Positive solutions for a fourth-order p-Laplacian boundary value problem with impulsive effects
© Zhang et al.; licensee Springer 2013
Received: 31 January 2013
Accepted: 15 April 2013
Published: 10 May 2013
This paper is devoted to study the existence and multiplicity of positive solutions for the fourth-order p-Laplacian 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.
Keywordsp-Laplacian boundary value problem with impulsive effects positive solution fixed point index concave function Jensen inequality
Here , , . Let be fixed, , where and denote the right and left limit of at , respectively.
Fourth-order boundary value problems, including those with the p-Laplacian operator, have their origin in beam theory [1, 2], ice formation [3, 4], fluids on lungs , 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 power-like stress and strain law, and a beam with two-sided links which satisfies a nonlinear power-like elasticity law can be described by fourth-order 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 [9–11]. 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 [12–26] and references therein devoted to the existence of solutions for the equations with p-Laplacian operator.
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 , Luo and Luo considered the existence, multiplicity, and nonexistence of symmetric positive solutions for (1.2) with a ϕ-Laplacian operator and the term f involving the first derivative.
In , 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 Guo-Krasnosel’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  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 [21–29].
with the norm . Then is also a Banach space.
and the function y satisfies the conditions , and the Dirichlet boundary conditions .
Lemma 2.1 (see )
then y is a solution of (1.1), where , . Note that if , , then is a completely continuous operator, and the existence of positive solutions for (1.1) is equivalent to that of positive fixed points of A.
implies y is concave on . Furthermore, () leads to , .
In what follows, we prove that .
Lemma 2.2 .
Therefore, , for any , as required. This completes the proof. □
We denote for in the sequel.
Lemma 2.3 (see )
If for all , then , where i is fixed point index on P.
If for all , then .
Lemma 2.4 (see )
If is a completely continuous operator. If there exists such that , , , then .
Lemma 2.5 (see )
If and is a completely continuous operator. If , , , then .
Lemma 2.7 (Jensen’s inequalities)
3 Main results
Let , , , , , , , . We now list our hypotheses.
Theorem 3.1 Suppose that (H1)-(H3) are satisfied. Then (1.1) has at least two positive solutions.
In what follows, we will distinguish three cases.
Therefore, (), and then , by Remark 2.1, which contradicts .
and thus , , which also contradicts .
where . Now we distinguish the following two cases.
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 3.2 Suppose that (H4)-(H6) are satisfied. Then (1.1) has at least two positive solutions.
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
There exist and , such that (H2) holds.
There exist and , such that (H3) holds.
Consequently, the problem (4.1) has at least two positive solutions by Theorem 3.1.
Research supported by the NNSF-China (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).
- Bernis F: Compactness of the support in convex and non-convex fourth order elasticity problem. Nonlinear Anal. 1982, 6: 1221-1243. 10.1016/0362-546X(82)90032-3MathSciNetView Article
- Zill D, Cullen M: Differential Equations with Boundary Value Problems. 5th edition. Brooks/Cole, Pacific Grove; 2001.
- Myers T, Charpin J: A mathematical model for atmospheric ice accretion and water flow on a cold surface. Int. J. Heat Mass Transf. 2004, 47: 5483-5500. 10.1016/j.ijheatmasstransfer.2004.06.037View Article
- Myers T, Charpin J, Chapman S: The flow and solidification of thin fluid film on an arbitrary three-dimensional surface. Phys. Fluids 2002, 12: 2788-2803.MathSciNetView Article
- Halpern D, Jensen O, Grotberg J: A theoretic study of surfactant and liquid delivery into the lungs. J. Appl. Physiol. 1998, 85: 333-352.
- Meméli F, Sapiro G, Thompson P: Implicit brain imaging. Hum. Brain Mapp. 2004, 23: 179-188.
- Toga A: Brain Warping. Academic Press, New York; 1998.
