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

  • Kexue Zhang1 and

    Affiliated with

    • Xinzhi Liu2Email author

      Affiliated with

      Journal of Inequalities and Applications20102010:490741

      DOI: 10.1155/2010/490741

      Received: 8 December 2009

      Accepted: 30 January 2010

      Published: 21 February 2010

      Abstract

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

      1. Introduction

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

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

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

      2. Preliminaries

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

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

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

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

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

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

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

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

         

      We will need the following lemma.

      Lemma 2.2 (see [1]).

      Asumme that

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

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

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

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

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

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

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

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

      Theorem 2.3.

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

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

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

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

      Proof.

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

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

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

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

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

      Casedi.

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

      Casedii.

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

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

      which is a contradiction.

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

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

      The following corollary is an easy consequence of Theorem 2.3.

      Corollary 2.4.

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

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

      Remark 2.5.

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

      Remark 2.6.

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

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

      Consider the linear boundary value problem (LBVP)

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

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

      By direct computation, we have the following result.

      Lemma 2.7.

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

      Lemma 2.8.

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

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

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

      3. Main Results

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

      Definition 3.1.

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

      For convenience, let us list the following conditions.

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

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

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

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

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

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

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

      Theorem 3.2.

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

      Proof.

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

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

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

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

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

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

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

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

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

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

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

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

      Theorem 3.3.

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

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

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

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

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

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

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

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

      Proof.

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

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

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

      4. Examples

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

      Example 4.1.

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

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

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

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

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

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

      Example 4.2.

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

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

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

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

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

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

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

      Authors’ Affiliations

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

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      Copyright

      © K. Zhang and X. Liu. 2010

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.