Open Access

Nonlinear Boundary Value Problem of First-Order Impulsive Functional Differential Equations

Journal of Inequalities and Applications20102010:490741

DOI: 10.1155/2010/490741

Received: 8 December 2009

Accepted: 30 January 2010

Published: 21 February 2010

Abstract

This paper investigates the nonlinear boundary value problem for a class of first-order impulsive functional differential equations. By establishing a comparison result and utilizing the method of upper and lower solutions, some criteria on the existence of extremal solutions as well as the unique solution are obtained. Examples are discussed to illustrate the validity of the obtained results.

1. Introduction

It is now realized that the theory of impulsive differential equations provides a general framework for mathematical modelling of many real world phenomena. In particular, it serves as an adequate mathematical tool for studying evolution processes that are subjected to abrupt changes in their states. Some typical physical systems that exhibit impulsive behaviour include the action of a pendulum clock, mechanical systems subject to impacts, the maintenance of a species through periodic stocking or harvesting, the thrust impulse maneuver of a spacecraft, and the function of the heart. For an introduction to the theory of impulsive differential equations, refer to [1].

It is also known that the method of upper and lower solutions coupled with the monotone iterative technique is a powerful tool for obtaining existence results of nonlinear differential equations [2]. There are numerous papers devoted to the applications of this method to nonlinear differential equations in the literature, see [39] and references therein. The existence of extremal solutions of impulsive differential equations is considered in papers [311]. However, only a few papers have implemented the technique in nonlinear boundary value problem of impulsive differential equations [5, 12]. In this paper, we will investigate nonlinear boundary value problem of a class of first-order impulsive functional differential equations. Such equations include the retarded impulsive differential equations as special cases [5, 1214].

The rest of this paper is organized as follows. In Section 2, we establish a new comparison principle and discuss the existence and uniqueness of the solution for first order impulsive functional differential equations with linear boundary condition. We then obtain existence results for extremal solutions and unique solution in Section 3 by using the method of upper and lower solutions coupled with monotone iterative technique. To illustrate the obtained results, two examples are discussed in Section 4.

2. Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq3_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq4_HTML.gif . We define that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq5_HTML.gif is continuous for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq6_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq8_HTML.gif exist and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq10_HTML.gif is continuously differentiable for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq11_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq13_HTML.gif exist and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq14_HTML.gif . It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq16_HTML.gif are Banach spaces with respective norms

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ1_HTML.gif
(2.1)

Let us consider the following nonlinear boundary value problem (NBVP):

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ2_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq17_HTML.gif is continuous in the second and the third variables, and for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq21_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq23_HTML.gif is continuous.

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq24_HTML.gif is called a solutions of NBVP (2.2) if it satisfies (2.2).

Remark 2.1.
  1. (i)
    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq25_HTML.gif and the impulses https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq26_HTML.gif depend only on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq27_HTML.gif , the equation of NBVP (2.2) reduces to the simpler case of impulsive differential equations:
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ3_HTML.gif
    (2.3)
     
which have been studied in many papers. In some situation, the impulse https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq28_HTML.gif depends also on some other parameters (e.g., the control of the amount of drug ingested by a patient at certain moments in the model for drug distribution [1, 3]).
  1. (ii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq29_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq30_HTML.gif , the equation of NBVP (2.2) can be regarded as retarded differential equation which has been considered in [5, 1214].

     

We will need the following lemma.

Lemma 2.2 (see [1]).

Asumme that

the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq32_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq33_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq34_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq36_HTML.gif is left continous at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq37_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq38_HTML.gif ,

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq41_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ4_HTML.gif
(2.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq44_HTML.gif are real constants.

Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ5_HTML.gif
(2.5)

In order to establish a comparison result and some lemmas, we will make the following assumptions on the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq45_HTML.gif .

(H1) There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq46_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ6_HTML.gif
(2.6)
(H2) The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq47_HTML.gif satisfies Lipschitz condition, that is, there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq48_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ7_HTML.gif
(2.7)

Inspired by the ideas in [5, 6], we shall establish the following comparison result.

Theorem 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq49_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ8_HTML.gif
(2.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq50_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq51_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq52_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq53_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq54_HTML.gif .

Suppose in addition that condition (H1) holds and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ9_HTML.gif
(2.9)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq55_HTML.gif .

Proof.

For simplicity, we let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq56_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq57_HTML.gif . Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq58_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ10_HTML.gif
(2.10)

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq59_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq60_HTML.gif .

To show https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq61_HTML.gif , we suppose, on the contrary, that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq62_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq63_HTML.gif . It is enough to consider the following cases.

