Minimal Nonnegative Solution of Nonlinear Impulsive Differential Equations on Infinite Interval

  • Xuemei Zhang1Email author,

    Affiliated with

    • Xiaozhong Yang1 and

      Affiliated with

      • Meiqiang Feng2

        Affiliated with

        Boundary Value Problems20102011:684542

        DOI: 10.1155/2011/684542

        Received: 20 May 2010

        Accepted: 19 July 2010

        Published: 2 August 2010

        Abstract

        The cone theory and monotone iterative technique are used to investigate the minimal nonnegative solution of nonlocal boundary value problems for second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times. All the existing results obtained in previous papers on nonlocal boundary value problems are under the case of the boundary conditions with no impulsive effects or the boundary conditions with impulsive effects on a finite interval with a finite number of impulsive times, so our work is new. Meanwhile, an example is worked out to demonstrate the main results.

        1. Introduction

        The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. The theory of impulsive differential equations has become an important area of investigation in the recent years and is much richer than the corresponding theory of differential equations. For an introduction of the basic theory of impulsive differential equations in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq1_HTML.gif see Lakshmikantham et al. [1], Bainov and Simeonov [2], and Samoĭlenko and Perestyuk [3] and the references therein.

        Usually, we only consider the differential equation, integrodifferential equation, functional differential equations, or dynamic equations on time scales on a finite interval with a finite number of impulsive times. To identify a few, we refer the reader to [413] and references therein. In particular, we would like to mention some results of Guo and Liu [5] and Guo [6]. In [5], by using fixed-point index theory for cone mappings, Guo and Liu investigated the existence of multiple positive solutions of a boundary value problem for the following second-order impulsive differential equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq2_HTML.gif is a cone in the real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq4_HTML.gif denotes the zero element of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq6_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq8_HTML.gif .

        In [6], by using fixed-point theory, Guo established the existence of solutions of a boundary value problem for the following second-order impulsive differential equation in a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq9_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ2_HTML.gif
        (1.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq11_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq12_HTML.gif .

        On the other hand, the readers can also find some recent developments and applications of the case that impulse effects on a finite interval with a finite number of impulsive times to a variety of problems from Nieto and Rodríguez-López [1416], Jankowski [1719], Lin and Jiang [20], Ma and Sun [21], He and Yu [22], Feng and Xie [23], Yan [24], Benchohra et al. [25], and Benchohra et al. [26].

        Recently, in [27], Li and Nieto obtained some new results of the case that impulse effects on an infinite interval with a finite number of impulsive times. By using a fixed-point theorem due to Avery and Peterson [28], Li and Nieto considered the existence of multiple positive solutions of the following impulsive boundary value problem on an infinite interval:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ3_HTML.gif
        (1.3)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq13_HTML.gif .

        At the same time, we also notice that there has been increasing interest in studying nonlinear differential equation and impulsive integrodifferential equation on an infinite interval with an infinite number of impulsive times; to identify a few, we refer the reader to Guo and Liu [29], Guo [3032], and Li and Shen [33]. It is here worth mentioning the works by Guo [31]. In [31], Guo investigated the minimal nonnegative solution of the following initial value problem for a second order nonlinear impulsive integrodifferential equation of Volterra type on an infinite interval with an infinite number of impulsive times in a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq14_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ4_HTML.gif
        (1.4)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq15_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq16_HTML.gif is a cone of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq17_HTML.gif .

        However, the corresponding theory for nonlocal boundary value problems for impulsive differential equations on an infinite interval with an infinite number of impulsive times is not investigated till now. Now, in this paper, we will use the cone theory and monotone iterative technique to investigate the existence of minimal nonnegative solution for a class of second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times.

        Consider the following boundary value problem for second-order nonlinear impulsive differential equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ5_HTML.gif
        (1.5)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq18_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq19_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq20_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq21_HTML.gif denotes the jump of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq22_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq23_HTML.gif , that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ6_HTML.gif
        (1.6)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq25_HTML.gif represent the right-hand limit and left-hand limit of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq26_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq27_HTML.gif , respectively. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq28_HTML.gif has a similar meaning for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq29_HTML.gif .

