Solutions to a boundary value problem of a fourth-order impulsive differential equation

Jingli Xie1 and Zhiguo Luo2*

Author Affiliations

1 College of Mathematics and Statistics, Jishou University, Jishou, Hunan, 416000, P.R. China

2 Department of Mathematics, Hunan Normal University, Changsha, Hunan, 410081, P.R. China

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Boundary Value Problems 2013, 2013:154  doi:10.1186/1687-2770-2013-154

 Received: 26 March 2013 Accepted: 15 June 2013 Published: 1 July 2013

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.

Abstract

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

1 Introduction

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:

(1.1)

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:

(1.2)

where , , and the operator Δ is defined as , where () denotes the right-hand (left-hand) limit of U at and is referred to as a control parameter.

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 [21]. Our results extend those ones in [21].

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

(2.1)

Let

and

Take and define

(2.2)

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. Then, .

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 as

(2.3)

where

(2.4)

and

(2.5)

with

Note that is Fréchet differentiable at any , and for any we have

(2.6)

Next we show that a critical point of the functional is a solution of system (1.2).

Lemma 2.2Ifis a critical point of, thenuis a solution of system (1.2).

Proof Suppose that is a critical point of . Then for any one has

(2.7)

For , choose such that for , then we have

Thus

Therefore, by (2.7) we have

Next we show that u satisfies

Suppose on the contrary that there exists some such that

Pick

then

Clearly, . Simple calculations show that , , and , . Thus

which is a contradiction. Similarly, one can show that , . Therefore, u is a solution of (1.2). □

For with , we define

(2.8)

and

(2.9)

For , we define

(2.10)

(2.11)

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 such that for each

(3.1)

and

(3.2)

Let and with given in (3.1) and given in (3.2). For constants , , and c satisfying

(3.3)

we define

(3.4)

and

(3.5)

where

(3.6)

Note that for every and we have . Since , then . Thus, if c and satisfy (3.3), then and and hence .

Theorem 3.1Assume that (H1) is satisfied. If there exist constants, , andcsatisfying (3.3) and

(3.7)

then, for each, system (1.2) admits at least one solutionuand, whereand.

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 [[10], 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

Note that . It then follows from (H1) that

(3.8)

and

(3.9)

By (3.3) we have

For satisfying , by Lemma 2.1, one has

which implies that

Hence

(3.10)

For with , one can similarly obtain

(3.11)

It follows from the definition of that

(3.12)

Note that . By (3.10) one has

Making use of , , and (3.8), we obtain

By (2.8), and note that , one has

By (3.11) we have

Note that (3.7) implies that

which, together with (3.8), gives

Therefore, . Thus all the conditions in [[10], 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 . □

In particular, if we take , then (3.4) and (3.5) become

and

Correspondingly, conditions (3.3) and (3.7) reduce to

(3.13)

and

(3.14)

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 constantscandsatisfying (3.13) and (3.14), then for eachsystem (1.2) admits at least one nontrivial solutionu.

Example 3.1 Consider the boundary value problem

(3.15)

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

Let and with given in (3.2). We define

(3.16)

where is given (2.10). Let

Theorem 3.3Assume that (H1), (H2), and (H3) are satisfied. If

(3.17)

holds, then for eachsystem (1.2) has an unbounded sequence of solutions inX.

Proof We apply [[5], Theorem 2.1] to show that the functional defined in (2.3) has an unbounded sequence of critical points.

We first show that . Let be a sequence of positive numbers such that as and

For any positive integer n, we let . For satisfying , similar to the proof of Theorem 3.1, one can show that

which implies that

Note that , thus we have

which, together with (3.16), gives us

This shows that . For any fixed , it follows from [[5], Theorem 2.1] that either has a global minimum or there is a sequence of critical points (local minima) of such that .

Next we show that the functional has no global minimum for . Since , we can choose a constant M such that, for each ,

Thus, there exists such that

Define as follows:

This, together with (H2), yields

It then follows from (H3) that

Note that . Thus the functional is unbounded from below and hence it has no global minimum and the proof is complete. □

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

(3.18)

holds, then for eachsystem (1.2) has a sequence of non-zero solutions inX, which weakly converges to 0.

Proof The proof is similar to that of Theorem 3.3 by showing that and 0 is not a local minimum of the functional . □

Example 3.2 Consider

(3.19)

where .

Here , , , and . Note that

so (H1), (H2), and (H3) are satisfied. Moreover, we have , and

Therefore, condition (3.17) holds and Theorem 3.3 applies: For , (3.19) admits an unbounded sequence of solutions in X.

Example 3.3 Consider the boundary value problem

(3.20)

where

In this example, , , and . The assumptions (H1), (H2), and (H3) clearly hold.

Direct calculations give

and

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).

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