CUBO A Mathematical Journal
Vol.21, No¯ 03, (75–91). December 2019

http: // dx. doi. org/ 10. 4067/ S0719-06462019000300075

Weak solutions to Neumann discrete nonlinear system of
Kirchhoff type

Rodrigue Sanou1, Idrissa Ibrango2, Blaise Koné1, Aboudramane Guiro2

1 Laboratoire d’analyse Mathématiques et d’Informatique (LaMI),

Institut des Sciences Exactes et Appliquées,

Université Joseph KI-Zerbo,

Ouagadougou, Burkina- Faso.

2 Laboratoire d’analyse Mathématiques et d’Informatique (LaMI),

UFR, Sciences et Technique,

Université Nazi Boni,

01 BP 1091 Bobo 01, Bobo Dioulasso, Burkina-Faso.

drigoaime@gmail.com, ibrango2006@yahoo.fr, leizon71@yahoo.fr, abouguiro@yahoo.fr

ABSTRACT

We prove the existence of weak solutions for discrete nonlinear system of Kirchhoff

type. We build some Hilbert spaces with suitable norms. We define the notion of weak

solution corresponding to the problem (1.1). The proof of the main result is based on

a minimization method of an energy functional J.

RESUMEN

Probamos la existencia de soluciones débiles para sistemas discretos no-lineales de tipo

Kirchhoff. Construimos algunos espacios de Hilbert con normas apropiadas. Definimos

la noción de solución débil correspondiente al problema (1.1). La demostración del

resultado principal se basa en un método de minimización de un funcional de enerǵıa J.

Keywords and Phrases: Nonlinear difference equations, anisotropic nonlinear discrete systems,

minimization methods, weak solutions.

2010 AMS Mathematics Subject Classification: 47A75; 35B38; 35P30; 34L05; 34L30.

http://dx.doi.org/10.4067/S0719-06462019000300075


76 R. Sanou, I. Ibrango, B. Koné, A. Guiro CUBO
21, 3 (2019)

1 Introduction

In this paper, we are going to investigate the existence of weak solutions for the following anisotropic

nonlinear discrete system.

For i = 1, · · · , n





−M (A(k − 1, ∆ui(k − 1))) ∆(a(k − 1, ∆ui(k − 1)))=fi(k, u(k)), k ∈ Z[1, T]

∆ui(0) = ∆ui(T) = 0

(1.1)

where ∆ui(k) = ui(k + 1) − ui(k) is the forward difference operator for any i = 1, · · · , n;

Z[1, T] = {1, . . . , T} for T ≥ 2 and a, fi are functions to be defined later.

In the last few years, great attention has been paid to the study of fourth-order nonlinear difference

equations. These equations have been widely used to study discrete models in many fields such as

computer science, economics, neural network, ecology, cybernetics, etc. For background and recent

results, we refer the reader to [2]-[12], [14] and the references therein.

Note that in recent years, much attention has been paid to problems not local since they appear in

physical phenomena like the theory of nonlinear elasticity, heat diffusion, etc. Among this problems,

we find Kirchhoff type problems, which are known by the presence of the term M(
∫
Ω
|∇u|2)∆u

in the continuous case. As far as we know, the first study which deals with anisotropic discrete

boundary value problems of p(.)-Kirchhoff type difference equation was done by Yucedag (see [11]).

The function M(A(k − 1, ∆u(k − 1))) which appear in the left-hand side of problem (1.1) is more

general.

The main operator ∆(a(k − 1, ∆u(k − 1))) in problem (1.1) can be seen as a discrete counterpart

of the anisotropic operator
N∑

i=1

∂

∂xi
a

(
x,

∂

∂xi
u

)
. The functional a derives from a potential with

a(k, ξ) = ∂
∂ξ

A(k, ξ).

Our goal is to use a minimization method in order to establish some existence results of solutions

of (1.1). The idea of the proof is to transfer the problem of the existence of solutions for (1.1) into

the problem of existence of a minimizer for some associated energy functional. This method was

successfully used by Bonanno et al. [1] for the study of an eigenvalue nonhomogeneous Neumann

problem, where, under an appropriate oscillating behaviour of the nonlinear term, they proved the

existence of a determined open interval of positive parameters for which the problem considered

admits infinitely many weak solutions that strongly converge to zero, in an appropriate Orlicz

Sobolev space.

