This paper is mainly concerned with the following second order impulsive differential equations with Dirichlet boundary conditions
where , , , , , are nonnegative constants with . . Here (respectively ) denotes the right limit (respectively left limit) of at , and is locally Lipschitz continuous for uniformly in .
The phenomena of sudden or discontinuous jumps are often seen in chemotherapy, population dynamics, optimal control, ecology, engineering, etc. The mathematical model that describes the phenomena is impulsive differential equations. Due to their significance, impulsive differential equations have been developing as an important area of investigation in recent years. For the theory and classical results, we refer to    . Considerable effort has been devoted to impulsive differential equations due to their theoretical challenge and potential applications, for example  - . We point out that in the motion of spacecraft one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position . The impulses occur not only on the velocity but also in impulsive mechanics . Second order impulsive differential Equation (1.1) of this paper happens to be the mathematical model of this kind of problem.
Motivated by the papers mentioned above, we study the existence of sign-changing solution for second order impulsive differential equations with Dirichlet boundary conditions. However, to the best of our knowledge, there are few papers concerned with the existence of sign-changing solution for impulsive differential equations. In this paper, we obtain the existence of a positive solution, a negative solution and a sign-changing solution of (1.1) by using critical point theory and variational methods. An example is presented to illustrate the application of our main result. In comparison with previous works such as  , this paper has several new features. Firstly, we consider the eigenvalues and the eigenfunctions for second order linear impulsive differential equations with Dirichlet boundary conditions. Secondly, we construct the lower and upper solutions of (1.1) by using the eigenfunction corresponding to the first eigenvalue. Finally, the existence of sign-changing solution of (1.1) is obtained by using critical point theory and variational methods.
Let H be a Hilbert space and E a Banach space such that E is imbedded in H. Let be a functional defined on H, that is, the differential of is locally Lipschitz continuous from H to H. Assume that and is also locally Lipschitz continuous as an operator from E to E. Assume also that . For , consider the initial value problem both in H and in E:
Let and be the unique solution of this initial value problem considered in H and E respectively, with and the right maximal interval of existence. Because of the imbedding , and for . We assume that and for , and if in H for some then in E.
Lemma 1.1  Assume that the statement made in the last paragraph is valid. Assume that satisfies the (PS)-condition on H and there are two open convex subsets and of E with the properties that , , and . If there exists a path such that
then has at least four critical points, one in , one in , one in , and one in . Here and mean respectively the boundary and the closure of D relative to E.
Now we state our main result
Theorem 1.1 Assume
(f1) uniformly in , where
(f2) there exists a constant such that is creasing in s,
(f3) there exist and such that
Then problem (1.1) has at least three solutions: one positive, one negative, and one sign-changing.
Let be the Sobolev space endowed the norm
which is equivalent to the usual norm . Let
with the norm
Clearly E is a Banach space and densely embedded in H.
As is well known, for , the following linear eigenvalue problem
possesses a sequence of positive eigenvalue and the algebraic multiplicity of is equal to 1. Moreover
the eigenfunction with respect to satisfies , in and the eigenfunctions corresponding to are sign-changing in and (see ).
Let M be as in (f2), we define new inner product of H as follows
The inner product induces the norm
Define the functional by
It is clear that . By , we know that the solution of problem (1.1) is equivalent to the critical point of , that is for , where
Lemma 2.1 Two norms and defined on H are equivalent, that is, there exist positive constants such that
Proof. For any , by Lemma 2.3 in , we have
From this it is easy to see that
Since , , we may assume that . From (2.6) we have
From (2.7) and (2.8), there exists a positive constant such that . On the other hand, it is obvious that . So, (2.5) holds.
Let be the Green’s function of
where . From , we have the following result.
Lemma 2.2  The Green function of (2.9) possesses the following properties
1) can be written by
2) is increasing and .
4) is decreasing and .
6) is a positive constant.
7) is continuous and symmetrical over .
8) has continuously partial derivative over .
9) For fixed , satisfies for and .
10) has discontinuous point of the first kind at , , .
Define an operator by
Lemma 2.3  is a critical point of the functional if and only if is a fixed point of the operator B.
From Lemma 2.3, the critical point set . Notice that is locally Lipschitz continuous for uniformly in . It is easy to obtain that B defined by (2.10) is locally Lipschitz continuous both as an operator from H to H and as one from E to E. Let and consider the initial value problem (1.2) in both H and E.
Similar to Lemma 4.2 in , we have
Lemma 2.4  1) and for .
2) if in H for some then in E.
3. Proof of Theorem
Lemma 3.1 The gradient of at a point can be expressed as
This result is necessary, for the reader’s convenience we present the proof in the Appendix.
Lemma 3.2 Assume that (f3) holds, then the functional satisfies (PS)-condition.
Proof. Suppose that is a (PS)-sequence, namely such that for some constant
This implies that there is a constant such that
Moreover, thanks to we know that is bounded on . By (3.1) we have
Then from (f3) we get
where and . By (3.2) and (3.3), one has
which of course implies that is bounded by means of Lemma 2.1.
Since H is a reflexive Banach space, we can assume that, up to a subsequence, there exists such that . By (2.4) we have
By and , we have
By in H, we see that uniformly converges to in . it is easy to obtain that
Then (3.4)-(3.6) and Lemma 2.1 yield that in H, that is, strongly converges to in H.
Proof of Theorem 1.1. By (f1), there exist and such that
Let , then
So, we have
Similarly, we also have
Hence, and are lower and upper solutions to (1.1), respectively.
Let and , it is clear that and are open convex sets in E and . If , then by (f2)
From Lemma 2.3 and (3.8), we easily know that
Since B defined by (2.10) is also an operator from E to E. Hence , by (3.10) and (3.11), and . Similarly, . (f3) implies that there exist two positive constants such that
Let , then is a finitely dimensional subspace of E, if , then we have, for some ,
We define by
where R will be determined later. Then we have , by (3.12),
we see that
if R is sufficiently large. Applying Lemma 2.4 and Lemma 1.1, problem (1.1) has at least four solutions, , , , and . It is clear that is positive, is negative, and is sign-changing.
4. An Example
To illustrate the application of our main result we present the following example.
Example 4.1 Consider the following second order impulsive differential equations with Dirichlet boundary condition:
Then (4.1) has at least three solutions: one positive, one negative, and one sign-changing.
Proof. It is clear that (4.1) has the form of (1.1). Let , , , , in (1.1). By (1.4) and (2.6), we can obtain that .
Taking and , by simple calculations, the conditions in Theorem 1.1 are satisfied.
Hence, (4.1) has at least three solutions: one positive, one negative, and one sign-changing.
Supported by the National Natural Science Foundation of China (61803236), Natural Science Foundation of Shandong Province (ZR2018MA022).
In this Appendix , for the reader’s convenience we give the proof of Lemma 3.1.
Proof of Lemma 3.1. For any ,
By the Lagrange Theorem there exists with such that
For any , by (2.1), we have
By integrating by parts and Lemma 2.2, we can obtain immediately
By Lemma 2.2
Substituting (A.3), (A.4) into (A.2), and using (A.5), one has
From (A.1) and (A.6)
From Definition 1.1 and Remarks 1.2 in , we conclude that .
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