Applied and Computational Mathematics
Volume 5, Issue 2, April 2016, Pages: 91-96

Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations

Xiufeng Guo*, Yuan Gu

College of Sciences, Hezhou University, Hezhou, Guangxi, China

(Xiufeng Guo)
(Yuan Gu)

*Corresponding author

Xiufeng Guo, Yuan Gu. Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations. Applied and Computational Mathematics. Vol. 5, No. 2, 2016, pp. 91-96. doi: 10.11648/j.acm.20160502.19

Received: May 3, 2016; Accepted: May 13, 2016; Published: May 30, 2016

Abstract: This paper concerned with the existence of solutions of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator. Firstly, the definition a pair of coupled lower and upper solutions of the problem is introduced. Then, under the approach of coupled upper and lower solutions together with Nagumo condition, we prove that there exists at least one solution of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator.

Keywords: Anti-periodic Boundary Value Problems, Impulsive Differential Equations, φ-Laplacian Operator, Coupled Lower and Upper Solutions

Contents

1. Introduction

In this paper, we will study a class of nonlinear impulsive differential equation with φ-Laplacian operator and anti-periodic boundary conditions as following:

(1)

(2)

(3)

where φ is an increasing homeomorphism from  to , and  is a Carathéodory function. , ,  are impulsive functions and will be given later.

Impulsive differential equations have become an important aspect in some mathematical models of real processes and phenomena in science. There has a significant development in impulse theory and impulsive differential equations (see [1, 2, 3]). Moreover, p-Laplacian operator arises in non-Newtonian fluid flows, turbulent filtration in porous media and in many other application areas (see [5, 7] and references therein). Usually, p-Laplacian operator is replaced by abstract and more general version φ-Laplacian operator, which lead to clearer expositions and a better understanding of the methods which ware employed to derive the existence results (see [12, 13]). Recently, Cabada and Tomecek [4] focussed on the φ-Laplacian differential equations (1) subject to impulsive functions (2) with non-local boundary conditions

Later, the paper [6] generalize the problem of [4] and considered a more general φ-Laplacian differential equations

coupled with the compatible impulses and boundary conditions

where , and  is a Carathéodory function, , . After make a study of the monotonicity of the boundary condition functions in [4, 6, 8], we found that their boundary problems didn’t include the anti-periodic boundary condition problems.

Motivated by above the mentioned papers, we try to find some appropriate conditions to make sure the existence solutions for anti-periodic problem (1)-(3). As far as we known, although the papers [4, 6, 8] study such a general φ-Laplacian problems with nonlinear boundary conditions, but its didn’t contain the anti-periodic problems. Besides, there are few dependent references for studying the φ-Laplacian impulsive functional differential equations with anti-periodic boundary condition. Furthermore, the anti-periodic problem is a very important area of research. The study of anti-periodic problem for nonlinear evolution equations is closely related to the study of periodic problem which was initiated by Okochi [14]. Anti-periodic problem has been extensively studied during the past decades, such as anti-periodic trigonometric polynomials ([15]). Moreover, anti-periodic boundary conditions also appear in difference and differential equations (see [16, 17] and references therein).

This paper is organized as follows: In section 2, we state some preliminaries that will be used throughout the paper. In section 3, we obtain existence solutions for problem (1)-(3) by the approach of coupled upper and lower solutions together with Nagumo condition. Finally, we give the conclusion of our main work in section 4.

2. Preliminaries

For a real valued measurable function  defined almost everywhere on and , let

For a given Banachspace , let  be the set of all continuous functions . Let  be the set of all functions  which are  times continuously differentiable on  with finite norm

For ,  denotes the set of Lebesgue measurable functions  on  such that  is finite. Let  be the set of absolutely continuous functions  on satisfy . denotes the set of functions  and  with finite norm

It is well known that  and  are Banach spaces and  is a usual Sobolev space.

Let . A finite subset  of the interval  defined by

Let  and  for all . For and , we denote

for all  there exist

It is easy to verify that the spaces  and are Banach spaces with the norms

We say that ) satisfies the restricted Carathéodory conditions on  if

(i) for each  the function  is measurable on ;

(ii) the function  is continuous on a.e. ;

(iii) for every compact set , there exists a nonnegative function  such that

In this paper, we use Car () to denote the set of functions satisfying the restricted Carathéodory conditions on . In what follows,  and  denote the Dini derivatives.

Definition 1.The functions  such that  are said to bea pair of coupled lower and upper solutions of problem (1)-(3) if satisfy the following conditions:

(i)  for all . Moreover, if  such that , then there exists  such that ,  and

(ii)  for all . Moreover, if  such that , then there exists  such that ,  and

(iii) For all ,  are injective and there exist , , ,  such that

and there exist , , ,  such that

Definition 2. Given a function  is called a solution of the problem (1)-(3) if  and  satisfies (1) and fulfills conditions (2) and (3).

Definition 3. Assume that the functionsCar () and  satisfying  for . We say that the function  satisfies a Nagumo condition with respect to  and  if, for , there exist functions  and , such that  on ,

Moreover, there exists a constant with , such that the following inequalities

(5)

hold, where  and  Moreover, any constant such will be called a Nagumo constant.

