Open Access

A Fourth-Order Boundary Value Problem with One-Sided Nagumo Condition

Boundary Value Problems20112011:569191

DOI: 10.1155/2011/569191

Received: 10 January 2011

Accepted: 9 March 2011

Published: 14 March 2011

Abstract

The aim of this paper is to study a fourth-order separated boundary value problem with the right-hand side function satisfying one-sided Nagumo-type condition. By making a series of a priori estimates and applying lower and upper functions techniques and Leray-Schauder degree theory, the authors obtain the existence and location result of solutions to the problem.

1. Introduction

In this paper we apply the lower and upper functions method to study the fourth-order nonlinear equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ1_HTML.gif
(1.1)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq1_HTML.gif being a continuous function.

This equation can be used to model the deformations of an elastic beam, and the type of boundary conditions considered depends on how the beam is supported at the two endpoints [1, 2]. We consider the separated boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ2_HTML.gif
(1.2)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq3_HTML.gif .

For the fourth-order differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ3_HTML.gif
(1.3)

the authors in [3] obtained the existence of solutions with the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq4_HTML.gif satisfies the two-sided Nagumo-type conditions. For more related works, interested readers may refer to [114]. The one-sided Nagumo-type condition brings some difficulties in studying this kind of problem, as it can be seen in [1518].

Motivated by the above works, we consider the existence of solutions when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq5_HTML.gif satisfies one-sided Nagumo-type conditions. This is a generalization of the above cases. We apply lower and upper functions technique and topological degree method to prove the existence of solutions by making a priori estimates for the third derivative of all solutions of problems (1.1) and (1.2). The estimates are essential for proving the existence of solutions.

The outline of this paper is as follows. In Section 2, we give the definition of lower and upper functions to problems (1.1) and (1.2) and obtain some a priori estimates. Section 3 will be devoted to the study of the existence of solutions. In Section 4, we give an example to illustrate the conclusions.

2. Definitions and A Priori Estimates

Upper and lower functions will be an important tool to obtain a priori bounds on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq7_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq8_HTML.gif . For this problem we define them as follows.

Definition 2.1.

The functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq9_HTML.gif verifying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ4_HTML.gif
(2.1)

define a pair of lower and upper functions of problems (1.1) and (1.2) if the following conditions are satisfied:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq11_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq13_HTML.gif ,

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq14_HTML.gif .

Remark 2.2.

By integration, from (iii) and (2.1), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ5_HTML.gif
(2.2)

that is, lower and upper functions, and their first derivatives are also well ordered.

To have an a priori estimate on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq15_HTML.gif , we need a one-sided Nagumo-type growth condition, which is defined as follows.

Definition 2.3.

Given a set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq16_HTML.gif , a continuous https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq17_HTML.gif is said to satisfy the one-sided Nagumo-type condition in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq18_HTML.gif if there exists a real continuous function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq19_HTML.gif , for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq20_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ6_HTML.gif
(2.3)
with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ7_HTML.gif
(2.4)

Lemma 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq21_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ8_HTML.gif
(2.5)
and consider the set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ9_HTML.gif
(2.6)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq22_HTML.gif be a continuous function satisfying one-sided Nagumo-type condition in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq23_HTML.gif .

Then, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq24_HTML.gif , there exists an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq25_HTML.gif such that for every solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq26_HTML.gif of problems (1.1) and (1.2) with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ10_HTML.gif
(2.7)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ11_HTML.gif
(2.8)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq27_HTML.gif and every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq28_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq29_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq30_HTML.gif be a solution of problems (1.1) and (1.2) such that (2.7) and (2.8) hold. Define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ12_HTML.gif
(2.9)
Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq31_HTML.gif , and suppose, for contradiction, that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq32_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq33_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq34_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq35_HTML.gif , then we obtain the following contradiction:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ13_HTML.gif
(2.10)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq36_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq37_HTML.gif , a similar contradiction can be derived. So there is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq38_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq39_HTML.gif . By (2.4) we can take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq40_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ14_HTML.gif
(2.11)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq41_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq42_HTML.gif , then we have trivially https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq43_HTML.gif . If not, then we can take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq44_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq45_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq46_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq47_HTML.gif . Suppose that the first case holds. By (2.7) we can consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq48_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ15_HTML.gif
(2.12)
Applying a convenient change of variable, we have, by (2.3) and (2.11),
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ16_HTML.gif
(2.13)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq49_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq50_HTML.gif can be taken arbitrarily as long as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq51_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq52_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq53_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq54_HTML.gif .

