Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations
Xiufeng Guo^{*}, Yuan Gu
College of Sciences, Hezhou University, Hezhou, Guangxi, China
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To cite this article:
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
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.
References