Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 115, pp. 1–12.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE AND MULTIPLICITY OF HOMOCLINIC

SOLUTIONS FOR A DIFFERENCE EQUATION

SHAPOUR HEIDARKHANI, FARIBA GHAREHGAZLOUEI, MAURIZIO IMBESI

Abstract. The aim of this article is to obtain homoclinic solutions for a

discrete problem involving p-Laplacian. We prove the existence of at least one,

two and three solutions for the problem. Our approach is based on variational
methods.

1. Introduction

In this article, we study the nonlinear second-order difference equation, depend-
ing on a real parameter λ > 0,

−∆(a(k)φp(∆u(k − 1))) + b(k)φp(u(k)) = λf(k, u(k)) + h(u(k)) ∀k ∈ Z,
u(k)→ 0 as |k| → ∞.

(1.1)

Here p > 1 is a real number, φp(t) = |t|p−2t for t ∈ R, and a, b : Z → (0,+∞),
where b(k) ≥ α > 0 for all k ∈ Z, and b(k) → +∞ as |k| → +∞. The function
f : Z×R→ R is continuous, while h : R→ R is Lipschitz continuous of order p− 1
with Lipschitzian constant L ≥ 0, i.e.

|h(ξ1)− h(ξ2)| ≤ L|ξ1 − ξ2|p−1, ∀ξ1, ξ2 ∈ R ,
and such that h(0) = 0. The forward difference operator is

∆u(k − 1) = u(k)− u(k − 1) ∀k ∈ Z.
A solution u = {u(k)} of problem (1.1) is called homoclinic if lim|k|→∞ u(k) = 0.

Discrete boundary value problems have been extensively studied in the previous
decade. Modeling of certain nonlinear problems from biological neural net-works,
economics, optimal control, electrical circuit analysis, dynamical systems, and other
areas of study have led to the rapid progression of the theory of difference equations;
see [1, 2, 8, 9, 11]. Most of the classical methods used in differential equations
can be used for difference equations. Variational methods are powerful tools in
such problems. Critical point theory has been used for proving the existence and
multiplicity solutions of discrete nonlinear problems.

The issue of finding solutions on unbounded intervals is more delicate. For a
study such problems by variational methods, see [13, 16]. Variational methods
for difference equations consist in seeking solutions as critical points for a suitable
energy functional defined on a convenient Banach space. In the first approaches

2010 Mathematics Subject Classification. 39A10, 47J30, 35B38.

Key words and phrases. Difference equations; discrete p-Laplacian; variational methods.
c©2020 Texas State University.

Submitted January 14, 2020. Published November 23, 2020.

1



2 S. HEIDARKHANI, F. GHAREHGAZLOUEI, M. IMBESI EJDE-2020/115

to the issue, the variational methods are applied to boundary value problems on
bounded discrete intervals, which lead to the study of an energy functional defined
on a finite-dimensional Banach space.

In the case of difference equations on unbounded discrete intervals (typically, on
the whole set of integers Z) solutions are sought in a subspace of the space `p which
is still infinite-dimensional but compactly embedded into `p (see [12, 13, 16]). A
standard way to deal with problems on unbounded domains consists in introducing
coercive weight functions. This method goes back to a celebrated result of Omana
and Willem [17] on homoclinic orbits for a class of Hamiltonian systems.

There is an increasing interest in the existence and multiplicity of homoclinic
solutions to discrete nonlinear problems. The existence and multiplicity of homo-
clinic solutions have been investigated using various methods by many authors; see
[3, 10, 13, 14, 15, 19, 20, 21, 22] and the references therein. For example, Iannizzotto
and Tersian [13] used critical point theory, and proved the existence of at least two
nontrivial homoclinic solutions for the nonlinear second-order difference equation

−∆(φp(∆u(k − 1))) + a(k)φp(u(k)) = λf(k, u(k)) ∀k ∈ Z,
u(k)→ 0 as |k| → ∞

(1.2)

where p > 1 is a real number, φp(t) = |t|p−2t for t ∈ R, a : Z → (0,+∞) is a
positive and coercive weight function and f : Z× R→ R is a continuous function.

