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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 527082, 6 pages
Multiple Solutions for a Second-Order Impulsive Sturm-Liouville Equation
1College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, China
2Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
Received 29 March 2013; Revised 27 May 2013; Accepted 18 June 2013
Academic Editor: Gennaro Infante
Copyright © 2013 Jingli Xie and Zhiguo Luo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the existence of solutions to a boundary value problem of a second-order impulsive Sturm-Liouville equation with a control parameter . By employing some existing critical point theorems, we find the range of the control parameter in which the boundary value problem admits at least three solutions. 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. Some examples are given to demonstrate the main results in this paper.
The aim of this paper is to investigate the existence of multiple solutions of the following Neumann boundary value problem with impulsive Sturm-Liouville type equation:
where , , , , , and is a positive real parameter. , where and denote the right and left limits, respectively, of at , .
In the last few years, the existence of multiple solutions to Neumann problems has been widely investigated [1–5]. But little research has focused on the existence of multiple solutions for impulsive Sturm-Liouville equations whose right-hand side nonlinear term is depending on a parameter . Processes subject to sudden changes in their states are modeled by the impulsive differential equations and have been investigated in various fields of science and technology. In the motion of spacecraft, one has to consider instantaneous impulses at a position with jump discontinuities in velocity, but no change in the position [6, 7]. This motivates us to consider problem (1).
In the 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, our focus is on the existence of solutions of problem (1) with being the parameter. The problem is first transformed into the existence of critical points of some variational structure. Then with the help of critical point theory, results on the existence of at least three solutions and infinitely many solutions are established.
Throughout we assume that and satisfy
Take and define
For the norm in , we put
We have the following relation.
Lemma 1. Let . Then
Proof. For any , it follows from the mean value theorem that
for some . Hence, for , using Hölder inequality and (2), we have
Define a functional as
Note that is Fréchet differentiable at any , and for any , we have
Proof. For any function , we get
By the regularity theory, the weak solution is a classical solution of problem (1).
Next we show that a critical point of the functional is a solution of problem (1).
Lemma 3. If is a critical point of , then is a classical solution of problem (1).
For , we define
3. Main Results
3.1. Existence of At Least Three Solutions
In this section we derive conditions under which problem (1) admits at least three solutions. For this purpose, we introduce the following assumptions. (H1) Assume that there exists a positive constant such that for each (H2) Assume that there exist positive constants , , and , such that
Let and let with given in (16). Clearly, . For constants , , we define
Theorem 4. Assume that (H1), (H2) are satisfied. If there exist two positive constants , satisfying , and
then, for each , problem (1) admits at least three solutions.
Proof. By Lemma 3, it suffices to show that the functional defined in (8) has at least three critical points. We prove this by verifying the conditions given in [21, Theorem 3.2]. Note that defined in (9) is a nonnegative Gâteaux differentiable, coercive, and sequentially weakly lower semicontinuous functional and its Gâteaux derivative admits a continuous inverse on . Moreover, defined in (10) is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Set
Note that . It then follows from (H1) that
Then, we have
For satisfying , by Lemma 1, one has
which implies that
So, one has
Making use of (19), we obtain
By (H1) and (H2), when , we easily obtain that the functional is coercive.
Thus all conditions in [21, Theorem 3.2] are verified, and hence for each , the functional admits at least three critical points. Consequently, problem (1) admits at least three solutions.
Example 5. Consider the boundary value problem
Here, , , , and . Note that (H1), (H2) are satisfied. Moreover, we have , , , , and
Choose , . Direct calculations give
3.2. Existence of Infinitely Many Solutions
In this section, we derive some conditions under which problem (1) admits infinitely many distinct solutions. To this end, we need the following assumption. (H3) Assume that
Theorem 6. Assume that (H3) is satisfied. If
holds, then for each , problem (1) has an unbounded sequence of solutions in .
Proof. We apply [22, Theorem 2.1] to show that the functional defined in (8) 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 , we let . For satisfying , similar to the proof of Theorem 4, one can show that
which implies that
Note that ; thus we have
which, together with (15), gives us
This shows that . For any fixed , it follows from [22, 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 such that, for each ,
Thus, there exists such that
Define as follows:
Note that . Thus the functional has no lower bound and hence it has no global minimum and the proof is complete.
Theorem 7. Assume that (H3) is satisfied. If
holds, then, for each , problem (1) has a sequence of nonzero solutions in , which weakly converges to .
Proof. The proof is similar to that of Theorem 6 by showing that and is not a local minimum of the functional .
Example 8. Consider the boundary value problem
Here, , , , and . Hence we have , , , , and
so (H3) is satisfied. Moreover, we have
Therefore, condition (34) holds and Theorem 6 applies: for , problem (47) admits an unbounded sequence of solutions in .
Example 9. Consider the boundary value problem
In this example, , , , and . Hence we have , , , , and
Hence, one has
Therefore (46) holds. Owing to Theorem 7, when problem (50) admits a sequence of pairwise distinct classical solutions strongly converging at 0 in .
The authors are very grateful to the referees for their valuable comments and suggestions, which greatly improved the presentation of this paper. This work is supported by Hunan Provincial Natural Science Foundation of China (no. 11JJ3012).
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