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ7_HTML.gif
        (1.7)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq30_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq31_HTML.gif where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ8_HTML.gif
        (1.8)
        Define a cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq32_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ9_HTML.gif
        (1.9)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq33_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq34_HTML.gif is called a nonnegative solution of (1.5), if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq36_HTML.gif satisfies (1.5).

        If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq37_HTML.gif , then boundary value problem (1.5) reduces to the following two point boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ10_HTML.gif
        (1.10)

        which has been intensively studied; see Ma [34], Agarwal and O'Regan [35], Constantin [36], Liu [37, 38], and Yan and Liu [39] for some references along this line.

        The organization of this paper is as follows. In Section 2, we provide some necessary background. In Section 3, the main result of problem (1.5) will be stated and proved. In Section 4, we give an example to illustrate how the main results can be used in practice.

        2. Preliminaries

        To establish the existence of minimal nonnegative solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq38_HTML.gif of problem (1.5), let us list the following assumptions, which will stand throughout this paper.

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq39_HTML.gif Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq40_HTML.gif and there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq41_HTML.gif and nonnegative constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq42_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ11_HTML.gif
        (2.1)

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq43_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq44_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq45_HTML.gif

        Lemma 2.1.

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq46_HTML.gif holds. Then for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq50_HTML.gif are convergent.

        Proof.

        By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq51_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ12_HTML.gif
        (2.2)
        Thus,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ13_HTML.gif
        (2.3)

        The proof is complete.

        Lemma 2.2.

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq52_HTML.gif holds. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq53_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq54_HTML.gif is a solution of problem (1.5) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq55_HTML.gif is a solution of the following impulsive integral equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ14_HTML.gif
        (2.4)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ15_HTML.gif
        (2.5)

        Proof.

        First, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq56_HTML.gif is a solution of problem (1.5). It is easy to see by integration of (1.5) that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ16_HTML.gif
        (2.6)
        Taking limit for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq57_HTML.gif , by Lemma 2.1 and the boundary conditions, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ17_HTML.gif
        (2.7)
        Thus,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ18_HTML.gif
        (2.8)
        Integrating (2.8), we can get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ19_HTML.gif
        (2.9)
        It follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ20_HTML.gif
        (2.10)

        So we have (2.4).

        Conversely, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq58_HTML.gif is a solution of (2.4). Evidently,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ21_HTML.gif
        (2.11)
        Direct differentiation of (2.4) implies, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq59_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ22_HTML.gif
        (2.12)

        So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq60_HTML.gif . It is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq61_HTML.gif . The proof of Lemma 2.2 is complete.

        Define an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq62_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ23_HTML.gif
        (2.13)

        Lemma 2.3.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq64_HTML.gif hold. Then operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq65_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq66_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq67_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ24_HTML.gif
        (2.14)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ25_HTML.gif
        (2.15)
        Moreover, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq68_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq69_HTML.gif one has
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ26_HTML.gif
        (2.16)

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq70_HTML.gif . From the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq72_HTML.gif , we can obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq73_HTML.gif is an operator from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq74_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq75_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ27_HTML.gif
        (2.17)
        Direct differentiation of (2.13) implies, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq76_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ28_HTML.gif
        (2.18)

        Thus we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq77_HTML.gif It follows that (2.14) is satisfied. Equation (2.16) is easily obtained by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq78_HTML.gif .

        3. Main Result

        In this section, we establish the existence of a minimal nonnegative solution for problem (1.5).

        Theorem 3.1.

        Let conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq79_HTML.gif be satisfied. Suppose further that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ29_HTML.gif
        (3.1)
        Then problem (1.5) has the minimal nonnegative solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq80_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq81_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq82_HTML.gif is defined as in Lemma 2.3. Here, the meaning of minimal nonnegative solution is that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq83_HTML.gif is an arbitrary nonnegative solution of (1.5), then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq84_HTML.gif Moreover, if we let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq85_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq86_HTML.gif with
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ30_HTML.gif
        (3.2)

        and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq88_HTML.gif converge uniformly to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq90_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq91_HTML.gif , respectively.