Kong [15] applied the variational method and a variant of the fountain theorem
to find new conditions under which the problem (1.1), in the case h ≡ 0, has
infinitely many solutions. Sun and Mai [22] employed Nehari manifold methods
and critical point theory to study the existence of nontrivial homoclinic solutions
of discrete p-Laplacian equations with a coercive weight function and superlinear
nonlinearity. Without assuming the classical Ambrosetti-Rabinowitz condition and
without any periodicity assumptions, they proved the existence and multiplicity
results of problem (1.2).

Stegliński [21] by using both the general variational principle of Ricceri and
the direct method introduced by Faraci and Kristály [9] obtained infinitely many
solutions for the problem (1.1), when h ≡ 0. In [3] using variational methods and
critical point theory, sufficient conditions for the existence of at least one homoclinic
solution for the problem (1.1), in the case h ≡ 0 have been presented.

We want to point out the main novelties of our results: we investigate the ex-
istence of homoclinic solutions of problem (1.1) when the nonlinearity f has sub-
critical growth. We use variational methods and critical point theory to study the
existence of at least one, two and three weak solutions whenever the parameter λ
belongs to a precise positive interval. We note that the existence of three solu-
tions to difference the equations in the set Z of integers has rarely been studied.
The main tools are critical point theorems established in [4, 5, 7]. Examples are
presented to demonstrate the applicability of our results.

The rest of this article is organized as follows. Section 2 includes some prelim-
inary results. Section 3 contains the main results, their proofs, and some applica-
tions.

2. Preliminaries

In this section, we introduce some definitions and notation which will be used
later. For all 1 ≤ p <∞, denote `p the set of all functions u : Z→ R such that



EJDE-2020/115 HOMOCLINIC SOLUTIONS FOR A DIFFERENCE EQUATION 3

‖u‖pp =
∑
k∈Z
|u(k)|p < +∞.

Let `∞ be the set of all functions u : Z→ R such that

‖u‖∞ = sup
k∈Z
|u(k)| < +∞.

Set
X =

{
u : Z→ R :

∑
k∈Z

(a(k)|∆u(k − 1)|p + b(k)|u(k)|p) <∞
}

equipped with the norm

‖u‖ =
(∑
k∈Z

(a(k)|∆u(k − 1)|p + b(k)|u(k)|p)
)1/p

.

Clearly, we have

‖u‖∞ ≤ ‖u‖p ≤ α−1/p‖u‖ ∀u ∈ X. (2.1)

As is shown in [13, Proposition 3], (X, ‖ · ‖) is a reflexive Banach space, and the
embedding X ↪→ `p is compact. We define

Φ(u) :=
1

p
‖u‖p −

∑
k∈Z

H(u(k)) ∀u ∈ X, (2.2)

Ψ(u) :=
∑
k∈Z

F (k, u(k)) ∀ u ∈ lp (2.3)

where F (k, t) =
∫ t

0
f(k, ξ)dξ for t ∈ R and k ∈ Z, H(t) =

∫ t
0
h(ξ)dξ for t ∈ R. Let

Iλ : X → R be the energy functional associated to the problem (1.1) defined by

Iλ(u) = Φ(u)− λΨ(u).

We suppose that the Lipschitz constant L > 0 of the function h satisfies the condi-
tion L < α.

Proposition 2.1. Let f : Z× R→ R be a continuous function satisfying

sup
|t|≤T

|f(·, t)| ∈ `1 ∀T > 0. (2.4)

Then,

(1) Ψ ∈ C1(`p) and Ψ ∈ C1(X);
(2) Φ ∈ C1(X);
(3) Iλ ∈ C1(X) and every critical point u ∈ X of Iλ is a homoclinic solution

of problem (1.1);
(4) Iλ is sequentially weakly lower semicontinuous functional on X.

Proof. Arguing as in the proof of [21, Proposition 2.2], Part (1) follows from [21,
Lemma 2.1]. Parts (2) and (3) can be proved essentially by the same way as [13,
Propositions 5 and 7], where H(u(k)) = 0, a(k) = 1 on Z and the norm on X is
slightly different. See also [14, Lemmas 2.4 and 2.6]. The proof of Part (4) is based
on the facts Ψ ∈ C(`p) and the compactness of X → `p, then it is standard. �

Definition 2.2. Let Φ and Ψ be two continuously Gâteaux differentiable function-
als defined on a real Banach space X and fix r ∈ R. The functional I = Φ − Ψ is
said to verify the Palais-Smale condition cut off upper at r (in short (PS)[r]) if any
sequence {un}n∈N in X such that



4 S. HEIDARKHANI, F. GHAREHGAZLOUEI, M. IMBESI EJDE-2020/115

(1) {I(un)} is bounded;
(2) limn→∞ ‖I ′(un)‖X∗ = 0;
(3) Φ(un) < r for each n ∈ N

has a convergent subsequence.

