq165_HTML.gif is completely continuous by using the Arzela-Ascoli theorem [3].

Proposition 3.2.

It holds that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq166_HTML.gif is a solution of (1.1)  if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq167_HTML.gif .

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq169_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ53_HTML.gif
(3.20)
and for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq170_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ54_HTML.gif
(3.21)

From [8, Lemma  3.1], we know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq171_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq172_HTML.gif . So we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq173_HTML.gif is the solution of BVP (1.1).

For convenience, we list the following notations and assumptions:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ55_HTML.gif
(3.22)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ56_HTML.gif
(3.23)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ57_HTML.gif
(3.24)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ58_HTML.gif
(3.25)

From condition (C2) and (3.12), we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq174_HTML.gif .

Theorem 3.3.

Assume that there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq175_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq176_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq178_HTML.gif such that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq180_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq181_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq183_HTML.gif .

If (C1), (C2), and (C3) hold, then the boundary value problem (1.1) has at least one positive solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq184_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ59_HTML.gif
(3.26)

Proof.

Define the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq185_HTML.gif by (3.6). From (i) and (3.23), it follows that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq186_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ60_HTML.gif
(3.27)
We claim that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ61_HTML.gif
(3.28)

If it is false, then there exists some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq187_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq188_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq189_HTML.gif which implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq190_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq191_HTML.gif .

Set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ62_HTML.gif
(3.29)
We know from (2.22) and (3.27) that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq192_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ63_HTML.gif
(3.30)
the first inequality of (C2) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ64_HTML.gif
(3.31)

Clearly, (3.31) contradicts (3.29). This means that (3.28) holds.

Next we will show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ65_HTML.gif
(3.32)

Suppose on the contrary that there exists some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq193_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq194_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq195_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq196_HTML.gif , from (i) and (3.24), there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq197_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ66_HTML.gif
(3.33)
and for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq198_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq199_HTML.gif , from (ii), such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ67_HTML.gif
(3.34)
Put
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ68_HTML.gif
(3.35)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq200_HTML.gif , then we take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq201_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq202_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq203_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq204_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq205_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq206_HTML.gif . From (3.33) and (3.34), we find that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ69_HTML.gif
(3.36)

yielding a contradiction with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq207_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq208_HTML.gif . This means that (3.32) holds. Therefore, from (3.28), (3.32) and Lemma 2.9, we conclude that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq209_HTML.gif has at least one fixed point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq210_HTML.gif . From the definition of the cone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq211_HTML.gif and (2.18), we see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq212_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq213_HTML.gif . Thus, Proposition 3.2 implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq214_HTML.gif is a solution of BVP (1.1). So we obtain the desired result.

Adopting the same argument as in Theorem 3.3, we obtain the following results.

Corollary 3.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq215_HTML.gif be as in Theorem 3.3.Suppose that (ii) of Theorem 3.3 holds and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq216_HTML.gif . If (C1), (C2), and (C3) holds, then boundary value problem (1.1) has at least one positive solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq217_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ70_HTML.gif
(3.37)

Theorem 3.5.

Assume that there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq218_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq220_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq221_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq222_HTML.gif such that

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq223_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq224_HTML.gif

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq225_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq226_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq227_HTML.gif .

If (C1), (C2), and (C3) hold, then boundary value problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq228_HTML.gif positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq229_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq230_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ71_HTML.gif
(3.38)

Example 3.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq231_HTML.gif . Consider the following singular three-point boundary value problems for delay four-order dynamic equations:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ72_HTML.gif
(3.39)
where, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq232_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq233_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq234_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq236_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq237_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq238_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ73_HTML.gif
(3.40)
Clearly, we know that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ74_HTML.gif
(3.41)
Simple computations yield
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ75_HTML.gif
(3.42)
Obviously,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ76_HTML.gif
(3.43)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq239_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ77_HTML.gif
(3.44)
Therefore, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ78_HTML.gif
(3.45)
From (3.25), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ79_HTML.gif
(3.46)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ80_HTML.gif
(3.47)
Therefore, by Theorem 3.3, the BVP (3.39) has at least one positive solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq240_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_Equ81_HTML.gif
(3.48)

Declarations

Acknowledgments

The authors would like to thank the referees for helpful comments and suggestions. The work was supported partly by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory of Modern Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).

