Open Access

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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq4_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq7_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq8_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq9_HTML.gif is an impulsive function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq10_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq11_HTML.gif denotes the jump of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq12_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq13_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq14_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq15_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq17_HTML.gif at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq19_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq20_HTML.gif and an upper solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq21_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq22_HTML.gif and nonlinear term https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq23_HTML.gif satisfies the monoton condition
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ3_HTML.gif
(1.3)
with a positive constant https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq25_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq27_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq30_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq31_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq34_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq35_HTML.gif is nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq37_HTML.gif is nonincreasing in https://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
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq41_HTML.gif -quasisolutions. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq42_HTML.gif the coupled upper and lower https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq46_HTML.gif the coupled upper and lower https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq49_HTML.gif and partial order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq50_HTML.gif whose positive cone  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq51_HTML.gif is normal with normal constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq52_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq53_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq54_HTML.gif is a constant; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq55_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq56_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq58_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq59_HTML.gif is continuous at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq60_HTML.gif , and left continuous at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq61_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq62_HTML.gif exists, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq63_HTML.gif Evidently, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq65_HTML.gif . An abstract function https://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 https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq68_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq70_HTML.gif exist, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq71_HTML.gif . For https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq73_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq74_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq75_HTML.gif exists and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq76_HTML.gif and set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq77_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq78_HTML.gif If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq80_HTML.gif

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq82_HTML.gif -value functions on interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq83_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq84_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq86_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq87_HTML.gif set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq88_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq89_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq90_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq91_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq92_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq95_HTML.gif is Lebesgue integral on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq96_HTML.gif and
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq98_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq99_HTML.gif

Lemma 2.3 (see [11]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq100_HTML.gif be equicontinuous. Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq101_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq102_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq103_HTML.gif be a Banach space and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq105_HTML.gif be condensing, then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq106_HTML.gif has a fixed point in https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq109_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ8_HTML.gif
(2.3)

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

Lemma 2.5.

For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq113_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq115_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq117_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ9_HTML.gif
(2.4)

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

Proof.

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

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

If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq126_HTML.gif namely
https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq128_HTML.gif be a constant. If functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq129_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ14_HTML.gif
(2.9)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ15_HTML.gif
(2.10)

we call https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq130_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq131_HTML.gif coupled lower and upper https://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 " https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq134_HTML.gif coupled https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq136_HTML.gif we call https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq138_HTML.gif as following:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ16_HTML.gif
(2.11)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ17_HTML.gif
(2.12)

Evidently, https://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 " https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq140_HTML.gif " reduced by the positive cone  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq141_HTML.gif . https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq143_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq144_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq145_HTML.gif we use https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq146_HTML.gif to denote the order interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq147_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq148_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq149_HTML.gif to denote the order interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq150_HTML.gif in https://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 https://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  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq153_HTML.gif is normal, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq154_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq155_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq157_HTML.gif -quasisolutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq158_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq159_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq162_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq163_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ18_HTML.gif
    (3.1)

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

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

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

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

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

     
  4. (H4)

    The sequences https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq179_HTML.gif are convergent, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq180_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq181_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq183_HTML.gif -quasisolutions between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq184_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq186_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq187_HTML.gif , respectively.

Proof.

By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq188_HTML.gif and Lemma 2.5, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq189_HTML.gif is continuous, and the coupled https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq191_HTML.gif Combining this with the assumptions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq193_HTML.gif , we know https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq195_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq196_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq197_HTML.gif by (2.9), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq198_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq199_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq200_HTML.gif By Lemma 2.5
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ21_HTML.gif
(3.4)

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

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

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

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

For https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq241_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq242_HTML.gif we have
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq243_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq244_HTML.gif In particular, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq245_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq246_HTML.gif this means that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq247_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq248_HTML.gif are precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq249_HTML.gif Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq250_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq251_HTML.gif are precompact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq252_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq253_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq254_HTML.gif

Now, for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq256_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ25_HTML.gif
(3.8)

Again by Belman inequality, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq257_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq258_HTML.gif from which we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq259_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq260_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq261_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq263_HTML.gif we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq264_HTML.gif in every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq265_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq266_HTML.gif

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

Letting https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq290_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq291_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq292_HTML.gif By the mixed monotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq293_HTML.gif it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq294_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq296_HTML.gif in https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq299_HTML.gif respectively.

In Theorem 3.1, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq300_HTML.gif is weakly sequentially complete, condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq301_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq302_HTML.gif hold automatically. In fact, by Theorem https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq304_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq305_HTML.gif are convergent on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq306_HTML.gif In particular, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq307_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq308_HTML.gif are convergent. So, condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq309_HTML.gif holds. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq310_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq312_HTML.gif then by condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq313_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq315_HTML.gif Hence, condition https://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 https://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  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq318_HTML.gif is normal, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq319_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq320_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq322_HTML.gif -quasisolutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq323_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq324_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq325_HTML.gif and conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq326_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq328_HTML.gif -quasisolutions between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq329_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq331_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq332_HTML.gif respectively.