Motivated by the work of [13] where J. Zhao proved the existence of positive solutions, the approach

presented in this article is different than the one given in the papers mentioned above. To the best of



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21, 3 (2019)

Weak solutions to Neumann discrete nonlinear system of Kirchhoff . . . 77

our knowledge , results on existence of weak solutions of system (1.1), using minimization method,

have not been found in the literature.

The remaining part of this paper is organized as follows. Section 2 is devoted to mathematical

preliminaries. The main existence result is proved in Section 3. In the Section 4, we give an

extension of our system.

2 Mathematical background

In the T-dimensional Hilbert space

H =
{
u : Z[0, T + 1] −→ Rn such that ∆u(0) = ∆u(T) = 0

}
,

with the inner product

〈u, v〉 =

n∑

i=1

T+1∑

k=1

∆ui(k − 1)∆vi(k − 1), ∀ u, v ∈ H,

we consider the norm

‖u‖ =

( n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
2

)1
2

. (2.1)

We denote

Hi =
{
ui : Z[0, T + 1] −→ R such that ∆ui(0) = ∆ui(T) = 0

}
, for i = 1, · · · , n

with the norm

|ui|h =

(
T+1∑

k=1

|∆ui(k − 1)|
2

)1
2

∀ ui ∈ Hi for i = 1, · · · , n. (2.2)

Moreover, we may consider Hi with the following norm

|ui|m =

(
T∑

k=1

|ui(k)|
m

) 1
m

∀ ui ∈ Hi, m ≥ 2 for i = 1, · · · , n. (2.3)

We have the following inequalities (see [2])

T(2−m)/(2m)|ui|2 ≤ |ui|m ≤ T
1/m|ui|2, ∀ ui ∈ Hi, m ≥ 2 for i = 1, · · · , n. (2.4)

Let the function

p : Z[0, T] −→ (2, +∞) (2.5)

denoted by

p− = min
k∈Z[0,T]

p(k) and p+ = max
k∈Z[0,T]

p(k).



78 R. Sanou, I. Ibrango, B. Koné, A. Guiro CUBO
21, 3 (2019)

For the data a and fi, we assume the following.

(H1).

{
a(k, .) : R → R, k ∈ Z[0, T] and there exists A(., .) : Z[0, T] × R → R

which satisfies a(k, ξ) = ∂
∂ξ

A(k, ξ) and A(k, 0) = 0, for all k ∈ Z[0, T].

(H2). For all k ∈ Z[0, T] and ξ 6= η

(a(k, ξ) − a(k, η)) .(ξ − η) > 0. (2.6)

(H3). For any k ∈ Z[0, T], ξ ∈ R, we have

A(k, ξ) ≥
1

p(k)
|ξ|p(k). (2.7)

(H4). For each k ∈ Z[0, T], the function fi(k, .) : R
n −→ R is jointly continuous and there exists

(αi(.))1≤i≤n : Z[0, T] −→ (0, +∞) and a function (ri(.))1≤i≤n : Z[0, T] −→ [2, +∞) such that

|fi(k, u)| ≤ αi(k)
(
1 + |ui(k)|

ri(k)−1
)

(2.8)

where 2 ≤ ri (k) < p
− for i = 1, · · · , n.

In what follows, we denote by :

r− = min
{(k,i)∈Z[0,T]×Z[1,n]}

ri(k) and r
+
= max

{(k,i)∈Z[0,T]×Z[1,n]}
ri(k).

For each i = 1, · · · , n, there exists hi ∈ R
n such that

∇Fi(k, u)(hi) = fi(k, u) ∀u ∈ H for i = 1, · · · , n. (2.9)

By (2.8) there exists (βi(.))1≤i≤n : Z[0, T] −→ (0, +∞) such that

|Fi(k, u)| ≤ βi(k)
(
1 + |ui(k)|

ri(k)
)

for i = 1, · · · , n (2.10)

where

0 < β = inf
{(k,i)∈ Z[0,T]×Z[1,n]}

βi(k) ≤ sup
{(k,i)∈ Z[0,T]×Z[1,n]}

βi(k) = β < +∞. (2.11)

(H5). We also assume that the function M : (0, +∞) −→ (0, +∞) is continuous and non-decreasing

and there exist positive numbers B1, B2 with B1 ≤ B2 and α > 1 such that

B1t
α−1 ≤ M(t) ≤ B2t

α−1 for t > t∗ > 0. (2.12)

Example 2.1.

