TY - JOUR

T1 - Regularity of the inversion problem for the Sturm-Liouville difference equation III. A criterion for regularity of the inversion problem

AU - Chernyavskaya, Nina

AU - Schiff, J.

AU - Shuster, Leonid

PY - 2005

Y1 - 2005

N2 - We consider a difference equation
−h
−2∆(2)yn + qn(h)yn = fn(h), n ∈ Z = {0, ±1, ±2, . . . }, (1)
where h ∈ (0, h0], h0 is a fixed positive number,
∆(2)yn = yn+1 − 2yn + yn−1, n ∈ Z; f = {fn(h)}n∈Z ∈ Lp(h), p ∈ [1, ∞),
Lp(h) = {f : kfkLp(h) < ∞}, kfk
p
Lp(h) =
X
n∈Z
|fn(h)|
ph, and
0 ≤ qn(h) < ∞,
Xn
k=−∞
qk(h) > 0,
X∞
k=n
qk(h) > 0, n ∈ Z.
We obtain necessary and sufficient conditions under which assertions I) - II) hold together:
I) for a given p ∈ [1, ∞), for any f ∈ Lp(h), (1) has a unique solution
y = {yn(h)}n∈Z ∈ Lp(h) (regardless of h), and y = (Gf)(h)
def = {(Gf)n(h)}n∈Z,
(Gf)n(h) = P
m∈Z
Gn,m(h)fm(h)h, n ∈ Z.
II) kykLp(h) ≤ c(p)kfkLp(h)
for any f ∈ Lp(h).
Here c(p) is an absolute positive constant, {Gn,m(h)}n,m∈Z is the difference Green
function corresponding to (

AB - We consider a difference equation
−h
−2∆(2)yn + qn(h)yn = fn(h), n ∈ Z = {0, ±1, ±2, . . . }, (1)
where h ∈ (0, h0], h0 is a fixed positive number,
∆(2)yn = yn+1 − 2yn + yn−1, n ∈ Z; f = {fn(h)}n∈Z ∈ Lp(h), p ∈ [1, ∞),
Lp(h) = {f : kfkLp(h) < ∞}, kfk
p
Lp(h) =
X
n∈Z
|fn(h)|
ph, and
0 ≤ qn(h) < ∞,
Xn
k=−∞
qk(h) > 0,
X∞
k=n
qk(h) > 0, n ∈ Z.
We obtain necessary and sufficient conditions under which assertions I) - II) hold together:
I) for a given p ∈ [1, ∞), for any f ∈ Lp(h), (1) has a unique solution
y = {yn(h)}n∈Z ∈ Lp(h) (regardless of h), and y = (Gf)(h)
def = {(Gf)n(h)}n∈Z,
(Gf)n(h) = P
m∈Z
Gn,m(h)fm(h)h, n ∈ Z.
II) kykLp(h) ≤ c(p)kfkLp(h)
for any f ∈ Lp(h).
Here c(p) is an absolute positive constant, {Gn,m(h)}n,m∈Z is the difference Green
function corresponding to (

UR - http://www.tandfonline.com/doi/abs/10.1080/10236190500044312?journalCode=gdea20

M3 - Article

VL - 11

SP - 245

EP - 260

JO - Journal of Difference Equations and their Applications

JF - Journal of Difference Equations and their Applications

IS - 3

ER -