The following theorems are the main tools for proving our results.

Theorem 2.3 ([5, Theorem 2.3], [4, Theorem 5.1]). Let X be a real Banach space,
Φ,Ψ : X → R be two continuously Gâteaux differentiable functionals such that
infX Φ = Φ(0) = Ψ(0) = 0. Assume that there exist r > 0 and v ∈ X, with
0 < Φ(v̄) < r such that

(1)
supΦ(u)≤r Ψ(u)

r < Ψ(v)
Φ(v) ;

(2) for all λ ∈ Λ :=
(Φ(v)

Ψ(v) ,
r

supΦ(u)≤r Ψ(u)

)
the functional Iλ := Φ− λΨ satisfies

the (PS)[r] condition.

Then, for each λ ∈ Λ there is u0,λ ∈ Φ−1(0, r) such that I ′λ(u0,λ) = ϑX∗ and
Iλ(u0,λ) < Iλ(u) for all u ∈ Φ−1(0, r).

Theorem 2.4 ([5, Theorem 3.2]). Let X be a real Banach space, Φ,Ψ : X → R
be two continuously Gâteaux differentiable functionals such that Φ is bounded from
below and infX Φ = Φ(0) = Ψ(0) = 0. Fix r > 0 and assume that, for each

λ ∈
(

0,
r

supu∈Φ−1(−∞,r) Ψ(u)

)
the functional Iλ := Φ− λΨ satisfies the (PS) condition and it is unbounded from
below. Then, for each

λ ∈
(

0,
r

supu∈Φ−1(−∞,r) Ψ(u)

)
the functional Iλ admits two distinct critical points.

Theorem 2.5 ([7, Theorem 3.6]). Let X be a reflexive real Banach space, Φ : X →
R be a coercive and continuously Gâteaux differentiable and sequentially weakly
lower semi-continuous functional whose Gâteaux derivative admits a continuous
inverse on X∗, Ψ : X → R be a continuously Gâteaux differentiable functional
whose derivative is compact, such that

inf
X

Φ = Φ(0) = Ψ(0) = 0.

Assume that there exist r > 0 and v ∈ X, with r < Φ(v) such that

(1)
supΦ(u)≤r Ψ(u)

r < Ψ(v)
Φ(v) ;

(2) for all λ ∈ Λr :=
(

Φ(v)
Ψ(v) ,

r
supΦ(u)≤r Ψ(u)

)
the functional Φ− λΨ is coercive.

Then, for each λ ∈ Λr the functional Φ − λΨ has at least three distinct critical
points in X.

Theorems 2.3–2.5 have been successfully applied to the existence and multiplicity
of weak solutions for elliptic Dirichlet problems with variable exponent in [6].



EJDE-2020/115 HOMOCLINIC SOLUTIONS FOR A DIFFERENCE EQUATION 5

3. Main results

We first discuss the existence of one solution for problem (1.1) .

Theorem 3.1. Assume that there exist two real constants τ > 0 and η and also
assume that f : Z×R→ R is a continuous function and F (l, η) > 0 for some l ∈ Z,
such that

(1) α+L
α−L

2a(l)+b(l)
αp |η|p < τp;

(2)
∑
k∈Z max|t|≤τ F (k,t)

(α−L)τp < αpF (l,η)
(2a(l)+b(l))(α+L)|η|p .

Then, for each

λ ∈ Λ :=
( (2a(l) + b(l))(α+ L)|η|p

αp2F (l, η)
,

(α− L)τp

p
∑
k∈Z max|t|≤τ F (k, t)

)
, (3.1)

problem (1.1) admits at least one nontrivial solution uλ ∈ X such that ‖uλ‖∞ ≤ τ .

