- Open Access
New results for perturbed second-order impulsive differential equation on the half-line
© Zhao et al.; licensee Springer. 2014
- Received: 18 September 2014
- Accepted: 12 November 2014
- Published: 28 November 2014
By using a variational method and some critical points theorems, we establish some results on the multiplicity of solutions for second-order impulsive differential equation depending on two real parameters on the half-line. In addition, two examples to illustrate our results are given.
- impulsive differential equation
- variational methods
- critical points
for all , satisfying .
Boundary value problems on the half-line, arising naturally in the study of radially symmetric solutions of nonlinear elliptic equations and various physical phenomena , have been studied extensively and a variety of new results can be found in the papers – and the references cited therein. Criteria for the existence of solutions or multiplicities of positive solutions are established for the boundary value problem on the half-line. The main tools used in the literature for such a problem are the coincidence degree theory of Mawhin, fixed point arguments together with the lower and upper solutions method. For example the readers are referred to – and the references therein.
where and for all ; and is continuous. By using the fixed point theorem, the existence of at least one solution for IBVP (1.3) is obtained.
where and is continuous. By using a fixed point theorem due to Avery and Peterson, the existence of at least three positive solutions is obtained.
On the other hand, critical point theory and variational methods are proved to be a powerful tool in studying the existence of solutions for the impulsive differential equations –. For some recent works on the theory of critical point theory and variational methods we refer the readers to .
In the case , Chen and Sun  studied and presented some results on the existence and multiplicity of solutions for IBVP (1.1) by using a variational method and a three critical points theorem due to Bonanno and Marano (see Theorem 2.1 of ). The result is as follows.
(, Theorem 3.1])
Suppose that the following conditions hold.
(H1), are nondecreasing, and, for any.
Then, for each, IBVP (1.1) has at least three classical solutions.
Soon after, in the case , by using the variant fountain theorems (see Theorem 2.2 of ), Dai and Zhang  obtained some existence theorems of solutions for IBVP (1.1) when the function g and the impulsive functions () satisfies the following superlinear growth conditions:
(H1′): (), satisfy , for any ; and there exist positive constants , () and , q () such that , , .
However, there is no work for IBVP (1.1) when the parameter and f is an -Carathéodory function. As a result, the goal of this paper is to fill the gap in this area. Our aim is to establish a precise open interval , for each , there exists a such that for each , IBVP (1.1) admits at least three classical solutions.
The remainder of the paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we will state and prove the main results of the paper, and also two examples are presented to illustrate our main results.
In this section, we first introduce some notations and some necessary definitions.
where , .
for any .
Recall that a function is said to be an -Carathéodory function, if
(S1): is measurable for every ;
(S2): is continuous for almost every ;
If we assume that the function f satisfies the further condition
then one has the following result.
Suppose that condition (S3′) holds. Thenis a compact operator. In particular, is a weakly sequentially continuous functional.
the sequence converges in and the compactness is proved.
Finally, it follows from Corollary 41.9 of , p.236] that Ψ is a weakly sequentially continuous functional. This completes the proof. □
for any .
for any .
Arguing in a standard way, it is easy to prove that the critical points of the functional are the weak solution of IBVP (1.1) and so they are classical solutions.
The main tools to prove our results in Section 3 are the following critical points theorems.
- (ii)for each λ in
the functionalis coercive. Then, for eachthe functionalhas at least three distinct critical points in X.
- (2)for each and for every , which are local minimum for the functional and such that and , one has
Assume that there are two positive constants, , and, with, such that
Then, for each, the functionalhas at least three distinct critical points which lie in.
Thus, (3.1) is proved. □
Now we can state and prove our main results.
Assume that (C0) andhold. Letbe an-Carathéodory function such that (S3′) satisfies. Furthermore, suppose that there exist two positive constants a and b such that
such that for each, IBVP (1.1) has at least three distinct classical solutions.
Obviously, under the condition (S3′), is weakly sequentially lower semicontinuous and Gâteaux differentiable functional.
So Φ is coercive.
that is, is coercive.
so is a strongly monotone operator. By , Theorem 26.A], one finds that exists and is Lipschitz continuous on . Hence the functionals Φ and Ψ satisfy the regularity assumptions of Theorem 2.3.
Next we will prove the coercivity of the functional .
the functional has at least three distinct critical points, i.e. IBVP (1.1) has at least three distinct weak solutions. This completes the proof. □
Assume thatholds, andbe an-Carathéodory function such that (S3′) satisfies, andfor all. Furthermore, suppose that there exist a functionand two positive constants, withsuch that
such that, for each, IBVP (1.1) has three distinct classical solutions, , with.
for all .
Consequently, for every .
for all .
So, the conditions (b1) and (b2) of Theorem 2.4 are satisfied. Then by means of Theorem 2.4, IBVP (1.1) admits at least three distinct weak solutions () in X, such that . This completes the proof. □
IBVP (3.12) admits at least three distinct classical solutions () with .
We observe that in Example 3.4 and Example 3.5 the functions f, g, and the impulsive term do not satisfy the conditions (H1), (H2) of Theorem 3.1 in  or the conditions of Theorem 3.2 in . Hence, the problem (3.11) and (3.12) cannot be dealt with by the results of , .
The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by Hunan Provincial Natural Science Foundation of China (No. 13JJ3106).
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