# Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Conditions and One-Dimensional -Laplacian

- Meiqiang Feng
^{1}Email author

**Received: **2 February 2010

**Accepted: **5 June 2010

**Published: **29 June 2010

## Abstract

By using the fixed point theory for completely continuous operator, this paper investigates the existence of positive solutions for a class of fourth-order impulsive boundary value problems with integral boundary conditions and one-dimensional -Laplacian. Moreover, we offer some interesting discussion of the associated boundary value problems. Upper and lower bounds for these positive solutions also are given, so our work is new.

## Keywords

## 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. For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al. [1]; for an overview of existing results and of recent research areas of impulsive differential equations, see Benchohra et al. [2]. The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see, e.g., [3–18] and references cited therein).

Moreover, the theory of boundary-value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to the nonlocal problems with integral boundary conditions. For boundary-value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo [19], Karakostas and Tsamatos [20], Lomtatidze and Malaguti [21], and the references therein. For more information about the general theory of integral equations and their relation with boundary-value problems, we refer to the book of Corduneanu [22] and Agarwal and O'Regan [23].

where is a positive parameter, , is the zero element of , and . The authors investigated the multiplicity of positive solutions to problem (1.1) by using the fixed point index theory in cone for strict set contraction operator.

where and may be singular at or ; are continuous and may be singular at , and ; , and , and , and are nonnegative, .

where denotes a linear functional on given by involving a Stieltjes integral, and is a function of bounded variation.

where , , and . By using upper and lower solution method, fixed-point theorems, and the properties of Green's function and , the authors give sufficient conditions for the existence of one positive solution.

Here is -Laplace operator, that is, , , (where is fixed positive integer) are fixed points with , where and represent the right-hand limit and left-hand limit of at , respectively, and is nonnegative.

For the case of , problem (1.6) reduces to the problem studied by Zhang et al. in [33]. By using the fixed point theorem in cone, the authors obtained some sufficient conditions for the existence and multiplicity of symmetric positive solutions for a class of -Laplacian fourth-order differential equations with integral boundary conditions.

For the case of , and , problem (1.6) is related to fourth-order two-points boundary value problem of ODE. Under this case, problem (1.6) has received considerable attention (see, e.g., [40–42] and references cited therein). Aftabizadeh [40] showed the existence of a solution to problem (1.6) under the restriction that is a bounded function. Bai and Wang [41] have applied the fixed point theorem and degree theory to establish existence, uniqueness, and multiplicity of positive solutions to problem (1.6). Ma and Wang [42] have proved that there exist at least two positive solutions by applying the existence of positive solutions under the fact that is either superlinear or sublinear on by employing the fixed point theorem of cone extension or compression.

Being directly inspired by [18, 34, 43], in the present paper, we consider some existence results for problem (1.6) in a specially constructed cone by using the fixed point theorem. The main features of this paper are as follows. Firstly, comparing with [39–43], we discuss the impulsive boundary value problem with integral boundary conditions, that is, problem (1.6) includes fourth-order two-, three-, multipoint, and nonlocal boundary value problems as special cases. Secondly, the conditions are weaker than those of [33, 34, 46], and we consider the case . Finally, comparing with [33, 34, 39–43, 46], upper and lower bounds for these positive solutions also are given. Hence, we improve and generalize the results of [33, 34, 39–43, 46] to some degree, and so, it is interesting and important to study the existence of positive solutions of problem (1.6).

The organization of this paper is as follows. We shall introduce some lemmas in the rest of this section. In Section 2, we provide some necessary background. In particular, we state some properties of the Green's function associated with problem (1.6). In Section 3, the main results will be stated and proved. Finally, in Section 4, we offer some interesting discussion of the associated problem (1.6).

To obtain positive solutions of problem (1.6), the following fixed point theorem in cones is fundamental, which can be found in [47, page 94].

Lemma 1.1.

Let and be two bounded open sets in Banach space , such that and . Let be a cone in and let operator be completely continuous. Suppose that one of the following two conditions is satisfied:

## 2. Preliminaries

In order to define the solution of problem (1.6), we shall consider the following space.

A function is called a solution of problem (1.6) if it satisfies (1.6).

To establish the existence of multiple positive solutions in of problem (1.6), let us list the following assumptions:

Lemma 2.1.

Proof.

The proof follows by routine calculations.

Write . Then from (2.9) and (2.10), we can prove that have the following properties.

Proposition 2.2.

Proposition 2.3.

Proposition 2.4.

Proof.

The proof of Proposition 2.4 is complete.

Remark 2.5.

Lemma 2.6.

Proof.

