CUBO A Mathematical Journal
Vol.19, No¯ 03, (57–67). October 2017

Existence of solutions for discrete boundary value problems
with second order dependence on parameters

Aboudramane Guiro and Idrissa Ibrango

Laboratoire de Mathmatiques et Informatique (LAMI)

Unit de Formation et de Recherche en Sciences et Techniques

Universit Nazi Boni

01 BP 1091 Bobo-Dioulasso 01

Bobo Dioulasso, Burkina Faso

abouguiro@yahoo.fr, ibrango2006@yahoo.fr

ABSTRACT

We prove the existence of non trivial solution for discrete nonlinear problems of Kirch-

hoff type. The proof of the main result is based on a mountain pass lemma.

RESUMEN

Demostramos la existencia de soluciones no triviales para problemas discretos no lin-

eales de tipo Kirchhoff. La demostración del resultado principal está basado en un lema

del paso de la montaña.

Keywords and Phrases: Kleinian Groups, NEC groups

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



58 Aboudramane Guiro and Idrissa Ibrango CUBO
19, 3 (2017)

1 Introduction

The aim of this paper is to extend the following discrete boundary value problem studied by Joanna

et al in [10] which is subject to certain parameter w ∈ W:






−∆
(

|∆u(k − 1)|p(k−1)−2∆u(k − 1)
)

+ f
(

k, u(k), w
)

= 0, k ∈ [1, N]

u(0) = u(N + 1) = 0,

(1.1)

where N is a positive integer with N ≥ 2, ∆u(k) = u(k + 1) − u(k) is the forward difference

operator, W is some topological space and [1, N] = {1, 2, . . . , N}. Using the mountain pass lemma

the authors in [10] proved an existence of non trivial solutions and gives conditions under which

the problem (1.1) has a unique solution. In this paper we deal with the following discrete boundary

value problem






−∆
(

a
(

k − 1, ∆u(k − 1)
))

= δ(k)f
(

k, u(k), w
)

, k ∈ [1, N]

∆u(0) = ∆u(N) = 0,

(1.2)

where a, f, δ are functions to be defined later. Remark that in this paper, we consider another

boundary conditions (Neumann boundary condition). Also, the function a(k−1, ∆u(k−1)) which

appear in the left-hand side of problem (1.2) is more general than the one which appear in [10].

Note that difference equations can be seen as a discrete counterpart of partial differential equations

and are usually studied in connection with numerical analysis. In this way, the main operator in

problem (1.2)

∆(a(k − 1, ∆u(k − 1)))

can be seen as a discrete counterpart of the anisotropic operator

N∑

i=1

∂

∂xi
a

(

x,
∂

∂xi
u

)

.

The theory of difference equations occupy a central position in applicable analysis (see [1]-[3] or

[6]-[9], [12, 13]) motivated by applications in various biological, physical and chemical models, such

has phase-transition, physical processes in which the variable transits from an unstable equilibrium

to a stable one. . . For example, using the three critical points theorem (see [7], Theorem 2.6 ),

Candito and D’Agui studied in [4] the existence of three solutions for the problem






−∆
(

Φp(∆uk−1)
)

+ qkΦp(u(k)) = λf(k, uk), k ∈ [1, N]

∆u0 = ∆uN = 0

(1.3)

where λ is a real positive parameter, qk > 0 for all k ∈ [1, N], Φk(s) = |s|
p−2s with 1 < p < +∞

and f : [1, N] × R −→ R is a continuous function. The authors obtain a suitable interval of



CUBO
19, 3 (2017)

Existence of solutions for discrete boundary value . . . 59

parameters for which problem (1.3) admits constant-sign solutions which are local minimizers of

the corresponding Euler-Lagrange functional.

Motivated by the paper [6] and the ideas in [10], we used the mountain pass lemma to prove an

existence of a non trivial solution under some hypothesis. With additional conditions we prove the

uniqueness of the non trivial solution.

In order to bring evidence our main result, in the Section 2 we deal with some preliminary materials.

In Section 3, using the mountain pass lemma, we prove the existence of an non trivial solution and

adding some conditions, we study the uniqueness of solution.

