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The Existence of Countably Many Positive Solutions for Nonlinear https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq7_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq9_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq10_HTML.gif and has countably many singularities in https://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 https://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 https://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 https://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 https://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:

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq18_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq19_HTML.gif and/or https://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 https://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:

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq22_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq23_HTML.gif and has countably many singularities in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq25_HTML.gif th-order ordinary differential equation

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq26_HTML.gif -point boundary conditions:

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

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

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

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq39_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq41_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq43_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq44_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq46_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq47_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq48_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq49_HTML.gif and has countably many singularities in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq51_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq54_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq56_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq57_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq58_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq59_HTML.gif

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

Assuming that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq64_HTML.gif satisfies the conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq65_HTML.gif - https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq66_HTML.gif (we cite [15, Example https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq67_HTML.gif ] to verify existence of https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq71_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq73_HTML.gif be a Banach space over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq74_HTML.gif . A nonempty convex closed set https://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) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq76_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq77_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq78_HTML.gif ;

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

Definition 2.2.

The map https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq82_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq83_HTML.gif is continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ6_HTML.gif
(2.1)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq84_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq85_HTML.gif Similarly, we say that the map https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq87_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq88_HTML.gif is continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ7_HTML.gif
(2.2)

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

Definition 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq91_HTML.gif be given and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq93_HTML.gif . Define the convex sets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq95_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq96_HTML.gif be a Banach space and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq97_HTML.gif be a cone. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq99_HTML.gif are bounded open subsets of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq100_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq101_HTML.gif . Suppose that
https://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) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq102_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq103_HTML.gif , or

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

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq106_HTML.gif has a fixed point in https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq110_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq111_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq112_HTML.gif . Suppose there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq113_HTML.gif such that

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

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

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

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

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq129_HTML.gif , is defined by

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq131_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq132_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq133_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq134_HTML.gif and, moreover
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ13_HTML.gif
(2.8)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq136_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq137_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq138_HTML.gif the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ15_HTML.gif
(3.1)
has a unique solution
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ17_HTML.gif
(3.3)
is given by
https://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 https://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) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq140_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq141_HTML.gif ;

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

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

Lemma 3.4.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq146_HTML.gif then for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq147_HTML.gif the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ21_HTML.gif
(3.7)
has a unique solution
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq148_HTML.gif can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ23_HTML.gif
(3.9)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq149_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq150_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq151_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq152_HTML.gif . Now we solve for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq153_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq154_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq155_HTML.gif , it follows that
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ26_HTML.gif
(3.12)

Lemma 3.5.

Suppose https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ27_HTML.gif
(3.13)
is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ28_HTML.gif
(3.14)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq158_HTML.gif , the Green's function https://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) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq160_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq161_HTML.gif ;

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

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

Proof.

https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ31_HTML.gif
(3.17)
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq168_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ33_HTML.gif
(3.19)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq169_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq170_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq172_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq174_HTML.gif . For a fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq175_HTML.gif , define the cone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq176_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ34_HTML.gif
(3.20)
Define the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq177_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ35_HTML.gif
(3.21)

Obviously, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq179_HTML.gif is a fixed point of operator https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq183_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq184_HTML.gif and that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq186_HTML.gif is completely continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq187_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq188_HTML.gif .

Proof.

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

Let https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ36_HTML.gif
(3.22)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq198_HTML.gif . Thus
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq199_HTML.gif is compact. Hence, the operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq201_HTML.gif and https://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

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

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq217_HTML.gif  for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq219_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq220_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq221_HTML.gif

Proof.

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

It is obvious that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq239_HTML.gif has at least one fixed point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq240_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq241_HTML.gif . Since https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq244_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq245_HTML.gif by

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq246_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq247_HTML.gif .

For convenience, we denote

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

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

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq266_HTML.gif  for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq268_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq269_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ48_HTML.gif
(4.11)

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

Proof.

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

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

Therefore, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq282_HTML.gif implies that condition https://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 https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ50_HTML.gif
(4.13)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq285_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq286_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq287_HTML.gif . By condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq288_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ51_HTML.gif
(4.14)

Therefore, condition https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq291_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq292_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ52_HTML.gif
(4.15)
Therefore, condition https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq294_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq295_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq296_HTML.gif for problem (1.5) such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ53_HTML.gif
(4.16)

for each https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ54_HTML.gif
(5.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq299_HTML.gif is defined by [15, Example https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq300_HTML.gif ] and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq301_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ55_HTML.gif
(5.2)

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

If we take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq306_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq307_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq308_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq309_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq310_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq311_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq312_HTML.gif = min https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq313_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq314_HTML.gif = https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ56_HTML.gif
(5.3)
so
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq316_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq317_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq318_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq319_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq320_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ58_HTML.gif
(5.5)
and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq321_HTML.gif satisfies the following growth conditions:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq322_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq323_HTML.gif for each https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ60_HTML.gif
(5.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq325_HTML.gif is defined by [15, Example https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq326_HTML.gif ] and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq327_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ61_HTML.gif
(5.8)

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

If we take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq332_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq333_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq334_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq335_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq336_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq337_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq338_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq339_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq340_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq341_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq342_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq343_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq344_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq345_HTML.gif = https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq347_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq348_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq349_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq350_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq351_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq352_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_Equ63_HTML.gif
(5.10)
and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq353_HTML.gif satisfies the following growth conditions:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq354_HTML.gif such that

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

for each https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq356_HTML.gif and/or https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq360_HTML.gif or https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F572512/MediaObjects/13661_2009_Article_862_IEq362_HTML.gif and https://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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