- Research Article
- Open access
- Published:
Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Conditions and One-Dimensional -Laplacian
Boundary Value Problems volume 2011, Article number: 654871 (2011)
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.
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].
On the other hand, boundary-value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint and nonlocal boundary-value problems as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention in the literature. To identify a few, we refer the reader to [24–46] and references therein. In particular, we would like to mention some results of Zhang et al. [34], Kang et al. [44], and Webb et al. [45]. In [34], Zhang et al. studied the following fourth-order boundary value problem with integral boundary conditions
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.
In [44], Kang et al. have improved and generalized the work of [34] by applying the fixed point theory in cone for a strict set contraction operator; they proved that there exist various results on the existence of positive solutions to a class of fourth-order singular boundary value problems with integral boundary conditions
where and may be singular at or ; are continuous and may be singular at , and ; , and , and , and are nonnegative, .
More recently, by using a unified approach, Webb et al. [45] considered the widely studied boundary conditions corresponding to clamped and hinged ends and many nonlocal boundary conditions and established excellent existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam
subject to various boundary conditions
where denotes a linear functional on given by involving a Stieltjes integral, and is a function of bounded variation.
At the same time, we notice that there has been a considerable attention on -Laplacian BVPs [18, 32, 35, 36, 38, 42] as -Laplacian appears in the study of flow through porous media (), nonlinear elasticity (), glaciology (), and so forth. Here, it is worth mentioning that Liu et al. [43] considered the following fourth-order four-point boundary value problem:
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.
Motivated by works mentioned above, in this paper, we consider the existence of positive solutions for a class of boundary value problems with integral boundary conditions of fourth-order impulsive differential equations:
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:
(a), and ;
(b), and .
Then, has at least one fixed point in .
2. Preliminaries
In order to define the solution of problem (1.6), we shall consider the following space.
Let , and
Then is a real Banach space with norm
where .
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:
;
with
Write
From , it is clear that .
We shall reduce problem (1.6) to an integral equation. To this goal, firstly by means of the transformation
we convert problem (1.6) into
Lemma 2.1.
Assume that and hold. Then problem (2.6) has a unique solution given by
where
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.
If holds, then we have
Proposition 2.3.
For , we have
Proposition 2.4.
If holds, then for , we have
where
Proof.
By (2.6) and (2.12), we have
On the other hand, noticing , we obtain
The proof of Proposition 2.4 is complete.
Remark 2.5.
From (2.9) and (2.13), we can obtain that
Lemma 2.6.
If and hold, then problem (2.7) has a unique solution and can be expressed in the following form:
where
and is defined in (2.10).
Proof.
First suppose that is a solution of problem (2.7).
If it is easy to see by integration of problem (2.7) that
If then integrate from to ,
Similarly, if we have
Integrating again, we can get
Letting in (2.23), we find
Substituting and (2.24) into (2.23), we obtain
where
Therefore, we have
Let
Then,
and the proof of sufficient is complete.
Conversely, if is a solution of (2.18).
Direct differentiation of (2.18) implies, for ,
Evidently,
The Lemma is proved.
Remark 2.7.
From (2.19), we can prove that the properties of are similar to that of .
Suppose that is a solution of problem (1.6). Then from Lemmas 2.6 and 2.1, we have
For the sake of applying Lemma 1.1, we construct a cone in via
where
It is easy to see that is a closed convex cone of .
Define an operator by
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]).
The set is said to be quasi-equicontinuous in if for any there exist such that if then
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 .
Write
where .
Lemma 2.10.
Suppose that and hold. Then and is completely continuous.
Proof.
For , it is clear that , and
From (2.35) and Remark 2.5, we obtain the following cases.
Case 1.
if , noticing , then we have
Case 2.
if , noticing and , then we have
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.
Let be a bounded set. Then, for all , we have
Therefore is uniformly bounded.
On the other hand, for all with , we have
and by the continuity of , we have
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 .
There exist numbers such that
where is defined in (2.4), are defined in (2.14), respectively, and write
where denotes or
Theorem 3.1.
Assume that hold. Then problem (1.6) has at least one positive solution with
Proof.