2 Preliminaries

In the N-dimensional Hilbert space

X = {x : [0, N + 1] −→ R; such that ∆x(0) = 0 = ∆x(N)}

with the inner product

(x, y) =

N+1∑

k=1

∆x(k − 1)∆y(k − 1), ∀ x, y ∈ X,

we consider the norm

‖x‖ =

( N+1∑

k=1

|∆x(k − 1)|2
)

1
2

. (2.1)

Let the function

p : [0, N] −→ [2, +∞) (2.2)

and denoted by

p− = min
k∈[0,T]

p(k) and p+ = max
k∈[0,T]

p(k).

For the data a, f and δ, we assume the following:

(H0).

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

which satisfies a(k, ξ) = ∂
∂ξ

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

(H1). For any k ∈ [0, N], ξ ∈ R there exist a constant c0 > 0 such that

|a(k, ξ)| ≤ c0

(

1 + |ξ|p(k)−1
)

. (2.3)

(H2). For any k ∈ [0, N], ξ ∈ R, we have

A(k, ξ) ≥
1

p(k)
|ξ|p(k). (2.4)



60 Aboudramane Guiro and Idrissa Ibrango CUBO
19, 3 (2017)

(H3). For each k ∈ [0, N], the function f(k, ., .) : R × W −→ R is jointly continuous and there

exists a constant c1 > 0 and a function r : [0, N] −→ (1, +∞) such that

|f(k, ξ, w)| ≤ c1
(

1 + |ξ|r(k)−1
)

(2.5)

where the space W is some topological space. We denoting

F(k, ξ, w) =

∫ξ

0

f(k, s, w)ds for (k, ξ, w) ∈ [0, N] × R × W (2.6)

and we deduce that there exist a constant c2 > 0 such that

|F(k, ξ, w)| ≤ c2
(

1 + |ξ|r(k)
)

. (2.7)

(H4). lim
ξ→0

f(k, ξ, w)

|ξ|p
+−1

= 0 uniformly for all k ∈ [1, N], w ∈ W.

(H5). δ : [0, N] −→ (0, +∞) such that for all k ∈ [0, N],

0 < δ = inf
k∈[0,N]

(δ(k)) ≤ δ(k) ≤ δ̄ = sup
k∈[0,N]

(δ(k)) < +∞. (2.8)

(H6). There exists constants c3, c4 > 0 and µ > p
+ such that

F(k, ξ, w) ≥ c3|ξ|
µ − c4, for all k ∈ [0, N], ξ ∈ R, w ∈ W. (2.9)

Example 2.1.

We give the following functions where a non trivial solution can be estimate as in [10]:

• A(k, ξ) =
1

p(k)
|ξ|p(k), where a(k, ξ) = |ξ|p(k)−2ξ, ∀ k ∈ [0, N] and ξ ∈ R,

• 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 ∈ [0, N],

ξ ∈ R,

• f(k, ξ, w) = 1 +
∣

∣ξ
∣

∣

p(k)−1
, ∀ k ∈ [0, N] and ξ ∈ R,

• δ(k) = 1, ∀ k ∈ [0, N].

The conditions (H0) − (H6) are fulfilled.

Moreover we may consider X with the following norm

|x|m =

( N∑

k=1

|x(k)|m
)

1
m

, ∀ x ∈ X and m ≥ 2. (2.10)

We have the following inequalities (see [3, 9])

N(2−m)/(2m)|x|2 ≤ |x|m ≤ N
1/m

|x|2, ∀ x ∈ X and m ≥ 2. (2.11)



CUBO
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Existence of solutions for discrete boundary value . . . 61

We defined the convex modular ρ : X −→ R by

ρ(x) =

N+1∑

k=1

|∆x(k − 1)|p(k−1)

and we introduce the Luxembourg norm

‖x‖p(.) = inf
{
λ > 0 : ρ(x/λ) ≤ 1

}
. (2.12)

Since X has a finite dimension there exist constants c5 > 0, c6 > 1 such that

c5‖x‖p(.) ≤ ‖x‖ ≤ c6‖x‖p(.). (2.13)

The followings inequalities holds:

min

{

‖x‖
p−

p(.)
; ‖x‖

p+

p(.)

}

≤ ρ(x) ≤ max

{

‖x‖
p−

p(.)
; ‖x‖

p+

p(.)

}

. (2.14)

We need the following auxiliary results throughout our paper (see [5]).