There are many functions satisfying both (H1) − (H4). Let us mention the following.



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Weak solutions to Neumann discrete nonlinear system of Kirchhoff . . . 79

• A(k, ξ) =
1

p(k)

((
1 + |ξ|2

)p(k)/2
− 1

)
, where a(k, ξ) =

(
1 + |ξ|2

)(p(k)−2)/2
ξ,

∀ k ∈ Z[0, T], ξ ∈ R,

• fi(k, ξ) = 1 +
∣∣ξi
∣∣p(k)−1, ∀ (k, i) ∈ Z[0, T] × Z[1, n] and ξ = (ξ1, · · · , ξn) ,

• M(t) = 1, ∀ t ∈ (0, +∞).

Moreover, we may consider H with the following norm

‖u‖m =
n∑

i=1

( T∑

k=1

|ui(k)|
m

) 1
m

, ∀ u ∈ H and m ≥ 2. (2.13)

Using the relation (2.4) we can prove the following lemma.

Lemma 2.2. We have the following inequalities

T(2−m)/(2m)‖u‖2 ≤ ‖u‖m ≤ T
1/m‖u‖2, ∀ u ∈ H and m ≥ 2. (2.14)

We need the following auxiliary results throughout our paper.

Lemma 2.3.

(1) There exist two positive constant C1, C2 such that

n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
p(k−1) ≥ C1

(
n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
2

)p−
2

− C2,

(2.15)

for all u ∈ H with |ui|h > 1.

(2) For any m ≥ 2 there exists a positive constant cm such that

n∑

i=1

T∑

k=1

|ui(k)|
m ≤ cm

n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
m, ∀ u ∈ H. (2.16)

Indeed,

(1) By [6], there exists the positive constants λi and µi for i = 1, · · · n

T+1∑

k=1

|∆ui(k − 1)|
p(k−1) ≥ λi

(
T+1∑

k=1

|∆ui(k − 1)|
2

)p−
2

− µi ∀ ui ∈ Hi and |ui|h > 1.

n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
p(k−1) ≥ min

1≤i≤n
(λi)

n∑

i=1

(
T+1∑

k=1

|∆ui(k − 1)|
2

)p−
2

− max
1≤i≤n

(µi) n.



80 R. Sanou, I. Ibrango, B. Koné, A. Guiro CUBO
21, 3 (2019)

Since the function x 7−→ x
p−

2 is convex because p− > 2, then we have

n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
p(k−1) ≥ min

1≤i≤n
(λi)

(
n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
2

)p−
2

− max
1≤i≤n

(µi) n.

We deduce that

n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
p(k−1) ≥ C1

(
n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
2

)p−
2

− C2.

(2) By [8], for any m ≥ 2 there exists a positive constant cm such that for i = 1, · · · , n

T∑

k=1

|ui(k)|
m ≤ cm

T+1∑

k=1

|∆ui(k − 1)|
m ∀ ui ∈ Hi.

Therefore
n∑

i=1

T∑

k=1

|ui(k)|
m ≤ cm

n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
m ∀ u ∈ H.

3 Existence of weak solutions

In this section, we study the existence of weak solution of problem (1.1).

Definition 3.1. A weak solutions of problem (1.1) is u ∈ H such that

n∑

i=1

[
M

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)
T+1∑

k=1

a(k − 1, ∆ui(k − 1))∆vi(k − 1)

]

=

n∑

i=1

T∑

k=1

fi(k, u(k))vi(k)

(3.1)

for all v ∈ H.

Note that, since H is a finite dimensional space, the weak solutions coincide with the classical

solution the problem (1.1).

Theorem 3.2. Assume that (H1)−(H5) holds. Then, there exists a weak solution of the problem

(1.1).



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Weak solutions to Neumann discrete nonlinear system of Kirchhoff . . . 81

To prove this, we define the energy functional J : H −→ R by

J(u) =

n∑

i=1

M̂

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)
−

n∑

i=1

T∑

k=1

Fi
(
k, u(k)

)
(3.2)

where M̂(t) =

∫t

0

M(s)ds.

Lemma 3.3. The functional J is well defined on H and is of class C1
(
H, R

)
with the derivative

given by

〈J′(u), v〉 =

n∑

i=1

[
M

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1)

)
T+1∑

k=1

a(k − 1, ∆ui(k − 1))∆vi(k − 1)

]

−

n∑

i=1

T∑

k=1

fi(k, u(k))vi(k),

(3.3)

for all u, v ∈ H.

