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# Solutions and nonnegative solutions for a weighted variable exponent impulsive integro-differential system with multi-point and integral mixed boundary value problems

Rong Dong and Qihu Zhang*

Author Affiliations

Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan, 450002, China

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Boundary Value Problems 2013, 2013:161  doi:10.1186/1687-2770-2013-161

 Received: 19 March 2013 Accepted: 18 June 2013 Published: 5 July 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper investigates the existence of solutions for a weighted -Laplacian impulsive integro-differential system with multi-point and integral mixed boundary value problems via Leray-Schauder’s degree; sufficient conditions for the existence of solutions are given. Moreover, we get the existence of nonnegative solutions.

MSC: 34B37.

##### Keywords:
weighted -Laplacian; impulsive integro-differential system; Leray-Schauder’s degree

### 1 Introduction

In this paper, we consider the existence of solutions and nonnegative solutions for the following weighted -Laplacian integro-differential system:

(1)

where , , , , with the following impulsive boundary value conditions:

(2)

(3)

(4)

where and , is called the weighted -Laplacian; , ; (); is nonnegative, ; , ; ; T and S are linear operators defined by , , , where .

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

Throughout the paper, means functions which are uniformly convergent to 0 (as ); for any , will denote the jth component of v; the inner product in will be denoted by , will denote the absolute value and the Euclidean norm on . Denote , , , , , where , . Denote by the interior of , . Let

satisfy , , and ,

For any , denote .

Obviously, is a Banach space with the norm , and is a Banach space with the norm . Denote with the norm

In the following, and will be simply denoted by PC and , respectively. We denote

The study of differential equations and variational problems with nonstandard -growth conditions has attracted more and more interest in recent years (see [1-4]). The applied background of these kinds of problems includes nonlinear elasticity theory [4], electro-rheological fluids [1,3], and image processing [2]. Many results have been obtained on these kinds of problems; see, for example, [5-15]. Recently, the applications of variable exponent analysis in image restoration have attracted more and more attention [16-19]. If (a constant), (1)-(4) becomes the well-known p-Laplacian problem. If is a general function, one can see easily in general, but , so represents a non-homogeneity and possesses more nonlinearity, thus is more complicated than . For example:

(a) If is a bounded domain, the Rayleigh quotient

is zero in general, and only under some special conditions (see [9]), when () is an interval, the results show that if and only if is monotone. But the property of is very important in the study of p-Laplacian problems, for example, in [20], the authors use this property to deal with the existence of solutions.

(b) If and (a constant) and , then u is concave, this property is used extensively in the study of one-dimensional p-Laplacian problems (see [21]), but it is invalid for . It is another difference between and .

In recent years, many results have been devoted to the existence of solutions for the Laplacian impulsive differential equation boundary value problems; see, for example, [22-29]. There are some methods to deal with these problems, for example, sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree. Because of the nonlinear property of , results on the existence of solutions for p-Laplacian impulsive differential equation boundary value problems are rare (see [30-33]). In [34], using the coincidence degree method, the present author investigates the existence of solutions for -Laplacian impulsive differential equation with multi-point boundary value conditions, when the problem is nonresonant. Integral boundary conditions for evolution problems have various applications in chemical engineering, thermo-elasticity, underground water flow and population dynamics. There are many papers on the differential equations with integral boundary value problems; see, for example, [35-38].

In this paper, when is a general function, we investigate the existence of solutions and nonnegative solutions for the weighted -Laplacian impulsive integro-differential system with integral and multi-point boundary value conditions. Results on these kinds of problems are rare. Our results contain both of the cases of resonance and nonresonance. Our method is based upon Leray-Schauder’s degree. The homotopy transformation used in [34] is unsuitable for this paper. Moreover, this paper will consider the existence of (1) with (2), (4) and the following impulsive condition:

(5)

where , the impulsive condition (5) is called a linear impulsive condition (LI for short), and (3) is called a nonlinear impulsive condition (NLI for short). In general, p-Laplacian impulsive problems have two kinds of impulsive conditions, including LI and NLI; but Laplacian impulsive problems only have LI in general. It is another difference between p-Laplacian impulsive problems and Laplacian impulsive problems. Moreover, since the Rayleigh quotient in general and the -Laplacian is non-homogeneity, when we deal with the existence of solutions of variable exponent impulsive problems like (1)-(4), we usually need the nonlinear term that satisfies the sub- growth condition, but for the p-Laplacian impulsive problems, the nonlinear term only needs to satisfy the sub- growth condition.

