Existence of Solutions for Superlinear Second-Order System with Noninstantaneous Impulses

Variational methods are used in order to establish the existence of nontrivial weak solution for superlinear second-order system with noninstantaneous impulses. The main result is obtained when a kind of definition of the weak solution for this system is introduced. Meanwhile, an example is presented to illustrate the main result.

This paper is mainly motivated by two facts. For the past thirty years, impulsive differential equations have been studied extensively in the literature. There are many approaches to investigate solutions for impulsive differential equations, such as the method of lower and upper solutions, fixed point theory, coincidence degree theory, and geometric approach (see for example [1][2][3][4][5][6][7][8]). In addition, variational methods have been also dealt with impulsive differential equations since papers [9,10] appeared. Meanwhile, many existence and multiplicity results of solutions for various impulsive differential equations have been obtained. We refer the reader to a large quantity of the references for papers [9,10]. It is worth noting that in all these references, most of impulsive effects are instantaneous. However, in [11,12], the authors considered a kind of noninstantaneous impulsive effects, which is motivated by paper [13]. On the other hand, in recent years, some new critical point theorems are used to study nonlinear differential equations with no impulsive effects (see for example, [14][15][16][17][18][19][20][21]). Based on these two facts, we investigate the existence of nontrivial solution for (1). This paper is organized as follows. In Section 2, we recall a critical point theorem in [16] and introduce a definition of the weak solution for (1), which generalizes Definition 2.3 in [12] to an N-dimensional case. In Section 3, we present our result and complete its proof via variational methods. The result is new in the field of studying noninstantaneous impulsive problem. In Section 4, we present an example in order to illustrate our result.

Preliminaries
The following critical point theorem is crucial in our arguments.
Consider the Hilbert space: with the inner product where ð·, · Þ is the inner product in R N . The corresponding norm is defined by Meanwhile, for every u, v ∈ H 1 0 , we define which is also an inner product in H 1 0 , whose corresponding norm is that Poincare's inequality [22] implies that where λ 1 = π 2 /T 2 is the first eigenvalue of the Dirichlet problem Hence, That is, k·k is equivalent to k·k H 1 0 . It is well known that Since the norms k·k H 1 0 and k· k are equivalent, there exists a positive number γ such that Following the idea of the variational approach for impulsive differential equation of [11,12] and combining with (1), one has that for every v ∈ H 1 0 , In addition, combing with (1) again, one has (12) and (13) show that Abstract and Applied Analysis On the other hand, Hence, (14) and (15), together with vðt m+1 Þ = vðTÞ = 0, imply that Similar to Definition 2.3 in [12], we can introduce the following concept of a weak solution for problem (1).

Main Result and Its Proof
From now on, we refer to M as the range of i, unless specifically stated. Let kb i k 1 be the usual norm in L 1 s i , t i+1 . The following theorem is our main result.
Theorem 3. Suppose that (F) There exist r i > 0 and μ i > 2 such that for jξj ≥ r i , hold. Then, for every b i ∈ Ls i , t i+1 \ f0g and for every (1) admits at least one nontrivial weak solution.
Proof. We complete the proof in four steps.
Step 2. I λ satisfies the (PS)-condition. That is, every fu n g such that I λ ðu n Þ is bounded and I λ ′ ðu n Þ ⟶ 0 as n ⟶ +∞ contains a convergent subsequence.
By (11), we have Let μ ≔ min fμ i : i ∈ Mg, then μ > 2. From ðFÞ and (25), we deduce that which implies that u n is bounded in H 1 0 . Since H 1 0 is a reflexive Banach space, passing to a subsequence if necessary, we may assume that there is a u 0 ∈ H 1 0 such that u n u 0 in H 1 0 . Then, fu n g converges uniformly to u 0 on ½0, T and u n ⟶ u 0 in L 2 ð0, T½ ; R N Þ. In addition, 3 Abstract and Applied Analysis Hence, the second term on the right hand of (27) converges to 0 as m, n ⟶ ∞ because of (28) and F i ∈ C 1 ðR N Þ. Moreover, I′ λ ðu n Þ ⟶ 0 as n ⟶ +∞ implies as m, n ⟶ ∞. (27)-(29) show that ku m − u n k ⟶ 0 as m, n ⟶ ∞. By the completeness of H 1 0 , we see that fu n g contains a convergent subsequence in H 1 0 .
Step 4. λ ∈ 0, sup c>0 Let c be such that and put By (18), one has Now, we apply Theorem 1 to conclude that (1) admits at least one nontrivial weak solution.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The author declares that he has no conflicts of interest.