First suppose that is a solution of problem (2.7).

and the proof of sufficient is complete.

Conversely, if is a solution of (2.18).

The Lemma is proved.

Remark 2.7.

From (2.19), we can prove that the properties of are similar to that of .

It is easy to see that is a closed convex cone of .

From (2.35), we know that is a solution of problem (1.6) if and only if is a fixed point of operator .

Definition 2.8 (see [1]).

We present the following result about relatively compact sets in which is a consequence of the Arzela-Ascoli Theorem. The reader can find its proof partially in [1].

Lemma 2.9.

is relatively compact if and only if is bounded and quasi-equicontinuous on .

Lemma 2.10.

Suppose that and hold. Then and is completely continuous.

Proof.

From (2.35) and Remark 2.5, we obtain the following cases.

Case 1.

Case 2.

Therefore, , that is, . Also, we have since . Hence we have .

Next, we prove that is completely continuous.

It is obvious that is continuous. Now we prove is relatively compact.

Therefore is uniformly bounded.

and then is quasi-equicontinuous. It follows that is relatively compact on by Lemma 2.9. So is completely continuous.

## 3. Main Results

In this section, we apply Lemma 1.1 to establish the existence of positive solutions of problem (1.6). We begin by introducing the following conditions on and .

Theorem 3.1.

Proof.

Applying (b) of Lemma 1.1 to (3.7) and (3.12) yields that has a fixed point with . Hence, since for we have , it follows that (3.4) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.1, we can prove the following results.

Corollary 3.2.

Assume that and hold. If , and , then, for being sufficiently small and being sufficiently large, BVP (1.6) has at least one positive solution with property (3.4).

Proof.

The proof is similar to that of Theorem of [6].

In Theorem 3.3, we assume the following condition on and .

Theorem 3.3.

Proof.

Applying (a) of Lemma 1.1 to (3.19) and (3.24) yields that has a fixed point with . Hence, since for we have , it follows that (3.17) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.3, we can prove the following results.

Corollary 3.4.

Assume that and hold. If and ; then, for being sufficiently small and being sufficiently large, BVP (1.6) has at least one positive solution with property (3.17).

Proof.

The proof is similar to that of Theorem of [6].

Theorem 3.5.

Assume that (3.1) of and (3.14) and (3.15) of hold. In addition, letting and satisfy the following condition:

Proof.

which implies that (3.30) holds.

Applying Lemma 1.1 to (3.28), (3.29), and (3.30) yields that has two fixed point with , and . Hence, since for we have , it follows that (3.26) holds. This and Lemma 2.9 complete the proof.

Remark 3.6.

Similar to the proof of that of [5], we can prove that problem (1.6) can be generalized to obtain many positive solutions.

## 4. Discussion

In this section, we offer some interesting discussions associated with problem (1.6).

Discussion.

Generally, it is difficult to obtain the upper and lower bounds of positive solutions for nonlinear higher-order boundary value problems (see, e.g., [33, 34, 39–43, 46, 48, 49] and their references).

Here is -Laplace operator, that is, , , (where is fixed positive integer) are fixed points with , where and represent the right-hand limit and left-hand limit of at , respectively, and is nonnegative.

Lemma 4.1.

Lemma 4.2.

It is not difficult to prove that and have the similar properties to that of and . But for , and have no property (2.13). In fact, if , then we can prove that and have the following properties.

Proposition 4.3.

Proof.

Similarly, we can prove that (4.10) holds, too.

Remark 4.4.

which implies that we cannot obtain the lower bounds for the positive solutions of problem (4.1).

## 5. Example

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

Example 5.1.

Conclusion.

Proof.

## Declarations

### Acknowledgments

The authors thank the referee for his/her careful reading of the manuscript and useful suggestions. This work is sponsored by 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), and Beijing Municipal Education Commission (71D0911003).

