- Research Article
- Open Access

# Existence of Solutions for a Weighted -Laplacian Impulsive Integrodifferential System with Multipoint and Integral Boundary Value Conditions

- Rong Dong
^{1}, - Yunrui Guo
^{2}, - Yuanzhang Zhao
^{3}and - Qihu Zhang
^{1}Email author

**2010**:392545

https://doi.org/10.1155/2010/392545

© Rong Dong et al. 2010

**Received:**5 October 2010**Accepted:**15 December 2010**Published:**21 December 2010

## Abstract

By the Leray-Schauder's degree, the existence of solutions for a weighted -Laplacian impulsive integro-differential system with multi-point and integral boundary value conditions is considered. The sufficient results for the existence are given under the resonance and nonresonance cases, respectively. Moreover, we get the existence of nonnegative solutions at nonresonance.

## Keywords

- Operator Equation
- Integral Boundary
- Nonnegative Solution
- Integral Boundary Condition
- Coincidence Degree

## 1. Introduction

where and , is called the weighted -Laplacian; , ; , and ; is nonnegative, with ; ; and are linear operators defined by , , , where .

If and , we say the problem is nonresonant, but if and , we say the problem is resonant.

For any , denote . Obviously, is a Banach space with the norm , and is a Banach space with the norm . Denote with the norm , for all , where .

In recent years, there has been an increasing interest in the study of differential equations with nonstandard -growth conditions. These problems have many interesting applications (see [1–4]). Many results have been obtained on these kinds of problems, for example [5–17]. If (a constant), (1.1)–(1.4) becomes the well known -Laplacian problem. If is a general function, one can see easily that in general, while , so represents a non-homogeneity and possesses more nonlinearity, thus is more complicated than . For example, we have the following.

(a)In general, the infimum of eigenvalues for the -Laplacian Dirichlet problems is zero, and only under some special conditions (see [10]). When ( ) is an interval, the results in [10] show that if and only if is monotone. But the property of is very important in the study of -Laplacian problems, for example, in [18], the authors use this property to deal with the existence of solutions.

(b)If and (a constant) and , then is concave, this property is used extensively in the study of one-dimensional -Laplacian problems (see [19]), but it is invalid for . It is another difference between and .

Recently, there are many works devoted to the existence of solutions to the Laplacian impulsive differential equation boundary value problems, for example [20–28]. Many methods had been applied to deal with these problems, for example sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree, variational principles (see [29]), and so forth. Because of the nonlinearity of , results about the existence of solutions for -Laplacian impulsive differential equation boundary value problems are rare (see [30]). In [31], using coincidence degree method, the present author investigate the existence of solutions for -Laplacian impulsive differential equation with multipoint boundary value conditions. Integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow and population dynamics, there are many papers on the differential equations with integral boundary value problems, for example, [32–35].

where , the impulsive condition (1.8) is called linear impulsive condition (LI for short), and (1.3) is called nonlinear impulsive condition (NLI for short). In generaly, -Laplacian impulsive problems have two kinds of impulsive conditions, that is, LI and NLI.

Let , the function is assumed to be Caratheodory, by this we mean the following:

(i)for almost every the function is continuous;

(ii)for each the function is measurable on ;

(iii)for each there is a such that, for almost every and every with , , , , one has

We say a function is a solution of (1.1) if with absolutely continuous on , , which satisfies (1.1) a.e. on .

where , and .

This paper is organized as four sections. In Section 2, we present some preliminary and give the operator equation which has the same solutions of (1.1)–(1.4). In Section 3, we give the existence of solutions and nonnegative solutions for system (1.1)–(1.4) at nonresonance. Finally, in Section 4, we give the existence of solutions for system (1.1)–(1.4) at resonance.

## 2. Preliminary

For any , denote . Obviously, has the following properties.

Lemma 2.1 (see [31]).

is a continuous function and satisfies the following.

It is clear that is continuous and sends bounded sets to bounded sets.

where ; .

We will discuss (2.4) with (1.4) in the cases of resonance and nonresonance, respectively.

### 2.1. The Case of Nonresonance

Denote with the norm , for all , then is a Banach space.

then is continuous. Throughout the paper, we denote . It is easy to see the following.

