EXISTENCE OF SOLUTION TO A SEMILINEAR DISCRETE

```No. 43/2015
pp. 45-51
doi: 10.17114/j.aua.2015.43.04
Acta Universitatis Apulensis
ISSN: 1582-5329
http://www.uab.ro/auajournal/
EXISTENCE OF SOLUTION TO A SEMILINEAR DISCRETE
PROBLEM INVOLVING P −LAPLACIAN
A. Ourraoui
Abstract. In this work, under appropriate assumptions, we present a result
on the existence of solutions of difference equation
−∆ |∆u(k − 1)|p−2 ∆u(k − 1) = f (k, u(k)) + g(k)|u(k)|q−2 u, k ∈ [1, T ].
Our method is based on the Ekeland’s variational principle.
2010 Mathematics Subject Classification: 47A75, 35B38, 35P30, 34L05, 34L30.
Keywords: Variational method; Discrete boundary value problem.
1. Introduction
Many papers devoted to study of the discrete equations have increased in recent
years, by using various methods and techniques as fixed point theorems, lower and
upper solutions, and variational approach, see for example [1, 2, 3, 4, 5, 6, 7, 10, 11,
12]...
Let T ≥ 2 be a fixed positive integer, [a, b] be the discrete interval {a, a + 1, ..., b}
where a and b are integers and a &lt; b.
Motivated by the above mentioned papers, we deal with the following discrete
boundary value problem
−∆ |∆u(k − 1)|p−2 ∆u(k − 1) = f (k, u(k)) + g(k)|u(k)|q−2 u, k ∈ [1, T ].
(1)
u(0) = u(T + 1) = 0,
where ∆u(k) = u(k + 1) − u(k) is the forward difference operator, 1 &lt; q &lt; p &lt; ∞
while f : [1, T ] &times; R → R is a continuous function.
Denoting by F : [1, T ] → R the primitive of f i.e.,
Z u
F (k, u) =
f (k, s)ds, u ∈ R.
0
45
A. Ourraoui – Existence of solution to a semilinear discrete problem . . .
There is by now a rich literature on problems like (1), which began to receive
much attention and are known to be mathematical models of various phenomena
arising in the study of elastic mechanics, control systems, artificial or biological
neural networks, computing, electrical circuit analysis, dynamical systems, economics,...For more detail, we may refer to [8, 13].
Here, we are concerned with investigating nonlinear discrete boundary value
problems by using variational approach.
The rest of this paper is arranged as follows. In section 2, we recall some basic
definitions and tools in order to prove our main result and at after all we give an
example.
2. Preliminaries and statement of main results
In this section, we recall some basic properties which will be used in the proof of
the precise result.
Solutions to (1) will be investigated in a space
n
o
W = u : [0, T + 1] → R s.t u(0) = u(T + 1) = 0 ,
which is a T -dimensional Hilbert space, see [2], with the inner product
(u, v) =
T
+1
X
∆u(k − 1)∆v(k − 1),
for all u, v ∈ W.
k=1
The associated norm is defined by
kuk =
+1
TX
|∆u(k − 1)|p
1
p
.
k=1
Moreover, it is useful to introduce other norms on W , namely
|u|m =
T
X
|u(k)|m
1
m
,
∀u ∈ W and m ≥ 2.
k=1
It can be verified (see [6]) that
T
2−m
2m
1
|u|2 ≤ |u|m ≤ T m |u|2 ,
∀u ∈ W and m ≥ 2.
Next, we list some inequalities that will be are used later.
46
(2)
A. Ourraoui – Existence of solution to a semilinear discrete problem . . .
Lemma 1. ([10]) For every u ∈ W , we have
(a)
T
X
m
|u(k)|
≤ T (T + 1)
m−1
k=1
(b)
T
+1
X
T
+1
X
|∆u(k − 1)|m ,
∀m ≥ 2.
k=1
p(k−1)
|∆u(k − 1)|
≤2
m
k=1
T
X
|u(k)|m ,
∀m &gt; 1.
k=1
We say that u ∈ W is a weak solution of problem (1) if
T
+1 X
p(k−1)−2
|∆u(k − 1)|
T
T
X
X
∆ϕ(k − 1) =
f (k, u(k))ϕ(k) + &micro;
g(k)|u(k)|p ϕ(k),
k=1
k=1
k=1
for any ϕ ∈ W .
Related to problem (1), define the functional φ : W → R given by
φ(u) =
T
+1 X
k=1
T
T
k=1
k=1
X
1X
1
F (k, u(k)) −
|∆u(k − 1)|p −
g(k)|u(k)|q .
p
q
We assume the following conditions.
