Research Article | Open Access

Guizhen Zhi, Yunrui Guo, Yan Wang, Qihu Zhang, "Existence of Solutions for a Class of Variable Exponent Integrodifferential System Boundary Value Problems", *Journal of Applied Mathematics*, vol. 2011, Article ID 814103, 40 pages, 2011. https://doi.org/10.1155/2011/814103

# Existence of Solutions for a Class of Variable Exponent Integrodifferential System Boundary Value Problems

**Academic Editor:**Alexandre Carvalho

#### Abstract

This paper investigates the existence of solutions for a class of variable exponent integrodifferential system with multipoint and integral boundary value condition in half line. When the nonlinearity term satisfies sub-() growth condition or general growth condition, we give the existence of solutions and nonnegative solutions via Leray-Schauder degree at nonresonance, respectively. Moreover, the existence of solutions for the problem at resonance has been discussed.

#### 1. Introduction

In this paper, we consider the existence of solutions for the following variable exponent integrodifferential system with the following nonlinear multipoint and integral boundary value condition where and are linear operators defined by where and are uniformly bounded with exists and is called the weighted -Laplacian; satisfies , for all , and and is nonnegative, and is a positive parameter.

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

The study of differential equations and variational problems with variable exponent growth conditions is a new and interesting topic. Many results have been obtained on these problems, for example, [1–23]. We refer to [3, 19, 23], for the applied background on these problems. If and (a constant), becomes the well-known -Laplacian. If is a general function, represents a nonhomogeneity and possesses more nonlinearity, and thus is more complicated than . For example, if is a bounded domain, the Rayleigh quotient is zero in general, and only under some special conditions (see [9, 16–18]), but the fact that is very important in the study of -Laplacian problems.

Integral boundary conditions for evolution problems have been applied variously in chemical engineering, thermoelasticity, underground water flow, and population dynamics. There are many papers on the differential equations with integral boundary value conditions, for example, [24–29]. On the existence of solutions for -Laplacian systems boundary value problems, we refer to [2, 4, 7, 8, 10–12, 20–22]. In [20], the present author deals with the existence and asymptotic behavior of solutions for (1.1) with the following linear boundary value conditions when and . But results on the existence of solutions for variable exponent integrodifferential systems with nonlinear boundary value conditions are rare. In this paper, when is a general function, we investigate the existence of solutions and nonnegative solutions for variable exponent integrodifferential systems with nonlinear multipoint and integral boundary value conditions, when the problem is at nonresonance. Moreover, we discuss the existence of solutions for the problem at resonance. Since the nonlinear multipoint boundary value condition is on the derivative of solution , we meet more difficulties than [20].

Let and , the function is assumed to be Caratheodory, by this we mean,(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

Throughout the paper, we denote

The inner product in will be denoted by will denote the absolute value and the Euclidean norm on . Let denote the space of absolutely continuous functions on the interval . For , we set . For any , we denote , and . Spaces and will be equipped with the norm and , respectively. Then, and are Banach spaces. Denote with the norm .

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

In this paper, we always use to denote positive constants, if it cannot lead to confusion. Denote

We say satisfies sub-() growth condition, if satisfies where and . We say satisfies general growth condition, if does not satisfy sub-() growth condition.

We will discuss the existence of solutions of (1.1)-(1.2) in the following three cases.Case (i): ;Case (ii): ;Case (iii): .

This paper is divided into five sections. In the second section, we present some preliminary and give the operator equations which have the same solutions of (1.1)-(1.2) in the three cases, respectively. In the third section, we will discuss the existence of solutions of (1.1)-(1.2) when , and we give the existence of nonnegative solutions. In the fourth section, we will discuss the existence of solutions of (1.1)-(1.2) when . In the fifth section, we will discuss the existence of solutions of (1.1)-(1.2) when .

#### 2. Preliminary

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

Lemma 2.1 (see [7]). * is a continuous function and satisfies the following.*(i)*For any , is strictly monotone, that is
*(ii)*There exists a function as , such that
*

It is well known that is a homeomorphism from to for any fixed . For any , denote by the inverse operator of , then

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

Let us now consider the following problem with boundary value condition (1.2) where .

If is a solution of (2.4) with (1.2), by integrating (2.4) from 0 to , we find that

Define operator as

By solving for in (2.5) and integrating, we find that

In the following, we will give the operator equations which have the same solutions of (1.1)-(1.2) in three cases, respectively.

##### 2.1. Case (i):

We denote in (2.7). It is easy to see that is dependent on , then we find that

The boundary value condition (1.2) implies that

For fixed , we define as

Lemma 2.2. * is continuous and sends bounded sets of to bounded sets of . Moreover,
*

*Proof. *Since consists of continuous operators, it is continuous. It is easy to see that

This completes the proof.

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

If is a solution of (2.4) with (1.2), we find that

We denote

We say a set be equi-integrable, if there exists a nonnegative , such that

Lemma 2.3. *The operator is continuous and sends equi-integrable sets in to relatively compact sets in .*

*Proof. *It is easy to check that , for all . Since and
it is easy to check that is a continuous operator from to .

Let now be an equi-integrable set in , then there exists a nonnegative , such that

We want to show that is a compact set.

Let be a sequence in , then there exists a sequence such that . Since where , we have
then for any with , we have
which together with (2.17) implies

Hence, the sequence is uniformly bounded. According to the absolute continuity of Lebesgue integral, for any , there exists a such that if , then we have . Thus, (2.20) means that is equicontinuous.

