1 Introduction

In this paper, we deal with the following impulsive differential system of the form

$$ \left \{ \textstyle\begin{array}{l} -u''(t)+g(t)u(t)=f_{u}(u,v), \quad \mbox{a.e. } t\in[0,T], \\ -v''(t)+h(t)v(t)=f_{v}(u,v),\quad \mbox{a.e. } t\in[0,T], \\ u(0)=u(T)=v(0)=v(T)=0, \\ \Delta u'(t_{k})=u'(t_{k}^{+})-u'(t_{k}^{-})=I_{k}(u(t_{k})), \\ \Delta v'(t_{k})=v'(t_{k}^{+})-v'(t_{k}^{-})=J_{k}(v(t_{k})),\quad k=1,2,\ldots,m, \end{array}\displaystyle \right . $$
(1.1)

where \(t_{0}=0< t_{1}< t_{2}<\cdots<t_{m}<t_{m+1}=T\), \(g,h\in L^{\infty}[0,T]\), \(f_{u},f_{v}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}\) are continuous, and \(I_{k},J_{k}:{\mathbb{R}}\rightarrow{\mathbb{R}}\), \(k=1,2,\ldots,m\), are continuous.

We point out that many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. Based on the significance, a lot of developments have been made in the theory and applications of impulsive differential systems by numerous mathematicians. We refer the reader to the classical monograph (see [1, 2]), the general works on the theory (see [3–10]) and applications of impulsive differential equations which occur in biology, control theory, optimization theory, population dynamics, medicine, mechanics, engineering and chaos theory, etc. (see [11–27]). These classical techniques contain fixed point theory, topological degree theory and comparison method (including monotone iterative method and upper and lower solutions methods).

For a second order differential equation \(u''=f(t,u,u')\), one usually considers, as impulsive, the position u and the velocity \(u'\). However, in the motion of spacecraft one has to deal with instantaneous impulses depending on the position that results in jump discontinuities in velocity, but no change in position (see [12, 28–30]). The impulses only on the velocity occur also in impulsive mechanics.

Many problems can be solved in terms of the minimization of a functional, usually related to the energy, in an appropriate space of functions. The purpose of this work is to investigate the variational structure under the impulsive differential system (1.1). Based on variational method, we introduce a different concept of solution, that is, a weak solution to problem (1.1). The critical points of the corresponding functional are indeed weak solutions of the impulsive problem (1.1). For the impulsive Dirichlet boundary value problems, the known results obtained by variational approach and critical point theory are as follows.

In [31], to the best of our knowledge, Tian and Ge firstly study the impulsive differential problem by variational method. They deal with the following problem:

$$ \left \{ \textstyle\begin{array}{l} (\rho(t)\phi_{p}(u'(t)))'+s(t)\phi_{p}(u(t))=f(t,u(t)), \quad \mbox{a.e. } t\in [a,b], \\ \Delta(\rho(t_{j})\phi_{p}(u'(t_{j})))=I_{j}(u(t_{j})),\quad j=1,2,\ldots,p, \\ \alpha u'(a)+\beta u(a)=A,\qquad \gamma u(b)+\sigma u'(b)=B, \end{array}\displaystyle \right . $$
(1.2)

and essentially prove that when f and \(I_{j}\) satisfy some conditions, problem (1.2) has at least two positive solutions via variational method.

Nieto and O’Regan [32] consider the impulsive linear problem

$$ \left \{ \textstyle\begin{array}{l} -u''(t)+\lambda u(t)=\sigma(t),\quad \mbox{a.e. } t\in[0,T], \\ u(0)=u(T)=0, \\ \Delta u'(t_{j})=u'(t_{j}^{+})-u'(t_{j}^{-})=d_{j},\quad j=1,2,\ldots,p, \end{array}\displaystyle \right . $$
(1.3)

and the impulsive nonlinear problem

$$ \left \{ \textstyle\begin{array}{l} -u''(t)+\lambda u(t)=f(t,u(t)), \quad \mbox{a.e. } t\in[0,T], \\ u(0)=u(T)=0, \\ \Delta u'(t_{j})=I_{j}(u(t_{j})),\quad j=1,2,\ldots,p, \end{array}\displaystyle \right . $$
(1.4)

where \(d_{j}\) are constants, \(I_{j}:{\mathbb{R}}\rightarrow{\mathbb{R}}\), \(j=1,2,\ldots,p\), are continuous, \(f:[0,T]\times{\mathbb{R}}\rightarrow{\mathbb{R}}\) is continuous. They exhibit the variational formulations for problems (1.3) and (1.4) and establish the existence and multiplicity of solutions using standard results of critical point theory. For more results, we refer the reader to [33–35].

