### Abstract

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effects

where

**MSC: **
34B37, 34N05.

##### Keywords:

nonautonomous second-order systems; time scales; impulse; variational approach### 1 Introduction

Consider the nonautonomous second-order system on time scales with impulsive effects

where

(A)
*t* for every
*x* for Δ-a.e.

for all
*x*.

For the sake of convenience, in the sequel, we denote

When

In [1], the authors study the Sobolev’s spaces on time scales and their properties. As applications, they present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for (1.2).

When

Problem (1.1) covers the second-order Hamiltonian system with impulsive effects (when

as well as the second-order discrete Hamiltonian system (when

In [2], the authors establish some sufficient conditions on the existence of solutions of
(1.3) by means of some critical point theorems when

Impulsive effects exist 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 (see [3-5]). Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory and so on (see [6-9]).

For a second-order differential equation
*u* and the velocity

The study of dynamical systems on time scales is now an active area of research. One of the reasons for this is the fact that the study on time scales unifies the study of both discrete and continuous processes, besides many others. The pioneering works in this direction are Refs. [13-17]. The theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis in 1988, providing a rich theory that unifies and extends discrete and continuous analysis [18,19]. The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks and social sciences (see [16]). For example, it can model insect populations that are continuous while in season (and may follow a difference scheme with variable step-size), die out in winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population.

There have been many approaches to study solutions of differential equations on time scales, such as the method of lower and upper solutions, fixed-point theory, coincidence degree theory and so on (see [1,20-29]). In [24], authors used the fixed point theorem of strict-set-contraction to study the existence of positive periodic solutions for functional differential equations with impulse effects on time scales. However, the study of the existence and multiplicity of solutions for differential equations on time scales using the variational method has received considerably less attention (see, for example, [1,29]). The variational method is, to the best of our knowledge, novel and it may open a new approach to deal with nonlinear problems, with some type of discontinuities such as impulses.

Motivated by the above, we research the existence of variational construction for problem (1.1) in an appropriate space of functions and study the existence of solutions for (1.1) by some critical point theorems in this paper. All these results are new.

### 2 Preliminaries and statements

In this section, we present some fundamental definitions and results from the calculus
on time scales and Sobolev’s spaces on time scales that will be required below. These
are a generalization to

**Definition 2.1** ([[17], Definition 1.1])

Let

while the backward jump operator

(supplemented by
*t* is called right-dense, and if
*t* is called left-dense. Points that are right-dense and left-dense at the same time
are called dense. The set
*m*, then

When

respectively. Note that
*b* is left-dense and
*b* is left-scattered. We denote
*b* is left-dense and
*b* is left-scattered.

**Definition 2.2** ([[17], Definition 1.10])

Assume that
*U* of *t* (*i.e.*,

We call
*f* at *t*. The function *f* is delta (or Hilger) differentiable on
*f* on

**Definition 2.3** ([[1], Definition 2.3])

Assume that

and let
*f* at *t*. The function *f* is delta (or Hilger) differentiable provided
*f* on

**Definition 2.4** ([[17], Definition 2.7])

For a function

**Definition 2.5** ([[1], Definition 2.5])

For a function

The Δ-measure

**Definition 2.6** ([[1], Definition 2.7])

Assume that
*A* be a Δ-measurable subset of
*f* is integrable on *A* if and only if
*A*, and

**Definition 2.7** ([[17], Definition 2.3])

Let
*B* is called a Δ-null set if
*P* holds Δ-almost everywhere (Δ-a.e.) on *B*, or for Δ-almost all (Δ-a.a.)
*P* holds for all

For

with the norm

We have the following theorem.

**Theorem 2.1** ([[1], Theorem 2.1])

*Let*
*be such that*
*Then the space*
*is a Banach space together with the norm*
*Moreover*,
*is a Hilbert space together with the inner product given for every*
*by*

*where*
*denotes the inner product in*

**Definition 2.8** ([[1], Definition 2.11])

A function
*f* is absolutely continuous on
*i.e.*,

Now, we recall the Sobolev space

**Definition 2.9** ([[1], Definition 2.12])

Let

For

It follows from Remark 2.2 in [1] that

is true for every

**Theorem 2.2** ([[1], Theorem 2.5])

*Suppose that*
*for some*
*with*
*and that* (2.1) *holds for*
*Then there exists a unique function*
*such that the equalities*

*are satisfied and*

By identifying

**Theorem 2.3** ([[25], Theorem 2.21])

*Assume*
*and*
*The set*
*is a Banach space together with the norm defined as*

*Moreover*, *the set*
*is a Hilbert space together with the inner product*

The Banach space

**Theorem 2.4** ([[25], Theorem 2.23])

*There exists*
*such that the inequality*

*holds for all*
*where*

*Moreover*, *if*
*then*

**Theorem 2.5** ([[25], Theorem 2.25])

*If the sequence*
*converges weakly to**u**in*
*then*
*converges strongly in*
*to**u*.

**Theorem 2.6** ([[25], Theorem 2.27])

*Let*
*be* Δ-*measurable in**t**for each*
*and continuously differentiable in*
*for* Δ-*almost every*
*If there exist*
*and*
*such that for* Δ-*almost*
*and every*
*one has*

*where*
*then the functional*
*defined as*

*is continuously differentiable on*
*and*

### 3 Variational setting

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we make a variational structure. From this variational structure, we can reduce the problem of finding solutions of (1.1) to the one of seeking the critical points of a corresponding functional.

If
*u* is absolutely continuous and

Take

by

Moreover, combining

Combining (3.1), we have

Considering the above, we introduce the following concept solution for problem (1.1).

