The Existence of Countably Many Positive Solutions for Nonlinear http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq1_HTML.gif th-Order Three-Point Boundary Value Problems

Boundary Value Problems20092009:572512

DOI: 10.1155/2009/572512

Received: 5 July 2009

Accepted: 30 October 2009

Published: 4 November 2009

Abstract

We consider the existence of countably many positive solutions for nonlinear http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq2_HTML.gif th-order three-point boundary value problem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq7_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq9_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq10_HTML.gif and has countably many singularities in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq11_HTML.gif . The associated Green's function for the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq12_HTML.gif th-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq13_HTML.gif which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone.

1. Introduction

The existence of positive solutions for nonlinear second-order and higher-order multipoint boundary value problems has been studied by several authors, for example, see [112] and the references therein. However, there are a few papers dealing with the existence of positive solutions for the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq14_HTML.gif th-order multipoint boundary value problems with infinitely many singularities. Hao et al. [13] discussed the existence and multiplicity of positive solutions for the following http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq15_HTML.gif th-order nonlinear singular boundary value problems:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq16_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq18_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq19_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq20_HTML.gif . Hao et al. established the existence of at least two positive solution for the boundary value problems if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq21_HTML.gif is either superlinear or sublinear by applying the Krasnosel'skii-Guo theorem on cone expansion and compression.

In [14], Kaufmann and Kosmatov showed that there exist countably many positive solutions for the two-point boundary value problems with infinitely many singularities of following form:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq22_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq23_HTML.gif and has countably many singularities in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq24_HTML.gif .

In [15], Ji and Guo proved the existence of countably many positive solutions for the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq25_HTML.gif th-order ordinary differential equation

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ3_HTML.gif
(1.3)

with one of the following http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq26_HTML.gif -point boundary conditions:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ4_HTML.gif
(1.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq27_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq28_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq29_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq31_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq33_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq35_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq36_HTML.gif and has countably many singularities in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq37_HTML.gif .

Motivated by the result of [1315], in this paper we are interested in the existence of countably many positive solutions for nonlinear http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq38_HTML.gif th-order three-point boundary value problem

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ5_HTML.gif
(1.5)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq41_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq44_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq46_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq48_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq49_HTML.gif and has countably many singularities in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq50_HTML.gif . We show that the problem (1.5) has countably many solutions if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq52_HTML.gif satisfy some suitable conditions. Our approach is based on the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem in cones.

Suppose that the following conditions are satisfied.

There exists a sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq54_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq55_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq56_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq57_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq58_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq59_HTML.gif

There exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq61_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq62_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq63_HTML.gif .

Assuming that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq64_HTML.gif satisfies the conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq65_HTML.gif - http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq66_HTML.gif (we cite [15, Example http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq67_HTML.gif ] to verify existence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq68_HTML.gif ) and imposing growth conditions on the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq69_HTML.gif , it will be shown that problem (1.5) has infinitely many solutions.

The paper is organized as follows. In Section 2, we provide some necessary background material such as the Krasnosel'skii fixed-point theorem and Leggett-Williams fixed point theorem in cones. In Section 3, the associated Green's function for the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq70_HTML.gif th-order three-point boundary value problem is first given and we also look at some properties of the Green's function associated with problem (1.5). In Section 4, we prove the existence of countably many positive solutions for problem (1.5) under suitable conditions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq72_HTML.gif . In Section 5, we give two simple examples to illustrate the applications of obtained results.

2. Preliminary Results

Definition 2.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq73_HTML.gif be a Banach space over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq74_HTML.gif . A nonempty convex closed set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq75_HTML.gif is said to be a cone provided that

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq76_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq77_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq78_HTML.gif ;

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq79_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq80_HTML.gif .

Definition 2.2.

The map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq81_HTML.gif is said to be a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq82_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq83_HTML.gif is continuous and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ6_HTML.gif
(2.1)
for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq85_HTML.gif Similarly, we say that the map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq86_HTML.gif is a nonnegative continuous convex functional on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq87_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq88_HTML.gif is continuous and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ7_HTML.gif
(2.2)

for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq90_HTML.gif

Definition 2.3.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq91_HTML.gif be given and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq92_HTML.gif be a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq93_HTML.gif . Define the convex sets http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq95_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ8_HTML.gif
(2.3)

The following Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem play an important role in this paper.

