Mixed Monotone Iterative Technique for Impulsive Periodic Boundary Value Problems in Banach Spaces

Boundary Value Problems20102011:421261

DOI: 10.1155/2011/421261

Received: 20 April 2010

Accepted: 15 September 2010

Published: 21 September 2010

Abstract

This paper deals with the existence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq1_HTML.gif -quasi-solutions for impulsive periodic boundary value problems in an ordered Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq2_HTML.gif . Under a new concept of upper and lower solutions, a new monotone iterative technique on periodic boundary value problems of impulsive differential equations has been established. Our result improves and extends some relevant results in abstract differential equations.

1. Introduction

The theory of impulsive differential equations is a new and important branch of differential equation theory, which has an extensive physical, chemical, biological, and engineering background and realistic mathematical model, and hence has been emerging as an important area of investigation in the last few decades; see [1]. Correspondingly, applications of the theory of impulsive differential equations to different areas were considered by many authors, and some basic results on impulsive differential equations have been obtained; see [25]. But many of them are about impulsive initial value problem; see [2, 3] and the references therein. The research on impulsive periodic boundary value problems is seldom; see [4, 5].

In this paper, we use a monotone iterative technique in the presence of coupled lower and upper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq3_HTML.gif -quasisolutions to discuss the existence of solutions to the impulsive periodic boundary value problem (IPBVP) in an ordered Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq4_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq7_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq8_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq9_HTML.gif is an impulsive function, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq10_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq11_HTML.gif denotes the jump of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq12_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq13_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq14_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq15_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq16_HTML.gif represent the right and left limits of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq17_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq18_HTML.gif , respectively.

The monotone iterative technique in the presence of lower and upper solutions is an important method for seeking solutions of differential equations in abstract spaces. Early on, Lakshmikantham and Leela [4] built a monotone iterative method for the periodic boundary value problem of first-order differential equation in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq19_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ2_HTML.gif
(1.2)
and they proved that, if PBVP(1.2) has a lower solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq20_HTML.gif and an upper solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq21_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq22_HTML.gif and nonlinear term http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq23_HTML.gif satisfies the monoton condition
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ3_HTML.gif
(1.3)
with a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq24_HTML.gif , then PBVP(1.2) has minimal and maximal solutions between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq26_HTML.gif which can be obtained by a monotone iterative procedure starting from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq27_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq28_HTML.gif , respectively. Later, He and Yu [5] developed the problem to impulsive differential equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ4_HTML.gif
(1.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq31_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq32_HTML.gif

But all of these results are in real spaces http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq33_HTML.gif We not only consider problems in Banach spaces, but also expand the nonlinear term to the case of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq34_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq35_HTML.gif is nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq37_HTML.gif is nonincreasing in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq38_HTML.gif then the monotonity condition (1.3) is not satisfied, and the results in [4, 5] are not right, in this case, we studied the IPBVP(1.1). As far as we know, no work has been done for the existence of solutions for IPBVP(1.1) in Banach spaces.

In order to apply the monotone iterative technique to the initial value problem without impulse
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ5_HTML.gif
(1.5)

Lakshmikantham et al. [6] and Guo and Lakshmikantham [7] obtained the existence of coupled quasisolutions of problem (1.5) by mixed monotone sequence of coupled quasiupper and lower solutions under the concept of quasiupper and lower solutions. In this paper, we improve and extend the above-mentioned results, and obtain the existence of the coupled minimal and maximal http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq39_HTML.gif -quasisolutions and the solutions between the coupled minimal and maximal http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq40_HTML.gif -quasisolutions of the problem (1.1) through the mixed monotone iterative about the coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq41_HTML.gif -quasisolutions. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq42_HTML.gif the coupled upper and lower http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq43_HTML.gif -quasisolutions are equivalent to coupled upper and lower quasisolutions of the IPBVP(1.1). If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq45_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq46_HTML.gif the coupled upper and lower http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq47_HTML.gif -quasisolutions are equivalent to upper and lower solutions of IPBVP(1.4).