**Definition 3.1** We say that a function

holds for any

Consider the functional

where

and

**Lemma 3.1***The functional**φ**is continuously differentiable on*
*and*

*Proof* Set
*ψ* is continuously differentiable on

for all

On the other hand, by the continuity of

for all
*φ* is continuously differentiable on

By Definition 3.1 and Lemma 3.1, the weak solutions of problem (1.1) correspond to
the critical points of *φ*.

Moreover, we need more preliminaries. For any

We see that

where

*I* denote an

The compact imbedding of
*K* is compact. By classical spectral theory, we can decompose

where

**Remark 3.1***K* has only finitely many eigenvalues
*K* is compact on
*I*. By a well-known theorem, we know that 0 is not in the essential spectrum of

To prove our main results, we need the following definitions and theorems.

**Definition 3.2** ([[30],

Let *X* be a real Banach space and
*I* is said to be satisfying (PS) condition on *X* if any sequence
*X*.

Firstly, we state the local linking theorem.

Let *X* be a real Banach space with a direct decomposition

such that

and

For every multi-index

**Definition 3.3** ([[31], Definition 2.2])

Let
*I* satisfies the

contains a subsequence which converges to a critical point of *I*.

**Theorem 3.1** [[31], Theorem 2.2]

*Suppose that*
*satisfies the following assumptions*:

(I_{1})
*and**I**has a local linking at* 0 *with respect to*
*that is*, *for some*

(I_{2}) *I**satisfies*
*condition*.

(I_{3}) *I**maps bounded sets into bounded sets*.

(I_{4}) *For every*
*as*

*Then**I**has at least two critical points*.

**Remark 3.2** Since
_{1}) of Theorem 3.1, 0 is the critical point of *I*. Thus, under the conditions of Theorem 3.1, *I* has at least one nontrivial critical point.

Secondly, we state another three critical point theorems.

**Theorem 3.2** ([[32], Theorem 5.29])

*Let**E**be a Hilbert space with*
*and*
*Suppose*
*satisfies* (PS) *condition*, *and*

(I_{5})
*where*
*and*
*is bounded and self*-*adjoint*,

(I_{6})
*is compact*, *and*

(I_{7}) *there exist a subspace*
*and sets*
*and constants*
*such that*

(i)
*and*

(ii) *Q**is bounded and*

(iii) *S**and**∂Q**link*.

*Then**I**possesses a critical value*

**Theorem 3.3** ([[32], Theorem 9.12])

*Let**E**be a Banach space*. *Let*
*be an even functional which satisfies the* (PS) *condition and*
*If*
*where**V**is finite dimensional*, *and**I**satisfies*

(I_{8}) *there are constants*
*such that*
*where*

(I_{9}) *for each finite dimensional subspace*
*there is an*
*such that*
*on*

*then**I**possesses an unbounded sequence of critical values*.

In order to state another critical point theorem, we need the following notions. Let
*X* and *Y* be Banach spaces with *X* being separable and reflexive, and set
*E* defined by

We denote by
*E* induced by a semi-norm family
*w* and

For a functional
*E*, one has
*i.e.*,

Suppose that

(

(

(

**Theorem 3.4** ([33])

*Assume that* Φ *is even and* (
*are satisfied*. *Then* Φ *has at least*
*pairs of critical points with critical values less than or equal to*
*provided* Φ *satisfies the*
*condition for all*

**Remark 3.3** In our applications, we take
*X* and the strong topology on *Y*.

### 4 Main results

**Lemma 4.1**
*is compact on*

*Proof* Let

The continuity of

First of all, we give two existence results.

**Theorem 4.1***Suppose that* (A) *and the following conditions are satisfied*.

(F_{1})
*uniformly for* Δ-*a*.*e*.

(F_{2})
*uniformly for* Δ-*a*.*e*.

(F_{3}) *there exist*
*and*
*such that*

*and*

(F_{4}) *there exists*
*such that*

(F_{5}) *there exist*
*and*
*such that*

(F_{6})
*for every*

(F_{7}) *there exists*
*such that*

*and*

*Then problem* (1.1) *has at least two weak solutions*. *The one is a nontrivial weak solution*, *the other is a trivial weak solution*.

*Proof* By Lemma 3.1,

Then we have

and

We divide our proof into four parts in order to show Theorem 4.1.

Firstly, we show that *φ* satisfies the

Let

then there exists a constant

for all large *n*. On the other hand, by (F_{3}), there are constants

for all

for all

for all

From (F_{5}) and (2.5), we have that

for all

for all large *n*, where
_{3}), there exist

for all

for all

for all
_{7}), there exists

Thus, by (4.1), (4.10) and (4.11), we obtain

for all large *n*. From (4.12),

Since

Since

This implies
*φ* satisfies the

Secondly, we show that *φ* maps bounded sets into bounded sets.

It follows from (3.2), (4.4), (4.5) and (4.6) that

for all
*φ* maps bounded sets into bounded sets.

Thirdly, we claim that *φ* has a local linking at 0 with respect to

Applying (F_{2}), for

for all
_{7}), for

Let

This implies that

On the other hand, it follows from (F_{6}) that

for all
_{4}), (2.5), (3.2), (3.5) and (4.17), we obtain

This implies that

Let
*φ* satisfies the condition

Finally, we claim that for every

For given

By (F_{1}), there exists

for all

for all

for all

where

Thus, by Theorem 3.1, problem (1.1) has at least one nontrivial weak solution. The proof is complete. □

**Example 4.1** Let