In this paper, a research has been done about the existence of solutions to the Dirichlet boundary value problem for

Fractional calculus involves arbitrary order derivatives and integration, so it plays a very important role in various fields such as physical engineering, medical image processing, mathematics, chemical engineering, and electricity. For this reason, many scholars did research on the theory of fractional differential equations continuously and have made enormous achievements; readers who are interested in these kinds of researches can refer to relevant literature (see [

For a long time, many scholars have conducted in-depth research on instantaneous impulsive differential equations. By using fixed-point theorems, critical point theorems, and variational methods, they obtained the existence of solutions (see [

In [

In [

Motivated by the above-mentioned work, the paper focuses on the existence of solutions for the following

The rest of the paper is organized as follows. In Section

In this section, we introduce some important definitions, lemmas, and theorems that are important to use later. For the definitions of the left and right fractional integrals and derivatives, we can refer to References [

Let

Let

Let

Let

Let

Moreover, if

Let

Let

A function

Let

If

For

In this section, we further discuss the variational structure of problem (

For each

There exist constants

There exist constants

A function

For

We have

As for problem (

Above all, problem (

A function

Define the function

From (

Under condition (

From (

According to the mean value theorem and condition (

By the Lebesgue dominated convergence theorem, it is easy to show that

Thus,

Therefore,

This yields that the critical points of

The functional

Let sequence

From Proposition

From the definition of classical solutions, without loss of generality, if

Since

Because of

Therefore,

Without loss of generality, we take the test function

By using (

Hence,

Then, combining with (

Therefore,

Suppose that (

From Proposition

Since

Therefore, problem (

Considering the following

In this paper, we use the variational method to discuss the existence of solutions for Dirichlet boundary value problems with instantaneous and noninstantaneous fractional impulsive differential equations, and the existence results of classical solution for our equations are shown. In addition to extending the linear differential operator to a nonlinear differential operator, it is more important to use the

No data were used to support this study.

The authors declare that they have no competing interest.

All authors read and approved the final manuscript.

This work is supported by the Natural Science Foundation of Xinjiang (2019D01A71), Scientific Research Programs of Colleges in Xinjiang (XJEDU2018Y033), National Natural Science Foundation of China (11961069), Outstanding Young Science and technology personnel Training program of Xinjiang.