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This paper discusses the existence of infinitely many periodic solutions for a semilinear fourth-order impulsive differential inclusion with a perturbed nonlinearity and two parameters. The approach is based on a critical point theorem for nonsmooth functionals.

The goal of this paper is to establish the existence of infinitely many periodic solutions for the following perturbed semilinear fourth-order impulsive differential inclusion:

there exist two constants

there exists a constant

where

We study the existence of solutions, that is, absolutely continuous on every

Fourth-order ordinary differential equations act as models for the bending or deforming of elastic beams, and, therefore, they have important applications in engineering and physical sciences. Boundary value problems for fourth-order ordinary differential equations have been of great concern in recent years (e.g., see [

Recently, multiplicity of solutions for differential inclusions via nonsmooth variational methods and critical point theory has been considered and here we cite the papers [

In the present paper, motivated by [

To the best of our knowledge, no investigation has been devoted to establishing the existence of infinitely many solutions to a problem such as (

A special case of our main result is the following theorem.

Assume that

Let

Let

Let

Let

Let

Let

We say that

Henceforth, we assume that

Under the above assumption on

For every

If

there is a sequence

If

there is a global minimum of

there is a sequence

Now we recall some basic definitions and notations. We consider the reflexive Banach space

A function

A solution

If

Let

we have

Now we introduce the functionals

Assume that

Let

Now we show that each critical point of

First for every

Now we formulate our main result using the following assumptions:

Assume that

putting

Our goal is to apply Theorem

Under the conditions

The following result is a special case of Theorem

Assume that

Now, we present the following example to illustrate Theorem

Let

Now we state the following consequence of Theorem

Assume that

Theorem

Here, we give a consequence of the main result.

Let

Then, for every function

Set

We observe that in Theorem

We end this paper by presenting the following example.

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.