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

(i) For almost every , the function is continuous;

(ii) For each , the function is measurable on J;

(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) if with absolutely continuous on , , which satisfies (1) a.e. on J.

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

We say f satisfies the sub- growth condition if f satisfies

where and .

We will discuss the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) in the following three cases:

Case (i): , ;

Case (ii): , ;

Case (iii): , .

This paper is organized as five sections. In Section 2, we present some preliminaries and give the operator equation which has the same solutions of (1)-(4) in the three cases, respectively. In Section 3, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when , . In Section 4, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when , . Finally, in Section 5, we give the existence of solutions and nonnegative solutions for system (1)-(4) or (1) with (2), (4) and (5) when , .

### 2 Preliminary

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

Lemma 2.1 (see [34])

φis a continuous function and satisfies:

(i) For any, is strictly monotone, i.e.,

(ii) There exists a function, assuch that

It is well known that is a homeomorphism from to for any fixed . Denote

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

In this section, we will do some preparation and give the operator equation which has the same solutions of (1)-(4) in three cases, respectively. At first, let us now consider the following simple impulsive problem with boundary value condition (4):

(6)

where ; .

Denote , . Obviously, .

We will discuss it in three cases, respectively.

#### 2.1 Case (i)

Suppose that and . If u is a solution of (6) with (4), we have

(7)

Denote . It is easy to see that is dependent on a, b and . Define the operator as

By solving for in (7) and integrating, we find

which together with boundary value condition (4) implies

and

Denote with the norm

then W is a Banach space.

For any , we denote

Denote . Then

Throughout the paper, we denote

Lemma 2.2Suppose thaton, () andon. Then the functionhas the following properties:

(i) For any fixed, the equation

(8)

has a unique solution.

(ii) The function, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any, we have

where the notationmeans

Proof (i) From Lemma 2.1, it is immediate that

and hence, if (8) has a solution, then it is unique.

Set .

Suppose that , it is easy to see that there exists some such that the absolute value of the th component of satisfies

Thus the th component of keeps sign on J, namely, for any , we have

Obviously, we have

then it is easy to see that the th component of keeps the same sign of . Thus,

Let us consider the equation

(9)

According to the preceding discussion, all the solutions of (9) belong to . Therefore

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 W and , as . Since is a bounded sequence, it contains a convergent subsequence . Suppose that as . Since , letting , we have , which together with (i) implies , it means is continuous. This completes the proof. □

Now we denote by the Nemytskii operator associated to f defined by

(10)

We define as

(11)

where , .

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

If u is a solution of (6) with (4), we have

For fixed , we denote as

Define as

Lemma 2.3 (i) The operatoris continuous and sends equi-integrable sets into relatively compact sets in.

(ii) The operatoris continuous and sends bounded sets into relatively compact sets in.

Proof (i) It is easy to check that , , . Since and

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

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

We want to show that is a compact set.

Let be a sequence in , then there exists a sequence such that . For any , we have

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

Since

it follows from the continuity of and the integrability of in that is convergent in PC. Thus 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 2.4Suppose that, ; on, () andon. Thenuis a solution of (1)-(4) if and only ifuis a solution of the following abstract operator equation:

(12)

Proof Suppose that u is a solution of (1)-(4). By integrating (1) from 0 to t, we find that

(13)

It follows from (13) and (4) that

(14)

Combining the definition of , we can see

Conversely, if u is a solution of (12), then (2) is satisfied. It is easy to check that

(15)

and

By the condition of the mapping , we have

Thus

(16)

It follows from (15) and (16) that (4) is satisfied.

From (12), we have

(17)

It follows from (17) that (3) is satisfied.

Hence u is a solution of (1)-(4). This completes the proof. □

#### 2.2 Case (ii)

Suppose that and . If u is a solution of (6) with (4), we have

Denote . It is easy to see that is dependent on a, b and . Boundary value condition (4) implies that

For any , we denote

Throughout the paper, we denote .

Lemma 2.5The functionhas the following properties:

(i) For any fixed, the equationhas a unique solution.

(ii) The function, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any, we have

where the notationmeans

Proof Similar to the proof of Lemma 2.2, we omit it here. □

We define as , where , .

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

For fixed , we denote as

Define as

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

Lemma 2.6 (i) The operatoris continuous and sends equi-integrable sets into relatively compact sets in.

(ii) The operatoris continuous and sends bounded sets into relatively compact sets in.

Let us defineas

It is easy to see thatis compact continuous.

