Abstract
This paper is concerned with the existence of solutions to a boundary value problem of a fourthorder impulsive differential equation with a control parameter λ. By employing some existing critical point theorems, we find the range of the control parameter in which the boundary value problem admits at least one solution. It is also shown that under certain conditions there exists an interval of the control parameter in which the boundary value problem possesses infinitely many solutions. The main results are also demonstrated with examples.
MSC: 34B15, 34B18, 34B37, 58E30.
Keywords:
critical point theorem; impulsive differential equations; boundary value problem1 Introduction
Fourthorder twopoint boundary value problems of ordinary differential equations are widely employed by engineers to describe the beam deflection with two simply supported ends [13]. One example is the following fourthorder twopoint boundary value problem:
where
where
We are mainly concerned with the existence of solutions of system (1.2). A function
The rest of this paper is organized as follows. In Section 2 we present some preliminary results. Our main results and their proofs are given in Section 3.
2 Preliminaries
Throughout we assume that A and B satisfy
Let
and
Take
Since A and B satisfy (2.1), it is straightforward to verify that (2.2) defines a norm for the Sobolev space X and this norm is equivalent to the usual norm defined as follows:
It follows from (2.1) that
For the norm in
we have the following relation.
Lemma 2.1Let
Proof The proof follows easily from Wirtinger’s inequality [22], Lemma 2.3 of [23] and Hölder’s inequality. The detailed argument is similar to the proof of Lemma 2.2 in [21], and we thus omit it here. □
Define a functional
where
and
with
Note that
Next we show that a critical point of the functional
Lemma 2.2If
Proof Suppose that
For
Thus
Therefore, by (2.7) we have
Next we show that u satisfies
Suppose on the contrary that there exists some
Pick
then
Clearly,
which is a contradiction. Similarly, one can show that
For
and
For
3 Main results
3.1 Existence of at least one solution
In this section we derive conditions under which system (1.2) admits at least one solution. For this purpose, we introduce the following assumption.
(H1) Assume that there exist two positive constants
and
Let
we define
and
where
Note that for every
Theorem 3.1Assume that (H1) is satisfied. If there exist constants
then, for each
Proof By Lemma 2.2, it suffices to show the functional
Note that
and
By (3.3) we have
For
which implies that
Hence
For
It follows from the definition of
Note that
Making use of
By (2.8), and note that
By (3.11) we have
Note that (3.7) implies that
which, together with (3.8), gives
Therefore,
In particular, if we take
and
Correspondingly, conditions (3.3) and (3.7) reduce to
and
If (3.13) and (3.14) hold, then
and
As a consequence, we have the following result.
Corollary 3.2Assume that (H1) is satisfied. If there exist two constantscand
Example 3.1 Consider the boundary value problem
Here,
3.2 Existence of infinitely many solutions
In this section, we derive some conditions under which system (1.2) admits infinitely many distinct solutions. To this end, we need the following assumptions.
(H2) Assume that
(H3) Assume that
Let
where
Theorem 3.3Assume that (H1), (H2), and (H3) are satisfied. If
holds, then for each
Proof We apply [[5], Theorem 2.1] to show that the functional
We first show that
For any positive integer n, we let
which implies that
Note that
which, together with (3.16), gives us
This shows that
Next we show that the functional
Thus, there exists
Define
This, together with (H2), yields
It then follows from (H3) that
Note that
Corollary 3.4Assume that (H1), (H2), and (H3) are satisfied. If
and
hold, then (1.2) has an unbounded sequence of solutions inX.
Let
Theorem 3.5Assume that (H1), (H2), and (H3) are satisfied. If
holds, then for each
Proof The proof is similar to that of Theorem 3.3 by showing that
Example 3.2 Consider
where
Here
so (H1), (H2), and (H3) are satisfied. Moreover, we have
Therefore, condition (3.17) holds and Theorem 3.3 applies: For
Example 3.3 Consider the boundary value problem
where
In this example,
Direct calculations give
Hence (3.18) holds. Therefore it follows from Theorem 3.5 that (3.20) admits a sequence
of distinct solutions in X provided that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors made equal contribution. Both authors read and approved the final manuscript.
Authors’ information
JX is with the College of Mathematics and Statistics, Jishou University, China and is a PhD candidate at the Department of Mathematics, Hunan Normal University, China. ZL is a professor at the Department of Mathematics, Hunan Normal University, China.
Acknowledgements
The authors are very grateful to the referees for their valuable comments and suggestions, which greatly improved the presentation of this paper. The work is partially supported by Hunan Provincial Natural Science Foundation of China (No: 11JJ3012).
References