In this paper we consider the impulsive nonlinear coupled differential system (1.1) motivated by the results [32–35]. Our main result extends the studies made in [32–35] in the sense that we are concerned with a class of problems that is not considered in the papers.

Throughout the paper, we need the following conditions.

(H1):

Assume that \(\alpha>-\lambda_{1}\), where \(\alpha=\min\{ \operatorname{ess}\inf_{t\in[0,T]}g(t), \operatorname{ess}\inf_{t\in[0,T]}h(t)\}\) and \(\lambda_{1}=\frac{\pi^{2}}{T^{2}}\) is the first eigenvalue of the problem

$$ \left \{ \textstyle\begin{array}{l} -u''(t)=\lambda u(t), \quad t\in[0,T], \\ u(0)=u(T)=0. \end{array}\displaystyle \right . $$
(H2):

There exist \(a,b>0\) and \(\gamma_{1},\gamma_{2} \in[0,1)\) such that

$$\bigl\vert f_{x}(x,y)\bigr\vert \leq a+b|x|^{\gamma_{1}}\quad \mbox{for every } (x,y)\in{\mathbb{R}}^{2} $$

and

$$\bigl\vert f_{y}(x,y)\bigr\vert \leq a+b|y|^{\gamma_{2}}\quad \mbox{for every } (x,y)\in{\mathbb{R}}^{2}. $$
(H3):

There exist \(a_{k},b_{k}>0\) and \(\beta_{k}\in[0,1)\) (\(k=1,2,\ldots,m\)) such that

$$\bigl\vert I_{k}(u)\bigr\vert \leq a_{k}+b_{k}|u|^{\beta_{k}} \quad \mbox{for every } u\in{\mathbb{R}} $$

and

$$\bigl\vert J_{k}(v)\bigr\vert \leq a_{k}+b_{k}|v|^{\beta_{k}} \quad \mbox{for every } v\in{\mathbb{R}}. $$

The main result of this paper is the following.

Theorem 1.1

Let assumptions (H1)-(H3) be satisfied. Then problem (1.1) has at least one nontrivial solution.

Obviously, Theorem 3.2 in [35] is a special case of Theorem 1.1 in this paper.

This paper is organized as follows. In Section 2, we introduce a Hilbert space \(X=H_{0}^{1}(0,T)\times H_{0}^{1}(0,T)\), on which the corresponding functional Φ of problem (1.1) is defined. Furthermore, we give some necessary notations and preliminaries. In Section 3, we prove the main result via variational approach.

2 Variational structure

Let \(L^{p}[0,T]\) be the space formed by functions which are p-times integrable on \([0,T]\) under the norm

$$\|u\|_{L^{p}}= \biggl(\int_{0}^{T}\bigl\vert u(t)\bigr\vert ^{p}\, dt \biggr)^{\frac{1}{p}} $$

and \(C[0,T]\) be the space of all continuous functions on \([0,T]\) with the norm

$$\|u\|_{\infty}=\max_{t\in[0,T]}\bigl\vert u(t)\bigr\vert . $$

In the Sobolev space \(H_{0}^{1}(0,T)\), we consider the inner conduct

$$(u,v)_{1}=\int_{0}^{T}u'(t)v'(t) \, dt $$

and

$$(u,v)_{2}=\int_{0}^{T}u(t)v(t)\, dt+ \int_{0}^{T}u'(t)v'(t)\, dt, $$

which induce the corresponding norms

$$\|u\|_{1}= \biggl(\int_{0}^{T}\bigl\vert u'(t)\bigr\vert ^{2}\, dt \biggr)^{\frac{1}{2}} $$

and

$$\|u\|_{2}= \biggl(\int_{0}^{T}\bigl\vert u(t)\bigr\vert ^{2}\, dt+\int_{0}^{T} \bigl\vert u'(t)\bigr\vert ^{2}\, dt \biggr)^{\frac{1}{2}}. $$

