Abstract
We will consider the nonhomogeneous ϕLaplacian differential equation
where
Keywords:
nonhomogeneous; ϕLaplacian; positive solution; fixed point; negative coefficient1 Introduction
We are concerned with the ϕLaplacian differential equation with the nonhomogeneous DirichletNeumann boundary conditions
where
Boundary value problems, including the ϕLaplacian operator, have received a lot of attention with respect to the existence and multiplicity of solutions. Since 2004, with a number of papers, Bereanu and Mawhin have considered such problems with Dirichlet, Neumann or periodic boundary conditions (see [14] and the references therein). In these papers, the various boundary value problems are reduced to the search for fixed points of some nonlinear operators defined on Banach spaces. In particular, they have studied some boundary value problems with nonhomogeneous boundary conditions and obtained the existence of solutions by the use of Schauder’s fixed point theorem (see [3,4]). Recently, Torres [5] has proved the existence of a solution of a forced Liénard differential equation with ϕLaplacian by means of Schauder’s fixed point theorem.
However, many nonlinear differential equations need to seek the existence of positive solutions because the positive solutions are very meaningful. The existence of positive solutions for homogeneous and nonhomogeneous boundary value problems have been studied by several authors and many interesting results have been obtained (only to mention some of them, see [69], their references and the papers citing them). The problems with negative coefficients for the boundary conditions (see [1012]) often occur in some heat flow problems, the deflection of a beam, and Floquet theory of the beam equation and have been considered by some experts (see [7,8,1012]). If the coefficient takes a negative value, then it is sometimes difficult to find an appropriate cone to guarantee the existence of a positive solution of the corresponding differential equation. Comparing with the previous result [24], the cone may be smaller.
The purpose of this paper is to establish the criteria of the existence of a positive
solution to the problem (1.1) by utilizing the Krasnosel’skii fixed point theorem,
even if some of the
This paper is organized as follows. In Section 2, we give some lemmas, which play an important role in the proof of the main theorem. In Section 3, we obtain the existence of a positive solution to the problem (1.1). Moreover, two examples are also given to illustrate the main results.
2 Preliminaries and lemmas
Let X denote the Banach space
Deriving on both sides of (2.1) leads to
i.e.,
Again, deriving in (2.3) implies
Moreover, from (2.1) and (2.3), we get that
For the sake of convenience, we give the following conditions.
(A) Denote
and
(F) The function
(B)
(H) There exists a
be true.
For the unboundedϕLaplacian (
Lemma 1Assume that the conditions (F) and (H) hold,
Proof From the representation (2.2) and the conditions (F)(H), we have
Again since
and
where
This completes the proof. □
Next, let us define a cone by
The definition of the cone is inspired by the results in [7,8]. To show our main results, the following lemma is essential.
Lemma 2Let the conditions (A), (F), and (H) hold and the nonlinear operatorSbe defined by (2.1). Then
Proof From the definition of the operator S, we find for any
The conditions (F) and (H) yield
Further, Lemma 1 shows
Consequently, we get that
Remark 1 If the coefficients
Lemma 3If
Lemma 4 (See [13])
LetXbe a Banach space and
(i)
(ii)
thenShas at least one fixed point in
3 The main result
Theorem 1Assume that the conditions (A), (F), and (H) hold and
Proof Lemma 2 shows that
Now, we choose a positive constant
and define
Thus, for any
From the hypothesis (A), we can let
and define
Then, for any
Based on Lemma 4, we get from (3.1) and (3.2) that the operator S has at least one fixed point. Thus, it follows that Eq. (1.1) has at least one positive solution. □
Remark 2 If the coefficients
(A′)
Applying the results in Remark 1 and Theorem 1, we get the following result.
Corollary 1Assume that the conditions (A′) and (F) hold and
If
Theorem 2Assume that the conditions (A), (F), (B), and (H) hold. Then Eq. (1.1) has at least one positive solution.
Proof Using Lemma 3 and the proof of Theorem 1, we get that the conclusion holds. □
Example 1 Consider the differential equation
subjected to the boundary conditions
Clearly, we find
Computing yields
Therefore, we conclude from Theorem 1 that Eq. (3.3)(3.4) has at least one positive solution.
Example 2 Consider the differential equation
subjected to the boundary conditions
Obviously, we obtain
It is easy to verify that the conditions (B), (F), and (H) hold. Consequently, we get from Theorem 2 that the equation (3.5)(3.6) has at least one positive solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors typed, read and approved the final manuscript.
Acknowledgements
The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of the presentation and quality of the work. The work was supported partly by NSFC of Tianyuan Youth Foundation (No.11126125), K.C. Wong Magna Fund of Ningbo University and Ningbo Natural Science Foundation (2012A610031).
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