Abstract
We treat the nonhomogeneous boundary value problems with ϕLaplacian operator
Keywords:
nonhomogeneous; ϕLaplacian; negative coefficient; positive solution1 Introduction
We are concerned with the nonhomogeneous boundary value problem with ϕLaplacian operator
where
According to the related literature [15], a ϕLaplacian operator is said to be singular when the domain of ϕ is finite (i.e.,
(1)
(2)
(3)
The study of the ϕLaplacian equations is a classical topic that has attracted the attention of many experts because of its interest in applications. Since 2004, in a number of papers, Bereanu and Mawhin have considered such problems with Dirichlet, Neumann or periodic boundary conditions (see, for example, [13] and the references therein). In these papers, the various boundary value problems are reduced to the search for fixed point of some nonlinear operators defined on Banach spaces. In particular, they have also considered some boundary value problems with nonhomogeneous boundary conditions, and they obtained the existence of solutions by the use of the Schauder fixed point theorem (see [2,3]). Recently, Torres [4] proved the existence of a solution of a forced Liénard differential equation with ϕLaplacian by means of Schauder fixed point theorem.
We note here that many nonlinear differential problems require the search of positive, meaningful, solutions. The existence of positive solutions for ordinary differential equations and pLaplacian equations have been studied by several authors and many interesting results has been obtained (only to mention some of them; see [57], and the references therein). If the coefficient occurring in the boundary conditions takes a negative value, then the existence of a positive solution for a BVP with ϕLaplacian operator is less considered because it is sometimes difficult to construct a corresponding cone for applying the fixed point theorem.
The purpose of this paper is to study the nonhomogeneous boundary value problem (1.1)
with ϕLaplacian operator even in the case where some of the τ and
(1) Some of the τ and
(2) In order to obtain a positive solution of the problem (1.1), we make a change
of variable and generate two firstorder differential equations and the corresponding
nonlinear operator B. The new method can be used for the differential domains and ranges of
This paper is organized as follows. In Section 2, we give some preliminaries and lemmas. In Section 3, we show some theorems on the existence of (positive) solutions of the differential equations (1.1). Moreover, an example is given to illustrate our results.
2 Preliminaries and lemmas
Firstly, we consider the following ϕLaplacian differential equation:
where
Next we make the change of variable
i.e.,
Together with (2.1), we get the following two firstorder differential equations:
and
Integration in (2.4) yields
Then we get
Thus, together with the boundary condition in (2.5), we obtain
where we denote
Integration leads to
Now, we define two nonlinear operators B and S, respectively, by
and
Remark 2.1 From the deduction of (2.2)(2.7), we find that if
To such a continuous function f, we associate its Nemytskii operator
It is easy to verify that
Lemma 2.1 (See [8])
LetXbe a Banach space and
(i)
(ii)
thenShas at least one fixed point in
3 The main result
Let X be the Banach space
For the sake of convenience, we give the following conditions:
(i) The nonlinearity
for any
(ii)
(iii) There exists a positive constant r such that
where
(iv)
Remark 3.1
(1) If
(2) If
where
Case I. Singular ϕLaplacian operator:
Theorem 3.1Iffis continuous, then the problem (1.1) has at least one solution.
Proof Define a set by
From the definition of
and
Thus
From the definition of S in (2.8), we have
i.e.,
Then we obtain
and
Consequently, we conclude that the fixed point of S is a solution of the problem (1.1). □
Remark 3.2 Theorem 3.1 shows that if ϕ is singular (
Case II. Bounded ϕLaplacian operator:
Theorem 3.2If the conditions (i)(iv) hold, then the nonlinear operatorBdefined by (2.7) has at least one fixed point. Further, the problem (1.1) has at least one positive solution.
Proof Define a set by
From the conditions (ii) and (iv), we obtain, for any
Clearly, the nonlinear operator
The condition (iii) implies that we find a constant
Define a set by
In virtue of the increasing property of
Thus, for any
We choose a small positive constant
Thus, for any
In addition, a standard argument involving the ArzelaAscoli theorem implies that
Consequently, we get from Remark 2.1 that the problem (1.1) has a positive solution
Remark 3.3 In order to prove the existence of a positive solution of the problem (1.1), we make
a change of variable and introduce a firstorder differential equation, and investigate
the existence of a fixed point of the corresponding nonlinear operator B. The technique can be used for the different domains and ranges of
Theorem 3.3If
for any
Proof Adopting a similar technique to (2.2)(2.6), we define a nonlinear operator by
Then we get, for any
Then the operator
Remark 3.4 Observe that the solution provided by Theorem 3.3 could be trivial or negative.
Theorem 3.4If the conditions (ii) and (iv) hold andfsatisfies the condition
for any
Proof Taking arbitrarily
Thus, it implies that there exists a neighborhood
Example 3.1 We consider the following nonhomogeneous boundary value problem with ϕLaplacian operator:
where
It is easy to see that the nonlinearity f satisfies the condition (i), that is,
for any
and there exists a positive constant
and
Then conditions (ii)(iv) also hold. Therefore, we find from Theorem 3.2 that the differential equation (3.3) has at least one positive solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would wish to express their appreciations to the anonymous referee for his/her valuable suggestions, which have greatly improved this paper. The work was partially supported by NSFC of China (No. 11201248), K.C. Wong Fund of Ningbo University.
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