By Poincaré’s inequality,

$$\lambda_{1}\int_{0}^{T}u^{2}(t) \, dt\leq\int_{0}^{T}\bigl\vert u'(t)\bigr\vert ^{2}\, dt \quad \mbox{for any } u\in H_{0}^{1}(0,T), $$

we easily obtain that the norms \(\|\cdot\|_{1}\) and \(\|\cdot\|_{2}\) are equivalent. Set \(X=H_{0}^{1}(0,T)\times H_{0}^{1}(0,T)\). In the Hilbert space X, for any \((u,v)\in X\), we set the norm

$$\bigl\Vert (u,v)\bigr\Vert ^{2}=\|u\|_{1}^{2}+ \|v\|_{1}^{2}. $$

By (H1), we also introduce the norm

$$\bigl\Vert (u,v)\bigr\Vert _{X}= \biggl(\int_{0}^{T} \bigl(\bigl\vert u'(t)\bigr\vert ^{2}+g(t)u^{2}(t) \bigr)\, dt+\int_{0}^{T}\bigl(\bigl\vert v'(t)\bigr\vert ^{2}+h(t)v^{2}(t)\bigr)\, dt \biggr)^{\frac{1}{2}}. $$

We have the following results.

Lemma 2.1

Assume that assumption (H1) holds, then, for the Sobolev space X, the norm \(\|\cdot\|\) and the norm \(\|\cdot\|_{X}\) are equivalent.

Proof

Since \(\alpha>-\lambda_{1}\), there exists \(c_{1}\in(0,1)\) such that \(-\alpha\leq\lambda_{1}(1-c_{1})\). Using Poincaré’s inequality, we have

$$\begin{aligned} (1-c_{1})\int_{0}^{T}\bigl\vert u'(t)\bigr\vert ^{2}\,dt \geq& (1-c_{1}) \lambda_{1}\int_{0}^{T}\bigl\vert u(t) \bigr\vert ^{2}\,dt \\ \geq& -\alpha\int_{0}^{T}\bigl\vert u(t)\bigr\vert ^{2}\,dt \end{aligned}$$

for any \(u\in H_{0}^{1}(0,T)\). Thus, we get

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{X}^{2} =& \int _{0}^{T}\bigl(\bigl\vert u'(t)\bigr\vert ^{2}+g(t)u^{2}(t)\bigr)\,dt+\int_{0}^{T} \bigl(\bigl|v'(t)\bigr|^{2}+h(t)v^{2}(t)\bigr)\,dt \\ \geq& c_{1}\bigl(\Vert u\Vert _{1}^{2}+ \Vert v\Vert _{1}^{2}\bigr)=c_{1}\bigl\Vert (u,v)\bigr\Vert ^{2}. \end{aligned}$$

Moreover, one has

$$\begin{aligned} \bigl\Vert (u,v)\bigr\Vert _{X}^{2} =& \int _{0}^{T}\bigl(\bigl\vert u'(t)\bigr\vert ^{2}+g(t)u^{2}(t)\bigr)\,dt+\int_{0}^{T} \bigl(\bigl\vert v'(t)\bigr\vert ^{2}+h(t)v^{2}(t) \bigr)\,dt \\ \leq& \Vert g\Vert _{\infty}\int_{0}^{T}u^{2}(t)\,dt+ \Vert h\Vert _{\infty}\int_{0}^{T}v^{2}(t)\,dt+ \int_{0}^{T}\bigl(\bigl\vert u'(t) \bigr\vert ^{2}+\bigl\vert v'(t)\bigr\vert ^{2}\bigr)\,dt \\ \leq& \biggl(\max\biggl\{ \frac{\Vert g\Vert _{\infty}}{\lambda_{1}},\frac{\Vert h\Vert _{\infty}}{\lambda _{1}}\biggr\} +1 \biggr) \bigl\Vert (u,v)^{2}\bigr\Vert . \end{aligned}$$

Thereby, the norm \(\|\cdot\|\) and the norm \(\|\cdot\|_{X}\) are equivalent. □

Lemma 2.2

For any \((u,v)\in X\), there exists \(c_{2}>0\) such that \(\|u\|_{\infty},\|v\|_{\infty}\leq c_{2} \|(u,v)\|_{X}\).

