Abstract
This paper is devoted to study the existence and multiplicity of positive solutions for the fourthorder pLaplacian boundary value problem involving impulsive effects
where
MSC: 34B18, 47H07, 47H11, 45M20, 26D15.
Keywords:
pLaplacian boundary value problem with impulsive effects; positive solution; fixed point index; concave function; Jensen inequality1 Introduction
In this paper, we mainly investigate the existence and multiplicity of positive solutions for the fourthorder pLaplacian boundary value problem with impulsive effects
Here
Fourthorder boundary value problems, including those with the pLaplacian operator, have their origin in beam theory [1,2], ice formation [3,4], fluids on lungs [5], brain warping [6,7], designing special curves on surfaces [6,8], etc. In beam theory, more specifically, a beam with a small deformation, a beam of a material
which satisfies a nonlinear powerlike stress and strain law, and a beam with twosided
links which satisfies a nonlinear powerlike elasticity law can be described by fourthorder
differential equations along with their boundary value conditions. For the case of
In [17], Zhang et al. studied the existence and nonexistence of symmetric positive solutions of the following fourthorder boundary value problem with integral boundary conditions:
where
In [16], Luo and Luo considered the existence, multiplicity, and nonexistence of symmetric positive solutions for (1.2) with a ϕLaplacian operator and the term f involving the first derivative.
Except that, many researchers considered and studied the existence of positive solutions for a lot of impulsive boundary value problems; see, for example, [2129] and the references therein.
In [21], Feng considered the problem (1.2) with impulsive effects and he obtained the existence and multiplicity of positive solutions. The fundamental tool in this paper is GuoKrasnosel’skii fixed point theorem on a cone. Moreover, the nonlinearity f can be allowed to grow both sublinear and superlinear. Therefore, he improved and generalized the results of [17] to some degree. However, we can easily find that these papers do only simple promotion based on their original papers, and no substantial changes.
Motivated by the works mentioned above, in this paper, we study the existence and multiplicity of positive solutions for (1.1). Nevertheless, our methodology and results in this paper are different from those in the papers cited above. The main features of this paper are as follows. Firstly, we convert the boundary value problem (1.1) into an equivalent integral equation. Next, we consider impulsive effect as a perturbation to the corresponding problem without the impulsive terms, so that we can construct an integral operator for an appropriate linear Dirichlet boundary value problem and obtain its first eigenvalue and eigenfunction. Our main results are formulated in terms of spectral radii of the linear integral operator, and our a priori estimates for positive solutions are derived by developing some properties of positive concave functions and using Jensen’s inequality. It is of interest to note that our nonlinearity f may grow superlinearly and sublinearly. The main tool used in the proofs is fixed point index theory, combined with the a priori estimates of positive solutions. Although our problem (1.1) merely involves Dirichlet boundary conditions, both our methodology and the results in this work improve and extend the corresponding ones from [2129].
2 Preliminaries
Let
with the norm
A function
and the function y satisfies the conditions
Lemma 2.1 (see [21])
Ifyis a solution of the integral equation
thenyis a solution of (1.1), where
Remark 2.1 By (2.1), we easily find y is concave on
implies y is concave on
Let P be a cone in
In what follows, we prove that
Lemma 2.2
Proof We easily see that
On the other hand,
Therefore,
We denote
Lemma 2.3 (see [30])
Suppose
1. If
2. If
Lemma 2.4 (see [30])
If
Lemma 2.5 (see [30])
If
Lemma 2.6Let
Lemma 2.7 (Jensen’s inequalities)
Let
3 Main results
Let
(H1) There is a
where
(H2) There exist
such that
where
(H3) There exist
such that
(H4) There is a
where
(H5) There exist
such that
(H6) There exist
such that
Theorem 3.1Suppose that (H1)(H3) are satisfied. Then (1.1) has at least two positive solutions.
Proof If
Now Lemma 2.3 yields
Let
where
Multiply both sides of the above by
Combining this and (3.1), we get
In what follows, we will distinguish three cases.
Case 1.
Therefore,
Case 2.
and thus
Case 3.
Therefore,
which contradicts (H2). So, we have
On the other hand, by (H3), we prove
where
Case 1.
Combining this and (3.11), we have
Therefore,
Case 2.
and thus
Therefore, we obtain the boundedness of
Combining (3.5), (3.10), and (3.12), we arrive at
Now A has at least two fixed points, one on
Theorem 3.2Suppose that (H4)(H6) are satisfied. Then (1.1) has at least two positive solutions.
Proof If
By (H4),
so that
Now Lemma 2.3 yields
Let
Let
Multiply both sides of the above by
Combining this and (3.3), we have
Consequently,
which contradicts (H5). This implies
On the other hand, by (H6), we prove
where
namely,
This proves the boundedness of
Combining (3.14), (3.17), and (3.18), we obtain
Hence A has at least two fixed points, one on
4 An example
Let us consider the problem
Taking
As a result, (H1) holds. On the other hand, by simple computation, we have
Therefore,
(i) There exist
(ii) There exist
Consequently, the problem (4.1) has at least two positive solutions by Theorem 3.1.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
KZ and JX obtained the results in a joint research. All the authors read and approved the final manuscript.
Acknowledgements
Research supported by the NNSFChina (10971046), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007, A2012402036), GIIFSDU (yzc12063), IIFSDU (2012TS020) and the Project of Shandong Province Higher Educational Science and Technology Program (J09LA55).
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