2. Preliminaries

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq48_HTML.gif be an ordered Banach space with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq49_HTML.gif and partial order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq50_HTML.gif whose positive cone  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq51_HTML.gif is normal with normal constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq52_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq53_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq54_HTML.gif is a constant; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq55_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq56_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq57_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq58_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq59_HTML.gif is continuous at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq60_HTML.gif , and left continuous at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq61_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq62_HTML.gif exists, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq63_HTML.gif Evidently, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq64_HTML.gif is a Banach space with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq65_HTML.gif . An abstract function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq66_HTML.gif is called a solution of IPBVP(1.1) if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq67_HTML.gif satisfies all the equalities of (1.1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq68_HTML.gif

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq70_HTML.gif exist, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq71_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq72_HTML.gif it is easy to see that the left derivative http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq73_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq74_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq75_HTML.gif exists and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq76_HTML.gif and set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq77_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq78_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq79_HTML.gif is a solution of IPBVP(1.1), by the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq80_HTML.gif

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq81_HTML.gif denote the Banach space of all continuous http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq82_HTML.gif -value functions on interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq83_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq84_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq85_HTML.gif denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [8]. For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq87_HTML.gif set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq88_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq89_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq90_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq91_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq93_HTML.gif

Now, we first give the following lemmas in order to prove our main results.

Lemma 2.1 (see [9]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq94_HTML.gif be a bounded and countable set. Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq95_HTML.gif is Lebesgue integral on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq96_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ6_HTML.gif
(2.1)

Lemma 2.2 (see [10]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq97_HTML.gif be bounded. Then exist a countable set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq98_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq99_HTML.gif

Lemma 2.3 (see [11]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq100_HTML.gif be equicontinuous. Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq101_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq102_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ7_HTML.gif
(2.2)

Lemma 2.4 (see [8]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq103_HTML.gif be a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq104_HTML.gif is a bounded convex closed set in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq105_HTML.gif be condensing, then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq106_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq107_HTML.gif

To prove our main results, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq108_HTML.gif we consider the periodic boundary value problem (PBVP) of linear impulsive differential equation in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq109_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ8_HTML.gif
(2.3)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq110_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq111_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq112_HTML.gif

Lemma 2.5.

For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq113_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq115_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq116_HTML.gif the linear PBVP(2.3) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq117_HTML.gif given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ9_HTML.gif
(2.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq118_HTML.gif

Proof.

For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq120_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq121_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq122_HTML.gif the linear initial value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ10_HTML.gif
(2.5)
has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq123_HTML.gif given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ11_HTML.gif
(2.6)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq124_HTML.gif is a constant [3].

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq125_HTML.gif is a solution of the linear initial value problem (2.5) satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq126_HTML.gif namely
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ12_HTML.gif
(2.7)
then it is the solution of the linear PBVP(2.3). From (2.7), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ13_HTML.gif
(2.8)

So, (2.4) is satisfied.

Inversely, we can verify directly that the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq127_HTML.gif defined by (2.4) is a solution of the linear PBVP(2.3). Therefore, the conclusion of Lemma 2.5 holds.

Definition 2.6.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq128_HTML.gif be a constant. If functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq129_HTML.gif satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ14_HTML.gif
(2.9)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ15_HTML.gif
(2.10)

we call http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq130_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq131_HTML.gif coupled lower and upper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq132_HTML.gif -quasisolutions of the IPBVP(1.1). Only choose " http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq133_HTML.gif " in (2.9) and (2.10), we call http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq134_HTML.gif coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq135_HTML.gif -quasisolution pair of the IPBVP(1.1). Furthermore, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq136_HTML.gif we call http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq137_HTML.gif a solution of the IPBVP(1.1).

Now, we define an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq138_HTML.gif as following:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ16_HTML.gif
(2.11)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ17_HTML.gif
(2.12)

Evidently, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq139_HTML.gif is also an ordered Banach space with the partial order " http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq140_HTML.gif " reduced by the positive cone  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq141_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq142_HTML.gif is also normal with the same normal constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq143_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq144_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq145_HTML.gif we use http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq146_HTML.gif to denote the order interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq147_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq148_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq149_HTML.gif to denote the order interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq150_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq151_HTML.gif .

