Homoclinic solutions of non-autonomous difference equations arising in hydrodynamics

• Published in: Nonlinear analysis: real world applications Vol. 12; no. 1; pp. 14 - 23
• Main Authors: Rachunek, Lukas, Rachunkova, Irena
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-02-2011
• Subjects: Computational fluid dynamics; Difference equations; Fluid flow; Homoclinic solutions; Hydrodynamics; Mathematical analysis; Mathematical models; Non-autonomous second-order difference equation; Nonlinearity; Strictly increasing solutions; Homoclinic solutions; Strictly increasing solutions; Non-autonomous second-order difference equation
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2010.05.031

The paper deals with the second-order non-autonomous difference equation
x
(
n
+
1
)
=
x
(
n
)
+
(
n
n
+
1
)
2
(
x
(
n
)
−
x
(
n
−
1
)
+
h
2
f
(
x
(
n
)
)
)
,
n
∈
N
,
where
h
>
0
is a parameter and
f
is Lipschitz continuous and has three real zeros
L
0
<
0
<
L
.
We provide conditions for
f
under which for each sufficiently small
h
>
0
there exists a homoclinic solution of the above equation. The homoclinic solution is a sequence
{
x
(
n
)
}
n
=
0
∞
satisfying the equation and such that
{
x
(
n
)
}
n
=
1
∞
is increasing,
x
(
0
)
=
x
(
1
)
∈
(
L
0
,
0
)
and
lim
n
→
∞
x
(
n
)
=
L
. The problem is motivated by some models arising in hydrodynamics.