Open Access

Positive Solutions of a Nonlinear Three-Point Integral Boundary Value Problem

Boundary Value Problems20102010:519210

DOI: 10.1155/2010/519210

Received: 4 August 2010

Accepted: 18 September 2010

Published: 20 September 2010

Abstract

We study the existence of positive solutions to the three-point integral boundary value problem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq6_HTML.gif . We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones.

1. Introduction

The study of the existence of solutions of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [1]. Then Gupta [2] studied three-point boundary value problems for nonlinear second-order ordinary differential equations. Since then, nonlinear second-order three-point boundary value problems have also been studied by several authors. We refer the reader to [319] and the references therein. However, all these papers are concerned with problems with three-point boundary condition restrictions on the slope of the solutions and the solutions themselves, for example,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ1_HTML.gif
(11)

and so forth.

In this paper, we consider the existence of positive solutions to the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ2_HTML.gif
(12)
with the three-point integral boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ3_HTML.gif
(13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq7_HTML.gif . We note that the new three-point boundary conditions are related to the area under the curve of solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq8_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq9_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq10_HTML.gif .

The aim of this paper is to give some results for existence of positive solutions to (1.2)-(1.3), assuming that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq12_HTML.gif is either superlinear or sublinear. Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ4_HTML.gif
(14)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq13_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq14_HTML.gif correspond to the superlinear case, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq16_HTML.gif correspond to the sublinear case. By the positive solution of (1.2)-(1.3) we mean that a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq17_HTML.gif is positive on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq18_HTML.gif and satisfies the problem (1.2)-(1.3).

Throughout this paper, we suppose the following conditions hold:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq20_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq22_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq23_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq24_HTML.gif .

The proof of the main theorem is based upon an application of the following Krasnoselskii's fixed point theorem in a cone.

Theorem 1.1 (see [20]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq25_HTML.gif be a Banach space, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq26_HTML.gif be a cone. Assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq27_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq28_HTML.gif are open subsets of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq29_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq30_HTML.gif , and let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ5_HTML.gif
(15)

be a completely continuous operator such that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq32_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq34_HTML.gif or

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq36_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq38_HTML.gif .

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq39_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq40_HTML.gif .

2. Preliminaries

We now state and prove several lemmas before stating our main results.

Lemma 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq41_HTML.gif . Then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq42_HTML.gif , the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ6_HTML.gif
(21)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ7_HTML.gif
(22)
has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ8_HTML.gif
(23)

Proof.

From (2.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ9_HTML.gif
(24)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq43_HTML.gif , integration from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq44_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq45_HTML.gif , gives
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ10_HTML.gif
(25)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq46_HTML.gif , integration from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq47_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq48_HTML.gif yields that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ11_HTML.gif
(26)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ12_HTML.gif
(27)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ13_HTML.gif
(28)
Integrating (2.7) from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq49_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq50_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq51_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ14_HTML.gif
(29)
From (2.2), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ15_HTML.gif
(210)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ16_HTML.gif
(211)
Therefore, (2.1)-(2.2) has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ17_HTML.gif
(212)

Lemma 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq52_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq54_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq55_HTML.gif , then the unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq56_HTML.gif of (2.1)-(2.2) satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq57_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq58_HTML.gif .

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq59_HTML.gif , then, by the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq60_HTML.gif and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq61_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq62_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq63_HTML.gif .

Moreover, we know that the graph of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq64_HTML.gif is concave down on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq65_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ18_HTML.gif
(213)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq66_HTML.gif is the area of triangle under the curve https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq67_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq68_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq69_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq70_HTML.gif .

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq71_HTML.gif . From (2.2), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ19_HTML.gif
(214)

By concavity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq73_HTML.gif , it implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq74_HTML.gif .

Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ20_HTML.gif
(215)

which contradicts the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq75_HTML.gif .

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq76_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq77_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq78_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq79_HTML.gif , then (2.1)-(2.2) has no positive solution.

Proof.