Proof. Our goal is to apply Theorem 2.3 to problem (1.1). To this end, take the
real Banach space X with the norm as defined in Section 2, and and let Φ,Ψ be
the functionals defined in (2.2) and (2.3). Taking into account that h is a (p− 1)-
Lipschitz continuous function with Lipschizian constant L > 0 and h(0) = 0, we
have

α− L
αp
‖u‖p ≤ 1

p
‖u‖p − L

p

∑
k∈Z
|u(k)|p

≤ Φ(u)

≤ 1

p
‖u‖p +

L

p

∑
k∈Z
|u(k)|p

≤ α+ L

αp
‖u‖p,

namely
α− L
αp
‖u‖p ≤ Φ(u) ≤ α+ L

αp
‖u‖p. (3.2)

From the first inequality of (3.2) it follows that Φ is coercive. From Proposition 2.1,
we observe that Φ,Ψ ∈ C1(X,R). Moreover, Ψ′ : X → X∗ is a compact operator.
Indeed, it is enough to show that Ψ′ is strongly continuous on X. For this end, for
fixed u ∈ X, let un → u weakly in X as n→∞, then un(x) converges uniformly to
u(x) on Z as n→∞; see [23]. Since f is continuous in R for every k ∈ Z, we have

f(k, un(k))→ f(k, u(k)),

as n→∞. Thus Ψ′(un)→ Ψ′(u) as n→∞. Hence, we proved that Ψ′ is a compact
operator by [23, Proposition 26.2]. This ensures that the functional Iλ = Φ − λΨ
satisfies the (PS)[r] condition for each r > 0 [4, see Proposition 2.1]. Our aim is to
apply Theorem 2.3 to the functional Iλ. We have infX Φ = Φ(0) = Ψ(0) = 0. Now,
it remains to verify condition (1) in Theorem 2.3. To this aim, put r := α−L

p τp.

For any l ∈ Z, we define el ∈ X with el(k) = δlk for all k ∈ Z (δlk = 1 if l = k,
δlk = 0 if l 6= k). Set

w(k) = ηel(k), ∀k ∈ Z. (3.3)

Simple calculations show that

‖w‖p = (
2a(l) + b(l)

p
)|η|p,



6 S. HEIDARKHANI, F. GHAREHGAZLOUEI, M. IMBESI EJDE-2020/115

and from (3.2) it follows that

(α− L)(2a(k) + b(k))

αp2
|η|p ≤ Φ(w) ≤ (α+ L)(2a(k) + b(k))

αp2
|η|p. (3.4)

The right hand side of (3.4) in conjunction with the condition (1) in Theorem 3.1
yields 0 < Φ(w) < r. From the definition of Φ and (2.1), the estimate Φ(u) ≤ r
implies that

|u(k)|p ≤ ‖u‖p∞ ≤
1

α
‖u‖p ≤ p

α− L
Φ(u) ≤ p

α− L
r = τp, ∀k ∈ Z,

from which it follows that

Φ−1(−∞, r] = {u ∈ X; Φ(u) ≤ r} ⊆ {u ∈ X; |u| ≤ τ}.

Accordingly, we have

sup
u∈Φ−1(−∞,r]

Ψ(u) = sup
u∈Φ−1(−∞,r]

∑
k∈Z

F (k, u(k)) ≤
∑
k∈Z

max
|t|≤τ

F (k, t). (3.5)

In view of (3.4) and (3.5), taking into account (2) of Theorem 3.1 (A2), we obtain

supu∈Φ−1(−∞,r] Ψ(u)

r
≤
p
∑
k∈Z max|t|≤τ F (k, t)

(α− L)τp

<
αp2F (l, η)

(2a(l) + b(l))(α+ L)|η|p

≤ Ψ(w)

Φ(w)
,

(3.6)

which means that, condition (1) of Theorem 2.3 is satisfied. Hence, applying The-

orem 2.3, for each λ ∈
]

Φ(w)
Ψ(w) ,

r
supΦ(u)≤r Ψ(u)

[
the functional Iλ admits at least one

critical point uλ such that

0 < Φ(uλ) < r, and ‖uλ‖∞ ≤ τ

which is a nontrivial solution of the problem (1.1). �

The following example is an application of Theorem 3.1.

Example 3.2. Consider the problem

−∆(φ5(∆u(k − 1))) + (
|k|+ 1

2
)φ5(u(k)) = λf(k, u(k)) + sin4(

u(k)

2
) ∀k ∈ Z,

u(k)→ 0 as |k| → ∞.

For all (k, t) ∈ Z× R put

f(k, t) =
1

k2 + 1
.

By the expression of f we have

F (k, t) =
t

k2 + 1
∀(k, t) ∈ Z× R.