Indeed, let’s

I(u) =

n∑

i=1

M̂

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)
and Λ(u) =

n∑

i=1

T∑

k=1

Fi
(
k, u(k)

)
.

Since M̂(.), A(k, .) and F(k, .) are continuous for all k ∈ Z[0, T], then

|I(u)| =

∣∣∣∣
n∑

i=1

M̂

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)∣∣∣∣ < +∞,

|Λ(u)| =

∣∣∣∣
n∑

i=1

T∑

k=1

Fi
(
k, u(k)

)∣∣∣∣ < +∞.

The energy functional J is well defined on H.

It is not difficult to see that the functional I derivative are give by

〈I′(u), v〉=

n∑

i=1

[
M

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1)

)
T+1∑

k=1

a(k − 1, ∆ui(k − 1))∆vi(k − 1)

]
(3.4)



82 R. Sanou, I. Ibrango, B. Koné, A. Guiro CUBO
21, 3 (2019)

On the other hand, for all u, v ∈ H, there exists hi ∈ R
n such that

〈Λ′(u), v〉 = lim
t→0+

Λ(u + tv) − Λ(u)

t

= lim
t→0+

n∑

i=1

T∑

k=1

Fi(k, u(k) + tv(k)) − Fi(k, u(k))

t

=

n∑

i=1

T∑

k=1

lim
t→0+

Fi(k, u(k) + tv(k)) − Fi(k, u(k))

t

=

n∑

i=1

T∑

k=1

∇Fi(k, u(k))(hi)vi(k)

=

n∑

i=1

T∑

k=1

fi(k, u(k))vi(k).

The functional J is clearly of class C1 �

Lemma 3.4. The functional J is lower semi-continuous.

Indeed since the functional Λ is completely continuous and weakly lower semi-continuous, we have

to prove the semi-continuity of I.

A is convex with respect to the second variable according (H1) and (H2). With the assumption

(H5) we conclude that I is convex. Thus, it is enough to show that I is lower semi-continuous.

For this, we fix u ∈ H and ε > 0. Since I is convex, we deduce that, for any v ∈ H.

I(v) ≥ I(u) + 〈I′(u), v − u〉

≥ I(u) −

n∑

i=1

[
M

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)

×

T+1∑

k=1

|a(k − 1, ∆ui(k − 1))||∆vi(k − 1) − ∆ui(k − 1)|

]

≥ I(u) − CM

(
n∑

i=1

T+1∑

k=1

|a(k − 1, ∆ui(k − 1))||∆vi(k − 1) − ∆ui(k − 1)|

)
,

where CM =

(
n∑

i=1

M

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1)

))



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Weak solutions to Neumann discrete nonlinear system of Kirchhoff . . . 83

By using Schwartz inequality, we get :

I(v) ≥ I(u) − CM

n∑

i=1

[(T+1∑

k=1

|a(k − 1, ∆ui(k − 1))|
2

)1
2

×

(
T+1∑

k=1

|∆vi(k − 1) − ∆ui(k − 1)|
2

)1
2 ]

≥ I(u) − CM




n∑

i=1

(
T+1∑

k=1

|a(k − 1, ∆ui(k − 1))|
2

)1
2




×




n∑

i=1

(
T+1∑

k=1

|∆vi(k − 1) − ∆ui(k − 1)|
2

)1
2




By (2.2)

I(v) ≥ I(u) − CM




n∑

i=1

(
T+1∑

k=1

|a(k − 1, ∆ui(k − 1))|
2

)1
2



[

n∑

i=1

|vi − ui|h

]
.

Since Hi is finite dimensional, there exist the positive constants θi for i = 1, · · · , n such that

|vi|h ≤ θi|vi|2 ∀ vi ∈ Hi. (3.5)

Then,

I(v) ≥ I(u) − CM




n∑

i=1

(
T+1∑

k=1

|a(k − 1, ∆ui(k − 1))|
2

)1
2



[

n∑

i=1

θi|vi − ui|2

]

≥ I(u) − max
1≤i≤n

(θi) CM




n∑

i=1

(
T+1∑

k=1

|a(k − 1, ∆ui(k − 1))|
2

)1
2



[

n∑

i=1

|vi − ui|2

]
.

Also, the space H is finite dimensional, there exists a positive constant γ such that:

‖u‖2 ≤ γ‖u‖ ∀ u ∈ H.