Theorem 2.4 ([16], Krasnosel'skii fixed point theorem).

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq96_HTML.gif be a Banach space and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq97_HTML.gif be a cone. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq98_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq99_HTML.gif are bounded open subsets of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq100_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq101_HTML.gif . Suppose that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ9_HTML.gif
(2.4)

is a completely continuous operator such that, either

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq102_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq103_HTML.gif , or

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq104_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq105_HTML.gif .

Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq106_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq107_HTML.gif .

Theorem 2.5 ([17], Leggett-Williams fixed point theorem).

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq108_HTML.gif be a completely continuous operator and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq109_HTML.gif be a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq110_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq111_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq112_HTML.gif . Suppose there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq113_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq115_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq116_HTML.gif  for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq117_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq119_HTML.gif  for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq120_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq122_HTML.gif  for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq123_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq124_HTML.gif .

Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq125_HTML.gif has at least three fixed points http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq126_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq127_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ10_HTML.gif
(2.5)

In order to establish some of the norm inequalities in Theorems 2.4 and 2.5 we will need Holder's inequality. We use standard notation of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq128_HTML.gif for the space of measurable functions such that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ11_HTML.gif
(2.6)

where the integral is understood in the Lebesgue sense. The norm on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq129_HTML.gif , is defined by

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ12_HTML.gif
(2.7)

Theorem 2.6 ([18], Holder's inequality).

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq131_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq132_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq133_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq134_HTML.gif and, moreover
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ13_HTML.gif
(2.8)
Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq135_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq136_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq137_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ14_HTML.gif
(2.9)

3. Preliminary Lemmas

To prove the main results, we need the following lemmas.

Lemma 3.1 (see [15]).

For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq138_HTML.gif the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ15_HTML.gif
(3.1)
has a unique solution
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ16_HTML.gif
(3.2)

Lemma 3.2 (see [15]).

The Green's function for the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ17_HTML.gif
(3.3)
is given by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ18_HTML.gif
(3.4)

Lemma 3.3 (see [15]).

The Green's function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq139_HTML.gif defined by (3.4) satisfies that

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq140_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq141_HTML.gif ;

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq142_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq143_HTML.gif and there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq144_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq145_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ19_HTML.gif
(3.5)
where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ20_HTML.gif
(3.6)

Lemma 3.4.

Suppose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq146_HTML.gif then for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq147_HTML.gif the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ21_HTML.gif
(3.7)
has a unique solution
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ22_HTML.gif
(3.8)

Proof.

The general solution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq148_HTML.gif can be written as
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ23_HTML.gif
(3.9)
Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq149_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq150_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq151_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq152_HTML.gif . Now we solve for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq153_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq155_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ24_HTML.gif
(3.10)
By solving the above equations, we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ25_HTML.gif
(3.11)
Therefore, (3.7) has a unique solution
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ26_HTML.gif
(3.12)

Lemma 3.5.

Suppose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq156_HTML.gif , the Green's function for the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ27_HTML.gif
(3.13)
is given by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ28_HTML.gif
(3.14)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq157_HTML.gif is defined by (3.4).

We omit the proof as it is immediate from Lemma 3.4 and (3.4).

Lemma 3.6.

Suppose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq158_HTML.gif , the Green's function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq159_HTML.gif defined by (3.14) satisfies that

(i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq160_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq161_HTML.gif ;

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq162_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq163_HTML.gif and there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq164_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq165_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ29_HTML.gif
(3.15)
where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ30_HTML.gif
(3.16)

Proof.

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq166_HTML.gif From Lemma 3.3 and (3.14), we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ31_HTML.gif
(3.17)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq167_HTML.gif From Lemma 3.3 and (3.14), we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ32_HTML.gif
(3.18)

Next, we prove that (3.15) holds.

From Lemma 3.3 and (3.14), for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq168_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ33_HTML.gif
(3.19)

for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq169_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq171_HTML.gif .