        Proof.

        By Lemma 2.3 and the definition of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq92_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq93_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ31_HTML.gif
        (3.3)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ32_HTML.gif
        (3.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ33_HTML.gif
        (3.5)
        By (3.3), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ34_HTML.gif
        (3.6)
        From (3.4), (3.5), and (3.6), we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq95_HTML.gif exist. Suppose that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ35_HTML.gif
        (3.7)
        By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq96_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ36_HTML.gif
        (3.8)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ37_HTML.gif
        (3.9)
        From (3.6), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ38_HTML.gif
        (3.10)
        It follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq97_HTML.gif is equicontinuous on every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq98_HTML.gif Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq99_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq100_HTML.gif . Which together with (3.4) imply that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq101_HTML.gif converges uniformly to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq102_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq103_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq104_HTML.gif On the other hand, by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq105_HTML.gif (3.6), and (3.9), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ39_HTML.gif
        (3.11)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq106_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq107_HTML.gif (M is a finite positive number), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq108_HTML.gif is equicontinuous on every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq109_HTML.gif Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq110_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq111_HTML.gif , which together with (3.5) imply that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq112_HTML.gif converges uniformly to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq113_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq114_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq115_HTML.gif From above, we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq116_HTML.gif exists and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq117_HTML.gif It follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq118_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ40_HTML.gif
        (3.12)

        Now we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq119_HTML.gif

        By the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq120_HTML.gif and the uniform convergence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq121_HTML.gif , we know that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ41_HTML.gif
        (3.13)
        On the other hand, by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq122_HTML.gif and (3.6) and (3.12), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ42_HTML.gif
        (3.14)
        Combining this with the dominated convergence theorem, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ43_HTML.gif
        (3.15)
        Moreover, we can see that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ44_HTML.gif
        (3.16)
        Now taking limits from two sides of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq123_HTML.gif and using (3.15)–(3.16), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ45_HTML.gif
        (3.17)

        By Lemma 2.2, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq124_HTML.gif is a nonnegative solution of (1.5).

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq125_HTML.gif is an arbitrary nonnegative solution of (1.5). Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq126_HTML.gif It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq127_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq128_HTML.gif By (2.16) we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq129_HTML.gif This means that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq130_HTML.gif Taking limit, we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq131_HTML.gif . The proof of Theorem 3.1 is complete.

        4. Example

        To illustrate how our main results can be used in practice, we present an example.

        Example 4.1.

        Consider the following boundary value problem of second-order impulsive differential equation on infinite interval
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ46_HTML.gif
        (4.1)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ47_HTML.gif
        (4.2)

        Evidently, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq132_HTML.gif is not the solution of (4.1).

        Conclusion

        Problem (4.1) has minimal positive solution.

        Proof.

        It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq134_HTML.gif is satisfied.

        By the inequality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq135_HTML.gif we see that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ48_HTML.gif
        (4.3)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ49_HTML.gif
        (4.4)
        Then, we easily obtain that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ50_HTML.gif
        (4.5)

        Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq136_HTML.gif is satisfied and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq137_HTML.gif . By Theorem 3.1, it follows that problem (4.1) has a minimal positive solution.

        Declarations

        Acknowledgments

        This work is supported by the National Natural Science Foundation of China (10771065), the Natural Sciences Foundation of Hebei Province (A2007001027), the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201008430), the Scientific Research Common Program of Beijing Municipal Commission of Education(KM201010772018), the 2010 level of scientific research of improving project (5028123900), and Beijing Municipal Education Commission(71D0911003). The authors thank the referee for his/her careful reading of the paper and useful suggestions.

        Authors’ Affiliations

        (1)
        Department of Mathematics and Physics, North China Electric Power University
        (2)
        School of Applied Science, Beijing Information Science and Technology University

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        © Xuemei Zhang et al. 2011

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