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Volume 2011, Article ID 654871, 26 pagesdoi:10.1155/2011/654871 Research Article Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Condi

Volume 2011, Article ID 654871, 26 pages

doi:10.1155/2011/654871

Research Article

Multiple Positive Solutions of Fourth-Order

Impulsive Differential Equations with Integral

Boundary Conditions and One-Dimensional

p-Laplacian

Meiqiang Feng

School of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China

Correspondence should be addressed to Meiqiang Feng,meiqiangfeng@sina.com

Received 2 February 2010; Revised 25 April 2010; Accepted 5 June 2010

Academic Editor: Gennaro Infante

Copyrightq 2011 Meiqiang Feng This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

By using the fixed point theory for completely continuous operator, this paper investigates theexistence of positive solutions for a class of fourth-order impulsive boundary value problems

with integral boundary conditions and one-dimensional p-Laplacian Moreover, we offer some

interesting discussion of the associated boundary value problems Upper and lower bounds forthese positive solutions also are given, so our work is new

1 Introduction

The theory of impulsive differential equations describes processes which experience a suddenchange of their state at certain moments Processes with such a character arise naturallyand often, especially in phenomena studied in physics, chemical technology, populationdynamics, biotechnology, and economics For an introduction of the basic theory of impulsivedifferential 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 Thetheory of impulsive differential equations has become an important area of investigation inrecent 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 conditionsfor ordinary differential equations arises in different areas of applied mathematics andphysics For example, heat conduction, chemical engineering, underground water flow,thermo-elasticity, and plasma physics can be reduced to the nonlocal problems with integralboundary conditions For boundary-value problems with integral boundary conditions and

comments on their importance, we refer the reader to the papers by Gallardo19, Karakostasand Tsamatos 20, Lomtatidze and Malaguti 21, and the references therein For moreinformation about the general theory of integral equations and their relation with boundary-value problems, we refer to the book of Corduneanu22 and Agarwal and O’Regan 23.

On the other hand, boundary-value problems with integral boundary conditionsconstitute a very interesting and important class of problems They include two, three,multipoint and nonlocal boundary-value problems as special cases The existence andmultiplicity of positive solutions for such problems have received a great deal of attention

in the literature To identify a few, we refer the reader to24–46 and references therein Inparticular, we would like to mention some results of Zhang et al.34, Kang et al 44, andWebb et al 45 In 34, Zhang et al studied the following fourth-order boundary valueproblem with integral boundary conditions

where λ is a positive parameter, f ∈ C0, 1 × P, P, θ is the zero element of E, and g, h ∈

L10, 1 The authors investigated the multiplicity of positive solutions to problem 1.1 byusing the fixed point index theory in cone for strict set contraction operator

In 44, Kang et al have improved and generalized the work of 34 by applyingthe fixed point theory in cone for a strict set contraction operator; they proved that thereexist various results on the existence of positive solutions to a class of fourth-order singularboundary value problems with integral boundary conditions

u4t  atft, u t, −ut btgt, u t, −ut, 0 < t < 1,

where a, b ∈ C0, 1, 0, ∞ and may be singular at t  0 or t  1; f, g : 0, 1 × P \ {θ} ×

P \ {θ} → P are continuous and may be singular at t  0, 1, u  0, and u  0; ai , b i , c i, and

d i ∈ 0, ∞, and ρi  ai c i  ai d i  bi c i > 0, and m i , n i ∈ L10, 1 are nonnegative, i  1, 2.

More recently, by using a unified approach, Webb et al 45 considered the widelystudied boundary conditions corresponding to clamped and hinged ends and manynonlocal boundary conditions and established excellent existence results for multiple positivesolutions of fourth-order nonlinear equations which model deflections of an elastic beam

u4t  gtft, ut, for almost every t ∈ 0, 1, 1.3

subject to various boundary conditions

At the same time, we notice that there has been a considerable attention on p-Laplacian

BVPs18,32,35,36,38,42 as p-Laplacian appears in the study of flow through porous media

p  3/2, nonlinear elasticity p ≥ 2, glaciology 1 ≤ p ≤ 4/3, and so forth Here, it is worth

mentioning that Liu et al.43 considered the following fourth-order four-point boundaryvalue problem:

Motivated by works mentioned above, in this paper, we consider the existence ofpositive solutions for a class of boundary value problems with integral boundary conditions

of fourth-order impulsive differential equations:

k represent the right-hand limit and left-hand

limit of xt at t  tk , respectively, and g, h ∈ L10, 1 is nonnegative.

For the case of Ik  0, k  1, 2, , m, problem 1.6 reduces to the problem studied

by Zhang et al in33 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 p-Laplacian fourth-order differential equations with integral boundary conditions For the case of I k  0, k  1, 2, , m, g  0, h  0, and p  2, problem 1.6 isrelated 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 f is a bounded function Bai and Wang41 have applied the fixed point theorem anddegree theory to establish existence, uniqueness, and multiplicity of positive solutions toproblem1.6 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 f t, u is either superlinear

or sublinear on u by employing the fixed point theorem of cone extension or compression.