3. Main Results

Theorem 3.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq152_HTML.gif be an ordered Banach space, whose positive cone  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq153_HTML.gif is normal, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq155_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq156_HTML.gif . Assume that the IPBVP(1.1) has coupled lower and upper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq157_HTML.gif -quasisolutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq159_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq160_HTML.gif . Suppose that the following conditions are satisfied:
  1. (H1)
    There exist constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq162_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq163_HTML.gif such that
    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ18_HTML.gif
    (3.1)

    for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq164_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq165_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq166_HTML.gif .

     
  2. (H2)
    The impulsive function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq168_HTML.gif satisfies
    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ19_HTML.gif
    (3.2)

    for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq169_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq171_HTML.gif

     
  3. (H3)
    There exist a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq173_HTML.gif such that
    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ20_HTML.gif
    (3.3)

    for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq174_HTML.gif and increasing or decreasing monotonic sequences http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq175_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq176_HTML.gif

     
  4. (H4)

    The sequences http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq178_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq179_HTML.gif are convergent, where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq180_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq181_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq182_HTML.gif

     

Then the IPBVP(1.1) has minimal and maximal coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq183_HTML.gif -quasisolutions between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq184_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq185_HTML.gif which can be obtained by a monotone iterative procedure starting from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq186_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq187_HTML.gif , respectively.

Proof.

By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq188_HTML.gif and Lemma 2.5, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq189_HTML.gif is continuous, and the coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq190_HTML.gif -quasisolutions of the IPBVP(1.1) is equivalent to the coupled fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq191_HTML.gif Combining this with the assumptions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq192_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq193_HTML.gif , we know http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq194_HTML.gif is a mixed monotone operator (about the mixed monotone operator, please see [6, 7]).

Next, we show http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq195_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq196_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq197_HTML.gif by (2.9), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq198_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq199_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq200_HTML.gif By Lemma 2.5
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ21_HTML.gif
(3.4)

namely, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq201_HTML.gif . Similarly, it can be show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq202_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq203_HTML.gif

Now, we define two sequences http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq204_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq205_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq206_HTML.gif by the iterative scheme
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ22_HTML.gif
(3.5)
Then from the mixed monotonicity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq207_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ23_HTML.gif
(3.6)

We prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq208_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq209_HTML.gif are uniformly convergent in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq210_HTML.gif

For convenience, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq211_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq212_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq213_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq214_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq215_HTML.gif . Since, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq216_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq217_HTML.gif by (2.11) and the boundedness of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq218_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq219_HTML.gif we easy see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq220_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq221_HTML.gif is equicontinuous in every interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq222_HTML.gif so, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq223_HTML.gif is equicontinuous in every interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq224_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq225_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq226_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq227_HTML.gif From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq228_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq229_HTML.gif it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq230_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq231_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq232_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq233_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq234_HTML.gif by Lemma 2.3, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq235_HTML.gif . Going from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq236_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq237_HTML.gif interval by interval we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq238_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq239_HTML.gif

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq240_HTML.gif from (2.11), using Lemma 2.1 and assumption http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq241_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq242_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ24_HTML.gif
(3.7)

Hence by the Belman inequality, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq243_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq244_HTML.gif In particular, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq245_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq246_HTML.gif this means that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq247_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq248_HTML.gif are precompact in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq249_HTML.gif Thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq250_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq251_HTML.gif are precompact in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq252_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq253_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq254_HTML.gif

Now, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq255_HTML.gif by (2.11) and the above argument for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq256_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ25_HTML.gif
(3.8)

Again by Belman inequality, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq257_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq258_HTML.gif from which we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq259_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq260_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq261_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq262_HTML.gif

Continuing such a process interval by intervai up to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq263_HTML.gif we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq264_HTML.gif in every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq265_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq266_HTML.gif