In a similar way, it can be proved that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq55_HTML.gif , for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq56_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq57_HTML.gif . Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ17_HTML.gif
(2.14)
Consider now the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq58_HTML.gif , and take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq59_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ18_HTML.gif
(2.15)
In a similar way, we may show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ19_HTML.gif
(2.16)

Taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq60_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq61_HTML.gif .

Remark 2.5.

Observe that the estimation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq62_HTML.gif depends only on the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq63_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq64_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq65_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq66_HTML.gif and it does not depend on the boundary conditions.

3. Existence and Location Result

In the presence of an ordered pair of lower and upper functions, the existence and location results for problems (1.1) and (1.2) can be obtained.

Theorem 3.1.

Suppose that there exist lower and upper functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq67_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq68_HTML.gif of problems (1.1) and (1.2), respectively. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq69_HTML.gif be a continuous function satisfying the one-sided Nagumo-type conditions (2.3) and (2.4) in
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ20_HTML.gif
(3.1)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq70_HTML.gif verifies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ21_HTML.gif
(3.2)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq71_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ22_HTML.gif
(3.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq72_HTML.gif means https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq74_HTML.gif , then problems (1.1) and (1.2) has at least one solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq75_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ23_HTML.gif
(3.4)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq76_HTML.gif .

Proof.

Define the auxiliary functions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ24_HTML.gif
(3.5)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq77_HTML.gif , consider the homotopic equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ25_HTML.gif
(3.6)
with the boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ26_HTML.gif
(3.7)
Take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq78_HTML.gif large enough such that, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq79_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ27_HTML.gif
(3.8)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ28_HTML.gif
(3.9)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ29_HTML.gif
(3.10)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ30_HTML.gif
(3.11)

Step 1.

Every solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq80_HTML.gif of problems (3.6) and (3.7) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ31_HTML.gif
(3.12)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq81_HTML.gif , for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq82_HTML.gif independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq83_HTML.gif .

Assume, for contradiction, that the above estimate does not hold for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq84_HTML.gif . So there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq85_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq86_HTML.gif , and a solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq87_HTML.gif of (3.6) and (3.7) such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq88_HTML.gif . In the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq89_HTML.gif define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ32_HTML.gif
(3.13)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq90_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq91_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq92_HTML.gif . Then, by (3.2) and (3.10), for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq93_HTML.gif , the following contradiction is obtained:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ33_HTML.gif
(3.14)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq94_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ34_HTML.gif
(3.15)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq95_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ35_HTML.gif
(3.16)
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq96_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq97_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq98_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq99_HTML.gif . Therefore, the above computations with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq100_HTML.gif replaced by 0 yield a contradiction. For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq101_HTML.gif , by (3.11), we get the following contradiction:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ36_HTML.gif
(3.17)

The case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq102_HTML.gif is analogous. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq103_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq104_HTML.gif . In a similar way, we may prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq105_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq106_HTML.gif .

By the boundary condition (3.7) there exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq107_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq108_HTML.gif . Then by integration we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ37_HTML.gif
(3.18)

Step 2.

There is an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq109_HTML.gif such that for every solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq110_HTML.gif of problems (3.6) and (3.7)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ38_HTML.gif
(3.19)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq111_HTML.gif independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq112_HTML.gif .

Consider the set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ39_HTML.gif
(3.20)
and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq113_HTML.gif the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq114_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ40_HTML.gif
(3.21)
In the following we will prove that the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq115_HTML.gif satisfies the one-sided Nagumo-type conditions (2.3) and (2.4) in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq116_HTML.gif independently of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq117_HTML.gif . Indeed, as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq118_HTML.gif verifies (2.3) in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq119_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ41_HTML.gif
(3.22)
So, defining https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq120_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq121_HTML.gif , we see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq122_HTML.gif verifies (2.3) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq123_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq124_HTML.gif replaced by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq126_HTML.gif , respectively. The condition (2.4) is also verified since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ42_HTML.gif
(3.23)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq127_HTML.gif satisfies the one-sided Nagumo-type condition in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq128_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq129_HTML.gif replaced by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq130_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq131_HTML.gif independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq132_HTML.gif .

Moreover, for
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ43_HTML.gif
(3.24)
every solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq133_HTML.gif of (3.6) and (3.7) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ44_HTML.gif
(3.25)
Define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ45_HTML.gif
(3.26)

The hypotheses of Lemma 2.4 are satisfied with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq134_HTML.gif replaced by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq135_HTML.gif . So there exists an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq136_HTML.gif , depending on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq137_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq138_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq139_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq140_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq142_HTML.gif do not depend on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq143_HTML.gif , we see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq144_HTML.gif is maybe independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq145_HTML.gif .