By choosing η = 1, τ = 2 and l = 1, through simple calculations we obtain

α+ L

α− L
2a(l) + b(l)

αp
|η|p =

54

35
< 32



EJDE-2020/115 HOMOCLINIC SOLUTIONS FOR A DIFFERENCE EQUATION 7

and ∑
k∈Z max|t|≤τ F (k, t)

(α− L)τp
=

1

7
(
∑
k∈Z

1

k2 + 1
) =

1

7
(1 + 2

∞∑
k=1

1

k2 + 1
)

=
1

7
(π coth(π)) = 0.45

<
αpF (l, η)

(2a(l) + b(l))(α+ L)|η|p
=

20

27
= 0.74.

Obviously, all assumptions of Theorem 3.1 are satisfied. Hence, it follows that for
each λ ∈ (0.27, 0.89) the above problem admits at least one nontrivial solution
uλ ∈ X such that ‖uλ‖∞ ≤ 2.

Now, we discuss the existence of two solutions for (1.1) applying Theorem 2.4.

Theorem 3.3. Let f : Z × R → R be a continuous function. Moreover, assume
that

(1) there exist µ > p and R > 0 such that

0 < µF (k, t) < tf(k, t)

for all k ∈ Z and |t| ≥ R.

Then, for each

λ ∈ Λ :=
(

0,
(α− L)τp

p
∑
k∈Z max|t|<τ F (k, t)

)
,

problem (1.1) admits at least two nontrivial solutions.

Proof. Let Φ,Ψ be the functionals defined in Theorem 3.1 which satisfy all regu-
larity assumptions requested in Theorem 2.4. Arguing as in the proof of Theorem
3.1, choosing r = α−L

p τp, for each λ ∈ Λ we obtain

supu∈Φ−1(−∞,r) Ψ(u)

r
≤
p
∑
k∈Z max|t|≤τ F (k, t)

(α− L)τp
<

1

λ
.

Now, from the condition (1), by standard computations, there is a positive constant
m such that

F (k, t) ≥ m|t|µ for all k ∈ Z. (3.7)

Hence, for every λ ∈ Λ, u ∈ X \ {0} and t > 1, we obtain

Iλ(tu(k)) = Φ(tu(k))− λ
∑
k∈Z

F (k, tu(k)) ≤ α+ L

p
tp‖u‖p −mλtµ

∑
k∈Z
|u(k)|µ.

Since µ > p, this condition guarantees that Iλ is unbounded from below. We recall
that Iλ is a Gâteaux differentiable functional whose Gâteaux derivative at the point
u ∈ X is the functional I ′λ(u) ∈ X∗ given by

I ′λ(u)(v) =
∑
k∈Z

a(k)φp(∆u(k − 1))∆v(k − 1) +
∑
k∈Z

b(k)φp(u(k))v(k)

−
∑
k∈Z

h(u(k))v(k)− λ
∑
k∈Z

f(k, u(k))v(k),



8 S. HEIDARKHANI, F. GHAREHGAZLOUEI, M. IMBESI EJDE-2020/115

for every v ∈ X. Finally, we verify that Iλ satisfies the (PS)-condition. Indeed, if
{un}n∈N ⊂ X such that {Iλ(un)}n∈N is bounded and I ′λ(un)→ 0 in X∗ as n→ +∞.
Then, there exists a positive constant s0 such that

|Iλ(un)| ≤ s0, ‖I ′λ(un)‖ ≤ s0 ∀n ∈ N.
For every n ∈ N we put

ϕ(un(k)) := f(k, un(k))un(k)− µF (k, un(k)),

Kn := {k ∈ Z : |un(k)| > R}.

Using also conditions (1) and the definition of I ′λ, we deduce that, for all n ∈ N,

s0 + s1‖un‖ ≥ µIλ(un)− I ′λ(un)un

≥ (
µ

p
− 1)(1− L

α
)‖un‖p + λ

∑
k∈Z

ϕ(un(k))

= (
µ

p
− 1)(1− L

α
)‖un‖p + λ

∑
k∈Kn

ϕ(un(k)) + λ
∑

k∈Z\Kn

ϕ(un(k))

≥ (
µ

p
− 1)(1− L

α
)‖un‖p − λs2,

for some s1, s2 > 0. Since µ > p and α > L it follows {un}n∈N is bounded.
Consequently, since X is a reflexive Banach space, up to a subsequence, we have

un ⇀ u in X.

