Solutions to a boundary value problem of a fourth-order impulsive differential equation
© Xie and Luo; licensee Springer. 2013
Received: 26 March 2013
Accepted: 15 June 2013
Published: 1 July 2013
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.
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.
we have the following relation.
Lemma 2.1 Let . 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. □
Next we show that a critical point of the functional is a solution of system (1.2).
Lemma 2.2 If is a critical point of , then u is a solution of system (1.2).
which is a contradiction. Similarly, one can show that , . Therefore, u is a solution of (1.2). □
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.
Note that for every and we have . Since , then . Thus, if c and satisfy (3.3), then and and hence .
then, for each , system (1.2) admits at least one solution u and , where and .
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 . □
As a consequence, we have the following result.
Corollary 3.2 Assume that (H1) is satisfied. If there exist two constants c and satisfying (3.13) and (3.14), then for each system (1.2) admits at least one nontrivial solution u.
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.
holds, then for each system (1.2) has an unbounded sequence of solutions in X.
Proof We apply [, Theorem 2.1] to show that the functional defined in (2.3) has an unbounded sequence of critical points.
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 .
Note that . Thus the functional is unbounded from below and hence it has no global minimum and the proof is complete. □
hold, then (1.2) has an unbounded sequence of solutions in X.
holds, then for each system (1.2) has a sequence of non-zero solutions in X, 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 . □
Therefore, condition (3.17) holds and Theorem 3.3 applies: For , (3.19) admits an unbounded sequence of solutions in X.
In this example, , , and . The assumptions (H1), (H2), and (H3) clearly hold.
Hence (3.18) holds. Therefore it follows from Theorem 3.5 that (3.20) admits a sequence of distinct solutions in X provided that .
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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