Lemma 2.2.

where , .

It is clear that is continuous and sends bounded sets of to bounded sets of , and hence it is compact continuous.

- (i)
The operator is continuous and sends equiintegrable sets in to relatively compact sets in .

- (ii)
The operator is continuous and sends bounded sets in to relatively compact sets in .

Proof.

it is easy to check that is a continuous operator from to .

We want to show that is a compact set.

Hence the sequence is uniformly bounded and equicontinuous. By Ascoli-Arzela theorem, there exists a subsequence of (which we rename the same) which is convergent in . According to the bounded continuous of the operator , we can choose a subsequence of (which we still denote by ) which is convergent in , then is convergent in .

- (ii)
It is easy to see from (i) and Lemma 2.2.

This completes the proof.

Let us define as .

It is easy to see that is compact continuous.

Lemma 2.4.

Proof.

Suppose is a solution of (1.1)–(1.4). From the definition of and , similar to the discussion before Lemma 2.2, we know that is a solution of (2.20).

Conversely, if is a solution of (2.20), then (1.2) is satisfied.

It follows from (2.21) that (1.3) is satisfied.

From (2.21) and the definition of , we have

It follows from (2.22) and (2.23) that (1.4) is satisfied.

Hence is a solutions of (1.1)–(1.4). This completes the proof.

### 2.2. The Case of Resonance

Denote , , . It is easy to see that is dependent on and .

Lemma 2.5.

The function has the following properties.

has unique solution .

and hence, if (2.27) has a solution, then it is unique.

Thus the th component of keeps sign on , then it is not hard to check that the th component of keeps the same sign of .

it means the existence of solutions of .

- (ii)

It only remains to prove the continuity of . Let is a convergent sequence in and , as . Since is a bounded sequence, it contains a convergent subsequence . Suppose as . Since , letting , we have , which together with (i) implies , it means is continuous. This completes the proof.

where , .

It is clear that is continuous and sends bounded sets of to bounded sets of , and hence it is a compact continuous mapping.

Similar to the proof of Lemma 2.3, we have the following lemma.

Lemma 2.6.

The operator is continuous and sends equiintegrable sets in to relatively compact sets in .

Lemma 2.7.

Proof.

Suppose is a solution of (1.1)–(1.4), it is clear that is a solution of (2.40).

Thus .

then .

Hence is a solutions of (1.1)–(1.4). This completes the proof.

## 3. Existence of Solutions in the Case of Nonresonance

In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions and nonnegative solutions for system (1.1)–(1.4) at nonresonance.

When satisfies sub- growth condition, we have the following.

Theorem 3.1.

then problem (1.1)−(1.4) has at least one solution.

Proof.

where is defined in (2.11).

It is easy to see that operator is compact continuous for any . It follows from Lemmas 2.2 and 2.3 that is compact continuous from to for any .

We claim that all the solutions of (3.3) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (3.3) such that as , and for any .

It follows from (3.8) and (3.11) that is uniformly bounded.

which implies that (1.1)–(1.4) has at least one solution. This completes the proof.

Theorem 3.2.

where , and , , then problem (1.1) with (1.2), (1.4), and (1.8) has at least one solution.

Proof.

Since , we have . Thus (3.16) is valid.

There are two cases.

Case 1 ( ).

Thus (3.16) is valid.

Case 2 ( ).

Thus (3.16) is valid. Thus problem (1.1) with (1.2), (1.4), and (1.8) has at least one solution. This completes the proof.

where , are Caratheodory.

We have the following.

Theorem 3.3.

then problem (3.21) with (1.2)–(1.4) has at least one solution when the parameter is small enough.

Proof.

where is defined in (2.11).

We know that (3.25) with (1.2)–(1.4) has the same solution of .

Since , from the proof of Theorem 3.1, we can see that the right hand side is nonzero. Thus (3.21) with (1.2)–(1.4) has at least one solution. This completes the proof.

Theorem 3.4.

where , and , , then problem (3.21) with (1.2), (1.4), and (1.8) has at least one solution when the parameter is small enough.

Proof.

