Multiple Solutions of Second-Order Damped Impulsive Differential Equations with Mixed Boundary Conditions

and Applied Analysis 3 Thus ∫ T 0 [−(e G(t) u 󸀠 (t)) 󸀠 + λe G(t) u (t) − e G(t) f (t, u (t))] V (t) dt


Introduction
Impulsive effect exists widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians [1][2][3][4][5][6]. Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory, and so on [7][8][9][10][11].
The characteristic of (1) is the presence of the damped term ( ) . Most of the results concerning the existence of solutions of these equations are obtained using upper and lower solutions methods, coincidence degree theory, and fixed point theorems [12][13][14][15]. On the other hand, when there is no presence of the damped term, some researchers have used variational methods to study the existence of solutions for these problems [16][17][18][19][20][21]. However, to the best of our knowledge, there are few papers concerned with the existence of solutions for impulsive boundary value problems like problem (1) by using variational methods.
For this nonlinear damped mixed boundary problem (1), the variational structure due to the presence of the damped term ( ) is not apparent. However, inspired by the work [22,23], we will be able to transform it into a variational formulation. In this paper, our aim is to study the existence of distinct pairs of nontrivial solutions of problem (1). Our main results extend the study made in [22,23], in the sense that we deal with a class of problems that is not considered in those papers. (2)
Consider the Hilbert spaces with the inner product inducing the norm We also consider the inner product inducing the norm Consider the problem As is well known, (7) possesses a sequence of eigenvalues The corresponding eigenfunctions are normalized so that Now multiply (2) by V ∈ and integrate on the interval [0, ]: Then, a weak solution of (2) is a critical point of the following functional: where ( , ) = ∫ 0 ( , ) . We say that ∈ [0, ] is a classical solution of IBVP (1) if it satisfies the following conditions: satisfies the first equation of (1) a.e. on [0, ]; the limits ( + ), ( − ), = 1, 2, . . . , , exist and impulsive condition of (1) holds; satisfies the boundary condition of (1).

Lemma 1.
If ∈ is a weak solution of (1), then is a classical solution of (1).

Proof. If ∈
is a weak solution of (1), then is a weak solution of (2), so ( ( ), V) = 0 holds for all V ∈ ; that is, By integrating by part, we have Abstract and Applied Analysis holds for all V ∈ . Without loss of generality, for any = {1, 2, . . . , } and V ∈ with V( ) ≡ 0, for every ∈ [0, ] ∪ [ +1 , ], then substituting V into (14), we get Hence satisfies the first equation of (2). Therefore, by (14) we have Next we will show that satisfies the impulsive and the boundary condition in (2). If the impulsive condition in (2) does not hold, without loss of generality, we assume that there exists ∈ {1, 2, . . . , } such that which contradicts (16). So satisfies the impulsive condition in (2) and (16) implies which contradicts (19), so satisfies the boundary condition. Therefore, is a solution of (1).

Lemma 2.
Let ∈ . Then there exists a constant > 0, such that where Proof. By Hölder inequality, for ∈ , Lemma 3 (see [24,Theorem 9.1]). Let be a real Banach space, ∈ 1 ( , ) with even, bounded from below, and satisfying P.S. condition. Suppose (0) = 0; there is a set ⊂ such that is homeomorphic to −1 by an odd map and sup < 0. Then possesses at least distinct pairs of critical points.

Main Results
Theorem 4. Suppose that the following conditions hold.
Proof. Set Abstract and Applied Analysis Next, we will verify that the solutions of problem (26) are solutions of problem (1).

Theorem 5. Suppose that the following conditions hold.
(H1) There exist 1 > 0, > / , which is the kth eigenvalue of (7) such that Proof. The proof is similar to the proof of Theorem 4, and therefore we omit it.

Example
To illustrate how our main results can be used in practice we present the following example.