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 , , (). Based on a priori estimates achieved by utilizing the properties of concave functions and Jensen’s inequality, we adopt fixed point index theory to establish our main results.
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 , , . Let be fixed, , where and denote the right and left limit of at , respectively.
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 , , and , problem (1.1) reduces to the differential equation subject to boundary value conditions , which can be used to model the deflection of elastic beams simply supported at the endpoints [911]. This explains the reason that the last two decades have witnessed an overgrowing interest in the research of such problems, with many papers in this direction published. We refer the interested reader to [1226] and references therein devoted to the existence of solutions for the equations with pLaplacian operator.
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 is nonnegative, symmetric on the interval (i.e., for ), , for all , and are nonnegative, symmetric on . The arguments are based upon a specially constructed cone and the fixed point theory for cones. Moreover, they also studied the nonexistence of a positive solution.
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 , . Then is a real Banach space. Let and introduce the following space:
with the norm . Then is also a Banach space.
A function is called a solution of (1.1) if it satisfies the differential equation
and the function y satisfies the conditions , and the Dirichlet boundary conditions .
Lemma 2.1 (see [21])
Ifyis a solution of the integral equation
thenyis a solution of (1.1), where, . Note that if, , thenis a completely continuous operator, and the existence of positive solutions for (1.1) is equivalent to that of positive fixed points ofA.
Remark 2.1 By (2.1), we easily find y is concave on . Indeed,
implies y is concave on . Furthermore, () leads to , .
Let P be a cone in which is defined as
In what follows, we prove that .
Proof We easily see that , . Consequently, on the one hand, we find
On the other hand,
Therefore, , for any , as required. This completes the proof. □
Lemma 2.3 (see [30])
Supposeis a completely continuous operator and has no fixed points on.
1. Iffor all, then, whereiis fixed point index onP.
Lemma 2.4 (see [30])
Ifis a completely continuous operator. If there existssuch that, , , then.
Lemma 2.5 (see [30])
Ifandis a completely continuous operator. If, , , then.
Lemma 2.7 (Jensen’s inequalities)
3 Main results
Let , , , , , , , . We now list our hypotheses.
(H1) There is a such that and imply
(H2) There exist and , satisfying
such that
(H3) There exist and , satisfying
such that
(H4) There is a such that and imply
(H5) There exist and , satisfying
such that
(H6) There exist and , satisfying
such that
Theorem 3.1Suppose that (H1)(H3) are satisfied. Then (1.1) has at least two positive solutions.
Proof If , it follows from (H1) that
Now Lemma 2.3 yields
where . Let , where . Next, from (H2), we prove . Indeed, implies . Lemma 2.6, together with this, leads to
Multiply both sides of the above by and integrate over and use (2.2) to obtain
Combining this and (3.1), we get
In what follows, we will distinguish three cases.
Case 1. . By (H2), we know . (3.9) implies
Therefore, (), and then , by Remark 2.1, which contradicts .
Case 2. . Equation (3.9) implies
and thus , , which also contradicts .
Case 3. . Since , we have by (3.6) and (3.9),
Therefore,
which contradicts (H2). So, we have for all and . Now, by virtue of Lemma 2.4, we obtain
On the other hand, by (H3), we prove is bounded in P. By (3.2) together with (3.8), we obtain
where . Now we distinguish the following two cases.
Combining this and (3.11), we have
Therefore,
and thus
Therefore, we obtain the boundedness of , as claimed. Taking , we have for all and . Now, by virtue of Lemma 2.4, we obtain
Combining (3.5), (3.10), and (3.12), we arrive at
Now A has at least two fixed points, one on and the other on . Hence (1.1) has at least two positive solutions. The proof is completed. □
Theorem 3.2Suppose that (H4)(H6) are satisfied. Then (1.1) has at least two positive solutions.
By (H4),
so that
Now Lemma 2.3 yields
Let . Next, from (H5), we prove . Indeed, if , we have
Multiply both sides of the above by and integrate over and use (2.2) to obtain
Combining this and (3.3), we have
Consequently,
which contradicts (H5). This implies , and thus for all and . Now Lemma 2.5 yields
On the other hand, by (H6), we prove is bounded in P. By (3.4) together with (3.16), we obtain
namely,
This proves the boundedness of , as required. Choosing and , we have for all and . Now Lemma 2.5 yields
Combining (3.14), (3.17), and (3.18), we obtain
Hence A has at least two fixed points, one on and the other on , and thus (1.1) has at least two positive solutions. The proof is completed. □
4 An example
Let us consider the problem
Taking in (H1), , and is chosen such that . Set , , . Therefore, , , and
As a result, (H1) holds. On the other hand, by simple computation, we have
Therefore,
(i) There exist and , such that (H2) holds.
(ii) There exist and , such that (H3) holds.
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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