Proof

For any \((u,v)\in X\), by the mean value theorem, there exists a constant \(\tau\in[0,T]\) such that

$$u(\tau)=\frac{1}{T}\int_{0}^{T}u(s)\, ds. $$

Furthermore, using Hölder’s inequality and Poincaré’s inequality, we have

$$\begin{aligned} \bigl|u(t)\bigr| =& \biggl\vert u(\tau)+\int_{\tau}^{t}u'(s) \, ds\biggr\vert \\ \leq& \frac{1}{T}\int_{0}^{T}\bigl\vert u(s)\bigr\vert \, ds+\int_{0}^{T}\bigl\vert u'(s)\bigr\vert \, ds \\ \leq& T^{-\frac{1}{2}}\|u\|_{L^{2}}+T^{\frac{1}{2}}\bigl\Vert u'\bigr\Vert _{L^{2}} \\ \leq& \bigl((\lambda_{1}T)^{-\frac{1}{2}}+T^{\frac{1}{2}}\bigr)\bigl\Vert u'\bigr\Vert _{L^{2}} \\ \leq& \bigl((\lambda_{1}T)^{-\frac{1}{2}}+T^{\frac{1}{2}}\bigr)\bigl\Vert (u,v)\bigr\Vert . \end{aligned}$$
(2.1)

Combining Lemma 2.1 and (2.1), there exists \(c_{2}>0\) such that

$$\|u\|_{\infty}\leq c_{2} \bigl\Vert (u,v)\bigr\Vert _{X}. $$

Similarly, we can get

$$\|v\|_{\infty}\leq c_{2} \bigl\Vert (u,v)\bigr\Vert _{X}. $$

 □

In the following, we are concerned with problem (1.1) subject to impulses in the derivative at the prescribed instants \(t_{k}\), \(k=1,2,\ldots,m\). We are interested in the solution \((u,v)\) of problem (1.1) satisfying the impulse conditions

$$ \Delta u'(t_{k})=u' \bigl(t_{k}^{+}\bigr)-u'\bigl(t_{k}^{-} \bigr)=I_{k}\bigl(u(t_{k})\bigr) $$
(2.2)

and

$$ \Delta v'(t_{k})=v' \bigl(t_{k}^{+}\bigr)-v'\bigl(t_{k}^{-} \bigr)=J_{k}\bigl(v(t_{k})\bigr),\quad k=1,2,\ldots,m. $$
(2.3)

For \(u,v \in H^{2}(0,T)\), we have that u, v, \(u'\) and \(v'\) are both absolutely continuous. Meanwhile, \(u'',v''\in L^{2}(0,T)\). Hence, \(u'(t^{+})=u'(t^{-})\) and \(v'(t^{+})=v'(t^{-})\) for any \(t\in[0,T]\). If \(u,v\in H_{0}^{1}(0,T)\), then u, v are absolutely continuous and \(u',v'\in L^{2}(0,T)\). In this case, the one-sided derivatives \(u'(t^{+})\), \(u'(t^{-})\), \(v'(t^{+})\) and \(v'(t^{-})\) may not exist. Thus, we need to introduce a concept of solution which is different from a classical solution. We say that \((u,v)\) is a classical solution of problem (1.1) if it satisfies the corresponding equations a.e. on \([0,T]\), the limits \(u'(t_{k}^{+})\), \(u'(t_{k}^{-})\), \(v'(t_{k}^{+})\) and \(v'(t_{k}^{-})\), \(k=1,2,\ldots,m\), exist and (2.2), (2.3) hold.