Lemma 2.2.

(1) For every x ∈ X and for every m ≥ 2 we have

N+1∑

k=1

|∆x(k − 1)|m ≤ 2m
N∑

k=1

|x(k)|m (2.15)

and
N+1∑

k=1

|∆x(k − 1)|m ≥ (N + 1)
2−m

2 ‖x‖m. (2.16)

(2) For every x ∈ X and for every m > 1 we have

N∑

k=1

|x(k)|m ≤ N(N + 1)m−1
N+1∑

k=1

|∆x(k − 1)|m. (2.17)

(3) For every x ∈ X and for every m ≥ 1 we have

N+1∑

k=1

|∆x(k − 1)|m ≤ (N + 1)‖x‖m. (2.18)

(4) For every x ∈ X we have

N+1∑

k=1

|∆x(k − 1)|p(k−1) ≤ (N + 1)‖x‖p
+

+ (N + 1). (2.19)



62 Aboudramane Guiro and Idrissa Ibrango CUBO
19, 3 (2017)

Definition 2.3. A C1 functional J satisfies the Palais-Smale condition (in short (PS) condition)

if any sequence {un} ⊂ X such that {J(un)} is bounded and

J′(un) −→ 0, as n −→ ∞

has a convergent subsequence.

From (PS) condition, it follows that the set of critical points for a bounded functional is compact.

This condition was a base for the modern development of critical point theory. It is needed for the

mountain pass lemma (see [11]).

Lemma 2.4. Let X be a real reflexive Banach space. Assume that J ∈ C1(X, R) and J satisfies the

(PS) condition. Suppose also that:

(1) J(0) = 0,

(2) there exist ρ > 0 and α > 0 such that J(u) ≥ α for all u ∈ X with ‖u‖ = ρ,

(3) there exists u1 ∈ X with ‖u1‖ > ρ such that J(u1) < α.

Then, J has a critical value c ≥ α. Moreover, c can be characterized as

inf
g∈Γ

(

max
u∈g([0,1])

J(u)
)

,

where Γ = {g ∈ C([0, 1], X) : g(0) = 0, g(1) = u1}.

Lemma 2.5 ([10], Lemma 2.2). Let X be a finite dimensional Banach space and let J ∈ C1(X, R)

be an anti-coercive functional (namely, lim
‖u‖−→+∞

J(u) = −∞). Then J satisfies the (PS) condition.

3 Existence of nontrivial solutions

We define the energy functional J : X −→ R by

J(u) =

N+1∑

k=1

A
(

k − 1, ∆u(k − 1)
)

−

N∑

k=1

δ(k)F
(

k, u(k), w
)

(3.1)

and we put

I(u) =

N+1∑

k=1

A
(

k − 1, ∆u(k − 1)
)

and Λ(u) =
N∑

k=1

δ(k)F
(

k, u(k), w
)

.

Lemma 3.1. The functional J is well defined on X and is of class C1
(

X, R
)

with the derivative

given by

〈J′(u), v〉 =

N+1∑

k=1

a(k − 1, ∆u(k − 1))∆v(k − 1) −

N∑

k=1

δ(k)f
(

k, u(k), w
)

v(k), (3.2)

for all u, v ∈ X.



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Existence of solutions for discrete boundary value . . . 63

Proof

We have |J(u)| ≤ |I(u)| + |Λ(u)|.

Since A(k, .) is continuous for all k ∈ [0, N], then

|I(u)| =

∣

∣

∣

∣

N+1∑

k=1

A
(

k − 1, ∆u(k − 1)
)

∣

∣

∣

∣

< +∞.

Also from (2.7) and Holder’s inequality we have

|Λ(u)| =

∣

∣

∣

∣

N∑

k=1

δ(k)F
(

k, u(k), w
)

∣

∣

∣

∣

≤

N∑

k=1

|δ(k)||F
(

k, u(k), w
)

|

≤ c2δ̄

N∑

k=1

(

1 + |u(k)|r(k)
)

≤ Nc2δ̄ + c2δ̄

N∑

k=1

|u(k)|r(k)

≤ 2Nc2δ̄ + c2δ̄

N∑

k=1

|u(k)|r
+

≤ 2Nc2δ̄ + c2δ̄

( N∑

k=1

∣

∣|u(k)|r
+∣

∣

2
r+

)
r+

2
( N∑

k=1

|1|
1

1− r
+

2

)1− r
+

2

≤ 2Nc2δ̄ + c2δ̄N
2−r+

2

( N∑

k=1

|u(k)|2
)

r+

2

< +∞.