As it is similar to the proof of Theorems 3.2 and 3.3, we omit it here.

In the following, we will consider the existence of nonnegative solutions. For any , the notation means for any .

Theorem 3.5.

Suppose , , we also assume

, for all ;

for any , , for all .

Then every solution of (1.1)–(1.4) is nonnegative.

Proof.

It follows from conditions (1^{0})-(2^{0}) and (3.36) that
is increasing on
, namely
, for all
with
. Thus the boundary value condition holds
, then
.

Since is increasing and , we have , for all .

Thus every solution of (1.1)–(1.4) is nonnegative. The proof is completed.

Corollary 3.6.

Under the conditions of Theorem 3.1, we also assume

, for all with ;

for any , , for all with ;

for any and , , .

Then (1.1)–(1.4) has a nonnegative solution.

Proof.

then satisfies Caratheodory condition, and for any .

It is not hard to check that

, for uniformly, where , and ;

, for all ;

, for all .

It follows from Theorems 3.1 and 3.5 that (3.41) have a nonnegative solution . Since , we have . Thus is a nonnegative solution of (1.1)−(1.4). This completes the proof.

## 4. Existence of Solutions in the Case of Resonance

In the following, we will consider the existence of solutions for system (1.1)–(1.4) at resonance.

Theorem 4.1.

Suppose and , is an open bounded set in such that the following conditions hold.

has no solution on .

has no solution on .

The Brouwer degree .

Then problem (1.1)–(1.4) have a solution on .

Proof.

It means that (4.1) and (4.3) have the same solutions for .

^{0}) it follows that (4.8) has no solutions for . For , (4.3) is equivalent to the following usual problem

^{0}) implies that . Thus we have proved that (4.8) has no solution on . Therefore the Leray-Schauder degree is well defined for , and from the homotopy invariant property of that degree we have

where the function
is defined in (4.2) and
denotes the Brouwer degree. By hypothesis (3^{0}), this last degree is different from zero. This completes the proof.

with (1.2), (1.3), and (1.4), where is Caratheodory, is continuous, and for any fixed , holds , for all , .

Theorem 4.2.

Suppose that the following conditions hold

for all and all , where satisfies ;

, for uniformly;

, for all , where ;

, for all , where ;

has no solution on , where ;

the Brouwer degree for large enough , where .

Then problem (4.18) with (1.2), (1.3), and (1.4) has at least one solution.

Proof.

where is defined in (2.39).

We claim that all the solutions of (4.21) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (4.21) such that as , and for any .

where , satisfies , .

In order to obtaining the existence of solutions (4.18) with (1.2), (1.3), and (1.4), we only need to prove that .

where .

Similar to the preceding discussion, for any , all the solutions of (4.35) are uniformly bounded.

By hypothesis (5^{0}), (4.35) has no solutions on
, from Theorem 4.1, we obtain that (4.18) with (1.2), (1.3), and (1.4) has at least one solution. This completes the proof.

Corollary 4.3.

If
is Caratheodory, conditions (2^{0}), (3^{0}) and (4^{0}) of Theorem 4.2 are satisfied, condition (3^{0}) of Corollary 3.6 is also satisfied,
, where
are positive functions satisfying
; then (4.18) with (1.2), (1.3), and (1.4) has at least one solution.

Proof.

According to Theorem 4.2, we get that (4.18) with (1.2), (1.3), and (1.4) has at least a solution. This completes the proof.

where are Caratheodory.

From Theorem 4.2, similar to the proof of Theorem 3.3, we have the following.

Theorem 4.4.

If conditions of (1^{0}) and (3^{0})–(6^{0}) of Theorem 4.2 are satisfied, then problem (4.43) with (1.2), (1.3), and (1.4) has at least one solution when the parameter
is small enough.

Theorem 4.5.

^{0})–(3

^{0}) and (5

^{0})-(6

^{0}) of Theorem 4.2 are satisfied, and satisfy

then problem (4.18) with (1.2), (1.3), and (1.8) has at least one solution.

Proof.

Similar to the proof of Theorem 3.2, the condition (4^{0}) of Theorem 4.2 is satisfied. Thus problem (4.18) with (1.2), (1.3) and (1.8) has at least a solution.

