1. Introduction
In recent years, equations involving the discrete
p-Laplacian operator, subjected to classical conditions, have been studied by many authors using various techniques. The variational method appears to be a very fruitful one. In this direction, we mention Refs. [
1,
2,
3,
4,
5,
6,
7,
8]. In [
3,
4,
5], using a variational approach, the authors obtained the existence of periodic solutions for systems involving a general discrete
-Laplacian operator. In addition, in recent years, boundary value problems with discrete
-Laplacian have been studied (we refer the reader to [
1,
7]). Existence results for the discrete
-Laplacian equations subjected to a general potential type boundary condition are obtained in [
1] using Szulkin’s critical point theory [
1]. By mountain pass type arguments and the Karush–Kuhn–Tucker theorem, in [
9], the existence of at least two positive solutions in the case of Dirichlet boundary conditions are established.
In all the aforementioned papers, discrete boundary value problems involving a variety of operators and boundary conditions are studied in a variational framework. The solutions are seen as critical points of a convenient energy functional, defined on a function space. In general, such function spaces have a finite dimension, which makes things easier (in comparison with the variational methods for differential equations).
There seems to be increasing interest in the existence of solutions to boundary value problems for finite difference equations with the
p-Laplacian operator. This is as a result of their applications in many fields. Recently, difference equations have attracted the interest of many researchers since they provide a natural description of several discrete models. Such discrete models are often investigated in various fields of science and technology such as computer science, economics, neural networks, ecology, cybernetics, optimal control and population dynamics. These studies cover many branches of difference equations, such as stability, attractiveness, periodicity, oscillation, and boundary value problems (see [
1,
2,
3,
4,
5,
6,
7,
8] and the references therein).
This article, using variational methods, aims at studying the existence of multiple solutions for the Lagrangian discrete boundary-value system of second-order difference equations involving the discrete
p-Laplacian operator having the following form:
where
, for all
is a homeomorphism given by the
for all
and
for all
such that
,
with
of class
strictly convex on
and given by
for all
denotes the forward difference operator defined by
and
is the discrete
p-Laplace operator
In this case,
is a fixed positive integer,
is a discrete interval
and the potential
where
is measurable with respect to t, for all
continuously differentiable in
for almost every
assuming that the functional
is a continuous function and
is a Lipschitz continuous function with the Lipschitz constant.
Inspired by the above works, in this article, we discuss the existence of multiple solutions for the second-order discrete Lagrangian boundary value system with a real parameter. The main tool used in ensuring the existence of multiple non-trivial solutions to the system in Equation (
1) is a version of Ricceri’s variational principle [
6]. We establish the existence of a precise interval
such that for every
, the system in Equation (
1) admits one nontrivial solution, which is in the space
W and is introduced below.
In detail, using the local minimum theorem (see Theorem 1) the existence of at least one nontrivial solution of Equation (
1) is proven. Under suitable conditions and Theorem 2, we get the existence of at least three solutions. To prove the main result, we introduce some suitable hypotheses. In Theorem 3, we establish the existence of at least one nontrivial weak solution for the system in Equation (
1). In Theorem 4, we prove the existence of at least three solutions of the system in Equation (
1).
3. Existence of at Least One Nontrivial Solution
In the following, by using the conditions of Theorem 1, we prove that the system in Equation (
1) has at least one nontrivial weak solution.
- (H1)
Suppose that
is a strictly monotone Lipschitz continuous function of order
p with
and Lipschitazian constant
L satisfying
, i.e.,
and
.
Theorem 3. Assume that holds and suppose that there is a positive real vector and positive constant α with and such that the following condition is satisfied.
- (H2)
where and for all
such that for eachThen, the system in Equation (1) has at least one nontrivial solution. Proof. To apply Theorem 1 to our problem, let us prove that the functionals J and satisfy the required conditions in Theorem 1. From 2, we can get that the functionals J and are Gâteaux differentiable function.
Since J and are continuous and since (every continuous real valued function on W is lower semi-continuous), they are lower semi-continuous, and since W is finite dimensional, they are weakly lower semi-continuous, thus it follows that the functional is lower semi-continues in W.
Now, we want to show that the functional
J is coercive on
W, taking into account the relations in Equations (
4) and (
5) and supposing that for any sequence
such that
thus, we get that
as
and so the functional
J is coercive.
For every
such that
. Then,
Now, we want to show that zero is not local minimum for the functional
; to this end, we claim that the mapping
is negative. From the definition of
, we observe that
Now, suppose that
such that
Moreover, since
defined as
Moreover, since
, we have
since
it follows that the functional
is negative and thus zero is not a local minimum for the functional
.