From this, we have

I(v) ≥ I(u) − γ max
1≤i≤n

(θi) CM




n∑

i=1

(
T+1∑

k=1

|a(k − 1, ∆ui(k − 1))|
2

)1
2


‖v − u‖

≥ I(u) −


1 + γ max

1≤i≤n
(θi) CM

n∑

i=1

(
T+1∑

k=1

|a(k − 1, ∆ui(k − 1))|
2

)1
2


‖v − u‖



84 R. Sanou, I. Ibrango, B. Koné, A. Guiro CUBO
21, 3 (2019)

Finally

I(v) ≥ I(u) − S(T, u)‖v − u‖ ≥ I(u) − ε, (3.6)

for all v ∈ H with ‖v − u‖ < δ = ε
S(T,u)

, where

S(T, u) = 1 + γ max
1≤i≤n

(θi) CM

n∑

i=1

(
T+1∑

k=1

|a(k − 1, ∆ui(k − 1))|
2

)1
2

.

We conclude that J is weakly lower semi-continuous.

Proposition 3.5. The functional J is coercive and bounded from below.

Indeed, according to (2.7), (2.10)-(2.12) we have

J(u) =

n∑

i=1

M̂

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)
−

n∑

i=1

T∑

k=1

Fi
(
k, u(k)

)

≥
B1

α(p+)α

[
n∑

i=1

(
T+1∑

k=1

|∆ui(k − 1)|
p(k−1)

)α]
−

n∑

i=1

T∑

k=1

Fi
(
k, u(k)

)

≥
B1

α(p+)α

[
n∑

i=1

(
T+1∑

k=1

|∆ui(k − 1)|
p(k−1)

)α]
−

n∑

i=1

T∑

k=1

βi(k)
(
1 + |ui(k)|

ri(k)
)

≥
B1

α(p+)α

[
n∑

i=1

(
T+1∑

k=1

|∆ui(k − 1)|
p(k−1)

)α]
− β

n∑

i=1

T∑

k=1

(
1 + |ui(k)|

ri(k)
)

≥
B1

α(p+)α

[
n∑

i=1

(
T+1∑

k=1

|∆ui(k − 1)|
p(k−1)

)α]
− β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − βnT.

There exist ηi and νi such that

J(u) ≥
B1

α(p+)α

[
min

1≤i≤n
(ηi)

(
n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
p(k−1)

)α
− max

1≤i≤n
(νi)

]

− β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − βnT. (3.7)

To prove the coerciveness of the functional J, we may assume that ||u|| > 1 and we deduce from

the above inequality (2.15) that



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Weak solutions to Neumann discrete nonlinear system of Kirchhoff . . . 85

J(u) ≥
B1

α(p+)α


 min
1≤i≤n

(ηi)


C1

(
n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
2

)p−
2

− C2




α

− max
1≤i≤n

(νi)




−β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − βnT.

There exist a function K(α, C) such that

J(u) ≥
B1

α(p+)α

(
min

1≤i≤n
(ηi)C

α
1 ||u||

αp− − min
1≤i≤n

(ηi)K(α, C)C
α
2 − max

1≤i≤n
(νi)

)

− β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − βnT.

Namely

J(u) ≥ A1||u||
αp− − β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − A2,

where

A1 =
B1

α(p+)α
min

1≤i≤n
(ηi)C

α
1

and

A2 =
B1

α(p+)α

(
min

1≤i≤n
(ηi)K(α, C)C

α
2 + max

1≤i≤n
(νi)

)
+ βnT.

So

J(u) ≥ A1||u||
αp− − β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − A2

≥ A1||u||
αp− − β

n∑

i=1

T∑

k=1

|ui(k)|
r+ − β

n∑

i=1

T∑

k=1

|ui(k)|
r− − A2.

Using (2.16)

J(u) ≥ A1||u||
αp− − (Cr−)β

n∑

i=1

T∑

k=1

|∆ui(k)|
r− − (Cr+)β

n∑

i=1

T∑

k=1

|∆ui(k)|
r+ − A2

By using (2.4) there exists the positive constants K1 and K2 such that

J(u) ≥ A1||u||
αp− − K1

n∑

i=1

(
T∑

k=1

|∆ui(k)|
2

)r−
2

− K2

n∑

i=1

(
T∑

k=1

|∆ui(k)|
2

)r+
2

− A2.