Being directly inspired by18,34,43, in the present paper, we consider some existenceresults for problem1.6 in a specially constructed cone by using the fixed point theorem Themain features of this paper are as follows Firstly, comparing with 39–43, we discuss theimpulsive boundary value problem with integral boundary conditions, that is, problem1.6includes fourth-order two-, three-, multipoint, and nonlocal boundary value problems asspecial cases Secondly, the conditions are weaker than those of33,34,46, and we consider

the case I k / 0 Finally, comparing with 33,34,39–43,46, upper and lower bounds for thesepositive solutions also are given Hence, we improve and generalize the results of33,34,39–

43,46 to some degree, and so, it is interesting and important to study the existence of positivesolutions of problem1.6

The organization of this paper is as follows We shall introduce some lemmas in the rest

of this section InSection 2, we provide some necessary background In particular, we statesome properties of the Green’s function associated with problem1.6 InSection 3, the mainresults will be stated and proved Finally, inSection 4, we offer some interesting discussion

of the associated problem1.6

To obtain positive solutions of problem 1.6, the following fixed point theorem incones is fundamental, which can be found in47, page 94

Lemma 1.1 Let Ω1 and Ω2 be two bounded open sets in Banach space E, such that 0 ∈ Ω1 and

Ω1 ⊂ Ω2 Let P be a cone in E and let operator A : P∩ Ω2\ Ω1 → P be completely continuous.

Suppose that one of the following two conditions is satisfied:

Then PC10, 1 is a real Banach space with norm

PC1 max ∞, x

where ∞ supt ∈J  ∞ supt ∈J |xt|.

A function x∈ PC10, 1 ∩ C4J is called a solution of problem 1.6 if it satisfies 1.6

To establish the existence of multiple positive solutions in PC10, 1∩C4J of problem

1.6, let us list the following assumptions:

FromH2, it is clear that μ ∈ 0, 1, ν ∈ 0, 1.

We shall reduce problem1.6 to an integral equation To this goal, firstly by means ofthe transformation

Proof The proof follows by routine calculations.

Write et  t1 − t Then from 2.9 and 2.10, we can prove that Ht, s and Gt, s

have the following properties

Proposition 2.2 If H2  holds, then we have

H t, s > 0, Gt, s > 0, for t, s ∈ 0, 1,

H t, s ≥ 0, Gt, s ≥ 0, for t, s ∈ J. 2.11

Proposition 2.3 For t, s ∈ 0, 1, we have

e tes ≤ Gt, s ≤ Gt, t  t1 − t  et ≤ e  max

On the other hand, noticing Gt, s ≤ s1 − s, we obtain

The proof ofProposition 2.4is complete

Remark 2.5 From2.9 and 2.13, we can obtain that

ρe s ≤ Hs, s ≤ γs1 − s  γes ≤ 1

4γ, s ∈ J. 2.17

Lemma 2.6 If H1  and H2 hold, then problem 2.7 has a unique solution y and y can be expressed

in the following form:

Proof First suppose that y∈ PC10, 1 ∩ C2J is a solution of problem 2.7

If t ∈ 0, t1, it is easy to see by integration of problem 2.7 that

yt  y0 −

t

0

φ q xsds. 2.20

If t ∈ t1, t2, then integrate from t1to t,

The Lemma is proved.

Remark 2.7 From 2.19, we can prove that the properties of H1t, s are similar to that of

From2.35, we know that y ∈ PC10, 1 is a solution of problem 1.6 if and only if y

is a fixed point of operator T.

Definition 2.8see 1 The set S ⊂ PC10, 1 is said to be quasi-equicontinuous in PC10, 1 if for any ε > 0 there exist δ > 0 such that if u ∈ S, s, t ∈ Jk k  1, 2, , m, |s − t| < δ, then

|us − ut| < ε,us − ut< ε. 2.36

We present the following result about relatively compact sets in PC1J which is a

consequence of the Arzela-Ascoli Theorem The reader can find its proof partially in1

Lemma 2.9 S ⊂ PC10, 1 is relatively compact if and only if S is bounded and quasi-equicontinuous

From2.35 andRemark 2.5, we obtain the following cases

Case 1 if μ, ν ∈ 0, 4/5, noticing 0 ≤ ρ ≤ 1, then we have

Case 2 if μ, ν ∈ 4/5, 1, noticing ρ ≥ 1 and γ q−1> ρ q−1, then we have

Next, we prove that T : K r,R → K is completely continuous.

It is obvious that T : K r,R → K is continuous Now we prove T is relatively compact Let Br  {x ∈ PC1

PC 1 ≤ r} be a bounded set Then, for all x ∈ Br, we have

4γ1φ q

1

4γM



 mA  r1.