Therefore, the assertion of Theorem 1 follows and the existence of one solution to our problem is established. □
Remark 3. In Theorem 3, if the functional for is nonnegative hypothesis assumes a simpler form Moreover, if for some , the obtained solution clearly non-zero.
Example 1. Assume that , , and suppose that and Consider the following system for the case of .for every the system in Equation (11) has at least one nontrivial solution by Theorem 3. 4. Existence of Three Solutions
In this section, our goal is to obtain the existence of three distinct weak solutions for the problem in Equation (
1). The following result is obtained by applying Theorem 2. We introduce the suitable hypothesis for calculating of the critical points of the system in Equation (
1) and give some auxiliary lemmas used in the proof of the main results.
Lemma 1. The functional J is sequentially weakly lower semi-continuous.
Proof. From the continuity of H, we observe that the functional
J Gâteaux differentiable whose Gâteaux derivative of the point
is the functional
given by
for every
We can assert that J is sequentially weakly lower semi-continuous. As a matter of fact, owing to
is continuous for all
. For any
,
weakly in W. Since the inner product is sequentially weakly lower semi-continuous in Banach space, we have
□
Lemma 2. The functional is a compact operator.
Proof. We want to prove that the Gâteaux derivative of
is compact operator. Indeed, it is enough to show that
is strongly continuous on
W. Let
be a bounded sequence in
W. Since
W is reflexive and since the embedding
in
is compact, there exist a subsequent
that converge in
. Without any loss of generality, we assume that
converge in
to an element
. According to Equation (
3), the functional
belongs to
W. By Equation (
3), the following inequality holds
Using the Lebesgue dominated convergence theorem, we conclude that converge to in , thus is compact operator. □
Theorem 4. Assume that there exists two positive real vectors and two positive constants α and ζ withwhere and Suppose that L satisfies the condition inwithsuch that the following conditions are satisfied. - (H3)
where - (H4)
Suppose thatfor and - (H5)
Suppose that thefor and .
Then, for eachthe system in Equation (1) at least three nontrivial weak solutions. Proof. For each
, the functionals
are given by Equations (
6) and (
7). Now, we set the functional
for each
. To apply Theorem 2 to our problem, let us prove that the functionals
J,
satisfy the required conditions in Theorem 2.
It follow from Lemma 1 that the functional J is sequentially weakly upper semi-continuous. Using Lemma 2, we get that the functional is a compact operator.
Now, set
for each
; it is easy to check that
and
Taking
into account and from
there exists
such that
Then, there exists a positive constant
such that
It follows that for each
,
Thus, we obtain that for all . Hence, the functional I is coercive.
To prove the other conditions of Theorem 2, for each
and for any
. In fact, taking into account that
and by the definition of
r, it follows that
hence
for every
such that
. Thus,
By considering the above computations and since
and
one has
and
It is clear that
and
Since
, for
with
, by assumption
, we obtain that
Hence, the conditions of Theorem 4 are fulfilled. The proof is complete. □
Example 2. Assume that and and suppose that and Consider the following system for the case of .for every by Theorem 4 the system in Equation (16) has at least three nontrivial solutions. Remark 4. In Theorem 4, if the functional for is nonnegative, hypothesisassumes a simpler form Moreover, if for some , , the obtained solution is clearly non-zero.
Remark 5. We observe that, in our results, no asymptotic conditions on F is needed and only algebraic conditions on F are imposed to guarantee the existence of solution. Moreover, in the conclusions of the above results, one of the three solutions may be trivial since the values of for are not determined.
Remark 6. Scalar case. As an application of Theorems 3 and 4, we consider the following problem:where is a homeomorphism such that , Δ denotes the forward difference operator defined by , , , is continuous function and is a strictly monotone Lipschitz continuous function of order with Lipschitzian condition andand . , respectively, as followsand All amputations of 3 and 4 are satisfied for the scalar case.
We here present the following consequence of Theorems 3 and 4.
Theorem 5. Assume that holds and suppose that there exists two positive constant α and ϵ with and .
(
H6)
where such that for eachThen, the problem in Equation (17) has at least one nontrivial solution. Theorem 6. Assume that there exists four positive constants β, α and ζ with and Suppose that L satisfies the condition with such that the following conditions are satisfied.
- (H7)
where - (H8)
Suppose thatfor and , - (H9)
Suppose thatfor and .
Then, for eachthe problem in Equation (17) has at least three nontrivial weak solutions. Now, we have the following example to illustrate the results of Theorems 5 and 6.
Example 3. Consider the following nonlinear discrete problem By considering and By using Theorems 5 and 6, we get that the problem in Equation (18) has at least one nontrivial solution for every and for all the problem has at least three solutions.