Let be the cone preserving, completely continuous operator that was defined by (2.35). For with , (2.13), (2.19), and (3.1) imply
where
Now if we let , then (3.5) shows that
Further, let
Then, and implies
that is,
Hence, for all . Therefore, for all , (3.2) implies
that is, implies
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 .
There exist numbers such that
where are defined in (2.14), and write
Theorem 3.3.
Assume that , and hold. Then problem (1.6) has at least one positive solution with
Proof.
For with , (3.13) implies
that is, implies
Next, we turn to (3.14) and (3.15). From (3.14), (3.15), and (3.16), we have
Let
Thus, for , we have
Then, (3.22) and (3.23) imply
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:
There is a such that and implies
Then, problem (1.6) has at least two positive solutions and with
where and satisfy
Proof.
If (3.1) of holds, similar to the proof of (3.7), we can prove that
If (3.14) and (3.15) of hold, similar to the proof of (3.23), we have
Finally, we show that
In fact, for with then by (2.18), we have
and it follows from that
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).
For example, we consider the following problems:
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.
By means of the transformation (2.5), we can convert problem (4.1) into
Using the similar proof of that of Lemmas 2.1 and 2.6, we can obtain the following results. In addition, if we replace by in , respectively, then we obtain , where
Lemma 4.1.
If and hold, then BVP (4.2) has a unique solution and can be expressed in the form
where
is defined in (2.10).
Lemma 4.2.
If and hold, then BVP (4.3) has a unique solution and can be expressed in the form
where
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.
If holds, then for , we have
where
Proof.
We only consider (4.9). By (4.3), we have
On the other hand, noticing , we obtain
Similarly, we can prove that (4.10) holds, too.
Remark 4.4.
Since , .
From (4.9) and (4.10), we can only define a cone by
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.
Consider the following boundary value problem:
where , and
Conclusion.
Equation(5.1) has at least one positive solution for with
Proof.
By simple computation, we have . Select then for , we have
By Theorem 3.1, (5.1) has a positive solution with .
References
Lakshmikantham V, BaÄnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Singapore; 1989:xii+273.
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.
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.2009.09.002
BaÄnov DD, Simeonov PS: Systems with Impulse Effect. Ellis Horwood, Chichester, UK; 1989:255.
SamoÄlenko AM, Perestyuk NA: Impulsive Differential Equations. Volume 14. World Scientific, Singapore; 1995:x+462.
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.2006.04.018
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.2008.01.024
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.1997.5207
Nieto JJ: Impulsive resonance periodic problems of first order. Applied Mathematics Letters 2002, 15(4):489-493. 10.1016/S0893-9659(01)00163-X
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)00356-9
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.1997.5688
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)00074-0
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)00110-7
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/s00010-004-2735-9
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.2005.07.076
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.2007.07.081
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.2008.02.040
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.2008.04.015
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/1021477351
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.
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.
Corduneanu C: Integral Equations and Applications. Cambridge University Press, Cambridge, UK; 1991:x+366.
Agarwal RP, O'Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+341.
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/s00030-007-4067-7
Webb JRL, Infante G: Non-local boundary value problems of arbitrary order. Journal of the London Mathematical Society 2009, 79(1):238-258.
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/S0024610706023179
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.
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.
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.
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.2007.11.003
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.2007.10.019
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.2006.11.009
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.2007.12.002
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.2007.09.020
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.2005.09.002
Ma R:Positive solutions for multipoint boundary value problem with a one-dimensional -Laplacian. Computational & Applied Mathematics 2001, 42: 755-765.
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.2004.10.010
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.2006.04.005
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.2007.03.006
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)80006-3
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)00071-9
Ma R, Wang H: On the existence of positive solutions of fourth-order ordinary differential equations. Applicable Analysis 1995, 59(1–4):225-231.
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.2006.03.029
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.2008.09.010
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.
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.2006.11.026
Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.
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.2004.05.009
Hao X, Liu L, Wu Y:Positive solutions for nonlinear th-order singular nonlocal boundary value problems. Boundary Value Problems 2007, 2007:-10.
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Feng, M. Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Conditions and One-Dimensional -Laplacian. Bound Value Probl 2011, 654871 (2011). https://doi.org/10.1155/2011/654871
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/654871