Denote . Obviously, is uniformly bounded and equicontinuous on for . By Ascoli-Arzela Theorem, there exists a subsequence of being convergent in , we may assume in . Since is uniformly bounded and equicontinuous on , there exists a subsequence of such that is convergent in , we may assume in . Obviously, , for any . Repeating the process, we get a subsequence of such that is convergent in , we may assume in . Obviously, for any . Select the diagonal element, we can see that is a subsequence of which satisfies that is convergent in and in . Thus, we get a function which is defined on such that for any , and in .

From (2.20), it is easy to see that for any , exists (we denote the limit by ), and, for any , there exists an integer such that , and then

Since is uniformly bounded, then is bounded. By choosing a subsequence, we may assume that

We claim that . In fact, for any , from (2.21), we have

Since and letting , the above inequality implies

Thus,

Next, we will prove that tend to uniformly.

Suppose . From (2.21) and (2.24), we have

From (2.22), there exists a such that for . Thus, for any ,

Suppose . Since in , there exists a such that

Thus,

This means that tend to uniformly, that is, tend to in .

According to the bounded continuous of the operator , we can choose a subsequence of (which we still denote ) which is convergent in , then is convergent in .

Since
it follows from the continuity of and the integrability of in that is convergent in . Thus, is convergent in . This completes the proof.

Let us define as It is easy to see that is compact continuous.

Throughout the paper, we denote the Nemytskii operator associated to defined by

Lemma 2.4. *In the Case (i), is a solution of (1.1)-(1.2) if and only if is a solution of the following abstract equation:
*

*Proof. *If is a solution of (1.1)-(1.2), by integrating (1.1) from 0 to , we find that
which implies that

From , we have

So we have

Conversely, if is a solution of (2.33), we have
which implies that
then

From (2.33), we can obtain

It follows from the condition of the mapping that
and then
thus

From (2.40) and (2.44), we obtain (1.2).

From (2.41), we have

Hence, is a solution of (1.1)-(1.2). This completes the proof.

##### 2.2. Case (ii):

We denote in (2.7). It is easy to see that is dependent on , and we have

The boundary value condition (1.2) implies that

For fixed , we define as

Similar to Lemma 2.2, we have the following.

Lemma 2.5. * is continuous and sends bounded sets of to bounded sets of . Moreover,
*

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

Let us define

Lemma 2.6. *The operator is continuous and sends equi-integrable sets in to relatively compact sets in .*

*Proof. *Similar to the proof of Lemma 2.3, we omit it here.

Lemma 2.7. *In Case (ii), is a solution of (1.1)-(1.2) if and only if is a solution of the following abstract equation:
*

*Proof. *If is a solution of (1.1)-(1.2), by integrating (1.1) from 0 to , we find that
which implies that

From and , we obtain
then

Conversely, if is a solution of (2.51), then

Thus, , and we have
then

Thus,

Similar to the proof of Lemma 2.4, we can have

From (2.59) and (2.60), we obtain (1.2).

From (2.51), we have
then

Hence, is a solution of (1.1)-(1.2). This completes the proof.

##### 2.3. Case (iii):

We denote in (2.7). It is easy to see that is dependent on , then we find that

The boundary value condition (1.2) implies that

For fixed , we denote

Throughout the paper, we denote

Lemma 2.8. *The function has the following properties.*(i)*For any fixed , the equation
**has a unique solution .*(ii)*The function , defined in (i), is continuous and sends bounded sets to bounded sets. Moreover,
**where the notation means
*

*Proof. *(i) From Lemma 2.1, it is immediate that
and, hence, if (2.67) has a solution, then it is unique.

Let . Since and , if , it is easy to see that there exists an such that the th component of satisfies . Thus, keeps sign on and

Obviously, , then

Thus, the th component of is nonzero and keeps sign, and then we have

Let us consider the equation

It is easy to see that all the solutions of (2.74) belong to . So, we have
and it means the existence of solutions of .

In this way, we define a function , which satisfies

(ii) By the proof of (i), we also obtain sends bounded sets to bounded sets, and

It only remains to prove the continuity of . Let be a convergent sequence in and as . Since is a bounded sequence, then it contains a convergent subsequence . Let as . Since , letting , we have . From (i), we get , it means that is continuous. This completes the proof.

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

Let us define where and satisfies . We denote as

Similar to Lemmas 2.3 and 2.7, we have the following

Lemma 2.9. *The operator is continuous and sends equi-integrable sets in to relatively compact sets in .*

Lemma 2.10. *In Case (iii), is a solution of (1.1)-(1.2) if and only if is a solution of the following abstract equation:
*

#### 3. Existence of Solutions in Case (i)

In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)-(1.2) when . Moreover, we give the existence of nonnegative solutions.

Theorem 3.1. *In Case (i), if satisfies sub-() growth condition, then problem (1.1)-(1.2) has at least a solution for any fixed parameter .*

*Proof. *Denote , where is defined in (2.32). We know that (1.1)-(1.2) has the same solution of
when .

It is easy to see that the operator is compact continuous. According to Lemmas 2.2 and 2.3, we can see that is compact continuous from to .

We claim that all the solutions of (3.1) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (3.1) such that as and for any .

From Lemma 2.2, we have
then we have

From (3.1), we have
then

Denote , from the above inequality, we have

It follows from (2.36) and (3.3) that

For any , we have
which implies that

Thus,

It follows from (3.6) and (3.10) that is bounded.

Thus, we can choose a large enough such that all the solutions of (3.1) belong to . Thus, the Leray-Schauder degree is well defined for each , and

Let
where is defined in (2.10), thus is the unique solution of .

It is easy to see that is a solution of if and only if is a solution of the following system

Obviously, system (*I *) possesses a unique solution . Note that , we have

Therefore, (1.1)-(1.2) has at least one solution when . This completes the proof.

Denote