If we replace the assumption https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq335_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq336_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ27_HTML.gif
    (3.10)

    for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq337_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq338_HTML.gif , https://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 https://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  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq341_HTML.gif is normal, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq342_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq343_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq345_HTML.gif -quasisolutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq346_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq347_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq348_HTML.gif and conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq349_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq350_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq351_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq353_HTML.gif -quasisolutions between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq354_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq356_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq357_HTML.gif respectively.

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq358_HTML.gif let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq359_HTML.gif be a increasing sequence and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq360_HTML.gif be a decreasing sequence. For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq361_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq362_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq363_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq364_HTML.gif
https://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  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq365_HTML.gif we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ30_HTML.gif
(3.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq366_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq367_HTML.gif is a increasing sequence and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq370_HTML.gif -quasisolutions between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq371_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq373_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq375_HTML.gif -quasisolutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq376_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq377_HTML.gif If we replace the assumptions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq378_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq381_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ31_HTML.gif
(3.14)
for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq382_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq383_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq384_HTML.gif and there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq385_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq386_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ32_HTML.gif
(3.15)

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

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

for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq392_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq393_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq394_HTML.gif are countable sets in https://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 https://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  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq397_HTML.gif is normal, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq398_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq399_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq401_HTML.gif -quasisolutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq402_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq403_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq404_HTML.gif such that assumptions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq405_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq406_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq407_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq409_HTML.gif -quasisolutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq410_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq411_HTML.gif between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq412_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq414_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq415_HTML.gif

Proof.

We can easily see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq416_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq418_HTML.gif -quasisolutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq419_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq420_HTML.gif between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq421_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq423_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq424_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq425_HTML.gif clearly, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq427_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq428_HTML.gif is bounded and equicontinuous for any https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq430_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ34_HTML.gif
(3.17)
By assumptions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq431_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq432_HTML.gif and Lemma 2.1,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ35_HTML.gif
(3.18)

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

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

Hence, the operator https://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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq437_HTML.gif has fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq438_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq439_HTML.gif

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

Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq443_HTML.gif in general, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq444_HTML.gif letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq445_HTML.gif we get https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq447_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq448_HTML.gif

Remark 3.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq449_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq451_HTML.gif and https://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 https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq454_HTML.gif -quasisolutions.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq455_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq456_HTML.gif with norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq457_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq458_HTML.gif Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq460_HTML.gif is normal cone  in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq462_HTML.gif In this case, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq463_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq464_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq465_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq466_HTML.gif in which
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_Equ39_HTML.gif
(4.2)

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq473_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq475_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq476_HTML.gif are coupled lower and upper https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F421261/MediaObjects/13661_2010_Article_39_IEq478_HTML.gif , https://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

References

  1. Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar
  2. Guo DJ, Liu X: Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces. Journal of Mathematical Analysis and Applications 1993, 177(2):538-552. 10.1006/jmaa.1993.1276View ArticleMathSciNetGoogle Scholar
  3. Li Y, Liu Z: Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(1):83-92. 10.1016/j.na.2005.11.013View ArticleMathSciNetGoogle Scholar
  4. Lakshmikantham V, Leela S: Existence and monotone method for periodic solutions of first-order differential equations. Journal of Mathematical Analysis and Applications 1983, 91(1):237-243. 10.1016/0022-247X(83)90102-6View ArticleMathSciNetGoogle Scholar
  5. He Z, Yu J: Periodic boundary value problem for first-order impulsive ordinary differential equations. Journal of Mathematical Analysis and Applications 2002, 272(1):67-78. 10.1016/S0022-247X(02)00133-6View ArticleMathSciNetGoogle Scholar
  6. Lakshmikantham V, Leela S, Vatsala AS: Method of quasi-upper and lower solutions in abstract cones. Nonlinear Analysis 1982, 6(8):833-838. 10.1016/0362-546X(82)90067-0View ArticleMathSciNetGoogle Scholar
  7. Guo DJ, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Analysis: Theory, Methods & Applications 1987, 11(5):623-632. 10.1016/0362-546X(87)90077-0View ArticleMathSciNetGoogle Scholar
  8. Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.View ArticleGoogle Scholar
  9. Heinz H-P: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Analysis: Theory, Methods & Applications 1983, 7(12):1351-1371. 10.1016/0362-546X(83)90006-8View ArticleMathSciNetGoogle Scholar
  10. Li YX: Existence of solutions to initial value problems for abstract semilinear evolution equations. Acta Mathematica Sinica 2005, 48(6):1089-1094.MathSciNetGoogle Scholar
  11. Guo DJ, Sun JX: Ordinary Differential Equations in Abstract Spaces. Shandong Science and Technology, Jinan, China; 1989.Google Scholar
  12. Du YH: Fixed points of increasing operators in ordered Banach spaces and applications. Applicable Analysis 1990, 38(1-2):1-20. 10.1080/00036819008839957View ArticleMathSciNetGoogle Scholar

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© Pengyu Chen. 2011

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