(g1 ) g ∈ L∞ ([1, T ]), g ≥ (6≡)0.
(f1 ) f (x, t) ∈ C([1, k] &times; R), f 6≡ 0.
f (k, t)
f (k, t)
&lt; λ1 and λ1 &lt; l2 := lim p−1 &lt; ∞
p−1
t→∞ t
t→0 |t|
uniformly in [1, T ] where λ1 &gt; 0 is the smallest positive eigenvalue of
−∆ |∆u(k − 1)|p−2 ∆u(k − 1) = λ|u(k)|p−2 u, k ∈ [1, T ],
(f2 ) 0 ≤ l1 := lim
(3)
u(0) = u(T + 1) = 0,
see [2] .
The following theorem is the main result of this paper.
Theorem 2. If (g1 ), (f1 ) and (f2 ) hold true, then the problem (1) has at least one
nontrivial weak solution.
Proof of Theorem 2. It is clear that the functional φ is differentiable and its derivative is given by
0
φ (u).v =
T
+1
X
|∆u(k−1)|
p−2
T
T
X
X
∆u(k−1)∆v(k−1)−
f (k, u(k))v(k)−
g(k)|u(k)|p−2 u(k)v(k),
k=1
k=1
47
k=1
A. Ourraoui – Existence of solution to a semilinear discrete problem . . .
for all u, v ∈ W.
For simplicity, we give an auxiliary result about f :
In view of (f1 ) and (f2 ) with l2 &lt; ∞ we have
lim
t→∞
f (k, t)
= 0, for p &lt; r &lt; ∞, uniformly in
tr−1
[1, T ].
Therefore, for any ε &gt; 0 there exists C(ε) &gt; 0 such that
f (k, t) ≤ (l1 + ε)tp−1 + C(ε)tr−1 for all t ≥ 0, k ∈ [1, T ],
and thus
r
F (k, t) ≤ l1p+ε tp + C(ε)
r t , for all t ≥ 0, k ∈ [1, T ].
On the other hand, according to (f2 ) and the fact that l1 &lt; λ1 , we may find ε1 &gt; 0
such that
l1 + ε1 &lt; λ1 ,
F (k, t) ≤
l 1 + ε1 p
t + C0 tr , for all t ≥ 0, k ∈ [1, T ]
p
with C0 &gt; 0 is nonnegative constant.
We present the following Lemma:
Lemma 3. There exist ρ &gt; 0 and α &gt; 0 such that
φ(u) ≥ α &gt; 0, for every u ∈ W, with kuk = ρ.
Proof of Lemma 3. By using of Lemma 1 we have
φ(u) =
T
+1 X
k=1
≥
T
T
k=1
k=1
X
1
1X
F (k, u(k)) −
|∆u(k − 1)|p −
g(k)|u(k)|q
p
q
T
T
T
1
C(ε) X
|g|∞ X
l1 + ε X
p
r
p
|u(k)| −
|u(k)| −
|u(k)|q
kuk −
p
p
r
q
k=1
≥
k=1
1
|g|∞
kukp −
T (T + 1)q−1
p
q
−C(ε)T (T + 1)r−1
T
+1
X
T
+1
X
k=1
k=1
T +1
X
l1 + ε1
|∆u(k − 1)| −
T (T + 1)p−1
|∆u(k − 1)|p
p
q
k=1
|∆u(k − 1)|r
k=1
≥ C1 kukp − |g|∞ C2 kukq − C3 kukr &gt; 0,
1
&gt; 0 and C2 , C3 are positive
for kuk is small enough, with C1 = p1 1 − l1λ+ε
1
constants.
48
(4)
A. Ourraoui – Existence of solution to a semilinear discrete problem . . .
Define
B ρ = {u ∈ E : kuk ≤ ρ}
endowed with metric
d(u, v) = ku − vk, u, v ∈ B ρ ,
and
∂Bρ = {u ∈ W : kuk = ρ}.
By the previous Lemma 3, we know that
φ(u)/∂Bρ ≥ α &gt; 0
and it is easy to see that the functional φ is bounded from below.
Let c∗ = minB ρ φ(u), so we have c∗ &lt; 0. In fact, fix a nonnegative function v ∈
W \ {0}, and note that for t &gt; 0 we have
φ(tv) =
T
+1 X
k=1
≤
tp
p
T
T
X
1
1X
|∆tv(k − 1)|p −
F (k, tv(k)) −
g(k)|tv(k)|q
p
q
kvkp −
k=1
tp
l1
p
T
X
v(k)p −
k=1
tq
q
k=1
T
X
g(k)v(k)q .
k=1
Since q &lt; p, the last inequality implies that for some t∗ &gt; 0 sufficiently small,
φ(t∗ v) &lt; 0, and t∗ v ∈ B ρ (0).