Then, the energy functional J is well defined on X.

As in [6], Lemma 3.4, we can prove that the functional I derivative is given by

〈I′(u), v〉 =

N+1∑

k=1

a
(

k − 1, ∆u(k − 1)
)

∆v(k − 1). (3.3)



64 Aboudramane Guiro and Idrissa Ibrango CUBO
19, 3 (2017)

On the other hand, for all u, v ∈ X, we have

〈Λ′(u), v〉 = lim
h→0+

Λ(u + hv) − Λ(u)

h

= lim
h→0+

N∑

k=1

δ(k)
F(k, u(k) + hv(k), w) − F(k, u(k), w)

h

=

N∑

k=1

δ(k) lim
h→0+

F(k, u(k) + hv(k), w) − F(k, u(k), w)

h

=

N∑

k=1

δ(k)f(k, u(k), w).

The functional J is clearly of class C1 �

Lemma 3.2. The functional J is anti-coercive.

Proof

Let ‖u‖p(.) > 1. According to (2.3) and the inequality (2.14) there exist a constant c7 > 0 such

that

I(u) ≤ c7(N + 1) + c7

N+1∑

k=1

|∆u(k − 1)|p(k−1)

≤ c7(N + 1) + c7‖u‖
p+

p(.)
. (3.4)

According to (2.9), (2.15) and (2.16) we have

−Λ(u) = −

N∑

k=1

δ(k)F(k, u(k), w)

≤ −

N∑

k=1

δ(k)

(

c3|u(k)|
µ − c4

)

≤ −δc3

N∑

k=1

|u(k)|µ + c4δN

≤ −δc32
−µ

N+1∑

k=1

|∆u(k − 1)|µ + c4δN

≤ −δc32
−µ(N + 1)

2−µ
2 ‖u‖µ + c4δN. (3.5)

Consequently, combining (3.4) and (3.5) we get

J(u) ≤ −δc32
−µ(N + 1)

2−µ
2 ‖u‖µ + c4δN + c7‖u‖

p+

p(.)
+ c7(N + 1). (3.6)



CUBO
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Existence of solutions for discrete boundary value . . . 65

Since the norms ‖.‖ and ‖.‖p(.) are equivalents and since µ > p
+ then the functional J is anti-

coercive �

We can deduce from Lemma 2.5 that the function J satisfies the (PS) condition.

Theorem 3.3. For any w ∈ W, the problem (1.2) has at least one non trivial solution.

Proof

According to (H4), for any ε > 0 there exists β > 0 such that for all ξ ∈ [−β, β], |f(k, ξ, w)| ≤

ε|ξ|p
+
−1 for all k ∈ [1, N], w ∈ W.

As in [10], it is obvious to see that for any ε ∈

(

0,
(N+1)

2−p+

2

δ̄N(N+1)p
+

)

there exists β > 0 such that for

all x ∈ [−β, β] we have

|F(k, ξ, w)| ≤ ε
|ξ|p

+

p+
, ∀ k ∈ [1, N], w ∈ W. (3.7)

Let u ∈ X ∩
{
|u(k)| ≤ β, ∀ k ∈ [1, N]

}
∩
{
‖u‖ ≤ 1

}
. We have ‖u‖ ≤ 2β(N + 1)

1
2 and according

to (2.16) we have

N+1∑

k=1

|∆u(k − 1)|p(k−1) ≥

N+1∑

k=1

|∆u(k − 1)|p
+

≥ (N + 1)
2−p+

2 ‖u‖p
+

. (3.8)

If we put η = min{1; 2β(N + 1)
1
2 }, then for any u ∈ X with ‖u‖ ≤ η, by (3.7), (3.8), (H2), (2.17)

and (2.18) it follows that

J(u) ≥
1

p+

N+1∑

k=1

|∆u(k − 1)|p(k−1) − δ̄ε
1

p+

N∑

k=1

|u(k)|p
+

(3.9)

≥
1

p+
(N + 1)