- Hofer M, Pottmann H: Energy-minimizing splines in manifolds. ACM Trans. Graph. 2004, 23: 284-293. 10.1145/1015706.1015716View Article
- Li Y: Existence and multiplicity positive solutions for fourth-order boundary value problems. Acta Math. Appl. Sin. 2003, 26: 109-116. (in Chinese)MathSciNet
- O’Regan D: Fourth (and higher) order singular boundary value problems. Nonlinear Anal. 1990, 14: 1001-1038. 10.1016/0362-546X(90)90066-PMathSciNetView Article
- O’Regan D: Solvability of some fourth (and higher) order singular boundary value problems. J. Math. Anal. Appl. 1991, 161: 78-116. 10.1016/0022-247X(91)90363-5MathSciNetView Article
- Graef J, Kong L: Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems. J. Math. Anal. Appl. 2007, 331: 1467-1484. 10.1016/j.jmaa.2006.09.046MathSciNetView Article
- Graef J, Kong L: Necessary and sufficient conditions for the existence of symmetric positive solutions of multi-point boundary value problems. Nonlinear Anal. 2008, 68: 1529-1552. 10.1016/j.na.2006.12.037MathSciNetView Article
- Li J, Shen J: Existence of three positive solutions for boundary value problems with p -Laplacian. J. Math. Anal. Appl. 2005, 311: 457-465. 10.1016/j.jmaa.2005.02.054MathSciNetView Article
- Zhao J, Wang L, Ge W: Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems. Nonlinear Anal. 2010, 72: 822-835. 10.1016/j.na.2009.07.036MathSciNetView Article
- Luo Y, Luo Z: Symmetric positive solutions for nonlinear boundary value problems with ϕ -Laplacian operator. Appl. Math. Lett. 2010, 23: 657-664. 10.1016/j.aml.2010.01.027MathSciNetView Article
- Zhang X, Feng M, Ge W: Symmetric positive solutions for p -Laplacian fourth-order differential equations with integral boundary conditions. J. Comput. Appl. Math. 2008, 222: 561-573. 10.1016/j.cam.2007.12.002MathSciNetView Article
- Zhao X, Ge W: Successive iteration and positive symmetric solution for a Sturm-Liouville-like four-point boundary value problem with a p -Laplacian operator. Nonlinear Anal. 2009, 71: 5531-5544. 10.1016/j.na.2009.04.060MathSciNetView Article
- Yang J, Wei Z: Existence of positive solutions for fourth-order m -point boundary value problems with a one-dimensional p -Laplacian operator. Nonlinear Anal. 2009, 71: 2985-2996. 10.1016/j.na.2009.01.191MathSciNetView Article
- Xu J, Yang Z: Positive solutions for a fourth order p -Laplacian boundary value problem. Nonlinear Anal. 2011, 74: 2612-2623. 10.1016/j.na.2010.12.016MathSciNetView Article
- Feng M: Multiple positive solutions of fourth-order impulsive differential equations with integral boundary conditions and one-dimensional p -Laplacian. Bound. Value Probl. 2011., 2011: Article ID 654871
- Xu J, Kang P, Wei Z: Singular multipoint impulsive boundary value problem with p -Laplacian operator. J. Appl. Math. Comput. 2009, 30: 105-120. 10.1007/s12190-008-0160-2MathSciNetView Article
- Zhang X, Ge W: Impulsive boundary value problems involving the one-dimensional p -Laplacian. Nonlinear Anal. 2009, 70: 1692-1701. 10.1016/j.na.2008.02.052MathSciNetView Article
- Feng M, Du B, Ge W: Impulsive boundary value problems with integral boundary conditions and one-dimensional p -Laplacian. Nonlinear Anal. 2009, 70: 3119-3126. 10.1016/j.na.2008.04.015MathSciNetView Article
- Bai L, Dai B: Three solutions for a p -Laplacian boundary value problem with impulsive effects. Appl. Math. Comput. 2011, 217: 9895-9904. 10.1016/j.amc.2011.03.097MathSciNetView Article
- Shi G, Meng X: Monotone iterative for fourth-order p -Laplacian boundary value problems with impulsive effects. Appl. Math. Comput. 2006, 181: 1243-1248. 10.1016/j.amc.2006.02.024MathSciNetView Article
- Zhang X, Yang X, Ge W: Positive solutions of n th-order impulsive boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 2009, 71: 5930-5945. 10.1016/j.na.2009.05.016MathSciNetView Article
- Zhang X, Feng M, Ge W: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces. J. Comput. Appl. Math. 2010, 233: 1915-1926. 10.1016/j.cam.2009.07.060MathSciNetView Article
- Lin X, Jiang D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 2006, 321: 501-514. 10.1016/j.jmaa.2005.07.076MathSciNetView Article
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando; 1988.
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.