Similar to the proof of Theorem 3.2 and Corollary 4.3, we have the following.

Corollary 4.6.

If
is Caratheodory, (4.45), (4.46) and conditions (2^{0}) and (3^{0}) of Theorem 4.2 are satisfied, condition (3^{0}) of Corollary 3.6 is also satisfied,
, where
are positive functions satisfying
; then (4.43) with (1.2), (1.3), and (1.8) has at least one solution when the parameter
is small enough.

## Declarations

### Acknowledgments

This paper is partly supported by the National Science Foundation of China (10701066, 10926075, and 10971087), China Postdoctoral Science Foundation funded project (20090460969), and the Natural Science Foundation of Henan Education Committee (2008-755-65).

## Authors’ Affiliations

## References

- Acerbi E, Mingione G: Regularity results for stationary electro-rheological fluids.
*Archive for Rational Mechanics and Analysis*2002, 164(3):213–259. 10.1007/s00205-002-0208-7MathSciNetView ArticleMATHGoogle Scholar - Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration.
*SIAM Journal on Applied Mathematics*2006, 66(4):1383–1406. 10.1137/050624522MathSciNetView ArticleMATHGoogle Scholar - Růžička M:
*Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics*.*Volume 1748*. Springer, Berlin, Germany; 2000:xvi+176.MATHGoogle Scholar - Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory.
*Mathematics of the USSR-Izvestiya*1986, 29: 33–36.View ArticleMathSciNetMATHGoogle Scholar - Acerbi E, Mingione G: Regularity results for a class of functionals with non-standard growth.
*Archive for Rational Mechanics and Analysis*2001, 156(2):121–140. 10.1007/s002050100117MathSciNetView ArticleMATHGoogle Scholar - Coscia A, Mingione G: Hölder continuity of the gradient of -harmonic mappings.
*Comptes Rendus de l'Académie des Sciences. Série I. Mathématique*1999, 328(4):363–368.MathSciNetMATHGoogle Scholar - Deng S-G: A local mountain pass theorem and applications to a double perturbed -Laplacian equations.
*Applied Mathematics and Computation*2009, 211(1):234–241. 10.1016/j.amc.2009.01.042MathSciNetView ArticleMATHGoogle Scholar - Fan X: Global regularity for variable exponent elliptic equations in divergence form.
*Journal of Differential Equations*2007, 235(2):397–417. 10.1016/j.jde.2007.01.008MathSciNetView ArticleMATHGoogle Scholar - Fan X: Boundary trace embedding theorems for variable exponent Sobolev spaces.
*Journal of Mathematical Analysis and Applications*2008, 339(2):1395–1412. 10.1016/j.jmaa.2007.08.003MathSciNetView ArticleMATHGoogle Scholar - Fan X, Zhang Q, Zhao D: Eigenvalues of -Laplacian Dirichlet problem.
*Journal of Mathematical Analysis and Applications*2005, 302(2):306–317. 10.1016/j.jmaa.2003.11.020MathSciNetView ArticleMATHGoogle Scholar - Harjulehto P, Hästö P, Latvala V: Harnack's inequality for -harmonic functions with unbounded exponent .
*Journal of Mathematical Analysis and Applications*2009, 352(1):345–359. 10.1016/j.jmaa.2008.05.090MathSciNetView ArticleMATHGoogle Scholar - Mihăilescu M, Rădulescu V: Continuous spectrum for a class of nonhomogeneous differential operators.
*Manuscripta Mathematica*2008, 125(2):157–167. 10.1007/s00229-007-0137-8MathSciNetView ArticleMATHGoogle Scholar - Mihăilescu M, Pucci P, Rădulescu V: Nonhomogeneous boundary value problems in anisotropic Sobolev spaces.
*Comptes Rendus de l'Académie des Sciences—Series I*2007, 345(10):561–566.MathSciNetMATHGoogle Scholar - Mihăilescu M, Rădulescu V, Repovš D: On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting.
*Journal de Mathématiques Pures et Appliquées*2010, 93(2):132–148.View ArticleMathSciNetMATHGoogle Scholar - Musielak J:
*Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics*.*Volume 1034*. Springer, Berlin, Germany; 1983:iii+222.Google Scholar - Samko SG: Density in the generalized Sobolev spaces .
*Doklady Rossiĭskaya Akademiya Nauk*1999, 369(4):451–454.MathSciNetMATHGoogle Scholar - Zhang Q: Existence of solutions for -Laplacian equations with singular coefficients in .