By I ′λ(un)→ 0 and un ⇀ u in X we observe that

(I ′λ(un)− I ′λ(u))(un − u)→ 0.

From the continuity of f and h we have∑
k∈Z

(f(k, un(k))− f(k, u(k)))(un(k)− u(k))→ 0, as n→ +∞,∑
k∈Z

(h(un(k))− h(u(k)))(un(k)− u(k))→ 0, as n→ +∞.

An easy computation shows that

(I ′λ(un)− I ′λ(u))(un − u)

=
∑
k∈Z

(
a(k)(φp(∆un(k − 1))− φp(∆un(k − 1)))(∆un(k − 1)−∆u(k − 1))

)
+
∑
k∈Z

b(k)(φp(un(k))− φp(u(k)))(un(k)− u(k))

−
∑
k∈Z

(h(un(k)− h(u(k))(un(k)− u(k))

− λ
∑
k∈Z

(f(k, un(k)− f(k, u(k)))(un(k)− u(k))

≥ (1− L

α
)‖un − u‖p.

Thus, since α > L, this implies that the sequence {un}n∈N converges strongly to u
in X. Therefore, Iλ satisfies the (PS)-condition, and so all hypotheses of Theorem
2.4 are verified. Hence, applying Theorem 2.4, for each λ ∈ Λ the function Iλ



EJDE-2020/115 HOMOCLINIC SOLUTIONS FOR A DIFFERENCE EQUATION 9

admits at least two distinct critical points that are the solutions of the problem
(1.1). �

Remark 3.4. Assume that f satisfies the condition

(1) there exist m1,m2 ∈ [0,+∞) such that for 1 < q < p,

|F (k, t)| ≤ m1 +m2|t|q−1∀(k, t) ∈ Z× R.
Setting r = 1. Indeed, using condition (1) in Theorem 3.3. there exist m3 > 0 and
m4 > 0 such that

F (k, t) ≥ m3|t|µ −m4 for all (k, t) ∈ Z× R.
For fixed ū ∈ X \ {0X}, for each t > 1 one has

Iλ(tū(k)) ≤ α+ L

p
tp‖ū‖p − λ

∑
k∈Z

(m3t
µ|ū(k)|µ −m4).

Since µ > p, this condition guarantees that Iλ is unbounded from below. Therefore,
Theorem 2.4 ensures the conclusion. As a general example of application of this
result we consider the function f satisfying the condition (1) by fixing p < µ < q

and r > max{( (µ−1)m1

(µ−q)m2
)

1
q−1 , (m1

m2
)

1
q−1 }, so that, for this type of f , the problem (1.1)

admits at least two nontrivial solutions.

Now, we discuss the existence of at least three solutions for problem (1.1).

Theorem 3.5. Assume that there exist two real constants τ > 0 and η, and assume
that f : Z × R → R is a continuous function and F (l, η) > 0 for some l ∈ Z, such
that

(1) 2a(l)+b(l)
αp |η|p > τp;

(2) lim sup|t|→+∞
F (k,t)
|t|p ≤ 0 for allk ∈ Z.

Suppose that (2) in Theorem 3.1 holds. Then for every

λ ∈ Λr :=
( (2a(l) + b(l))(α+ L)|η|p

αp2F (l, η)
,

(α− L)τp

p
∑
k∈Z max|t|≤τ F (k, t)

)
,

problem (1.1) admits at least three solutions.

Proof. Our aim is to apply Theorem 2.5. We consider the functionals Φ and Ψ as
before, which satisfy the regularity assumptions requested in Theorem 2.5. Now,
arguing as in the proof of Theorem 3.1, putting w(k) as in (3.4) and r = α−L

p τp,

bearing in mind (1), we derive

Φ(w) > r > 0.

Therefore, (3.6) holds, and the assumption (1) of Theorem 2.5 is satisfied. Now,
we prove that, for each λ ∈ Λr the functional Iλ is coercive. By using the condition
(2), for all 0 < ε < α−L

λp there exists T > 0 such that

F (k, t) ≤ ε|t|p for all k ∈ Z, |t| > T.