By the Ekeland’s variational principle in [9], we may find a sequence (un )n ⊂ B ρ (0)
such that
φ(un ) → c∗ i.e c∗ ≤ φ(un ) ≤ c∗ + n1 , ∀n &gt; 0
and
1
φ(v) ≥ φ(un ) − kun − vk, ∀v 6= un .
n
Once that φ is differentiable, it follows from the last inequality that
φ0 (un ) → 0.
Indeed, let u ∈ W such that kuk = 1, putting ωn = un + λu. Fix n &gt; 1 we have
kωn k ≤ kun k + λ &lt; ρ
with λ &gt; 0 is small enough. It yields
φ(un + λu) ≥ φ(un ) −
49
λ
kuk
n
A. Ourraoui – Existence of solution to a semilinear discrete problem . . .
then
φ(un + λu) − φ(un )
1
≥ − kuk,
λ
n
tending λ → 0 so we get
φ0 (un ).u ≥ −
1
n
that is,
|φ0 (un ).u| ≤
1
n
with kuk = 1. So φ0 (un ) → 0.
Since the sequence (un ) is bounded in W, there exists u1 ∈ W such that, up to
a subsequence, (un ) converges to u1 in W. Therefore,
φ(u1 ) = c∗ , and φ0 (u1 ) = 0,
and thus u1 is a non trivial solution of problem (1).
Example 1. Taking β &gt; 0 and l2 &gt; λ1 , the function f such that
tβ+1
f (x, t) = l21+t
β when t ≥ 0 and f (x, t) = 0 if t ≤ 0,
satisfies the conditions (f1 ) and (f2 ) provided l2 &lt; ∞.
f (k, t)
&lt; λ1 , k ∈ [1, T ] then, it is easy to
p−2 t
|t|→∞ |t|
check that there exists 0 ≤ λ &lt; λ1 such that
1
λ
φ(u) ≥
1−
kukp − C4 kuk − C5 kukq ,
(5)
p
λ1
Remark 1. If we assume that lim sup
where C4 and C5 are positive constants.
Thus φ is coercive and bounded from below so it has a global minimizer.
References
[1] R.P. Agarwal, D. O’Regan, Boundary value problems for discrete equations,
Appl. Math. Lett 10 (1997), 83-89.
[2] R. P. Agarwal, K. Perera, D. O’Regan, Multiple positive solutions of singular
and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (2004),
69-73.
[3] R. P. Agarwal, K. Perera, and D. O’Regan, Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Advances in Difference
Equations, no. 2 (2005), 93-99.
50
A. Ourraoui – Existence of solution to a semilinear discrete problem . . .
[4] A. Ayoujil, On class of nonhomogeneous discrete Dirichlet problems, Acta Universitatis Apulensis, No. 39/2014 pp. 1-15. (in press)
[5] A. Ayoujil, On class of discrete boundary value problem via variational methods,
Afrika Matematika. (DOI 10.1007/s13370-014-0293-4.)
[6] X. Cai, J. Yu, Existence theorems for second-order discrete boundary value problems. J. Math. Anal. Appl. 320 ((2006)), 649-661.
[7] P. Candito, G. D’Agui, Three solutions to a perturbed nonlinear discrete Dirichlet problem, Journal of Mathematical Analysis and Applications, vol. 375, no. 2(
2011), 594-601.
[8] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in
image processing, SIAM J. Appl. Math. 66 (2006), No. 4, 1383-1406.
[9] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 3 (1979)
443-474.
[10] M. Galewski, R. Wieteska, Existence and multiplicity of positive solutions for
discrete anisotropic equations, Turk. J. Math, DOI: 10.3906/mat-1303-6.
[11] L. Kong, Homoclinic solutions for a second order difference equation with pLaplacian, Applied Mathematics and Computation 247 (2014) 1113-1121.
[12] B. Kone, S. Ouaro, Weak solutions for anisotropic discrete boundary value problems, J. Difference Equ. Appl., 18 February (2010).
[13] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity
theory, Math. USSR Izv. 29 (1987), 33-66.
Anass Ourraoui
Department of Mathematics, ENSAH,
University of Mohamed I,
Oujda, Morocco
email: anas.our@hotmail.com
51
```