Abstract
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.
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1 Introduction
In this paper we consider the following boundary value problem with impulsive effects:
where m is a non-zero constant; λ and μ are referred to as two control parameters, for , , ; is an -Carathéodory function, and be a Lipschitz continuous function with the Lipschitz constant ; i.e.
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 [1], have been studied extensively and a variety of new results can be found in the papers [2]–[6] 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 [1]–[8] and the references therein.
Recently, many researchers pay more attention to the impulsive boundary value problems, such as Dirichlet boundary value problem, periodic boundary value problem, two point boundary value problem and so on (see for example, [9], [10] and references therein). The existence or multiplicity of solutions for impulsive boundary value problems (IBVP) on the half-line has been studied by many authors [11]–[14]. Kaufmann et al.[13] investigated the following impulsive boundary value problem:
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.
In [14], Li and Nieto considered the existence of multiple positive solutions of the following IBVP on the half-line:
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 [15]–[28]. For some recent works on the theory of critical point theory and variational methods we refer the readers to [29].
In the case , Chen and Sun [17] 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 [30]). The result is as follows.
Theorem 1.1
([17], Theorem 3.1])
Suppose that the following conditions hold.
(H1), are nondecreasing, and, for any.
(H2)There exist positive constants a, l with, andsuch that
where.
(H3)There exist two constantssuch that
(H4).
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 [31]), Dai and Zhang [25] 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.
2 Preliminaries
In this section, we first introduce some notations and some necessary definitions.
Set
Denote the Sobolev space X by
with the inner product
which induces the norm
Obviously, X is a reflexive Banach space. We define the norm in as , and let , with the norm . Then is a Banach space. In addition, X is continuously embedded into , and therefore, there exist two constants , such that
and
Suppose that . By a classical solution of IBVP (1.1), we mean a function
that satisfies the equation in IBVP (1.1) a.e. on , the limits , , exist and the impulsive conditions in IBVP (1.1) hold; , exist, and the boundary conditions in IBVP (1.1) also are met.
For each , put
where , .
It is clear that Ψ is differentiable at any and
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 ;
(S3): for every there exists a function such that
If we assume that the function f satisfies the further condition
(S3′): there exists a function such that
then one has the following result.
Lemma 2.1
Suppose that condition (S3′) holds. Thenis a compact operator. In particular, is a weakly sequentially continuous functional.
Proof
Let Ω be a bounded set in X and let be a sequence in . Then there is a sequence in Ω such that and for all . Due to X being reflexive, there exists a subsequence converging weakly to . We can prove that has a subsequence which converges almost everywhere in to the function u. In fact, given a number , let . Then we easily infer that is bounded in . Pick , the Rellich-Kondrachov theorem [32], Theorem IX.16] yields a subsequence of such that at most all points . Applying this argument again, with 1 replaced by 2, we also obtain a sequence of such that at almost all points . Thus, the sequence clearly complies with the conclusion. Without loss of generality we write as . Therefore, converges to a.e. on . From (2.1), (2.2), and (2.5), we have
for all with . Hence, from (S3′), the Lebesgue dominated convergence theorem and continuity of show that the sequence converges to in . Therefore, taking into account that
the sequence converges in and the compactness is proved.
Finally, it follows from Corollary 41.9 of [33], p.236] that Ψ is a weakly sequentially continuous functional. This completes the proof. □
By standard arguments, we find that Φ is a Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative at the point is the functional , given by
for any .
Definition 2.2
A function is said to be a weak solution IBVP (1.1) if u satisfies
for any .
It is easy to verify that is a weak solution to IBVP (1.1) if and only if u is a classical solution of IBVP (1.1).
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.
Theorem 2.3
([18])
Let X be a reflexive real Banach space; be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on, be a sequentially weakly upper semicontinuous, continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, such that. Assume that there existand, withsuch that
-
(i)
,
-
(ii)
for each λ in
the functionalis coercive. Then, for eachthe functionalhas at least three distinct critical points in X.
Theorem 2.4
([34])
Let X be a reflexive real Banach space; be a convex, coercive and continuously Gâteaux differentiable functionals whose derivative admits a continuous inverse on, be a continuously Gâteaux differentiable functionals whose derivative is compact, such that
-
(1)
;
-
(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
(b1);
(b2).
Then, for each, the functionalhas at least three distinct critical points which lie in.