(i)there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq64_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq65_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq66_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq67_HTML.gif ;

(ii)there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq68_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq70_HTML.gif .

Casedi.

By (2.10), we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq71_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq73_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq74_HTML.gif , hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq75_HTML.gif is nonincreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq76_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq77_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq78_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq79_HTML.gif , which is a contradiction. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq80_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq81_HTML.gif which implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq82_HTML.gif . But from (2.10), we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq83_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq84_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq85_HTML.gif . It is again a contradiction.

Casedii.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq86_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq87_HTML.gif . For some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq88_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq89_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq90_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq91_HTML.gif . We only consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq92_HTML.gif , as for the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq93_HTML.gif , the proof is similar.

From (2.10) and condition (H1), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ11_HTML.gif
(2.11)
Consider the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ12_HTML.gif
(2.12)
By Lemma 2.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ13_HTML.gif
(2.13)
that is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ14_HTML.gif
(2.14)
First, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq94_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq95_HTML.gif in (2.14), then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ15_HTML.gif
(2.15)
Noting that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq96_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ16_HTML.gif
(2.16)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ17_HTML.gif
(2.17)

which is a contradiction.

Next, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq97_HTML.gif . By Lemma 2.2 and (2.10), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ18_HTML.gif
(2.18)
then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ19_HTML.gif
(2.19)
Setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq98_HTML.gif in (2.14), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ20_HTML.gif
(2.20)
with (2.19), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ21_HTML.gif
(2.21)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ22_HTML.gif
(2.22)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ23_HTML.gif
(2.23)

which is a contradiction. The proof of Theorem 2.3 is complete.

The following corollary is an easy consequence of Theorem 2.3.

Corollary 2.4.

Assume that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq99_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq100_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq101_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq102_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq103_HTML.gif satisfies (2.8) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq104_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ24_HTML.gif
(2.24)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq105_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq106_HTML.gif .

Remark 2.5.

Setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq107_HTML.gif , Corollary 2.4 reduces to the Theorem 2.3 of Li and Shen [6]. Therefore, Theorem 2.3 and Corollary 2.4 develops and generalizes the result in [6].

Remark 2.6.

We show some examples of function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq108_HTML.gif satisfying (H1).

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq109_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq110_HTML.gif , satisfies (H1) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq111_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ25_HTML.gif
(2.25)
(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq112_HTML.gif , satisfies (H1) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq113_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ26_HTML.gif
(2.26)

Consider the linear boundary value problem (LBVP)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ27_HTML.gif
(2.27)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq114_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq115_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq116_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq117_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq118_HTML.gif .

By direct computation, we have the following result.

Lemma 2.7.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq119_HTML.gif
is a solution of LBVP (2.27) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq120_HTML.gif is a solution of the impulsive integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ28_HTML.gif
(2.28)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq121_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq123_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ29_HTML.gif
(2.29)

Lemma 2.8.

Let (H2) hold. Suppose further
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ30_HTML.gif
(2.30)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq126_HTML.gif , then LBVP (2.27) has a unique solution.

By Lemma 2.7 and Banach fixed point theorem, the proof of Lemma 2.8 is apparent, so we omit the details.

3. Main Results

In this section, we use monotone iterative technique to obtain the existence results of extremal solutions and the unique solution of NBVP (2.2). We shall need the following definition.

Definition 3.1.

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq127_HTML.gif is said to be a lower solution of NBVP (2.2) if it satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ31_HTML.gif
(3.1)
Analogously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq128_HTML.gif is an upper solution of NBVP (2.2) if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ32_HTML.gif
(3.2)

For convenience, let us list the following conditions.

(H3) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq130_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ33_HTML.gif
(3.3)

wherever https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq131_HTML.gif .

(H4) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq132_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq133_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ34_HTML.gif
(3.4)

wherever https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq134_HTML.gif .

(H5) The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq135_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ35_HTML.gif
(3.5)
(H6) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq136_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq137_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq138_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ36_HTML.gif
(3.6)

wherever https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq139_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq140_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq141_HTML.gif . Now we are in the position to establish the main results of this paper.

Theorem 3.2.

Let ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq142_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq143_HTML.gif ) and inequalities (2.9) and (2.30) hold. Assume further that there exist lower and upper solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq145_HTML.gif of NBVP (2.2), respectively, such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq146_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq147_HTML.gif . Then there exist monotone sequences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq148_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq149_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq150_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq151_HTML.gif uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq152_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq153_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq154_HTML.gif are minimal and maximal solutions of NBVP (2.2) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq155_HTML.gif , respectively.