*Journal of Mathematical Analysis and Applications*2008, 348(1):38–50. 10.1016/j.jmaa.2008.06.026MathSciNetView ArticleMATHGoogle Scholar - Kim I-S, Kim Yun-Ho: Global bifurcation of the -Laplacian in .
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(7):2685–2690. 10.1016/j.na.2008.03.055MathSciNetView ArticleMATHGoogle Scholar - Ahmad B, Nieto JJ: The monotone iterative technique for three-point second-order integrodifferential boundary value problems with -Laplacian.
*Boundary Value Problems*2007, -9.Google Scholar - Jiao J, Chen L, Li L: Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations.
*Journal of Mathematical Analysis and Applications*2008, 337(1):458–463. 10.1016/j.jmaa.2007.04.021MathSciNetView ArticleMATHGoogle Scholar - Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations.
*Journal of Mathematical Analysis and Applications*2007, 325(1):226–236. 10.1016/j.jmaa.2005.04.005MathSciNetView ArticleMATHGoogle Scholar - Liu L, Hu L, Wu Y: Positive solutions of two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(11):3774–3789. 10.1016/j.na.2007.10.012MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, O'Regan D: Variational approach to impulsive differential equations.
*Nonlinear Analysis: Real World Applications*2009, 10(2):680–690. 10.1016/j.nonrwa.2007.10.022MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ: Impulsive resonance periodic problems of first order.
*Applied Mathematics Letters*2002, 15(4):489–493. 10.1016/S0893-9659(01)00163-XMathSciNetView ArticleMATHGoogle Scholar - Di Piazza L, Satco B: A new result on impulsive differential equations involving non-absolutely convergent integrals.
*Journal of Mathematical Analysis and Applications*2009, 352(2):954–963. 10.1016/j.jmaa.2008.11.048MathSciNetView ArticleMATHGoogle Scholar - Shen J, Wang W: Impulsive boundary value problems with nonlinear boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(11):4055–4062. 10.1016/j.na.2007.10.036MathSciNetView ArticleMATHGoogle Scholar - Yao M, Zhao A, Yan J: Periodic boundary value problems of second-order impulsive differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(1):262–273. 10.1016/j.na.2007.11.050MathSciNetView ArticleMATHGoogle 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.2008.04.015MathSciNetView ArticleMATHGoogle Scholar - Kristály A, Rădulescu V, Varga C:
*Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136*. Cambridge University Press, Cambridge, UK; 2010.View ArticleMATHGoogle Scholar - Cabada A, Tomeček J: Extremal solutions for nonlinear functional -Laplacian impulsive equations.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 67(3):827–841. 10.1016/j.na.2006.06.043MathSciNetView ArticleMATHGoogle Scholar - Zhang Q, Qiu Z, Liu X: Existence of solutions and nonnegative solutions for weighted -Laplacian impulsive system multi-point boundary value problems.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 71(9):3814–3825. 10.1016/j.na.2009.02.040MathSciNetView ArticleMATHGoogle Scholar - Yang Z: Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(1):216–225. 10.1016/j.na.2006.10.044MathSciNetView ArticleMATHGoogle Scholar - Li Y, Li F: Sign-changing solutions to second-order integral boundary value problems.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(4):1179–1187. 10.1016/j.na.2007.06.024MathSciNetView ArticleMATHGoogle Scholar - Ma R, An Y: Global structure of positive solutions for nonlocal boundary value problems involving integral conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 71(10):4364–4376. 10.1016/j.na.2009.02.113MathSciNetView ArticleMATHGoogle Scholar - Zhang X, Yang X, Ge W: Positive solutions of th-order impulsive boundary value problems with integral boundary conditions in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 71(12):5930–5945. 10.1016/j.na.2009.05.016MathSciNetView ArticleMATHGoogle Scholar

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