86 R. Sanou, I. Ibrango, B. Koné, A. Guiro CUBO
21, 3 (2019)

There exist the positive constants A3, A4, A5 and A6 such that

J(u) ≥ A1||u||
αp− − K1A3

(
n∑

i=1

T∑

k=1

|∆ui(k)|
2

)r−
2

− K1 A4 − A5 K2

(
n∑

i=1

T∑

k=1

|∆ui(k)|
2

)r+
2

− K2 A6 − A2.

Consequently, there exist the positive constants A7 , A8 and A9 such that

J(u) ≥ A1||u||
αp− − A7||u||

r− − A8||u||
r+ − A9. (3.8)

Recall that p− >
r+

α
≥

r−

α
. Then J is coercive.

Besides, for ||u|| ≤ 1, we have with (3.7)

J(u) ≥
B1

α(p+)α

[
min

1≤i≤n
(ηi)

(
n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
p(k−1)

)α
− max

1≤i≤n
(νi)

]

− β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − βnT

≥ −
B1

α(p+)α
max

1≤i≤n
(νi) − β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − βnT

≥ −
B1

α(p+)α
max

1≤i≤n
(νi) − β

n∑

i=1

T∑

k=1

|ui(k)|
r− − β

n∑

i=1

T∑

k=1

|ui(k)|
r+ − βnT.

Using (2.16)

J(u) ≥ − B1
α(p+)α

max
1≤i≤n

(νi) − (Kr−)β

n∑

i=1

T∑

k=1

|∆ui(k)|
r− − (Kr+)β

n∑

i=1

T∑

k=1

|∆ui(k)|
r+ − βnT.

By using (2.14) there exists the positives constants K′1 and K
′
2 such that

J(u) ≥ −
B1

α(p+)α
max

1≤i≤n
(νi) − K

′
1

n∑

i=1

(
T∑

k=1

|∆ui(k)|
2

)r−
2

− K′2

n∑

i=1

(
T∑

k=1

|∆ui(k)|
2

)r+
2

− βnT.



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Weak solutions to Neumann discrete nonlinear system of Kirchhoff . . . 87

There exist the positive constants C′3, C
′
4, C

′
5 and C

′
6 such that

J(u) ≥ −
B1

α(p+)α
max

1≤i≤n
(νi) − K

′
1C

′
3

(
n∑

i=1

T∑

k=1

|∆ui(k)|
2

)r−
2

− K′1 C
′
4 − C

′
5 K

′
2

(
n∑

i=1

T∑

k=1

|∆ui(k)|
2

)r+
2

− K′2 C
′
6 − βnT.

Consequently, there exist the positive constants C′7 and C
′
8 such that

J(u) ≥ −
B1

α(p+)α
max

1≤i≤n
(νi) − C

′
7||u||

r−
− K′1 C

′
4 − C

′
8||u||

r+
− K′2 C

′
6 − βnT

≥ −
B1

α(p+)α
max

1≤i≤n
(νi) − C

′
7 − K

′
1 C

′
4 − C

′
8 − K

′
2 C

′
6 − βnT.

Thus, J is bounded from below �

Since J is weakly lower semi-continuous, bounded from below and coercive on H, using the re-

lation between critical points of J and problem (1.1), we deduce that J has a minimizer which is a

weak solution to problem (1.1).

4 An extension

In this section we are going to show that the existence result obtained for system (1.1) can be

extended. Let’s consider the following system.

For i = 1, · · · , n






−M (A(k − 1, ∆ui(k − 1))) ∆(a(k − 1, ∆ui(k − 1))) + σi(k)φ(k, ui(k))

= δi(k)fi(k, u(k)), ∀ k ∈ Z[1, T]

∆ui(0) = ∆ui(T) = 0,

(4.1)

where T ≥ 2 is a fixed integer, and we shall use the following assumption.

(H6). σi : Z[1, T] −→ R and δi : Z[1, T] −→ R are such that σi(k) ≥ σ0 > 0 for

(k, i) ∈ Z[1, T] × Z[1, n] and 0 < δi(k) ≤ sup
{(k,i)∈Z[1,T]×Z[1,n]}

|δi(k)| = δ0.

(H7). φ(k, t) = |t|
p(k)−2t for (k, t) ∈ Z[0, T] × R.



