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Minimal Nonnegative Solution of Nonlinear Impulsive Differential Equations on Infinite Interval

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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 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:

(1.1)

where is a cone in the real Banach space , denotes the zero element of , for and .

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

(1.2)

where , , and .

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:

(1.3)

where .

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 :

(1.4)

where as is a cone of .

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:

(1.5)

where with . denotes the jump of at , that is,

(1.6)

where and represent the right-hand limit and left-hand limit of at , respectively. has a similar meaning for .

Let

(1.7)

Let with the norm where

(1.8)

Define a cone by

(1.9)

Let . is called a nonnegative solution of (1.5), if and satisfies (1.5).

If , then boundary value problem (1.5) reduces to the following two point boundary value problem:

(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 of problem (1.5), let us list the following assumptions, which will stand throughout this paper.

Suppose that and there exist and nonnegative constants such that

(2.1)

for

Lemma 2.1.

Suppose that holds. Then for all , , and are convergent.

Proof.

By we have

(2.2)

Thus,

(2.3)

The proof is complete.

Lemma 2.2.

Suppose that holds. If , then is a solution of problem (1.5) if and only if is a solution of the following impulsive integral equation:

(2.4)

where

(2.5)

Proof.

First, suppose that is a solution of problem (1.5). It is easy to see by integration of (1.5) that

(2.6)

Taking limit for , by Lemma 2.1 and the boundary conditions, we have

(2.7)

Thus,

(2.8)

Integrating (2.8), we can get

(2.9)

It follows that

(2.10)

So we have (2.4).

Conversely, suppose that is a solution of (2.4). Evidently,

(2.11)

Direct differentiation of (2.4) implies, for

(2.12)

So . It is easy to verify that . The proof of Lemma 2.2 is complete.

Define an operator

(2.13)

Lemma 2.3.

Assume that and hold. Then operator maps into , and

(2.14)

where

(2.15)

Moreover, for with one has

(2.16)

Proof.

Let . From the definition of and , we can obtain that is an operator from into , and

(2.17)

Direct differentiation of (2.13) implies, for

(2.18)

Thus we have It follows that (2.14) is satisfied. Equation (2.16) is easily obtained by .

3. Main Result

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

Theorem 3.1.

Let conditions be satisfied. Suppose further that

(3.1)

Then problem (1.5) has the minimal nonnegative solution with , where is defined as in Lemma 2.3. Here, the meaning of minimal nonnegative solution is that if is an arbitrary nonnegative solution of (1.5), then Moreover, if we let then with

(3.2)

and and converge uniformly to and on , respectively.

Proof.

By Lemma 2.3 and the definition of operator , we have , and

(3.3)
(3.4)
(3.5)

By (3.3), we have

(3.6)

From (3.4), (3.5), and (3.6), we know that and exist. Suppose that

(3.7)

By the definition of we have

(3.8)
(3.9)

From (3.6), we obtain

(3.10)

It follows that is equicontinuous on every Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to on . Which together with (3.4) imply that converges uniformly to on , and On the other hand, by (3.6), and (3.9), we have

(3.11)

Since is bounded on (M is a finite positive number), is equicontinuous on every Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to on , which together with (3.5) imply that converges uniformly to on , and From above, we know that exists and It follows that and

(3.12)

Now we prove that

By the continuity of and the uniform convergence of , we know that

(3.13)

On the other hand, by and (3.6) and (3.12), we have

(3.14)

Combining this with the dominated convergence theorem, we have

(3.15)

Moreover, we can see that

(3.16)

Now taking limits from two sides of and using (3.15)–(3.16), we have

(3.17)

By Lemma 2.2, is a nonnegative solution of (1.5).

Suppose that is an arbitrary nonnegative solution of (1.5). Then It is clear that . Suppose that By (2.16) we have This means that Taking limit, we have . 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

(4.1)

where

(4.2)

Evidently, is not the solution of (4.1).

Conclusion

Problem (4.1) has minimal positive solution.

Proof.

It is clear that and is satisfied.

By the inequality we see that

(4.3)

Let

(4.4)

Then, we easily obtain that

(4.5)

Thus, is satisfied and . By Theorem 3.1, it follows that problem (4.1) has a minimal positive solution.

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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.

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Correspondence to Xuemei Zhang.

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Keywords

  • Finite Interval
  • Nonlocal Boundary
  • Impulsive Effect
  • Impulsive Differential Equation
  • Multiple Positive Solution