For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq267_HTML.gif if we modify the value of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq268_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq269_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq270_HTML.gif via http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq271_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq272_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq273_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq274_HTML.gif and it is equicontinuous. Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq275_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq276_HTML.gif is precompact in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq277_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq278_HTML.gif By the Arzela-Ascoli theorem, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq279_HTML.gif is precompact in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq280_HTML.gif Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq281_HTML.gif has a convergent subsequence in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq282_HTML.gif Combining this with the monotonicity (3.6), we easily prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq283_HTML.gif itself is convergent in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq284_HTML.gif In particular, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq285_HTML.gif is uniformly convergent over the whole of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq286_HTML.gif Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq287_HTML.gif is uniformly convergent in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq288_HTML.gif Set
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ26_HTML.gif
(3.9)

Letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq289_HTML.gif in (3.5) and (3.6), we see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq290_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq291_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq292_HTML.gif By the mixed monotonicity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq293_HTML.gif it is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq294_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq295_HTML.gif are the minimal and maximal coupled fixed points of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq296_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq297_HTML.gif and therefore, they are the minimal and maximal coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq298_HTML.gif -quasisolutions of the IPBVP(1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq299_HTML.gif respectively.

In Theorem 3.1, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq300_HTML.gif is weakly sequentially complete, condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq301_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq302_HTML.gif hold automatically. In fact, by Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq303_HTML.gif in [12], any monotonic and order-bounded sequence is precompact. By the monotonicity (3.6) and the same method in proof of Theorem 3.1, we can easily see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq304_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq305_HTML.gif are convergent on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq306_HTML.gif In particular, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq307_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq308_HTML.gif are convergent. So, condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq309_HTML.gif holds. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq310_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq311_HTML.gif be increasing or decreasing sequences obeying condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq312_HTML.gif then by condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq313_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq314_HTML.gif is a monotonic and order-bounded sequence, so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq315_HTML.gif Hence, condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq316_HTML.gif holds. From Theorem 3.1, we obtain the following corollary.

Corollary 3.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq317_HTML.gif be an ordered and weakly sequentially complete Banach space, whose positive cone  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq318_HTML.gif is normal, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq319_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq320_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq321_HTML.gif If the IPBVP(1.1) has coupled lower and upper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq322_HTML.gif -quasisolutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq323_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq324_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq325_HTML.gif and conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq326_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq327_HTML.gif are satisfied, then the IPBVP(1.1) has minimal and maximal coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq328_HTML.gif -quasisolutions between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq329_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq330_HTML.gif which can be obtained by a monotone iterative procedure starting from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq331_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq332_HTML.gif respectively.

If we replace the assumption http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq333_HTML.gif by the following assumption:
  1. (H5)
    There exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq335_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq336_HTML.gif such that
    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ27_HTML.gif
    (3.10)

    for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq337_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq338_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq339_HTML.gif

     

We have the following result.

Theorem 3.3.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq340_HTML.gif be an ordered Banach space, whose positive cone  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq341_HTML.gif is normal, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq342_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq343_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq344_HTML.gif If the IPBVP(1.1) has coupled lower and upper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq345_HTML.gif -quasisolutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq346_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq347_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq348_HTML.gif and conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq349_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq350_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq351_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq352_HTML.gif hold, then the IPBVP(1.1) has minimal and maximal coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq353_HTML.gif -quasisolutions between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq354_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq355_HTML.gif which can be obtained by a monotone iterative procedure starting from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq356_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq357_HTML.gif respectively.

Proof.

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq358_HTML.gif let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq359_HTML.gif be a increasing sequence and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq360_HTML.gif be a decreasing sequence. For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq361_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq362_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq363_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq364_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ28_HTML.gif
(3.11)
By this and the normality of cone  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq365_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ29_HTML.gif
(3.12)
From this inequality and the definition of the measure noncompactness, it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ30_HTML.gif
(3.13)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq366_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq367_HTML.gif is a increasing sequence and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq368_HTML.gif is a decreasing sequence, the above inequality is also valid. Hence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq369_HTML.gif holds.

Therefore, by Theorem 3.1, the IPBVP(1.1) has minimal and maximal coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq370_HTML.gif -quasisolutions between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq371_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq372_HTML.gif which can be obtained by a monotone iterative procedure starting from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq373_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq374_HTML.gif , respectively.