88 R. Sanou, I. Ibrango, B. Koné, A. Guiro CUBO
21, 3 (2019)

In the T−dimensional Hilbert space H with the inner product

〈u, v〉 =

n∑

i=1

T+1∑

k=1

∆ui(k − 1)∆vi(k − 1) +

n∑

i=1

T+1∑

k=1

ui(k)vi(k),

we consider the norm

‖u‖ =
√
〈u, u〉 =

(
n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
2
+

n∑

i=1

T∑

k=1

|ui(k)|
2

)1
2

.

Definition 4.1. A weak solution of problem (4.1) is a function u ∈ H such that

n∑

i=1

[
M

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1)

)
T+1∑

k=1

a(k − 1, ∆ui(k − 1))∆vi(k − 1)

]

+

n∑

i=1

T∑

k=1

σi(k)|ui(k)|
p(k)−2ui(k)vi(k) =

n∑

i=1

T∑

k=1

δi(k)fi(k, u(k))vi(k).

for all v ∈ H.

Theorem 4.2. Under the assumptions (H1)- (H6) the problem (4.1) has a least weak solution in

H.

Indeed, for u ∈ H we define the energy functional corresponding to system (4.1) by

J(u) =

n∑

i=1

M̂

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)
+

n∑

i=1

T∑

k=1

σi(k)

p(k)
|ui(k)|

p(k) −

n∑

i=1

T∑

k=1

δi(k)Fi
(
k, u(k)

)
.

Obviously, J is class C1 (H, R) and is weakly lower semicontinuous, and we show that

〈J′(u), v〉 =

n∑

i=1

[
M

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1)

)
T+1∑

k=1

a(k − 1, ∆ui(k − 1))∆vi(k − 1)

]

+

n∑

i=1

T∑

k=1

σi(k)|ui(k)|
p(k)−2ui(k)vi(k) −

n∑

i=1

T∑

k=1

δi(k)fi
(
k, u(k)

)
vi(k).

for all u, v ∈ H.

This implies that the weak solution of system(4.1) coincides with the critical points of the func-

tional J. It suffices to prove that J is bounded below and coercive in order to complete the proof.



CUBO
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Weak solutions to Neumann discrete nonlinear system of Kirchhoff . . . 89

J(u) =

n∑

i=1

M̂

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)
+

n∑

i=1

T∑

k=1

σi(k)

p(k)
|ui(k)|

p(k) −

n∑

i=1

T∑

k=1

δi(k)Fi
(
k, u(k)

)

≥

n∑

i=1

M̂

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)
−

n∑

i=1

T∑

k=1

δi(k)Fi
(
k, u(k)

)

≥
n∑

i=1

M̂

(
T+1∑

k=1

A(k − 1, ∆ui(k − 1))

)
− δ0

n∑

i=1

T∑

k=1

Fi
(
k, u(k)

)
.

We obtain

J(u) ≥ B1
α(p+)α

[
min

1≤i≤n
(ηi)

(
n∑

i=1

T+1∑

k=1

|∆ui(k − 1)|
p(k−1)

)α
− max

1≤i≤n
(νi)

]

− δ0β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − δ0βnT.

(4.2)

For ‖u‖ > 1, by the same procedure, we prove that

J(u) ≥ A′1‖u‖
αp−

− A′7‖u‖
r−

− A′8‖u‖
r+

− A′9,

where A′1, A
′
7, A

′
8 and A

′
9 are the positive constants.

Hence p− >
r+

α
≥

r−

α
, J is coercive.

If ||u|| ≤ 1 by (4.2) we have

J(u) ≥ −
B1

α(p+)α
max

1≤i≤n
(νi) − δ0β

n∑

i=1

T∑

k=1

|ui(k)|
ri(k) − δ0βnT.

By the same reasoning

J(u) ≥ −D1 − δ0βnT

where D1 > 0.

Thus, J is bounded from below �

Since J is weakly lower semi-continuous, bounded from below and coercive on H, using the relation

between critical points of J and problem (4.1), we deduce that J has a minimizer which is a weak

solution to problem (4.1).



90 R. Sanou, I. Ibrango, B. Koné, A. Guiro CUBO
21, 3 (2019)

Competing interests

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

Acknowledgment

The authors express their deepest thanks to the editor and anonymous referee for their comments

and suggestions on the article.



CUBO
21, 3 (2019)

Weak solutions to Neumann discrete nonlinear system of Kirchhoff . . . 91

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	Introduction
	Mathematical background
	Existence of weak solutions
	 An extension