I nternat. J. Math. & ath. Si.
Vol. 3 No.
(1980)69-77
69
EXISTENCE THEOREM FOR THE DIFFERENCE EQUATION
2
Y
-h f(y)
-2Y -I-Y
n
n-I
n
n+l
F. WElL
Department of Physics-Mathematics
Universit de Moncton
Moncton, N. B., Canada
(Received November 27, 1978)
ABSTRACT.
For the difference equation
(Yn+l
2Y
n
+
Yn-i
h2
sufficient conditions are shown such that for a given
unique value of
Y1
for which the sequence
Yn
Y0
f(Y
n
there is either a
strictly monotonically approaches
a constant as n approaches infinity or a continuum interval of such values.
It has been shown previously that the first alternative is related to the
existence of a Peierls barrier energy in the dislocation model of Frenkel and
Kont o rova.
KEY WORDS AND PHRASES.
Existence Theorem,
Difference equations.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
I.
A3910
INTRODUCTION.
In this paper we discuss the conditions for the existence and uniqueness
70
F. WElL
of the solution to the nonlinear difference equation
Y
2Y
n+l
n
+Y n-i
f (Y)
n
h2
introduced in
of Hobart (1965).
3
_
conditions are
(i.I)
As stated there, the boundary
(i. 2a)
n _> 0
0
Y
The function
< Y <
0
f
Y!
<
(l.2b)
sm ;
n
as
n
(s.m.
strictly
mono toni cally
/
is odd, twice differentiable, negative on the interval
of Hobart (1965).
3
h2
assumption gives us reason not to incorporate
2.
We assume also that
and zero at the ends of this interval.
has been standardized according to
f
(i. 2c)
This
f.
in
DEFINITIONS.
Consider first the difference equation (i.i) together with the
conditions (l.2a) and (l.2b) only
0 _<
arbitrarily a value of
Y
n
Y!
For a given
<
we can choose
Y0
and then, step by step calculate
Only three cases occur
Y0
<
Y1 <
Y0 < Y1 <
<
YN -<
< Y
n
<
> Y
<
< Y
n
<
Yn+l
Y
with
n+l
or
Y0 <<Y
(2.1a)
YN+I
<
n
<
,
(2.1b)
Yo >- Y1
<
(2.1c)
EXISTENCE THEOREM FOR DIFFERENCE EQUATION
We shall call a value of
["small"
holds, a "large"
for which (2.1a)
Y1
"correct"] Y1"
or
[(2.1b)
or
(2.1c)]
Furthermore a "too
["too small"] YI is a large [small] Y!
[smaller] Y1 is large [small].
large"
We notice that the Y’s
n
71
for which any larger
corresponding to a correct
strictly monotonically increasing sequence bounded by
Y’s have a limit
n
+
24
satisfy
as
approaches infinity.
n
h 2 f(%)
Z
f(E)
or
form a
Y1
Thus the
E
The limit
must
for which, according to
0
the restrictions stated in the introduction, the only suitable root is
THEOREM I.
3.
If
Y
2Y
n+l
d 2 f (Y)
+
n
Y
h 2 f(Y
n-I
f (Y)
n
exists,
dy 2
is constant,
Y0
1
+
h2
(Y)
> 0
4 _< n _< m
implies
f(Y) > 0
on
0
<
0 < Y <
dY
then
dY
> 0
for each
_
Yn-2 < Yn-i
2 _< n
<
and
m.
n-I
dY
PROOF.
We first show
----n--n
dY
n-i
> 0
for
n _- 2
and
n
3
72
F. WElL
dY 2
2
dY
+
dY
(YI
h2
> 1 >
(3.1)
0
dY
(3.2)
h
and
dY 3
-----> i
dY
so
+
h2
(Y2) > 0.
(3.3)
2
dY
Now we show
dYn_ 1
and
dY
n
-------->
0
4 _< n _< m
for
dYn_ 1
assuming
dYn 2
-------->
dYn_ 3
0
> 0
n-2
dY
n
dY
dYn_ 2 +
dY
: 2
n-I
h2
n-i
(Yn-i
(3.4)
dY
Thus
dY
> 0
if
n-i
h2
(Yn_l
>
2
+
(2
Since
dYn_ 1
dY
< Y
4 < n <_ m
n-2
+
h
(Yn_2)
> 0 has been assumed
dYn-3
0
dY
n-2
(3.5)
2
n-2
dY
But
dYn_ 3
< Y<
Thus
resulting in
so
.