Taking \((\varphi,\psi)\in X\) and multiplying the two sides of the equalities

$$-u''(t)+g(t)u(t)=f_{u}(u,v) $$

and

$$-v''(t)+h(t)v(t)=f_{v}(u,v) $$

by φ and ψ respectively, then integrating from 0 to T, we have

$$ -\int_{0}^{T} u''(t) \varphi \, dt +\int_{0}^{T} g(t)u(t)\varphi(t) \, dt=\int_{0}^{T} f_{u}\bigl(u(t),v(t) \bigr)\varphi(t) \, dt $$
(2.4)

and

$$ -\int_{0}^{T} v''(t) \psi \,dt +\int_{0}^{T} h(t)v(t)\psi(t) \,dt=\int _{0}^{T} f_{v}\bigl(u(t),v(t)\bigr) \psi(t) \,dt. $$
(2.5)

The first terms of (2.4) and (2.5) are now

$$\begin{aligned} -\int_{0}^{T} u''(t) \varphi \,dt&= -\sum_{k=0}^{m}\int _{t_{k}}^{t_{k+1}}u''(t)\varphi(t) \,dt \\ &= \sum_{k=1}^{m}I_{k} \bigl(u(t_{k})\bigr)\varphi(t_{k})+\int_{0}^{T}u'(t) \varphi'(t)\,dt \end{aligned}$$
(2.6)

and

$$\begin{aligned} -\int_{0}^{T} v''(t) \psi(t) \,dt&= -\sum_{k=0}^{m}\int _{t_{k}}^{t_{k+1}}v''(t)\psi(t) \,dt \\ &= \sum_{k=1}^{m}J_{k} \bigl(v(t_{k})\bigr)\psi(t_{k})+\int_{0}^{T}v'(t) \psi'(t)\,dt. \end{aligned}$$
(2.7)

In connection with (2.4), (2.5), (2.6) and (2.7), we have

$$\begin{aligned}& \int_{0}^{T} u'(t) \varphi'(t) \,dt+ \int_{0}^{T}v'(t) \psi'(t)\,dt+\int_{0}^{T}g(t)u(t) \varphi(t)\,dt \\& \qquad {}+ \int_{0}^{T}h(t)v(t)\psi(t)\,dt+\sum _{k=1}^{m}I_{k} \bigl(u(t_{k})\bigr)\varphi(t_{k})+\sum _{k=1}^{m}J_{k}\bigl(v(t_{k}) \bigr)\psi(t_{k}) \\& \quad = \int_{0}^{T}f_{u}(u,v) \varphi(t)\,dt+\int_{0}^{T}f_{v}(u,v) \psi(t)\,dt. \end{aligned}$$
(2.8)

Based on equality (2.8), we introduce the concept of weak solution for problem (1.1). We say that a pair of functions \((u,v) \in X\) is a weak solution for problem (1.1) if identity (2.8) holds for any \((\varphi,\psi)\in X\). The corresponding energy functional Φ to problem (1.1) is defined by

$$\begin{aligned} \begin{aligned}[b] \Phi(u,v)={}& \frac{1}{2}\int_{0}^{T} \bigl(\bigl\vert u'(t)\bigr\vert ^{2}+g(t)u^{2}(t) \bigr)\,dt+\frac{1}{2}\int_{0}^{T}\bigl(\bigl\vert v'(t)\bigr\vert ^{2}+h(t)v^{2}(t)\bigr) \,dt \\ &{}+ \sum_{k=1}^{m}\int _{0}^{u(t_{k})}I_{k}(t)\,dt+\sum _{k=1}^{m}\int_{0}^{v(t_{k})}J_{k}(t) \,dt-\int_{0}^{T}f(u,v)\,dt \\ ={}& \frac{1}{2}\bigl\Vert (u,v)\bigr\Vert _{X}^{2}+ \sum_{k=1}^{m}\int_{0}^{u(t_{k})}I_{k}(t) \,dt+\sum_{k=1}^{m}\int_{0}^{v(t_{k})}J_{k}(t) \,dt-\int_{0}^{T}f(u,v)\,dt. \end{aligned} \end{aligned}$$
(2.9)