Lemma 2.7Suppose that, , thenuis a solution of (1)-(4) if and only ifuis a solution of the following abstract operator equation:

Proof Similar to the proof of Lemma 2.4, we omit it here. □

#### 2.3 Case (iii)

Suppose that and . If u is a solution of (6) with (4), we have

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

From , we have

(18)

From , we obtain

(19)

For fixed , we denote

From (18) and (19), we have .

Obviously, can be rewritten as

Denote . Moreover, we also have

Lemma 2.8Suppose that, g, hsatisfy one of the following:

(10) , ;

(20) on, () andon.

Then the functionhas the following properties:

(i) For any fixed, the equationhas a unique solution.

(ii) The function, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any, we have

where the notationmeans

Proof Similar to the proof of Lemma 2.2, we omit it here. □

We define as , where , .

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

For fixed , we denote as

Define as

Similar to the proof of Lemma 2.3, we have

Lemma 2.9 (i) The operatoris continuous and sends equi-integrable sets into relatively compact sets in.

(ii) The operatoris continuous and sends bounded sets into relatively compact sets in.

Let us defineas

It is easy to see thatis compact continuous.

Lemma 2.10Suppose that, and, g, hsatisfy one of the following:

(10) , ;

(20) on, () andon.

Thenuis a solution of (1)-(4) if and only ifuis a solution of the following abstract operator equation:

Proof Similar to the proof of Lemma 2.4, we omit it here. □

### 3 Existence of solutions in Case (i)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when , .

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

Theorem 3.1Suppose that, ; on, () andon; fsatisfies the sub-growth condition; and operatorsAandBsatisfy the following conditions:

(20)

then problem (1)-(4) has at least a solution.

Proof First we consider the following problem:

Denote

where is defined in (10).

Obviously, () has the same solution as the following operator equation when :

(21)

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

We claim that all the solutions of (21) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (21) such that as and for any  .

From Lemma 2.2, we have

Thus

(22)

From (), we have

It follows from (11) and Lemma 2.2 that

Denote . If the above inequality holds then

(23)

It follows from (14), (20) and (22) that

For any , we have

which implies that

Thus

(24)

It follows from (23) and (24) that is uniformly bounded.

Thus, we can choose a large enough such that all the solutions of (21) belong to . Therefore the Leray-Schauder degree is well defined for , and

It is easy to see that u is a solution of if and only if u is a solution of the following usual differential equation:

Obviously, system () possesses a unique solution . Since , we have

which implies that (1)-(4) has at least one solution. This completes the proof. □

Theorem 3.2Suppose that, ; on, () andon; fsatisfies the sub-growth condition; and operatorsAandsatisfy the following conditions:

where, and, .

Then problem (1) with (2), (4) and (5) has at least a solution.

Proof Obviously, .

From Theorem 3.1, it suffices to show that

(25)

(a) Suppose that , where is a large enough positive constant. From the definition of D, we have

Since , we have . Thus (25) is valid.

(b) Suppose that , we can see that

There are two cases: Case (i): ; Case (ii): .

Case (i): Since , we have , and

Thus (25) is valid.

Case (ii): Since , we have , and

Thus (25) is valid.

Thus problem (1) with (2), (4) and (5) has at least a solution. This completes the proof. □

Let us consider

(26)

where ε is a parameter, and

where are Caratheodory. We have the following theorem.

Theorem 3.3Suppose that, ; on, () andon; fsatisfies the sub-growth condition; and we assume that

then problem (26) with (2)-(4) has at least one solution when parameterεis small enough.

Proof Denote

We consider the existence of solutions of the following equation with (2)-(4)

(27)

Denote

where is defined in (10).

We know that (27) with (2)-(4) has the same solution of .

Obviously, . So . As in the proof of Theorem 3.1, we know that all the solutions of are uniformly bounded, then there exists a large enough such that all the solutions of belong to . Since is compact continuous from to , we have

(28)

Since f and h are Caratheodory, we have

Thus

Obviously, . We obtain

Thus, when ε is small enough, from (28), we can conclude that

Thus has no solution on for any , when ε is small enough. It means that the Leray-Schauder degree is well defined for any , and

Since , from the proof of Theorem 3.1, we can see that the right-hand side is nonzero. Thus (26) with (2)-(4) has at least one solution when ε is small enough. This completes the proof. □

Theorem 3.4Suppose that, ; on, () andon; fsatisfies the sub-growth condition; and we assume that

where, and, , then problem (26) with (2), (4) and (5) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □

### 4 Existence of solutions in Case (ii)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when , .

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

Theorem 4.1Suppose that, ; fsatisfies the sub-growth condition; and operatorsAandBsatisfy the following conditions:

then problem (1)-(4) has at least a solution.