3 Main results
Lemma 3.1
Suppose that
(C0)there exist constants, and () such that
Then, for each, we have
Proof
By the condition (C0), we have
Thus, (3.1) is proved. □
Now we can state and prove our main results.
Theorem 3.2
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
(C1);
(C2), and.
Then, for each λ in
there exists
such that for each, IBVP (1.1) has at least three distinct classical solutions.
Proof
Obviously, under the condition (S3′), is weakly sequentially lower semicontinuous and Gâteaux differentiable functional.
Note that as (1.2) holds for every and , one has
Furthermore, for any , one has
and
So Φ is coercive.
Next, we show that admits a Lipschitz continuous inverse. For any , it follows from (2.6) and (1.2) that
By the assumption , it turns out that
that is, is coercive.
For any ,
so is a strongly monotone operator. By [35], 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.
Furthermore for any fixed λ, and μ as in (3.2), (3.3). Put . Taking (2.1) into account, for every such that , we have . Therefore, it follows from (2.4), (3.1), and the Hölder inequality that
which implies
Since , one has
Put . Obviously, , . By a similar reasoning to (3.4) and (3.5), we get
From (C1) we get . It follows from (2.4) that
then
Since , one has
Combining with (3.6) and (3.7), condition (i) of Theorem 2.3 is fulfilled.
Next we will prove the coercivity of the functional .
Taking into account (S3′) and the Hölder inequality, one has
combining with (2.2) and Lemma 3.1, it follows that
Since , , the above inequality implies that , so is coercive. According to Theorem 2.3, it follows that, for each
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. □
Let
and
Theorem 3.3
Assume thatholds, andbe an-Carathéodory function such that (S3′) satisfies, andfor all. Furthermore, suppose that there exist a functionand two positive constants, withsuch that
(D1)
(D2)
Then, for each λ in
and for every negative continuous function, , andfor all, there exists
such that, for each, IBVP (1.1) has three distinct classical solutions, , with.
Proof
In order to apply Theorem 2.4 to IBVP (1.1), we take the functionals as given in (2.3) and (2.4). Obviously, Φ and Ψ satisfy the conditions (1) of Theorem 2.4. Now we will prove that the functional satisfies the assumption (2) of Theorem 2.4. Let and be two local minima for . Then and are critical points for , which implies that , are weak solutions of IBVP (1.1). In particular and are nonnegative. Indeed, with no loss of generality we may assume that be a weak solution of IBVP (1.1), and the set is nonempty and of positive measure. Furthermore, taking into account that is a weak solution, one has
for all .
Put for all . Clearly and we deduce that
which implies for . Hence, on Ω, which is absurd. Then we obtain , for all . So, one has for every , which implies that
and
Consequently, for every .
for all .
Note that by the condition , we get . It follows from the definition of Φ, (2.1), and (3.10) that
So, we have
Therefore, due to the assumption (D1), one gets
From the assumption (D2), one infers that
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. □
Example 3.4
Consider the following boundary value problem with impulsive effect:
where , . Choose , , , then the condition (C0) is satisfied. is a positive constant defined in (2.1). When lies in different intervals, we can choose different f and g satisfying the conditions. Hence we only consider one case. If , we may choose and
where is given in (2.1). Then
Obviously, f satisfies (S3′) with , and g satisfies (1.2) with , and . Take , . By simple calculations (C1) and (C2) are satisfied. Applying Theorem 3.2, IBVP (3.11) admits at least three distinct classical solutions for each , and for each
Example 3.5
Consider the following problem:
where , . Then for any . is a positive constant defined in (2.1). When lies in different intervals, we can choose different f and g satisfying the conditions. For example, if , we may choose and
where is given in (2.1). Then
Obviously, , for all and satisfies (S3′) with . g satisfies (1.2) with , and for any . Take , , and . Then , . By simple calculations (D1) and (D2) are satisfied. Applying Theorem 3.2, for each and for each
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 [17] or the conditions of Theorem 3.2 in [25]. Hence, the problem (3.11) and (3.12) cannot be dealt with by the results of [17], [25].
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Acknowledgements
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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Zhao, Y., Wang, X. & Liu, X. New results for perturbed second-order impulsive differential equation on the half-line. Bound Value Probl 2014, 246 (2014). https://doi.org/10.1186/s13661-014-0246-8
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DOI: https://doi.org/10.1186/s13661-014-0246-8