Combining the continuity of \(f_{u}\), \(f_{v}\), \(I_{k}\) and \(J_{k}\), \(k=1,2,\ldots,m\), by standard arguments [30], we can show that the functional \(\Phi \in C^{1}(X,{\mathbb{R}})\). Furthermore, we have

$$\begin{aligned} \Phi'(u,v) (\varphi,\psi) =& \int _{0}^{T}u'(t)\varphi'(t) \,dt+\int_{0}^{T}g(t)u(t)\varphi(t)\,dt \\ &{}+ \int_{0}^{T}v'(t) \psi'(t)\,dt+\int_{0}^{T}h(t)v(t) \psi(t)\,dt+\sum_{k=1}^{m}I_{k} \bigl(u(t_{k})\bigr)\psi(t_{k}) \\ &{}+ \sum_{k=1}^{m}J_{k} \bigl(v(t_{k})\bigr)\psi(t_{k})-\int_{0}^{T}f_{u}(u,v) \varphi(t)\,dt-\int_{0}^{T}f_{v}(u,v) \psi(t)\,dt. \end{aligned}$$
(2.10)

Indeed, we reduce the problem of finding weak solutions of (1.1) to the one of seeking the critical points of the corresponding functional Φ. To this end, we recall some known results from variational method. We say that a minimizing sequence for a functional \(F:X\rightarrow{\mathbb{R}}\) is a sequence \(\{(u_{i},v_{i})\}\) such that

$$F(u_{i},v_{i})\rightarrow\inf F \quad \mbox{whenever } i \rightarrow \infty. $$

Lemma 2.3

[36]

Let X be a reflexive Banach space and \(F:X\rightarrow{\mathbb{R}}\) be continuously Fréchet-differentiable. If F is weakly lower semi-continuous and has a bounded minimizing sequence, then F has a minimum on X.

3 Main result

Lemma 3.1

Assume that conditions (H1)-(H3) are satisfied. Then the functional Φ defined by (2.9) is continuously Fréchet-differentiable and weakly lower semi-continuous.

Proof

First, using the continuity of \(f_{u}\), \(f_{v}\), \(I_{k}\) and \(J_{k}\), \(k=1,2,\ldots,m\), we easily obtain the continuity and differentiability of Φ and \(\Phi':X=H_{0}^{1}(0,T)\times H_{0}^{1}(0,T)\rightarrow{\mathbb{R}}\) defined by (2.10).

In the following, we prove that Φ is weakly lower semi-continuous. If \(\{(u_{i},v_{i})\}\subset X\) with \((u_{i},v_{i})\rightharpoonup(u,v)\), then, by Lemma 2.2, we get that \(\{u_{i}\}\) and \(\{v_{i}\}\) converge uniformly to u and v on \([0,T]\) respectively. In connection with the fact that \(\liminf_{i\rightarrow\infty}\|(u_{i},v_{i})\|_{X} \geq\|(u,v)\|_{X}\), one has

$$\begin{aligned} \liminf_{i\rightarrow\infty}\Phi(u_{i},v_{i}) = & \liminf_{i\rightarrow \infty}\Biggl\{ \frac{1}{2}\bigl\Vert (u_{i},v_{i})\bigr\Vert _{X}^{2}+ \sum_{k=1}^{m}\int_{0}^{u_{i}(t_{k})}I_{k}(t) \,dt \\ &{}+ \sum_{k=1}^{m}\int _{0}^{v_{i}(t_{k})}J_{k}(t)\,dt-\int _{0}^{T}f(u_{i},v_{i})\,dt \Biggr\} \\ \geq& \frac{1}{2}\bigl\Vert (u,v)\bigr\Vert _{X}^{2}+ \sum_{k=1}^{m}\int_{0}^{u(t_{k})}I_{k}(t) \,dt \\ &{}+\sum_{k=1}^{m}\int_{0}^{v(t_{k})}J_{k}(t) \,dt-\int_{0}^{T}f(u,v)\,dt \\ =& \Phi(u,v). \end{aligned}$$

