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:
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 . 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
Take and define
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 , 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. □
Define a functional as
Note that is Fréchet differentiable at any , and for any we have
Next we show that a critical point of the functional is a solution of system (1.2).
Lemma 2.2If is a critical point of , thenuis a solution of system (1.2).
Proof Suppose that is a critical point of . Then for any one has
For , choose such that for , then we have
Therefore, by (2.7) we have
Next we show that u satisfies
Suppose on the contrary that there exists some such that
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
For , we define
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
Let and with given in (3.1) and given in (3.2). For constants , , and c satisfying
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
then, for each , system (1.2) admits at least one solutionuand , where and .
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
Note that . It then follows from (H1) that
By (3.3) we have
For satisfying , by Lemma 2.1, one has
which implies that
For with , one can similarly obtain
It follows from the definition of that
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 [, 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
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.
Corollary 3.2Assume that (H1) is satisfied. If there exist two constantscand satisfying (3.13) and (3.14), then for each system (1.2) admits at least one nontrivial solutionu.
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
Let and with given in (3.2). We define
where is given (2.10). Let
Theorem 3.3Assume that (H1), (H2), and (H3) are satisfied. If
holds, then for each system (1.2) has an unbounded sequence of solutions inX.
Proof We apply [, 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 [, 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
hold, then (1.2) has an unbounded sequence of solutions inX.
Theorem 3.5Assume that (H1), (H2), and (H3) are satisfied. If
holds, then for each system (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
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
In this example, , , and . The assumptions (H1), (H2), and (H3) clearly hold.
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 .
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).
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