Throughout this paper, we impose the following hypotheses:

(H) The function  is a continuous and strictly increasing.

(H) The problem (1)-(3) has a pair of coupled lower and upper solutions and .

(H) The functionCar() and satisfies a Nagumo condition with respect to and.

(H) For , the functions  are non-decreasing in the first variable,  are non-increasing in the third variable and non-decreasing in the fourth and fifth variables.

3. Existence Results

In this section, we will deduce that there exists at least one solution of the problem (1)-(3) lying between a pair of coupled lower and upper solutions.

Firstly, we state the following existence and uniqueness result for a problem with linear right-hand side.

Lemma 1. (Lemma 7 of [4]).Let and  for each . Suppose that  is a strictly increasing function satisfies . Then the non homogeneous impulsive Dirichlet problem

has a unique solution , which can be written in the form

where is the unique solution of the equation

Next, let us consider the following functions

where  is the constant introduced in definition 2.3,

for

coupled with functionals  given by

Moreover, for each  we consider a function  defined by

The function  is well defined according to the following result (by redefining function  as zero when it does not exist). It can be proved in a similar way to Lemma 2 in [10].

Lemma 2. For given  such that in , then

(i) exists for a.e. ;

(ii) for a.e. .

Now, define a strictly increasing homeomorphism  by:

In what follows, we are in a position to prove the existence theorem.

Theorem 3. Assume that the hypotheses (H)-(H) hold. Then there exists at least one solution  of the problem (1)-(3) such that

where  is the constant introduced in Definition 2.3.

Proof.First, we consider the following auxiliary problem

From the definitions of functions and , Lemma 3.2 and hypothesis (H), for , we have . Define an operator  by

for each , where  is given as the unique solution of equation

The proof of the existence of a fixed point  of operator  follows in a similar way to the one given in Theorem 9 of [4]. In view of Lemma 3.1, we know that the fixed piont  is a solution of the problem .

Next, we prove that on .

Let . From the definition of , it follows that

We will show that  is non-negative on . Assume that there exists  such that

and there exists  such that

(6)

Then . As , we have . Together with the definition 2.1, there exists  such that . Thus we have that , . As a consequence, from the definition of and , we have

and

for all . In view of the monotonicity of  we get , , which contradict to (5).

Similarly, we can prove that  is non-negative on .

Therefore, any solution of  satisfies

To see any solution  of  satisfies  for all . By contradiction, we assume

Then there exists  such that

Since , we have , then there exists  such that  or . If , there exist  such that

where  or . In what follows, we only consider the case  since the other case can be treated similarly. By variable substitution, we have

Then the Hölder’s inequality implies

where  is defined by definition 2.3. This contradict to (4). If , by a similar argument, we obtain a contradiction too. Therefore,  holds for .

To verify that condition (2) is fulfilled, we can proved in a similar way to the step 4 in Theorem 8 of [11].

Finally, we prove the solution of () satisfies the anti-periodic boundary condition. It is suffices to prove that

(7)

(8)

We will prove the first inequality of (6), by contradiction, if , by the fact , then , which contradicts to . The second inequality in (6) can be proved similarly. Hence, we obtain .

Let the first inequality of (7) be not satisfied, as a consequence, we have  and . By , we have . Together with , we can verify that  and . Hence

(9)

which is a contradiction. The second inequality of (7) can be proved similarly. Therefore, we have .

The proof is completed.

Remark. If, instead of the implicit conditions (2), we consider the following ones

Assume that there exists a pair of coupled lower and upper solutions which satisfy the conditions in Definition 2.1 but assuming

and the reversed conditions in , together with

For all , functions  are continuous and the functions  and  are monotone.

For all , functions

are continuous and the functions  and  are monotone.

As a consequence, we can deduce similar existence results to the ones obtained in Theorem 3.3. And we note that Remark 11 in [4], the conditions  are non-decreasing and  are non-increasing can be replaced by monotone.

4. Conclusions

In this paper, we mainly discuss the existence of solutions of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator. To give the existence results for the problem (1)-(3), we first introduce a pair of coupled lower and upper solutions (Definition 1) of the problem (1)-(3). Then, under the approach of coupled upper and lower solutions together with Nagumo condition (Definition 3), we deduce that there exists at least one solution of the anti-periodic boundary value problems (1)-(3) lying between a pair of coupled lower and upper solutions by the auxiliary non homogeneous impulsive Dirichlet problem.

Acknowledgments

The work was partially supported by NNSF of China Grants No.11461021, NNSF of Guangxi Grants No. 2014GXNSFAA118028, the Scientific Research Foundation of Guangxi Education Department No. KY2015YB306, Guangxi Colleges and Universities Key Laboratory of Symbolic Computation and Engineering Data Processing, and the Scientific Research Project of Hezhou University Nos. 2015ZZZK16, 2016HZXYSX07.

The authors also would like to thank the anonymous reviewer for their valuable comments and suggestions which improved the quality of the paper.

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