We use inequality (3.15) to define our cones. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq172_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq173_HTML.gif is a Banach space with the norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq174_HTML.gif . For a fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq175_HTML.gif , define the cone http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq176_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ34_HTML.gif
(3.20)
Define the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq177_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ35_HTML.gif
(3.21)

Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq178_HTML.gif is a solution of (1.5) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq179_HTML.gif is a fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq180_HTML.gif .

Theorems 2.4 and 2.5 require the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq181_HTML.gif to be completely continuous and cone preserving. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq182_HTML.gif is continuous and compact, then it is completely continuous. The next lemma shows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq183_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq184_HTML.gif and that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq185_HTML.gif is continuous and compact.

Lemma 3.7.

The operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq186_HTML.gif is completely continuous and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq187_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq188_HTML.gif .

Proof.

Fix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq189_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq190_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq191_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq192_HTML.gif and since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq193_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq194_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq195_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq196_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq197_HTML.gif , by (3.15) and (3.21) we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ36_HTML.gif
(3.22)
for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq198_HTML.gif . Thus
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ37_HTML.gif
(3.23)

Clearly operator (3.21) is continuous. By the Arzela-Ascoli theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq199_HTML.gif is compact. Hence, the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq200_HTML.gif is completely continuous and the proof is complete.

4. Main Results

In this section we present that problem (1.5) has countably many solutions if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq201_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq202_HTML.gif satisfy some suitable conditions.

For convenience, we denote

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ38_HTML.gif
(4.1)

Theorem 4.1.

Suppose conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq203_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq204_HTML.gif hold, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq205_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq206_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq207_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq208_HTML.gif be such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ39_HTML.gif
(4.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq209_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq210_HTML.gif . Furthermore, for each natural number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq211_HTML.gif , assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq212_HTML.gif satisfies the following two growth conditions:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq214_HTML.gif  for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq215_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq217_HTML.gif  for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq218_HTML.gif .

Then problem (1.5) has countably many positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq219_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq220_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq221_HTML.gif

Proof.

Consider the sequences http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq222_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq223_HTML.gif of open subsets of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq224_HTML.gif defined by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ40_HTML.gif
(4.3)
Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq225_HTML.gif be as in the hypothesis and note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq226_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq227_HTML.gif . For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq228_HTML.gif , define the cone http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq229_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ41_HTML.gif
(4.4)
Fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq230_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq231_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq232_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ42_HTML.gif
(4.5)
By condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq233_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ43_HTML.gif
(4.6)
Now let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq234_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq235_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq236_HTML.gif . By condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq237_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ44_HTML.gif
(4.7)

It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq238_HTML.gif . Therefore, by Theorem 2.4, the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq239_HTML.gif has at least one fixed point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq240_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq241_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq242_HTML.gif was arbitrary, Theorem 4.1 is completed.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq243_HTML.gif is defined by Theorem 4.1. We define the nonnegative continuous concave functionals http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq244_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq245_HTML.gif by

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ45_HTML.gif
(4.8)

We observe here that, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq246_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq247_HTML.gif .

For convenience, we denote

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ46_HTML.gif
(4.9)

Theorem 4.2.

Suppose conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq248_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq249_HTML.gif hold, let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq250_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq251_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq252_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq253_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq254_HTML.gif be such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ47_HTML.gif
(4.10)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq255_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq256_HTML.gif . Furthermore,  for each natural number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq257_HTML.gif , assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq258_HTML.gif satisfies the following growth conditions:

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq260_HTML.gif  for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq261_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq263_HTML.gif  for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq264_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq266_HTML.gif  for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq267_HTML.gif .

Then problem (1.5) has three infinite families of solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq268_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq269_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ48_HTML.gif
(4.11)

for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq270_HTML.gif

Proof.

We note first that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq271_HTML.gif is completely continuous operator. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq272_HTML.gif , then from properties of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq273_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq274_HTML.gif , and by Lemma 3.7, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq275_HTML.gif . Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq276_HTML.gif .

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq277_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq278_HTML.gif , and by condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq279_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ49_HTML.gif
(4.12)

Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq280_HTML.gif . Standard applications of Arzela-Ascoli theorem imply that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq281_HTML.gif is completely continuous operator.

In a completely analogous argument, condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq282_HTML.gif implies that condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq283_HTML.gif of Theorem 2.5 is satisfied.

