Existence of solutions of a second-order impulsive differential equation
© Luo et al.; licensee Springer. 2014
Received: 10 November 2013
Accepted: 4 April 2014
Published: 6 May 2014
This paper is concerned with the existence of solutions of a second-order impulsive differential equation with mixed boundary condition. We obtain sufficient conditions for the existence of a unique solution, at least one solution, at least two solutions and infinitely many solutions, respectively, by using critical point theorems. The main results are also demonstrated with examples.
MSC:34B15, 34B18, 34B37, 58E30.
where , , , , , α, β are constants with , , and the operator Δ is defined as , where denotes the right-hand (left-hand) limit of at .
In recent years, some classical tools such as some fixed point theorems in cones, topological degree theory and the upper and lower solutions method combined with the monotone iterative technique [9–14] have been widely used to get solutions of impulsive differential equations. On the other hand, in the last few years, some researchers have studied the existence of solutions for impulsive differential equations with boundary conditions via variational methods [15–19]. In this paper, we consider (1.1) by using critical point theory and variational methods.
The rest of this paper is organized as follows. In Section 2 we present several important lemmas. In Section 3, we present existence results of equation (1.1) by using critical point theory and variational methods.
In the following, we first introduce some notations and some necessary definitions.
Definition 2.1 
Let X be a real reflexive Banach space. For any sequence , if is bounded and as possesses a convergent subsequence, then we say that φ satisfies the Palais-Smale condition (PS condition).
Definition 2.2 
Let be differentiable and . We say that φ satisfies the condition if the existence of a sequence in X, such that , as , implies that c is a critical value of φ.
It is clear that the PS condition implies the condition for each .
Lemma 2.3 
Moreover, if a is also symmetric, then the functional defined by attains its minimum at u.
Lemma 2.4 
If φ is weakly lower semi-continuous on a reflexive Banach space X and has a bounded minimizing sequence, then φ has a minimum on X. The existence of a bounded minimizing sequence will be in particular insured when φ is coercive, i.e., such that if .
Lemma 2.5 
X is a real reflexive Banach space;
M is bounded and weakly sequentially closed;
F is weakly sequentially lower semi-continuous on M, i.e., by definition, for each sequence in M such that as , we have .
Lemma 2.6 
If φ satisfies the , then c is a critical value of φ and .
Lemma 2.7 
There exist constants such that ;
For each finite dimensional subspace , there is an such that for every with ;
then φ possesses an unbounded sequence of critical values.
For , we have that u and are both absolutely continuous and , hence for any . If , then u is absolutely continuous and . In this case, the one-side derivatives and may not exist. So, by a classical solution of (1.1), we mean a function satisfying the following conditions: For every , ; u satisfies the boundary condition of (1.1) and the first equation of (1.1); and , , exist and the impulsive conditions of (1.1) hold.
for any . Obviously, is continuous, and a critical point of φ gives a weak solution of (1.1).
Lemma 2.8 If the function is a critical point of the functional φ, then u is a solution of system (1.1).
Thus, u satisfies the equation in (1.1).
If , then .
This contradicts (2.10), so u satisfies the boundary condition. Therefore, u is a solution of system (1.1). □
Lemma 2.9 If , then , where .
Proof The proof follows easily from the Hölder inequality. The detailed argument is similar to the proof of Lemma 2.2 in , and we thus omit it here. □
3 Main results
3.1 Existence of a unique solution
In this section we derive conditions under which system (1.1) admits a unique solution.
Theorem 3.1 Assume that () are fixed constants, and (), then system (1.1) has a unique solution u, and u minimizes the functional (2.2).
It is evident that a is continuous and symmetric and l is bounded. Moreover, a is coercive. By Lemma 2.3, system (1.1) has a unique solution u, and u minimizes the functional (2.2). □
Here , , , , , , . Applying Theorem 3.1, problem (3.1) has a unique solution. By simple calculations, we obtain , , , .
3.2 Existence of at least one solution
In this section we derive conditions under which system (1.1) admits at least one solution. For this purpose, we introduce the following assumption.
Theorem 3.2 Assume that (H1) is satisfied, then system (1.1) has at least one solution u, and u minimizes the functional (2.2).
for all . This implies that φ is coercive.
Let be a weakly convergent sequence to u in H, then converges uniformly to u in .
then . So is weakly sequentially continuous. Clearly, is continuous and convex, which implies that is weakly sequentially lower semi-continuous. Therefore, φ is weakly sequentially lower semi-continuous on H.
By Lemma 2.4, the functional φ has a minimum which is a critical point of φ. Hence, system (1.1) has at least one solution. □
Here , , , , , , . Clearly, (H1) is satisfied. Applying Theorem 3.2, problem (3.2) has at least one solution.
3.3 Existence of at least two distinct solutions
In this section, we derive some sufficient conditions under which the functional φ admits at least two distinct critical points; consequently, (1.1) admits at least two distinct solutions. We first introduce some assumptions.
(H2) uniformly for , .
Theorem 3.3 Assume that (H2) and (H3) are satisfied. Then system (1.1) has at least two solutions.
Proof The proof will be given in three steps.
Step 1. The functional φ satisfies the PS condition.
Hence, is bounded in H.
So (3.5), (3.6) and (3.7) yield in H, i.e., strongly converges to u in H. Therefore, the functional φ satisfies the PS condition.
Step 2. We show that there exists such that the functional φ has a local minimum .
Firstly, we claim that is bounded and weakly sequentially closed.
such that in H. and , since is a closed convex set.
Secondly, we claim that the functional φ is weakly sequentially lower semi-continuous on .
then . By on H, we see that uniformly converges to u in . So is weakly sequentially continuous. Clearly, is continuous and convex, which implies that is weakly sequentially lower semi-continuous. Therefore, φ is weakly sequentially lower semi-continuous on .
Thirdly, we claim that φ has a minimum .
In fact, H is a reflexive Banach space, is bounded and weakly sequentially closed and φ is weakly sequentially lower semi-continuous on . So, by Lemma 2.4, there exists such that .
Finally, we claim that .
Choose , , then for any . Besides, for any . So . Hence, φ has a local minimum .
Step 3. We prove that there exists with such that .
Since , (3.10) implies . Therefore, we can choose with sufficiently large such that .
By Lemma 2.6, c is a critical value of φ, that is, there exists a critical point . Therefore, , are two critical points of φ, and they are solutions of (1.1). □
Here , , , , , , . Let , . Clearly, (H2) and (H3) are satisfied. By Theorem 3.3, problem (3.11) has at least two solutions.
3.4 Existence of infinitely many solutions
In this section, we derive some conditions under which system (1.1) admits infinitely many distinct solutions. To this end, we need the following assumption.
(H4) and , , are odd about u.
Theorem 3.4 Assume that (H2), (H3) and (H4) are satisfied. Then system (1.1) has infinitely many solutions.
Proof We apply Lemma 2.7 to finish the proof. Clearly, is even since and are odd about u, and . The arguments of Theorem 3.3 show that the functional φ satisfies the PS condition. In the same way as in Theorem 3.3, we can easily verify that conditions (i) and (ii) of Lemma 2.7 are satisfied. According to Lemma 2.7, φ possesses infinitely many critical points, i.e., system (1.1) has infinitely many solutions. □
Here , , , , , , . Obviously, , are odd about u. Let , . Clearly, (H2) and (H3) are satisfied. Applying Theorem 3.4, problem (3.12) has infinitely many solutions.
The authors are very grateful to the referees for their very helpful comments and suggestions, which greatly improved the presentation of this paper. This work is supported by the Scientific Research Fund of Hunan Provincial Education Department (No: 13K029).
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