## Authors’ Affiliations

## References

- Lakshmikantham V, Baĭnov DD, Simeonov PS:
*Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics*.*Volume 6*. World Scientific, Singapore; 1989:xii+273.View ArticleGoogle Scholar - Benchohra M, Henderson J, Ntouyas S:
*Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications*.*Volume 2*. Hindawi Publishing Corporation, New York, NY, USA; 2006:xiv+366.View ArticleGoogle Scholar - Ahmad B, Sivasundaram S: Existence of solutions for impulsive integral boundary value problems of fractional order.
*Nonlinear Analysis: Hybrid Systems*2010, 4(1):134-141. 10.1016/j.nahs.2009.09.002Google Scholar - Baĭnov DD, Simeonov PS:
*Systems with Impulse Effect*. Ellis Horwood, Chichester, UK; 1989:255.Google Scholar - Samoĭlenko AM, Perestyuk NA:
*Impulsive Differential Equations*.*Volume 14*. World Scientific, Singapore; 1995:x+462.Google Scholar - Yan J: Existence of positive periodic solutions of impulsive functional differential equations with two parameters.
*Journal of Mathematical Analysis and Applications*2007, 327(2):854-868. 10.1016/j.jmaa.2006.04.018View ArticleGoogle Scholar - Feng M, Xie D: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations.
*Journal of Computational and Applied Mathematics*2009, 223(1):438-448. 10.1016/j.cam.2008.01.024View ArticleGoogle Scholar - Nieto JJ: Basic theory for nonresonance impulsive periodic problems of first order.
*Journal of Mathematical Analysis and Applications*1997, 205(2):423-433. 10.1006/jmaa.1997.5207View ArticleGoogle Scholar - Nieto JJ: Impulsive resonance periodic problems of first order.
*Applied Mathematics Letters*2002, 15(4):489-493. 10.1016/S0893-9659(01)00163-XView ArticleGoogle Scholar - Guo D: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space.
*Applied Mathematics and Computation*2003, 143(2-3):233-249. 10.1016/S0096-3003(02)00356-9View ArticleGoogle Scholar - Liu X, Guo D: Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces.
*Journal of Mathematical Analysis and Applications*1997, 216(1):284-302. 10.1006/jmaa.1997.5688View ArticleGoogle Scholar - Agarwal RP, O'Regan D: Multiple nonnegative solutions for second order impulsive differential equations.
*Applied Mathematics and Computation*2000, 114(1):51-59. 10.1016/S0096-3003(99)00074-0View ArticleGoogle Scholar - Liu B, Yu J:Existence of solution of
-point boundary value problems of second-order differential systems with impulses.
*Applied Mathematics and Computation*2002, 125(2-3):155-175. 10.1016/S0096-3003(00)00110-7View ArticleGoogle Scholar - Agarwal RP, Franco D, O'Regan D: Singular boundary value problems for first and second order impulsive differential equations.
*Aequationes Mathematicae*2005, 69(1-2):83-96. 10.1007/s00010-004-2735-9View ArticleGoogle Scholar - Lin X, Jiang D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations.
*Journal of Mathematical Analysis and Applications*2006, 321(2):501-514. 10.1016/j.jmaa.2005.07.076View ArticleGoogle Scholar - Jankowski T: Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments.
*Applied Mathematics and Computation*2008, 197(1):179-189. 10.1016/j.amc.2007.07.081View ArticleGoogle Scholar - Jankowski T: Positive solutions to second order four-point boundary value problems for impulsive differential equations.
*Applied Mathematics and Computation*2008, 202(2):550-561. 10.1016/j.amc.2008.02.040View ArticleGoogle Scholar - Feng M, Du B, Ge W:Impulsive boundary value problems with integral boundary conditions and one-dimensional
-Laplacian.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(9):3119-3126. 10.1016/j.na.2008.04.015View ArticleGoogle Scholar - Gallardo JM: Second-order differential operators with integral boundary conditions and generation of analytic semigroups.
*Rocky Mountain Journal of Mathematics*2000, 30(4):1265-1291. 10.1216/rmjm/1021477351View ArticleGoogle Scholar - Karakostas GL, Tsamatos PCh: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems.
*Electronic Journal of Differential Equations*2002, 2002(30):1-17.Google Scholar - Lomtatidze A, Malaguti L: On a nonlocal boundary value problem for second order nonlinear singular differential equations.
*Georgian Mathematical Journal*2000, 7(1):133-154.Google Scholar - Corduneanu C:
*Integral Equations and Applications*. Cambridge University Press, Cambridge, UK; 1991:x+366.View ArticleGoogle Scholar - Agarwal RP, O'Regan D:
*Infinite Interval Problems for Differential, Difference and Integral Equations*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+341.View ArticleGoogle Scholar - Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems involving integral conditions.
*Nonlinear Differential Equations and Applications*2008, 15(1-2):45-67. 10.1007/s00030-007-4067-7View ArticleGoogle Scholar - Webb JRL, Infante G: Non-local boundary value problems of arbitrary order.