Now, we discuss the existence of the solution to the IPBVP(1.1) between the minimal and maximal coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq375_HTML.gif -quasisolutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq376_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq377_HTML.gif If we replace the assumptions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq378_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq379_HTML.gif by the following assumptions:

The impulsive function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq381_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ31_HTML.gif
(3.14)
for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq382_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq383_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq384_HTML.gif and there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq385_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq386_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ32_HTML.gif
(3.15)

for any countable sets http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq387_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq388_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq389_HTML.gif

There exist a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq391_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ33_HTML.gif
(3.16)

for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq392_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq393_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq394_HTML.gif are countable sets in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq395_HTML.gif

We have the following existence result.

Theorem 3.4.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq396_HTML.gif be an ordered Banach space, whose positive cone  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq397_HTML.gif is normal, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq398_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq399_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq400_HTML.gif If the IPBVP(1.1) has coupled lower and upper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq401_HTML.gif -quasisolutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq402_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq403_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq404_HTML.gif such that assumptions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq405_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq406_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq407_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq408_HTML.gif hold, then the IPBVP(1.1) has minimal and maximal coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq409_HTML.gif -quasisolutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq410_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq411_HTML.gif between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq412_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq413_HTML.gif and at least has one solution between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq414_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq415_HTML.gif

Proof.

We can easily see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq416_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq417_HTML.gif Hence, by the Theorem 3.1, the IPBVP(1.1) has minimal and maximal coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq418_HTML.gif -quasisolutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq419_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq420_HTML.gif between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq421_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq422_HTML.gif Next, we prove the existence of the solution of the equation between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq423_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq424_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq425_HTML.gif clearly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq426_HTML.gif is continuous and the solution of the IPBVP(1.1) is equivalent to the fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq427_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq428_HTML.gif is bounded and equicontinuous for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq429_HTML.gif by Lemma 2.2, there exist a countable set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq430_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ34_HTML.gif
(3.17)
By assumptions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq431_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq432_HTML.gif and Lemma 2.1,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ35_HTML.gif
(3.18)

Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq433_HTML.gif is equicontinuous, by Lemma 2.3, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq434_HTML.gif . Combing (3.17) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq435_HTML.gif .

We have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ36_HTML.gif
(3.19)

Hence, the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq436_HTML.gif is condensing, by the Lemma 2.4, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq437_HTML.gif has fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq438_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq439_HTML.gif

Lastly, since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq440_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq441_HTML.gif by the mixed monotonity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq442_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ37_HTML.gif
(3.20)

Similarly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq443_HTML.gif in general, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq444_HTML.gif letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq445_HTML.gif we get http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq446_HTML.gif Therefore, the IPBVP(1.1) at least has one solution between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq447_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq448_HTML.gif

Remark 3.5.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq449_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq450_HTML.gif then Theorems 3.1, 3.3 and 3.4 are generalizations of the main results of [5] in Banach spaces.

Remark 3.6.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq451_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq452_HTML.gif then Theorems 3.1, 3.3 and 3.4 are generalizations of the Theory http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq453_HTML.gif of [4] in Banach spaces.

4. An Example

Consider the PBVP of infinite system for nonlinear impulsive differential equations:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ38_HTML.gif
(4.1)

4.1. Conclusion

IPBVP(4.1) has minimal and maximal coupled http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq454_HTML.gif -quasisolutions.

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq455_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq456_HTML.gif with norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq457_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq458_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq459_HTML.gif is a weakly sequentially complete Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq460_HTML.gif is normal cone  in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq461_HTML.gif IPBVP(4.1) can be regarded as an PBVP of the form (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq462_HTML.gif In this case, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq463_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq464_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq465_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq466_HTML.gif in which
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ39_HTML.gif
(4.2)

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq467_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq468_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq469_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq470_HTML.gif

Evidently, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq471_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq472_HTML.gif Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ40_HTML.gif
(4.3)

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq473_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq474_HTML.gif Then it is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq475_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq476_HTML.gif are coupled lower and upper http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq477_HTML.gif -quasisolutions of the IPBVP(4.1), and conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq478_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq479_HTML.gif hold. Hence, our conclusion follows from Corollary 3.2.

Declarations

Acknowledgments

This paper was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNU-KJCXGC-3-47.

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University

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© Pengyu Chen. 2011

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