Proof.

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq156_HTML.gif , consider LVBP (2.27) with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ37_HTML.gif
(3.7)

By Lemma 2.8, we know that LBVP (2.27) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq157_HTML.gif . Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq158_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq159_HTML.gif , then the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq160_HTML.gif has the following properties:

(a) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq161_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq162_HTML.gif

(b) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq163_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq164_HTML.gif

To prove (a), let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq166_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ38_HTML.gif
(3.8)

By Theorem 2.3, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq167_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq168_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq169_HTML.gif . Similarly, we can show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq170_HTML.gif .

To prove (b), set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq171_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq173_HTML.gif . Using (H3), (H4) and (H6), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ39_HTML.gif
(3.9)

By Theorem 2.3, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq174_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq175_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq176_HTML.gif , then (b) is proved.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq177_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq178_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq179_HTML.gif By the properties (a) and (b), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ40_HTML.gif
(3.10)
By the definition of operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq180_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq181_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq182_HTML.gif are uniformly bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq183_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq184_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq185_HTML.gif are uniformly bounded and equicontinuous in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq186_HTML.gif . By Arzela-Ascoli Theorem and (3.10), we know that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq187_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq188_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq189_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ41_HTML.gif
(3.11)

Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq190_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq191_HTML.gif are solutions of NBVP (2.2) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq192_HTML.gif .

To prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq194_HTML.gif are extremal solutions of NBVP (2.2), let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq195_HTML.gif be any solution of NBVP (2.2), that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ42_HTML.gif
(3.12)

By Theorem 2.3 and Induction, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq196_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq197_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq198_HTML.gif which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq199_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq200_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq201_HTML.gif are minimal and maximal solution of NBVP (2.2) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq202_HTML.gif , respectively. The proof is complete.

Theorem 3.3.

Let the assumptions of Theorem 3.2 hold and assume the following.

(H7) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq203_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq204_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ43_HTML.gif
(3.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq205_HTML.gif .

(H8) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq206_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq207_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ44_HTML.gif
(3.14)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq208_HTML.gif .

(H9) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq209_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq210_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq211_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ45_HTML.gif
(3.15)

whenever https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq212_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq213_HTML.gif .

Then NBVP (2.2) has a unique solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq214_HTML.gif .

Proof.

By Theorem 3.2, we know that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq215_HTML.gif , which are minimal and maximal solutions of NBVP (2.2) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq216_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq217_HTML.gif . Using (H7), (H8), and (H9), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ46_HTML.gif
(3.16)

By Theorem 2.3, we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq218_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq219_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq220_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq221_HTML.gif , this completes the proof.

4. Examples

To illustrate our main results, we shall discuss in this section some examples.

Example 4.1.

Consider the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ47_HTML.gif
(4.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq223_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq224_HTML.gif .

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ48_HTML.gif
(4.2)

Setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq226_HTML.gif , it is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq227_HTML.gif is a lower solution, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq228_HTML.gif is an upper solution with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq229_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq230_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq231_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ49_HTML.gif
(4.3)
Setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq232_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq233_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq234_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq236_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq237_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq238_HTML.gif , then conditions (H1)–(H6) are all satisfied:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ50_HTML.gif
(4.4)

then inequalities (2.9) and (2.30) are satisfied. By Theorem 3.2, problem (4.1) has extremal solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq239_HTML.gif .

Example 4.2.

Consider the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ51_HTML.gif
(4.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq240_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq241_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq242_HTML.gif .

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ52_HTML.gif
(4.6)

Setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq243_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq244_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq245_HTML.gif is a lower solution, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq246_HTML.gif is an upper solution with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq247_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq248_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq249_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq250_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq251_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq252_HTML.gif . Setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq253_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq254_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq255_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq256_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq257_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq258_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq259_HTML.gif , then conditions (H1)–(H6) are all satisfied:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_Equ53_HTML.gif
(4.7)

then inequalities (2.24) and (2.30) are satisfied. By Corollary 2.4 and Theorem 3.2, problem (4.5) has extremal solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq260_HTML.gif .

Moreover, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq261_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq262_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq263_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq264_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq265_HTML.gif . It is easy to see that conditions (H7)–(H9) are satisfied. By Corollary 2.4 and Theorem 3.3, problem (4.5) has an unique solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F490741/MediaObjects/13660_2009_Article_2168_IEq266_HTML.gif .

Authors’ Affiliations

(1)
School of Control Science and Engineering, Shandong University
(2)
Department of Applied Mathematics, University of Waterloo

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© K. Zhang and X. Liu. 2010

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