This paper is concerned with the existence of solutions to a boundary value problem of a fourth-order 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 problem
Fourth-order two-point boundary value problems of ordinary differential equations are widely employed by engineers to describe the beam deflection with two simply supported ends [1-3]. One example is the following fourth-order two-point boundary value problem:
where , , are the fourth, third, and second derivatives of with respect to t, respectively, , A and B are two real constants. System (1.1) has been studied in [4-7] and the references therein. For a beam, and in (1.1) refer to the two ends of the beam. At other locations of the beam, , there may be some sudden changes in loads placed on the beam, or some unexpected forces working on the beam. These sudden changes may result in impulsive effects for the governing differential equation. This motivates us to consider the following boundary value problem for a fourth-order impulsive differential equation:
We are mainly concerned with the existence of solutions of system (1.2). A function is said to be a (classical) solution of (1.2) if satisfies (1.2). In literature, tools employed to establish the existence of solutions of impulsive differential equations include fixed point theorems, the upper and lower solutions method, the degree theory, critical point theory and variational methods. See, for example, [8-20]. In this paper, we establish the existence of solutions of (1.2) by converting the problem to the existence of critical points of some variational structure. In this paper we regard λ as a parameter and find the ranges in which (1.2) admits at least one and infinitely many solutions, respectively. Note that when system (1.2) reduces to the one studied in . Our results extend those ones in .
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.
Throughout we assume that A and B satisfy
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
we have the following relation.
Proof The proof follows easily from Wirtinger’s inequality , Lemma 2.3 of  and Hölder’s inequality. The detailed argument is similar to the proof of Lemma 2.2 in , and we thus omit it here. □
Therefore, by (2.7) we have
Next we show that u satisfies
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.
Proof By Lemma 2.2, it suffices to show the functional defined in (2.3) has at least one critical point. We prove this by verifying the conditions given in [, Theorem 5.1]. Note that Φ defined in (2.4) is a nonnegative Gâteaux differentiable, coercive, and sequentially weakly lower semicontinuous functional, and its Gâteaux derivative admits a continuous inverse on . Moreover, Ψ defined in (2.5) is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Set
By (3.3) we have
which implies that
By (3.11) we have
Note that (3.7) implies that
which, together with (3.8), gives
Therefore, . Thus all the conditions in [, Theorem 5.1] are verified, and hence for each the functional admits at least one critical point u such that . Consequently, system (1.2) admits at least one solution u and . □
Correspondingly, conditions (3.3) and (3.7) reduce to
If (3.13) and (3.14) hold, then
As a consequence, we have the following result.
Example 3.1 Consider the boundary value problem
Here, , , , and . It is easy to verify that (H1) is satisfied with and . Direct calculations give , , and . Let , , , then , c, satisfy (3.3) and . Thus, it follows from Theorem 3.1 that system (3.15) has at least one solution for .
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
Theorem 3.3Assume that (H1), (H2), and (H3) are satisfied. If
Proof We apply [, Theorem 2.1] to show that the functional defined in (2.3) has an unbounded sequence of critical points.
which implies that
which, together with (3.16), gives us
This shows that . For any fixed , it follows from [, Theorem 2.1] that either has a global minimum or there is a sequence of critical points (local minima) of such that .
This, together with (H2), yields
It then follows from (H3) that
Corollary 3.4Assume that (H1), (H2), and (H3) are satisfied. If
hold, then (1.2) has an unbounded sequence of solutions inX.
Theorem 3.5Assume that (H1), (H2), and (H3) are satisfied. If
Example 3.2 Consider
Example 3.3 Consider the boundary value problem
Direct calculations give
The authors declare that they have no competing interests.
Both authors made equal contribution. Both authors read and approved the final manuscript.
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.
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).
Aftabizadeh, AR: Existence and uniqueness theorems for fourth-order 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 fourth-order two-point boundary value problems. J. Math. Anal. Appl.. 215, 415–422 (1997). Publisher Full Text
Bonanno, G, Dibella, B, Regan, DO: Non-trivial solutions for nonlinear fourth-order elastic beam equations. Comput. Math. Appl.. 62, 1862–1869 (2011). Publisher Full Text
Han, G, Xu, Z: Multiple solutions of some nonlinear fourth-order beam equations. Nonlinear Anal.. 68, 3646–3656 (2008). 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
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 CISM-ICMS, 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
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 non-quadratic 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 fourth-order impulsive differential equations. J. Appl. Math. Comput.. 35, 323–340 (2011). Publisher Full Text