In addition, by (2.4) there exists w1 ∈ `1 such that

F (k, t) ≤ w1(k) ∀k ∈ Z, |t| ≤ T.
For each u ∈ X we have

Iλ(u) ≥ α− L
αp
‖u‖p − λ

∑
|u(k)|≤T

F (k, u(k))− λ
∑

|u(k)|>T

F (k, u(k))



10 S. HEIDARKHANI, F. GHAREHGAZLOUEI, M. IMBESI EJDE-2020/115

≥ α− L
αp
‖u‖p − λ‖w1‖1 − λε‖u‖pp

≥ (
α− L
αp

− λε

α
)‖u‖p − λ‖w1‖1,

in which Iλ → +∞ as ‖u‖ → +∞. Hence the functional Iλ is coercive, also
condition the (2) of theorem 2.5 holds. So, for each λ ∈ Λr, Theorem 2.5 implies
that the functional Iλ admits at least three critical points in X that are solutions
of problem (1.1). �

Now, we present an example to illustrate Theorem 3.5.

Example 3.6. Consider the problem

−∆(φ31(∆u(k − 1))) + (|k|+ 2)φ31(u(k)) = λf(k, u(k)) + sin30(u(k)) ∀k ∈ Z,
u(k)→ 0 as |k| → ∞.

For (k, t) ∈ Z× R, we put

f(k, t) = 30t29 sin(
k2π

4
).

From the definition of f we have

F (k, t) = t30 sin(
k2π

4
), ∀(k, t) ∈ Z× R.

Clearly, F satisfies assumption (2) of Theorem 3.5. By choosing η = 2, τ = 1 and

l = 1, by simple calculations we obtain 2a(l)+b(l)
αp |η|p > 1 and∑

k∈Z max|t|≤τ F (k, t)

(α− L)τp
=
∑
k∈Z

sin(
k2π

4
) =
√

2 <
αpF (l, η)

(2a(l) + b(l))(α+ L)|η|p
=

31
√

2

30
.

Thus, all assumptions of Theorem 3.5 are satisfied. It follows that for each λ ∈
(0.0220, 0.0228) the above problem admits at least three solutions.

We finish the discussion by pointing out a simple consequence of Theorem 3.5.

Theorem 3.7. Let f : R → R be a continuous function. Assume that F (η) > 0
for some η > 0 and F (ξ) ≥ 0 in [0, η] and

lim inf
ξ→0

F (ξ)

ξp
= lim sup

ξ→+∞

F (ξ)

ξp
= 0.

Then, there is λ∗ > 0 such that for each λ > λ∗ the problem

−∆(a(k)φp(∆u(k − 1))) + b(k)φp(u(k)) = λf(u(k)) + h(u(k)) ∀k ∈ Z,
u(k)→ 0 as |k| → ∞,

(3.8)

admits at least three nontrivial solutions.

Proof. Our aim is to use Theorem 3.5 with b(l) = (|l| + 1)/2 and a(l) = 1/4 for
some l ∈ Z. We fix

λ > λ∗ :=
( |l|+2

2 )(α+ L)|η|p

αpF (η)
.

Recalling that lim infξ→0 F (ξ)/ξp = 0, there is a sequence {θn} ⊂ (0,+∞) such
that limn→∞ θn = 0 and

lim
n→∞

sup|ξ|≤θn F (ξ)

θpn
= 0.



EJDE-2020/115 HOMOCLINIC SOLUTIONS FOR A DIFFERENCE EQUATION 11

Indeed, one has

lim
n→∞

sup|ξ|≤θn F (ξ)

θpn
= lim
n→∞

sup|ξ|≤θn F (ξθn)

ξpθn

ξpθn
θpn

= 0,

where F (ξθn) = sup|ξ|≤θn F (ξ). Hence, there exists θ̄ > 0 such that

sup|ξ|≤θ̄ F (ξ)

θ̄p
< min

{ (α− L)F (η)

( |l|+2
2 )(α+ L)|η|P

,
α− L
λαp

}
and θ̄ < η. The conclusion follows from Theorem 3.5. �

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Shapour Heidarkhani

Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah,
Iran

Email address: s.heidarkhani@razi.ac.ir

Fariba Gharehgazlouei

Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah,

Iran
Email address: f.gharehgazloo@yahoo.com

Maurizio Imbesi
Department of Mathematical and Computer Sciences, Physical and Earth Sciences,

University of Messina, Viale F. Stagno d’Alcontres 31, 98166 Messina, Italy

Email address: imbesim@unime.it


	1. Introduction
	2. Preliminaries
	3. Main results
	References