< 2 + h2
n-2
0 <
(Yn-2
Yn-2 < Yn-i
(3.6)
and
f (Y) > 0
on
(Yn-2)
< f ()
(3.7)
dY--n-2 <
2+h
(3.8)
EXISTENCE THEOREM FOR DIFFERENCE EQUATION
And also
73
< f (Yn-I)
(Yn-2)
(3.9)
dY
Using (3.8) and (3.9)we can modify (3.5) to
h2
if
(Yn_2
>-2
dYn_ 1
> 0
i
+
(3 .i0)
{ (Yn-2
(h2
dY
> 0
Noting (3.7) we obtain
(Yn_2)] a +
[h a
(I
+
2h 2
The two roots of this polynomial in
r_ <-
2
()
ha
and
<
r_ <
r+.
if
ha
(2
())
ha
<
(7))
(Yn_2)
[h a
(3.11)
0
(Yn_2
are such that
With (3.7) we obtain
(Yn_2) < r+
ha
(3.12)
on which range the polynomial is negative so (3.11) is satisfied.
dY
n >
0
Thus
for each 2 _< n _< m.
1
dYn_
4.
THEOREM II.
Under the assumptions of Theorem I and the additional assumption
f(Y) < 0
that
on
0 < Y
<
,
we show that large implies too large
and small implies too small.
PROOF.
of
YI
and
I
Y0 <
YIL <
Y!
Assume a constant value for
L
YI
that is large.
through
<
YLN -< YLN+I
is increased,
<
YN+I
and an initial value
This means there is an
2Yn + Yn-I
Yn+l
Y0
h
f
(Yn)
N
such that
Y0
give
By Theorem I (with
must also increase until
m
YN
N
+
i)
By
if
74
F. WElL
Theorem I (with
=
m
N)
YN-I
also increase until
L
Y!
Y!
Yl
During each step
is large, all
Y1
L
>
Y!
Yl*
If
and an initial value of
Y0
Y0
S
<
yS
<
<
YI
YI
with
Yn+l
0 < Y
<
yS
be a first
yS
n
>_
for
and
Yn+l
< 0
for some
0 < n
0 S
2Y
n
< p.
+
Y
Yn
Y
from
prior
Y
< 0 (n
if
S
0).
+
1
+
if
{_
5.
Thus if
Y! = Y1
is large)
is small, all
YI
<
that
f(Y) < 0
,
YI
is correct and for all
Y1
as
There is a contradiction in assuming a correct
YI.
such that
give that there must
YnS } with no
f(Yn
For neither case can such a
S
Y0
there were a correct
Y1
n
Y1
M
<_
Y
(If
Y’s increase or at least one
n
either all
n
1
< x
h2
n-i
If below
+
N
to and including
were correct, we would also
The assumptions that
0 < p < M
then by Theorem I (valid for all
n
S
Y!
YI
Y!
is increased
Y
n
YI
>
with no
be small.
below a small
are small.
ALGORITHM.
If
f(Y) < 0
on
0
< Y <
and zero for the end points, and if
the assumptions of Theorem I are satisfied, we can by the algorithm
described in
value of
YI
2
of the paper by Hobart (1965) construct a correct
for a given
bounded above by a large
S
Y!
that is either
is small, there is an
small implies too small is trivial).
on
YI*
If
S
Since
S
n
Y1*
Y!
YI
were large, this would contradict the argu-
ment that large implies too large.
have a contradiction.
Thus if
is large.
Yl
Now suppose there is a smaller
large or correct.
must also
Y2
are large.
Assume a constant valu for
that is small.
must
is finite, repetition of
N
Since
W
this process can be continued giving finally that
increase until
YN
is further increased, now
Y
if
Y0
YI
This involves choosing an interval
and below by a small
Y!
(initially
EXISTENCE THEOREM FOR DIFFERENCE EQUATION
0
<
< )
!
