1. Introduction
We will study the following nonlinear periodic boundary value problems of order
:
where
is an integer,
,
; the forward difference operator
is defined by
,
,
for
…
; and
is a continuous function, i.e., for any fixed
, a function
is continuous.
The fulfills , and it is the standard definition of a solution of .
Consider the following
-th order
p-Laplacian functional differential equation as a discrete analog of
.
Discrete nonlinear equations are crucial for describing a variety of physical issues, including nonlinear elasticity theory, mechanics, engineering topics, artificial or biological control systems, neural networks, and economics. For more information, see the citations provided by W. G. Kelly and P. D. Panagiotopolos [
1,
2]. According to the monographs cited by F. M. Atici et al. [
3,
4,
5,
6], some authors have studied the existence and multiplicity of solutions to some discrete
p-Laplacian problems in recent years.
It is common knowledge that critical point theory and variational approaches are useful tools for investigating the existence and variety of answers to a broad range of problems. In particular, El Amrouss and Hammouti showed the existence and multiplicity of solutions for the following problem
where
.
In [
7], Dimitrov obtained the existence of at least three solutions to the following problem:
where
,
, and
are real parameters and
f and
g are continuous.
In [
8], Saavedra and Tersian proved the existence and multiplicity of solutions for the following equation
and the boundary condition
where
is a fixed positive integer,
is a positive integer,
,
,
for
,
V is a
N- periodic positive function, and
f is continuous function about the second variable.
From the point of view of orders for equations, the earliest outstanding work comes from the research team of Yu and Zhou [
9]. In recent years, our group has also done further generalization and expansion without the
p-Laplacian; our methods have mainly included the topological degree theory and fixed point theorem, see [
10,
11].
The existence and multiplicity of nontrivial solutions to the discrete 2n-th order periodic boundary value problem with the
-Laplacian have been thoroughly examined by a large number of researchers employing a range of approaches and strategies. Refs. [
10,
11,
12,
13] is a recent work on this topic that the reader should consult. For instance, the following issue was researched by the authors in [
12],
by using the variational principle and critical point theory, some existence and multiplicty results of an anisotropic discrete nonlinear problem with variable exponents were obtained.
Inspired by the above literature, in the present paper, we will investigate the existence and multiplicity of nontrivial solutions to a discrete -th order periodic boundary value problem with -Laplacian; as far as we know, discrete cases are anaylzed less than continuous cases.
The main results of our problem involve two main theorems. Here, using a kind of variational method together with the Linking Theorem, we show that the problem admits at least two solutions. We also point out that our hypotheses here are more general under the previous conditions.
We consider the following linear eigenvalue problem:
where
and
. The following theorems are the main points of this paper:
Theorem 1. Let be a positive integer and . If , then the problem has exactly N real eigenvalues , , which satisfies Remark 1. PutWe will see later that Theorem 2. Let be a positive integer. Assume that
- ()
γ exists with such that where
for all and
- ()
.
- ()
for any such that withwhere
Then, the equation admits at least two nontrivial solutions.
Example 1. Takethe function given bywe have,we obtain , and for any such that . Then, G satisfies the conditions , , and .
The article is structured as follows. Several introductory lemmas are found in
Section 2.
Section 3 and
Section 4 provide proof of the main findings.
2. Preliminary Lemmas
In this study, we take into account the vector space
as specified in (
2).
has the inner product
and the norm
as follows:
Furthermore, we define the norm
on
by:
By the Hölder inequality, we have
where
Remark 2. It is evident that we have for any , Clearly, since
is isomorphic to a finite dimensional, it is an
N-dimensional Hilbert space. We understand that
can be extended to the vector
when
.
Lemma 1 (see [
14])
. Set be defined on . For any we havewhere the symbol is used to denote a binomial coefficient. Lemma 2. Set . For any , we have: Proof. For
, using
and
, we have
Assume that (
5) is true for
, and we aim to prove that is also true for
, i.e.,
By using this equality
and
we obtain
This implies
The evidence is conclusive. □
For
, let
be the functional denoted by
It is easy to see that
and
By Lemma 2,
can be expressed as
Finding the solution to the equation is the same as discovering the critical point of the function .
We denote is an open ball in E with radius and center 0.
4. Proof of Theorem 2
Proof of Theorem 2. From
, for
there is an
such that
For any
and
, we have
for any
.
Using Lemma 6 and (
19), we have
Take
. Therefore,
Additionally, we have established that constants and exist such that . In other words, the Linking Theorem’s condition is satisfied by .
We must validate all of the Linking Theorem’s additional assumptions before we can use it to improve critical point theory.
From
,
exists such that:
where
satisfies,
Therefore,
Moreover, by means of (
22) and the continuity of
,
exists such that
Thus, we have for any
Take
. For all
and
, let
; one has
Since
and
for any
, then a constant
exists such that
where
From (
21) and (
23), we obtain
as
. Thus,
is anti-coercive; then, for any
sequence
is bounded. It is clear that
satisfies the
condition since the dimension of
is finite.
According to the Linking Theorem [
15],
has a critical value
, where
,
Let be a critical point and .
Consequently, the nontrivial solution to the problem is .
Since is anti-coercive and bounded from above, then has a maximum point , i.e.,
The previous equality and (
20) allow us to achieve
Therefore, is nontrivial solution to the problem .
If , then we have two nontrivial solutions and .
Otherwise, suppose
; then,
, that is
Choosing , we obtain . Since the option of in Q is arbitrary, we can use .
Similar to this, there is a positive number
such that for any
,
where
Thus, possesses a critical value by the Linking Theorem. Once more, has a critical value of , where ,
If , then the case is established.
If , then . Due to the fact that and , attains its maximum at some points in the interior of Q and . However, and for any , which suggests that , in contrast to . The proof of Theorem 2 is finished. □
5. Conclusions
In our work, we used one critical paint theorem (the Linking Theorem) to obtain the new results that ensure the existence of at least two nontrivial solutions to the problem under discussion, namely, .
The discrete problem involving p-Laplacian has strong theoretical significance and application value.
Furthermore, our problem’s use of the term g makes it more difficult to look into the uniqueness and convergence of solutions. As such, we leave this subject as an open question for specialists in this domain.