2.41

Therefore TBr is uniformly bounded

On the other hand, for all t, s ∈ Jk k  0, 1, , m with t < s, we have

and by the continuity of H1t, s, we have

and then T Br is quasi-equicontinuous It follows that TBr is relatively compact on

PC10, 1 byLemma 2.9 So T is completely continuous.

3 Main Results

In this section, we applyLemma 1.1to establish the existence of positive solutions of problem

1.6 We begin by introducing the following conditions on ft, y and Iky.

H3 There exist numbers 0 < r < R < ∞ such that

f

t, y

≤ φp

1

Proof Let T be the cone preserving, completely continuous operator that was defined by

q1

Applyingb ofLemma 1.1to3.7 and 3.12 yields that T has a fixed point y ∈ Ω2\Ω1

with r PC1 ≤ R1 Hence, since for y ∈ K we have yt ≥ ρ1ρ q−1/γ1γ q−1ys, t, s ∈ J, it

follows that3.4 holds This andLemma 2.9complete the proof

As a special case ofTheorem 3.1, we can prove the following results

Corollary 3.2 Assume that H1  and H2 hold If f0 0, I0k  0, and f∞ ∞, then, for r > 0

being sufficiently small and R > 0 being sufficiently large, BVP 1.6 has at least one positive solution

y t, t ∈ J with property 3.4.

Proof The proof is similar to that of Theorem 3.1 of6

InTheorem 3.3, we assume the following condition on f t, y and Iky.

H4 There exist numbers 0 < r < R < ∞ such that

that is, y ∈ ∂Ω1implies

φ q

1 0

q−1

φ q

1 0



M  φp

14

q−1

2ηq−1

 m × 14m



1− μ y

≤ 34

Ty

PC 1≤ y

PC 1, y ∈ ∂Ω3. 3.24

Applyinga ofLemma 1.1to3.19 and 3.24 yields that T has a fixed point y ∈ Ω3\

Ω1with r PC1 ≤ R2 Hence, since for y ∈ K we have yt ≥ ρρ q−1/γ1γ q−1ys, t, s ∈ J,

it follows that3.17 holds This andLemma 2.9complete the proof

As a special case ofTheorem 3.3, we can prove the following results

Corollary 3.4 Assume that H1  and H2 hold If f0  ∞ and f∞ 0, I∞k  0; then, for r > 0

being sufficiently small and R > 0 being sufficiently large, BVP 1.6 has at least one positive solution

y t, t ∈ J with property 3.17.

Proof The proof is similar to that of Theorem 3.2 of6

Theorem 3.5 Assume that H1 , H2, 3.1 of H3 and 3.14 and 3.15 of H4 hold In

addition, letting f and I k satisfy the following condition:

H5 There is a ξ > 0 such that ρ1ρ q−1/γ1γ q−1ξ ≤ y ≤ ξ and t ∈ J implies

Then, problem1.6 has at least two positive solutions y∗t and y∗∗t with

which implies that3.30 holds

ApplyingLemma 1.1 to 3.28, 3.29, and 3.30 yields that T has two fixed point

y∗ and y∗∗with y∗∈ K l,ξ PC1< ξ }, and x∗∗∈ K ξ,L PC1≤ L}.

Hence, since for y∗ ∈ K we have y∗t ≥ ρρ q−1/γ1γ q−1y∗s, t, s ∈ J, it follows that 3.26holds This andLemma 2.9complete the proof.

Remark 3.6 Similar to the proof of that of 5, we can prove that problem 1.6 can begeneralized to obtain many positive solutions

4 Discussion

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

Discussion 1 Generally, it is difficult to obtain the upper and lower bounds of positivesolutions for nonlinear higher-order boundary value problemssee, e.g., 33,34,39–43,46,

48,49 and their references

For example, we consider the following problems:

k represent the right-hand limit and left-hand limit of

xt at t  tk , respectively, and g, h ∈ L10, 1 is nonnegative.

By means of the transformation2.5, we can convert problem 4.1 into

Using the similar proof of that of Lemmas 2.1and2.6, we can obtain the following

results In addition, if we replace μ, ν by μ∗, ν∗ inH2, respectively, then we obtain H∗

2,where

It is not difficult to prove that H∗t, s and H∗

1t, s have the similar properties to that

of Ht, s and H1t, s But for t ∈ J, H∗t, s and H∗

1t, s have no property 2.13 In fact, if

t ∈ t1, t m, then we can prove that H∗t, s and H∗

1t, s have the following properties.

5 Example

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

Example 5.1 Consider the following boundary value problem:

256 − 5668704



, 1 ≤ y ≤ 2, 5668704y2, y ≥ 2.

Proof By simple computation, we have μ  1/2, ν  1/2, γ  2, ρ  1/6, γ1  2, and ρ1 

1/6 Select r  1, and R  2, then for 0 < r < R < ∞, we have