Aftabizadeh, AR: Existence and uniqueness theorems for fourthorder boundary value problems. J. Math. Anal. Appl.. 116, 415–426 (1986). Publisher Full Text

Gupta, CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal.. 26, 289–304 (1988). Publisher Full Text

Ma, R, Zhan, J, Fu, S: The method of lower and upper solutions for fourthorder twopoint boundary value problems. J. Math. Anal. Appl.. 215, 415–422 (1997). Publisher Full Text

Bonanno, G, Dibella, B, Regan, DO: Nontrivial solutions for nonlinear fourthorder elastic beam equations. Comput. Math. Appl.. 62, 1862–1869 (2011). Publisher Full Text

Bonanno, G, Molica Bisci, G: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl.. 113, 1–20 (2009)

Han, G, Xu, Z: Multiple solutions of some nonlinear fourthorder beam equations. Nonlinear Anal.. 68, 3646–3656 (2008). Publisher Full Text

Liu, X, Li, W: Existence and multiplicity of solutions for fourthorder boundary values problems with parameters. J. Math. Anal. Appl.. 327, 362–375 (2007). PubMed Abstract  Publisher Full Text

Agarwal, RP, Rgean, DO: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem. Appl. Math. Comput.. 161, 433–439 (2005). Publisher Full Text

Agarwal, RP, Franco, D, Regan, DO: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math.. 69, 83–96 (2005). Publisher Full Text

Bonanno, G: A critical point theorem via the Ekeland variational principle. Nonlinear Anal.. 75, 2992–3007 (2012). Publisher Full Text

Chen, J, Nieto, JJ: Impulsive periodic solutions of firstorder singular differential equations. Bull. Lond. Math. Soc.. 40, 902–912 (2007)

De Coster, C, Habets, P: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. In: Zanolin F (ed.) Nonlinear Analysis and Boundary Value Problem for Ordinary Differential Equations CISMICMS, pp. 1–78. Springer, New York (1996)

Marek, G, Szymon, G: On the discrete boundary value problem for anisotropic equation. J. Math. Anal. Appl.. 386, 956–965 (2012). Publisher Full Text

Nieto, JJ, Regan, DO: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl.. 10, 680–690 (2009). Publisher Full Text

Qin, D, Li, X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl.. 303, 288–303 (2005). Publisher Full Text

Shen, JH, Wang, BW: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal.. 69, 4055–4062 (2008). Publisher Full Text

Tian, Y, Ge, W: Applications of variational methods to boundaryvalue problem for impulsive differential equations. Proc. Edinb. Math. Soc.. 51, 509–527 (2008)

Zhang, H, Li, Z: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal., Real World Appl.. 11, 67–78 (2010). Publisher Full Text

Zhang, X, Tang, X: Subharmonic solutions for a class of nonquadratic second order Hamiltonian systems. Nonlinear Anal., Real World Appl.. 13, 113–130 (2012). Publisher Full Text

Zhang, Z, Yuan, R: Applications of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl.. 11, 155–162 (2010). Publisher Full Text

Sun, J, Chen, H, Yang, L: Variational methods to fourthorder impulsive differential equations. J. Appl. Math. Comput.. 35, 323–340 (2011). Publisher Full Text

Dym, H, McKean, H: Fourier Series and Integrals, Academic Press, New York (1985)

Peletier, LA, Troy, WC, van der Vorst, RCAM: Stationary solutions of a fourth order nonlinear diffusion equation. Differ. Equ.. 31, 301–314 (1995)