2−p+

2 ‖u‖p
+

− δ̄ε
1

p+
N(N + 1)p

+

‖u‖p
+

(3.10)

≥ ‖u‖p
+ 1

p+

(

(N + 1)
2−p+

2 − δ̄εN(N + 1)p
+

)

. (3.11)

Remark that ε is such that (N + 1)
2−p+

2 − δ̄εN(N + 1)p
+

> 0. So there exists positive numbers

0 < ρ = ‖u‖ ≤ η and

α =
ρp

+

p+

(

(N + 1)
2−p+

2 − δ̄εN(N + 1)p
+

)

> 0

such that J(u) ≥ α for all u ∈ X with ‖u‖ = ρ. Also we have J(0) = 0 and since J is anti-coercive,

there exists u1 ∈ X with ‖u1‖ > ρ such that J(u1) < α.

By the mountain pass lemma (see Lemma 2.4), the functional J has a critical value c∗ > 0, i.e,

there exists u∗ ∈ X such that J(u∗) = c∗ and 〈J′(u∗), v〉 = 0, ∀ v ∈ X.

Since J(0) = 0 it is obvious to see that u∗ 6= 0. The critical value c∗ can be characterized as

c∗ = inf
v∈Γ

(

max
t∈[0,1]

J(v(t))
)

,



66 Aboudramane Guiro and Idrissa Ibrango CUBO
19, 3 (2017)

where Γ = {v ∈ C([0, 1], X) : v(0) = 0, v(1) = u1}. Then, the proof of the existence of solution to

problem (1.2) for any parameter w ∈ W is complete �

Now, in what follow, we examine conditions under which our problem (1.2) has a unique non

trivial solution. Assume that for any k ∈ [0, N], ξ1, ξ2 ∈ R and w ∈ W the followings additional

conditions are satisfies:

(H7). there exist a constant c8 > 0 such that

(

a(k, ξ1) − a(k, ξ2)
)

≥ c8|ξ1 − ξ2|
p+, (3.12)

(H8). there exist a constant 0 < d <
c8

δ̄N(N+1)
3p+−2

2

such that

|f(k, ξ1, w) − f(k, ξ2, w)| ≤ d|ξ1 − ξ2|
p+−1. (3.13)

Theorem 3.4. If u ∈ X is a non trivial solution of problem (1.2), if (H7) and (H8) are holds,

then for every fixed parameter w ∈ W, the solution u is unique.

Proof

Let u, v ∈ X two nonzero solutions of problem (1.2). Then

N+1∑

k=1

a(k − 1, ∆u(k − 1))∆(u − v)(k − 1) =

N∑

k=1

δ(k)f
(

k, u(k), w
)

(u − v)(k) (3.14)

and
N+1∑

k=1

a(k − 1, ∆v(k − 1))∆(v − u)(k − 1) =

N∑

k=1

δ(k)f
(

k, v(k), w
)

(v − u)(k). (3.15)

Upon addition, we get

N+1∑

k=1

[

a(k − 1, ∆u(k − 1)) − a(k − 1, ∆v(k − 1))

]

∆(u − v)(k − 1)

=

N∑

k=1

δ(k)

[

f
(

k, u(k), w
)

− f
(

k, v(k), w
)

]

(u − v)(k). (3.16)

Therefore, according to (H7) and (H8) we have

c8

N+1∑

k=1

|∆(u − v)(k − 1)|p
+

≤ dδ̄

N∑

k=1

|(u − v)(k)|p
+

. (3.17)

Since p+ ≥ 2, using the inequalities (2.16)-(2.18) we can write

c8(N + 1)
2−p+

2 ‖u − v‖p
+

≤ dδ̄N(N + 1)p
+
−1

N+1∑

k=1

|∆(u − v)(k − 1)|p
+

≤ dδ̄N(N + 1)p
+

‖u − v‖p
+

,



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Existence of solutions for discrete boundary value . . . 67

namely
[

c8

δ̄N(N + 1)
3p+−2

2

− d

]

‖u − v‖p
+

≤ 0. (3.18)

Recall that the constant d is such that
c8

δ̄N(N + 1)
3p+−2

2

− d > 0. Consequently,

‖u − v‖p
+

= 0, thus u = v �

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	Introduction
	Preliminaries
	Existence of nontrivial solutions