We now show that condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq284_HTML.gif of Theorem 2.5 is satisfied. Clearly,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ50_HTML.gif
(4.13)
If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq285_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq286_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq287_HTML.gif . By condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq288_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ51_HTML.gif
(4.14)

Therefore, condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq289_HTML.gif of Theorem 2.5 is satisfied.

Finally, we show that condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq290_HTML.gif of Theorem 2.5 is also satisfied.

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq291_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq292_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ52_HTML.gif
(4.15)
Therefore, condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq293_HTML.gif is also satisfied. By Theorem 2.5, There exist three infinite families of solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq294_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq295_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq296_HTML.gif for problem (1.5) such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ53_HTML.gif
(4.16)

for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq297_HTML.gif Thus, Theorem 4.2 is completed.

5. Example

In this section, we cite an example (see [15]) to verify existence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq298_HTML.gif , and two simple examples are presented to illustrate the applications for obtained conclusion of Theorems 4.1 and 4.2.

Example 5.1.

As an example of problem (1.5), we mention the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ54_HTML.gif
(5.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq299_HTML.gif is defined by [15, Example http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq300_HTML.gif ] and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq301_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ55_HTML.gif
(5.2)

We notice that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq302_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq303_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq304_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq305_HTML.gif .

If we take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq306_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq307_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq308_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq309_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq310_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq311_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq312_HTML.gif = min http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq313_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq314_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq315_HTML.gif

It follows from a direct calculation that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ56_HTML.gif
(5.3)
so
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ57_HTML.gif
(5.4)
In addition, if we take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq316_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq317_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq318_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq319_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq320_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ58_HTML.gif
(5.5)
and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq321_HTML.gif satisfies the following growth conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ59_HTML.gif
(5.6)

Then all the conditions of Theorem 4.1 are satisfied. Therefore, by Theorem 4.1 we know that problem (5.1) has countably many positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq322_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq323_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq324_HTML.gif

Example 5.2.

As another example of problem (1.5), we mention the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ60_HTML.gif
(5.7)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq325_HTML.gif is defined by [15, Example http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq326_HTML.gif ] and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq327_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ61_HTML.gif
(5.8)

We notice that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq328_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq329_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq330_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq331_HTML.gif .

If we take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq332_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq333_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq334_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq335_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq336_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq337_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq338_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq339_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq340_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq341_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq342_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq343_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq344_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq345_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq346_HTML.gif

It follows from a direct calculation that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ62_HTML.gif
(5.9)
In addition, if we take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq347_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq348_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq349_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq350_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq351_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq352_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ63_HTML.gif
(5.10)
and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq353_HTML.gif satisfies the following growth conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ64_HTML.gif
(5.11)

Then all the conditions of Theorem 4.2 are satisfied. Therefore, by Theorem 4.2 we know that problem (5.7) has countably many positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq354_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ65_HTML.gif
(5.12)

for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq355_HTML.gif

Remark 5.3.

In [812], the existence of solutions for local or nonlocal boundary value problems of higher-order nonlinear ordinary (fractional) differential equations that has been treated did not discuss problems with singularities. In [13], the singularity only allowed to appear at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq356_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq357_HTML.gif , the existence and multiplicity of positive solutions were asserted under suitable conditions on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq358_HTML.gif . Although, [14, 15] seem to have considered the existence of countably many positive solutions for the second-order and higher-order boundary value problems with infinitely many singularities in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq359_HTML.gif . However, in [15], only the boundary conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq360_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq361_HTML.gif have been considered. It is clear that the boundary conditions of Examples 5.1 and 5.2 are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq362_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq363_HTML.gif . Hence, we generalize second-order and higher-order multipoint boundary value problem.

Declarations

Acknowledgments

The project is supported by the Natural Science Foundation of Hebei Province (A2009000664), the Foundation of Hebei Education Department (2008153), the Foundation of Hebei University of Science and Technology (XL2006040), and the National Natural Science Foundation of PR China (10971045).

Authors’ Affiliations

(1)
College of Sciences, Hebei University of Science and Technology

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© Y. Ji and Y. Guo. 2009

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