*Journal of the London Mathematical Society*2009, 79(1):238-258.View ArticleGoogle Scholar - Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach.
*Journal of the London Mathematical Society*2006, 74(3):673-693. 10.1112/S0024610706023179View ArticleGoogle Scholar - Ahmad B, Nieto JJ:The monotone iterative technique for three-point second-order integrodifferential boundary value problems with
-Laplacian.
*Boundary Value Problems*2007, 2007:-9.Google Scholar - Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions.
*Boundary Value Problems*2009, 2009:-11.Google Scholar - Ahmad B, Alsaedi A, Alghamdi BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 9(4):1727-1740.View ArticleGoogle Scholar - Feng M, Ji D, Ge W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces.
*Journal of Computational and Applied Mathematics*2008, 222(2):351-363. 10.1016/j.cam.2007.11.003View ArticleGoogle Scholar - Zhang X, Feng M, Ge W:Multiple positive solutions for a class of
-point boundary value problems.
*Applied Mathematics Letters*2009, 22(1):12-18. 10.1016/j.aml.2007.10.019View ArticleGoogle Scholar - Feng M, Ge W:Positive solutions for a class of
-point singular boundary value problems.
*Mathematical and Computer Modelling*2007, 46(3-4):375-383. 10.1016/j.mcm.2006.11.009View ArticleGoogle Scholar - Zhang X, Feng M, Ge W:Symmetric positive solutions for
-Laplacian fourth-order differential equations with integral boundary conditions.
*Journal of Computational and Applied Mathematics*2008, 222(2):561-573. 10.1016/j.cam.2007.12.002View ArticleGoogle Scholar - Zhang X, Feng M, Ge W: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(10):3310-3321. 10.1016/j.na.2007.09.020View ArticleGoogle Scholar - Yang Z: Positive solutions of a second-order integral boundary value problem.
*Journal of Mathematical Analysis and Applications*2006, 321(2):751-765. 10.1016/j.jmaa.2005.09.002View ArticleGoogle Scholar - Ma R:Positive solutions for multipoint boundary value problem with a one-dimensional
-Laplacian.
*Computational & Applied Mathematics*2001, 42: 755-765.View ArticleGoogle Scholar - Bai Z, Huang B, Ge W:The iterative solutions for some fourth-order
-Laplace equation boundary value problems.
*Applied Mathematics Letters*2006, 19(1):8-14. 10.1016/j.aml.2004.10.010View ArticleGoogle Scholar - Liu B, Liu L, Wu Y: Positive solutions for singular second order three-point boundary value problems.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 66(12):2756-2766. 10.1016/j.na.2006.04.005View ArticleGoogle Scholar - Zhang X, Liu L:A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with
-Laplacian.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(10):3127-3137. 10.1016/j.na.2007.03.006View ArticleGoogle Scholar - Aftabizadeh AR: Existence and uniqueness theorems for fourth-order boundary value problems.
*Journal of Mathematical Analysis and Applications*1986, 116(2):415-426. 10.1016/S0022-247X(86)80006-3View ArticleGoogle Scholar - Bai Z, Wang H: On positive solutions of some nonlinear fourth-order beam equations.
*Journal of Mathematical Analysis and Applications*2002, 270(2):357-368. 10.1016/S0022-247X(02)00071-9View ArticleGoogle Scholar - Ma R, Wang H: On the existence of positive solutions of fourth-order ordinary differential equations.
*Applicable Analysis*1995, 59(1–4):225-231.Google Scholar - Liu L, Zhang X, Wu Y:Positive solutions of fourth order four-point boundary value problems with
-Laplacian operator.
*Journal of Mathematical Analysis and Applications*2007, 326(2):1212-1224. 10.1016/j.jmaa.2006.03.029View ArticleGoogle Scholar - Kang P, Wei Z, Xu J: Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces.
*Applied Mathematics and Computation*2008, 206(1):245-256. 10.1016/j.amc.2008.09.010View ArticleGoogle Scholar - Webb JRL, Infante G, Franco D: Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions.
*Proceedings of the Royal Society of Edinburgh*2008, 138(2):427-446.Google Scholar - Ma H: Symmetric positive solutions for nonlocal boundary value problems of fourth order.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(3):645-651. 10.1016/j.na.2006.11.026View ArticleGoogle Scholar - Guo D, Lakshmikantham V:
*Nonlinear Problems in Abstract Cones*.*Volume 5*. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar - Eloe PW, Ahmad B:Positive solutions of a nonlinear
th order boundary value problem with nonlocal conditions.
*Applied Mathematics Letters*2005, 18(5):521-527. 10.1016/j.aml.2004.05.009View ArticleGoogle Scholar - Hao X, Liu L, Wu Y:Positive solutions for nonlinear
th-order singular nonlocal boundary value problems.
*Boundary Value Problems*2007, 2007:-10.Google Scholar

## Copyright

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.