Existence and multiplicity of solutions for second-order impulsive differential equations on the half-line
© Yan et al.; licensee Springer. 2013
Received: 9 July 2013
Accepted: 29 August 2013
Published: 7 November 2013
In this paper, we use variational methods to investigate the solutions of impulsive differential equations on the half-line. The conditions for the existence and multiplicity of solutions are established. The main results are also demonstrated with examples.
Impulsive differential equations arising from the real world describe the dynamics of processes in which sudden, discontinuous jumps occur. Such processes are naturally seen in biology, medicine, mechanics, engineering, chaos theory and so on. Due to their significance, a great deal of work has been done in the theory of impulsive differential equations [1–8].
where , for , , .
In recent years, boundary value problems (BVPs) for impulsive differential equations in an infinite interval have been studied extensively and many results for the existence of solutions, positive solutions, multiple solutions have been obtained [9–12]. The main methods used for the infinite interval problems are upper and lower solutions techniques, fixed point theorems and the coincidence degree theory of Mawhin in a special Banach space. On the other hand, many researchers used variational methods to study the existence of solutions for impulsive boundary value problems on the finite intervals [13–19].
where λ is a positive parameter. By using a variational method and a three critical points theorem, the authors proved the existence and multiplicity of solutions for IBVP (1.2).
Motivated by the above work, in this paper we use critical point theory and variational methods to investigate the existence and multiple of solutions of IBVP (1.1), in particular, its multiple solutions generated from the impulsive. Here, a solution for problem (1.1) is said to be generated from the impulsive if this solution emerges when the impulsive is not zero, but disappears when the impulsive is zero. For example, if problem (1.1) possesses at most one solution when the impulsive is zero, but it possesses three solutions when the impulsive is not zero, then problem (1.1) has at least two solutions generated from the impulsive. Our method is different from problem (1.2) and the main results extend the study made in .
2 Preliminaries and statements
Firstly, we introduce some notations and some necessary definitions.
Obviously, X is a reflexive Banach space.
Suppose that . Moreover, assume that for every , belongs to and belongs to . We say that u is a classical solution of BVP (1.1) if it satisfies the following conditions: u satisfies the first equation of (1.1) a.e. on ; the limits , , , exist and the impulsive condition of Eq. (1.1) holds; , exists, and the boundary conditions in Eq. (1.1) hold.
In order to study problem (1.1), we assume that the following conditions are satisfied:
for any .
Since and , by applying (2.3) and Leibniz formula of differentiation, we obtain for any . That is, is well defined on X.
Lemma 2.1 If is a critical point of φ, then u is a classical solution of IBVP (1.1).
for any .
By a similar argument, we can get that and , exist. Therefore, u satisfies the equation in IBVP (1.1) a.e. on .
which contradicts (2.9). So u satisfies the impulsive conditions of (1.1).
for all . Since v is arbitrary, (2.10) shows that . Therefore, u is a classical solution of IBYP (1.1). □
To this end, we state some basic notions and celebrated results from critical points theory.
Definition 2.1 (see )
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 (denoted by the P.S. condition for short).
Lemma 2.2 (see )
is a critical value of φ.
Definition 2.2 (see )
If X is a real Banach space, we denote by the class of all functionals possessing the following property: if is a sequence in X converging weakly to and , then has a subsequence converging strongly to u.
Lemma 2.3 (see )
assume that . Then, for each compact interval (with the conventions , ), there exists with the following property: for every and every functional with compact derivative, there exists such that for each , the equation has at least three solutions whose norms are less than σ.
3 Main results
Now we get the main results of this paper.
Theorem 3.1 Suppose that (H1) and hold. Then IBVP (1.1) has at least one solution if the following conditions hold:
Since , , the above inequality implies that . So φ is a functional bounded from below.
In view of (3.2), (3.3) and (3.4), we obtain as . Then φ satisfies the P.S. condition. According to Lemma 2.2, φ has at least one critical point, i.e., IBVP (1.1) has at least one classical solution for . □
Theorem 3.2 Suppose that (H1) and the following conditions hold, then there exist constants , such that for each , IBVP (1.1) possesses at least three solutions, and their norms are less than σ. Moreover, two of them are generated from the impulsive.
(H3) is nonincreasing about u for all .
(H4) There exists a constant such that .
Proof We apply Lemma 2.3 to prove this theorem.
So is uniformly monotone. By , we know that exists and is continuous on .
For any , we have . Suppose that , then on . By , we have as . So is strongly continuous which implies is a compact operator by .
From the continuity of , we can obtain that converges uniformly to as . That is, as . So is strongly continuous on X, which shows that is a compact operator by . Moreover, is continuous since it is strongly continuous. In addition, ϕ has a strict local minimum 0 with .
Therefore, all the fundamental assumptions hold.
Next we show that .
Therefore, we obtain .
By Lemma 2.3, we can choose such that , there exists with the following property: for every , there exists such that for each , the equation has at least three solutions in X whose norms are less than σ. Hence, IBVP (1.1) has at least three solutions in X whose norms are less than σ.
which implies that , i.e., IBVP (1.1) has at most one solution when the impulsive are zero. Therefore, we obtain that IBVP (1.1) has at least two solutions generated from the impulsive.
This completes the proof. □
To illustrate how our main results can be used in practice, we present the following example.
, with , then it shows that (H1) is satisfied.
It is easy to see that the impulsive function has sublinear growth, then condition (H2) holds.
Applying Theorem 3.1, problem (4.1) possesses at least one solution.
, with . Then it shows that (H1) is satisfied.
It is easy to see that is nonincreasing about u for all , then (H3) holds.
so (H4) holds.
By a simple computation, one has , which implies that condition (H5) is satisfied.
Applying Theorem 3.2, problem (4.2) possesses at least three solutions, and two of them are generated from the impulsive.
This work is partially supported by the National Natural Science Foundation of China (No. 71201013), the Humanities and Social Sciences Project of the Ministry of Education of China (No. 12YJC630118), the Innovation Platform Open Funds for Universities in Hunan Province of China (No. 13K059), the Provincial Natural Science Foundation of Hunan (No. 11JJ3012).
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