Step 3.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq146_HTML.gif , the problems (3.6) and (3.7) has at least one solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq147_HTML.gif .

Define the operators
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ46_HTML.gif
(3.27)
by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ47_HTML.gif
(3.28)
and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq148_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq149_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ48_HTML.gif
(3.29)
with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ49_HTML.gif
(3.30)
Observe that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq150_HTML.gif has a compact inverse. Therefore, we can consider the completely continuous operator
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ50_HTML.gif
(3.31)
given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ51_HTML.gif
(3.32)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq151_HTML.gif given by Step 2, take the set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ52_HTML.gif
(3.33)
By Steps 1 and 2, degree https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq152_HTML.gif is well defined for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq153_HTML.gif and by the invariance with respect to a homotopy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ53_HTML.gif
(3.34)
The equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq154_HTML.gif is equivalent to the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ54_HTML.gif
(3.35)
and has only the trivial solution. Then, by the degree theory,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ55_HTML.gif
(3.36)
So the equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq155_HTML.gif has at least one solution, and therefore the equivalent problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ56_HTML.gif
(3.37)

has at least one solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq156_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq157_HTML.gif .

Step 4.

The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq158_HTML.gif is a solution of the problems (1.1) and (1.2).

The proof will be finished if the above function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq159_HTML.gif satisfies the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ57_HTML.gif
(3.38)
Assume, for contradiction, that there is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq160_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq161_HTML.gif , and define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ58_HTML.gif
(3.39)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq162_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq164_HTML.gif . Therefore, by (3.2) and Definition 2.1, we obtain the contradiction
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ59_HTML.gif
(3.40)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq165_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ60_HTML.gif
(3.41)
By Definition 2.1 this yields a contradiction
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ61_HTML.gif
(3.42)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq166_HTML.gif and, by similar arguments, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq167_HTML.gif . Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ62_HTML.gif
(3.43)
Using an analogous technique, it can be deduced that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq168_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq169_HTML.gif . So we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ63_HTML.gif
(3.44)
On the other hand, by (1.2),
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ64_HTML.gif
(3.45)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ65_HTML.gif
(3.46)
Applying the same technique, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ66_HTML.gif
(3.47)
and then by Definition 2.1 (iii), (3.44) and (3.46), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ67_HTML.gif
(3.48)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ68_HTML.gif
(3.49)
Since, by (3.44), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq170_HTML.gif is nondecreasing, we have by (3.49)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ69_HTML.gif
(3.50)
and, therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq171_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq172_HTML.gif . By the monotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq173_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ70_HTML.gif
(3.51)

and so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq174_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq175_HTML.gif .

The inequalities https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq176_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq177_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq178_HTML.gif can be proved in the same way. Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq179_HTML.gif is a solution of problems (1.1) and (1.2).

4. An Example

The following example shows the applicability of Theorem 3.1 when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq180_HTML.gif satisfies only the one-sided Nagumo-type condition.

Example 4.1.

Consider now the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ71_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ72_HTML.gif
(4.2)
with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq181_HTML.gif . The nonlinear function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ73_HTML.gif
(4.3)
is continuous in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq182_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq183_HTML.gif , then the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq184_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ74_HTML.gif
(4.4)
are, respectively, lower and upper functions of (4.1) and (4.2). Moreover, define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ75_HTML.gif
(4.5)

Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq185_HTML.gif satisfies condition (3.2) and the one-sided Nagumo-type condition with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq186_HTML.gif , in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq187_HTML.gif .

Therefore, by Theorem 3.1, there is at least one solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq188_HTML.gif of Problem (4.1) and (4.2) such that, for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_IEq189_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ76_HTML.gif
(4.6)
Notice that the function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F569191/MediaObjects/13661_2011_Article_46_Equ77_HTML.gif
(4.7)

does not satisfy the two-sided Nagumo condition.

Declarations

Acknowledgments

The authors would like to thank the referees for their valuable comments on and suggestions regarding the original manuscript. This work was supported by NSFC (10771085), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education, and by the 985 Program of Jilin University.

Authors’ Affiliations

(1)
Institute of Mathematics, Jilin University
(2)
Institute of Applied Mathematics, Jilin University of Finance and Economics

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© W. Song and W. Gao. 2011

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