This implies that the functional Φ is weakly lower semi-continuous. □

Proof of Theorem 1.1

For any \((u,v)\in X\), using assumptions (H2), (H3) and Lemma 2.2, we have

$$\begin{aligned} \Phi(u,v) =& \frac{1}{2}\bigl\Vert (u,v)\bigr\Vert _{X}^{2}+\sum_{k=1}^{m} \int_{0}^{u(t_{k})}I_{k}(t)\,dt+\sum _{k=1}^{m}\int_{0}^{v(t_{k})}J_{k}(t) \,dt-\int_{0}^{T}f(u,v)\,dt \\ \geq& \frac{1}{2}\bigl\Vert (u,v)\bigr\Vert _{X}^{2}- \sum_{k=1}^{m}\int_{0}^{u(t_{k})} \bigl(a_{k}+b_{k}|t|^{\beta_{k}}\bigr)\,dt-\sum _{k=1}^{m}\int_{0}^{v(t_{k})} \bigl(a_{k}+b_{k}|t|^{\beta_{k}}\bigr)\,dt \\ &{}- \int_{0}^{T} \bigl(a|u|+a|v|+b|u|^{\gamma_{1}+1}+b|v|^{\gamma_{2}+1} \bigr)\,dt \\ \geq& \frac{1}{2}\bigl\Vert (u,v)\bigr\Vert _{X}^{2}-m A \Vert u\Vert _{\infty}-B\sum_{k=1}^{m} \Vert u\Vert _{\infty}^{\beta_{k}+1}-m A \Vert v\Vert _{\infty}\\ &{}- B \sum_{k=1}^{m}\Vert v\Vert _{\infty}^{\beta_{k}+1}-aT\bigl(\Vert u\Vert _{\infty}+\Vert v \Vert _{\infty}\bigr)-bT\bigl(\Vert u\Vert _{\infty}^{\gamma_{1}+1}+ \Vert v\Vert _{\infty}^{\gamma_{2}+1}\bigr) \\ \geq& \frac{1}{2}\bigl\Vert (u,v)\bigr\Vert _{X}^{2}-2m A c_{2} \bigl\Vert (u,v)\bigr\Vert _{X}-2B\sum _{k=1}^{m}c_{2}^{\beta_{k}+1} \bigl\Vert (u,v)\bigr\Vert _{X}^{\beta_{k}+1} \\ &{}- 2aT\bigl\Vert (u,v)\bigr\Vert _{X}-bTc_{2}^{\gamma_{1}+1} \bigl\Vert (u,v)\bigr\Vert _{X}^{\gamma_{1}+1}-bTc_{2}^{\gamma _{2}+1} \bigl\Vert (u,v)\bigr\Vert _{X}^{\gamma_{2}+1}, \end{aligned}$$

where \(A=\max\{a_{1},a_{2},\ldots,a_{k}\}\), \(B=\max\{b_{1},b_{2},\ldots,b_{k}\}\).

In connection with \(\gamma_{1},\gamma_{2},\beta_{k}\in [0,1)\), \(k=1,2,\ldots,m\), it follows that the functional Φ is coercive on X. Furthermore, by Lemma 3.1 and Lemma 2.3, we have that Φ has a minimum point on X. Hence, problem (1.1) has at least one nontrivial solution. □

Corollary 3.1

Assume that \(f_{u}\), \(f_{v}\), \(I_{k}\) and \(J_{k}\), \(k=1,2,\ldots,m\), are bounded. Then problem (1.1) has at least one solution.

4 Example

Let \(T=\pi\), \(t_{1}=1\). We consider the following problem with impulses:

$$ \left \{ \textstyle\begin{array}{l} -u''(t)+(1+t)u(t)=t^{2}+\sqrt{u(t)}, \\ -v''(t)+(t+t^{2})v(t)=t+\sqrt[3]{v(t)}, \\ u(0)=u(\pi)=v(0)=v(\pi)=0, \\ \Delta u'(t_{1})=u'(t_{1}^{+})-u'(t_{1}^{-})=2+\sqrt[3]{u(t_{1})}, \\ \Delta v'(t_{1})=v'(t_{1}^{+})-v'(t_{1}^{-})=t_{1}+\sqrt[4]{v(t_{1})}. \end{array}\displaystyle \right . $$
(4.1)

First we can see that \(g(t)=1+t\), \(h(t)=t+t^{2}\), and \(\alpha=0 >-\frac {\pi^{2}}{T^{2}}\), then (H1) holds. Next, taking \(a=\pi^{2}\), \(b=1\), \(\gamma _{1}=\frac{1}{2}\), and \(\gamma_{2}=\frac{1}{3}\), (H2) holds. Finally, taking \(a_{1}=2\), \(b=1\), and \(\beta_{1}=\frac{1}{3}\), (H3) holds. Then, by Theorem 1.1, the impulsive problem (4.1) has at least one nontrivial solution.