Assume (2.1)-(2.2) has a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq80_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq81_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq82_HTML.gif , it implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq83_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ21_HTML.gif
(216)

which contradicts the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq84_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq85_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq86_HTML.gif , this is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq87_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq88_HTML.gif . If there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq89_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq90_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq91_HTML.gif , which contradicts the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq92_HTML.gif . Therefore, no positive solutions exist.

In the rest of the paper, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq93_HTML.gif . Moreover, we will work in the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq94_HTML.gif , and only the sup norm is used.

Lemma 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq95_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq97_HTML.gif , then the unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq98_HTML.gif of the problem (2.1)-(2.2) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ22_HTML.gif
(217)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ23_HTML.gif
(218)

Proof.

Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq99_HTML.gif . We divide the proof into three cases.

Case 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq101_HTML.gif , then the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq102_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ24_HTML.gif
(219)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ25_HTML.gif
(220)

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq104_HTML.gif , then (2.2), (2.13), and the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq105_HTML.gif implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ26_HTML.gif
(221)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ27_HTML.gif
(222)

Case 3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq106_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq107_HTML.gif . Using the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq108_HTML.gif and (2.2), (2.13), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ28_HTML.gif
(223)
This implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ29_HTML.gif
(224)

This completes the proof.

3. Main Results

Now we are in the position to establish the main result.

Theorem 3.1.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq109_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq110_HTML.gif hold. Then the problem (1.2)-(1.3) has at least one positive solution in the case

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq111_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq112_HTML.gif (superlinear), or

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq113_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq114_HTML.gif (sublinear).

Proof.

It is known that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq115_HTML.gif . From Lemma 2.1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq116_HTML.gif is a solution to the boundary value problem (1.2)-(1.3) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq117_HTML.gif is a fixed point of operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq118_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq119_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ30_HTML.gif
(31)
Denote that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ31_HTML.gif
(32)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq120_HTML.gif is defined in (2.18).

It is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq121_HTML.gif is a cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq122_HTML.gif . Moreover, by Lemmas 2.2 and 2.4, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq123_HTML.gif . It is also easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq124_HTML.gif is completely continuous.

Superlinear Case ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq126_HTML.gif )

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq127_HTML.gif , we may choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq128_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq129_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq130_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq131_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ32_HTML.gif
(33)
Thus, if we let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ33_HTML.gif
(34)
then, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq132_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ34_HTML.gif
(35)

Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq133_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq134_HTML.gif .

Further, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq135_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq136_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq137_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq138_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq139_HTML.gif is chosen so that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ35_HTML.gif
(36)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq140_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq141_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq142_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ36_HTML.gif
(37)
and so
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ37_HTML.gif
(38)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq143_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq144_HTML.gif . By the first past of Theorem 1.1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq145_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq146_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq147_HTML.gif .

Sublinear Case ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq148_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq149_HTML.gif )

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq150_HTML.gif , choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq151_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq152_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq153_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq154_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ38_HTML.gif
(39)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ39_HTML.gif
(310)
then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq155_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ40_HTML.gif
(311)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq156_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq157_HTML.gif . Now, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq158_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq159_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq160_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq161_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq162_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ41_HTML.gif
(312)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq163_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ42_HTML.gif
(313)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq164_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ43_HTML.gif
(314)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_Equ44_HTML.gif
(315)

Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq165_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq166_HTML.gif . By the second part of Theorem 1.1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq167_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq168_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq169_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F519210/MediaObjects/13661_2010_Article_933_IEq170_HTML.gif . This completes the sublinear part of the theorem. Therefore, the problem (1.2)-(1.3) has at least one positive solution.

Declarations

Acknowledgments

The authors would like to thank the referee for their comments and suggestions on the paper. Especially, the authors would like to thank Dr. Elvin James Moore for valuable advice. This research is supported by the Centre of Excellence in Mathematics, Thailand.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok
(2)
Centre of Excellence in Mathematics, CHE

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Copyright

© J. Tariboon and T. Sitthiwirattham. 2010

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