Authors’ Affiliations

(1)
Department of Mathematics, University of Science and Technology of China
(2)
School of Mathematical Sciences, Fudan University
(3)
Department of Mathematics, Shanghai Jiao Tong University

References

  1. Atici FM, Guseinov GSh: On Green's functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics 2002, 141(1-2):75-99. 10.1016/S0377-0427(01)00437-XMATHMathSciNetView ArticleGoogle Scholar
  2. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Application. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar
  3. Agarwal RP, Bohner M, Rehák P: Half-linear dynamic equations. In Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday. Vol. 1, 2. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:1-57.View ArticleGoogle Scholar
  4. Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions of singular Dirichlet problems on time scales via variational methods. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(2):368-381. 10.1016/j.na.2006.05.014MATHMathSciNetView ArticleGoogle Scholar
  5. Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations. Advances in Difference Equations 2008, 2008:-13.Google Scholar
  6. Ahmad B, Nieto JJ:The monotone iterative technique for three-point second-order integrodifferential boundary value problems with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F937064/MediaObjects/13661_2009_Article_889_IEq241_HTML.gif -Laplacian. Boundary Value Problems 2007, 2007:-9.Google Scholar
  7. Anderson DR: Solutions to second-order three-point problems on time scales. Journal of Difference Equations and Applications 2002, 8(8):673-688. 10.1080/1023619021000000717MATHMathSciNetView ArticleGoogle Scholar
  8. Anderson DR, Karaca IY: Higher-order three-point boundary value problem on time scales. Computers & Mathematics with Applications 2008, 56(9):2429-2443. 10.1016/j.camwa.2008.05.018MATHMathSciNetView ArticleGoogle Scholar
  9. Anderson DR, Smyrlis G: Solvability for a third-order three-point BVP on time scales. Mathematical and Computer Modelling 2009, 49(9-10):1994-2001. 10.1016/j.mcm.2008.11.009MATHMathSciNetView ArticleGoogle Scholar
  10. Aulbach B, Neidhart L: Integration on measure chains. In Proceedings of the 6th International Conference on Difference Equations, 2004, Boca Raton, Fla, USA. CRC Press; 239-252.
  11. Boey KL, Wong PJY: Positive solutions of two-point right focal boundary value problems on time scales. Computers & Mathematics with Applications 2006, 52(3-4):555-576. 10.1016/j.camwa.2006.08.025MATHMathSciNetView ArticleGoogle Scholar
  12. Cabada A, Cid JÁ: Existence of a solution for a singular differential equation with nonlinear functional boundary conditions. Glasgow Mathematical Journal 2007, 49(2):213-224. 10.1017/S0017089507003679MATHMathSciNetView ArticleGoogle Scholar
  13. DaCunha JJ, Davis JM, Singh PK: Existence results for singular three point boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004, 295(2):378-391. 10.1016/j.jmaa.2004.02.049MATHMathSciNetView ArticleGoogle Scholar
  14. Gatica JA, Oliker V, Waltman P: Singular nonlinear boundary value problems for second-order ordinary differential equations. Journal of Differential Equations 1989, 79(1):62-78. 10.1016/0022-0396(89)90113-7MATHMathSciNetView ArticleGoogle Scholar
  15. Henderson J, Tisdell CC, Yin WKC: Uniqueness implies existence for three-point boundary value problems for dynamic equations. Applied Mathematics Letters 2004, 17(12):1391-1395. 10.1016/j.am1.2003.08.015MATHMathSciNetView ArticleGoogle Scholar
  16. Kaufmann ER, Raffoul YN: Positive solutions for a nonlinear functional dynamic equation on a time scale. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(7):1267-1276. 10.1016/j.na.2005.04.031MATHMathSciNetView ArticleGoogle Scholar
  17. Khan RA, Nieto JJ, Otero-Espinar V: Existence and approximation of solution of three-point boundary value problems on time scales. Journal of Difference Equations and Applications 2008, 14(7):723-736. 10.1080/10236190701840906MATHMathSciNetView ArticleGoogle Scholar
  18. Liang J, Xiao T-J, Hao Z-C: Positive solutions of singular differential equations on measure chains. Computers & Mathematics with Applications 2005, 49(5-6):651-663. 10.1016/j.camwa.2004.12.001MATHMathSciNetView ArticleGoogle Scholar
  19. Yaslan İ: Multiple positive solutions for nonlinear three-point boundary value problems on time scales. Computers & Mathematics with Applications 2008, 55(8):1861-1869. 10.1016/j.camwa.2007.07.005MATHMathSciNetView ArticleGoogle Scholar
  20. Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar

Copyright

© Liang-Gen Hu et al. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.