75
testing the midpoint for large or small, retaining the
(half) interval bounded as the original, and repeating the process on
this interval.
If the midpoint is at no step correct, this process leads to a
unique limit point which we shall now argue must be a correct point
and the only correct point.
Certainly there are no correct points to
be found on the discarded intervals for if the midpoint is large
[small],
[small].
the discarded interval contains only points which are large
Since
f
is differentiable, it is continuous.
points in an infinitesimal neighbourhood of a large
be large
[small].
Thus all
[small]
point must
But the limit point has in its neighbourhood both
large and small points, so it must therefore be a correct point and
the only correct point.
If the midpoint is at some step a correct point, either it is the
only correct point or there is a continuous interval of correct points.
Two correct points cannot be separated by a large
above
[below]
points.
a large
[small]
[small]
point since
point there can be only large
A test can be made which distinguishes between an isolated
correct point and a continuum interval of correct points:
midpoint
[small]
Yl
C
Yl
If a
is correct, apply the algorithm described above
separately to the intervals
C
Y!
<
Y
<
0 <
and
no step for either the midpoint is correct, then
Y!
C
Y
<
C
Yl
If at
is unique.
If
the midpoint for either is at some step correct, then there is a
continuum interval.
6.
THEOREM III.
Unless the result of the algorithm is a continuous interval of
correct values of
Y1
it defines a unique
Y1
g(Y0
for each
Y0
76
F. WElL
0 _<
on
Y
<
Y0
0
For application to the Frenkel-Kontorova model, we
is small.
.
0
each
Y
If
n
then all
0,
(YI)
by showing that for
0
Yn+l =
Relabel
f
YI
so that
>
Y0 > -8.
8
< 8
are small for
is the unique
7.
<
Y
is small.
is small for
0
Y1
for
Noting that
0.
Y0
it follows that
f(Y) < 0
0
on
< Y <
g(Y0)
Y
there is one and only one correct
<
Yl = 0
correct Y!
in this domain
and
are satisfied, then either for each
Y
to include
g
If the assumptions of Theorem I and the additional
one and only one correct
-
Y0
Yl
Y2 = -Y0
0
assumptions that
<
0
THEOREM III.
or
and note that necessarily
g(0)
We now extend the domain of definition
-8 < Y0 <
Y0
8
Define
need the domain extended.
8 <
is large and
YI
since it is easily verified that
f(Y)
0 <_
and for each
g(Yo)
Yl
or
for
0
Y
0
Y0 <
there is
-g(0) <
Y0
< 0
for some
Y
there is a continuous interval of correct values of
APPLICATION.
is odd and that for the function
f
Assuming the function
function
g
values, we can use the
YI
chosen there is no continuum of correct
f
to define the path of configurations connecting II with I
in the Frenkel-Kontorova (1938) model as generalized in Hobart (1965).
For a given
,
as
n
y
s.m.__r
n
/
-g(0) <
that is
as
n
Y0 <
g(0),
Y1
YI
g(Y0
and
Y_I
that is
Y_!
/
,
equation (i.I) is satisfied for all
is chosen so that
n
g(Y0)
(y0)
Y
n
...s.m.
is chosen so that
-g(-Y0).
The difference
except zero for which
2Y0
f(y0
h
2
g(-Y0)
C7.1)
EXISTENCE THEORKM FOR DIFFERENCE EQUATION
77
is the nonzero external force only on the zeroth atom which is
necessary to hold static a general intermediate configuration.
configurations I and II are given by the conditions that
and
II
Y0
Y0
given by
II <
Y0
-g(Y
I)
< 0
I
Y0"
Y0
Y0
The
Y
0
The connecting path is
respectively.
The barrier energy is
I
V(I)
V(II)
JII
() d
(7.2)
REFERENCES.
i.
Frenkel, J. and Kontorova, T.
On the theory of plastic deforma-
tion and twinning, Phys. Z. Sowjet 13 (1938) i-i0.
2.
Frenkel, J. and Kontorova, T.
and twinning,
3.
Hobart, R.
Series of plastic deformation
J_ hys. (U.S.S:_R.)
i (1939) 137-149.
Peierls stress dependence on dislocation width,
j..App!. phys.
36
(1965) 1944-1948.