Proof Similar to the proof of Theorem 3.1, we omit it here. □

Theorem 4.2Suppose that, ; fsatisfies the sub-growth condition; and operatorsAandsatisfy the following conditions:

where

then problem (1) with (2), (4) and (5) has at least a solution.

Proof Similar to the proof of Theorem 3.2, we omit it here. □

Theorem 4.3Suppose that, ; fsatisfies the sub-growth condition; and we assume that

then problem (26) with (2)-(4) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.3, we omit it here. □

Theorem 4.4Suppose that, ; fsatisfies the sub-growth condition; and we assume that

where, and, , then problem (26) with (2), (4) and (5) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □

### 5 Existence of solutions in Case (iii)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions and nonnegative solutions for system (1)-(4) or (1) with (2), (4) and (5) when , .

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

Theorem 5.1Suppose that, and, g, hsatisfy one of the following:

(10) , ;

(20) on, () andon;

whenfsatisfies the sub-growth condition; and operatorsAandBsatisfy the following conditions:

then problem (1)-(4) has at least a solution.

Proof Similar to the proof of Theorem 3.1, we omit it here. □

Theorem 5.2Suppose that, and, g, hsatisfy one of the following:

(10) , ;

(20) on, () andon;

whenfsatisfies the sub-growth condition; and operatorsAandsatisfy the following conditions:

where

then problem (1) with (2), (4) and (5) has at least a solution.

Proof Similar to the proof of Theorem 3.2, we omit it here. □

Theorem 5.3Suppose that, and, g, hsatisfy one of the following:

(10) , ;

(20) on, () andon;

whenfsatisfies the sub-growth condition; and we assume that

then problem (26) with (2)-(4) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.3, we omit it here. □

Theorem 5.4Suppose that, and, g, hsatisfy one of the following:

(10) , ;

(20) on, () andon;

whenfsatisfies the sub-growth condition; and we assume that

where, and, , then problem (26) with (2), (4) and (5) has at least one solution when parameterεis small enough.

Proof Similar to the proof of Theorem 3.2 and Theorem 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 5.5Suppose that, , , . We also assume:

(10) , ;

(20) For any, , ;

(30) For any, , , ;

(40) .

Then every solution of (1)-(4) is nonnegative.

Proof Let u be a solution of (1)-(4). From Lemma 2.10, we have

We claim that . If it is false, then there exists some such that .

It follows from (10) and (20) that

(29)

Thus (29) and condition (30) hold

(30)

Similar to the proof before Lemma 2.8, from the boundary value conditions, we have

(31)

From (29) and (30), we get a contradiction to (31). Thus .

We claim that

(32)

If it is false, then there exists some such that

It follows from (10) and (20) that

(33)

Thus (33) and condition (30) hold

(34)

From (33), (34), we get a contradiction to (31). Thus (32) is valid.

Denote , .

Obviously, , , and is decreasing, i.e., for any with . For any , there exist such that

It follows from condition (30) that is increasing on and is decreasing on . Thus , .

For any fixed , if

(35)

from (4) and (35), we have . Then .

If

(36)

from (4), (36) and condition (40), we have . Then .

Thus , . The proof is completed. □

Corollary 5.6Under the conditions of Theorem 5.1, we also assume:

(10) , with;

(20) For any, , with;

(30) For any, , , with;

(40) ;

(50) For anyand, , .

Then (1)-(4) has a nonnegative solution.

Proof Define , where

Denote

then satisfies the Caratheodory condition, and for any .

For any , we denote

then and are continuous and satisfy

It is not hard to check that

(20)′ for uniformly, where , and ;

(30)′ , ;

(40)′ , .

Let us consider

(37)

It follows from Theorem 5.1 and Theorem 5.5 that (37) has a nonnegative solution u. Since , we have , and then

Thus u is a nonnegative solution of (1)-(4). This completes the proof. □

Note (i) Similarly, we can get the existence of nonnegative solutions of (26) with (2)-(4).

(ii) Similarly, under the conditions of Case (ii), we can discuss the existence of nonnegative solutions.

### 6 Examples

Example 6.1 Consider the existence of solutions of (1)-(4) under the following assumptions:

where , , , .

Obviously, ; when ; (); then the conditions of Theorem 3.1 are satisfied, then (1)-(4) has a solution.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors typed, read and approved the final manuscript.

### Acknowledgements

Partly supported by the National Science Foundation of China (10701066 & 10971087).

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