Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
FIBRE
CONTRACTION
PRINCIPLE
IN
COMPLETE
b
-METRIC
SPACES
∗
Adrian
Petru¸
sel
†
Gabriela
Petru¸
sel
‡
Jen-Chih
Yao
§
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.47
Abstract
In
the
framework
of
a
complete
b
-metric
space,
we
will
discuss
ex-
istence
and
uniqueness
results
for
the
fixed
points
of
triangular
opera-
tors.
Using
various
classical
metric
conditions
on
the
operators,
we
will
obtain
a
generalization
of
the
well-known
fibre
contraction
principle
in
complete
metric
spaces.
Keywords:
fixed
point,
complete
b
-metric
space,
contraction,
Picard
operator,
Ulam-Hyers
stability,
well-posedness
in
the
sense
of
Reich-Zaslavski.
MSC:
47H10,
54H25.
1
Introduction
Let
(
X,
d,
s
)
be
a
b
-metric
space
with
coefficient
s
≥
1.
(see
[
1
],
[
3
],
[
4
],
[
18
]).
The
only
difference
between
a
metric
and
a
b
-metric
is
given
by
the
triangle
inequality.
In
the
b
-metric
case,
the
third
axiom
of
the
metric
(the
triangle
inequality)
takes
the
following
form:
d
(
x,
y
)
≤
s
(
d
(
x,
z
)
+
d
(
z,
y
))
,
for
all
x,
y,
z
∈
X.
∗
Accepted
for
publication
on
March
10,
2026
†
adrian.petrusel@ubbcluj.ro
,
Department
of
Mathematics,
Babe¸
s-Bolyai
University
Cluj-Napoca
and
Academy
of
Romanian
Scientists,
50044
Bucharest,
Romania
‡
gabriela.petrusel@ubbcluj.ro
,
Department
of
Business,
Babe¸
s-Bolyai
University
Cluj-Napoca,
Romania
§
yaojc@mail.cmu.edu.tw
,
Center
for
General
Education,
China
Medical
University,
Taichung
404,
Taiwan
and
Academy
of
Romanian
Scientists,
50044
Bucharest,
Romania
47
Fibre
contraction
principle
48
A
special
case
of
b
-metric
spaces
is
the
strong
b
-metric
space,
when
the
triangle
inequality
has
the
following
relaxed
form:
d
(
x,
y
)
≤
d
(
x,
z
)
+
sd
(
z,
y
)
,
for
all
x,
y,
z
∈
X.
For
example,
if
(
X,
d
)
is
a
metric
space
and
p
>
1,
then
d
p
is
a
b
-metric
with
constant
2
p
,
i.e.,
(
X,
d
p
,
2
p
)
is
a
b
-metric
space,
see
[
1
].
Another
well-known
example
is
the
space
L
p
([0
,
1])
,
0
<
p
<
1,
where
L
p
([0
,
1])
=
f
:
[0
,
1]
→
R
measurable
:
1
0
|
f
(
t
)
|
p
dt
<
∞
.
Define
d
p
(
f,
g
)
=
(
1
0
|
f
(
t
)
−
g
(
t
)
|
p
dt
)
1
p
.
Then
d
p
is
a
b
-metric
on
L
p
([0
,
1])
with
coefficient
2
1
p
.
For
other
examples,
see
[
7
].
In
this
paper,
we
will
prove
a
fibre
contraction
principle
in
the
framework
of
a
complete
b
-metric
space.
A
saturated
form
of
this
principle
is
also
presented.
Some
stability
properties
are
also
obtained.
Our
results
are
in
connection
to
the
research
directions
proposed
in
[
10
].
For
more
details
on
fibre
contraction
principle
in
metric
structures
(theory
and
applications),
see
[
5
],
[
11
],
[
15
]–
[
19
].
2
Preliminary
notions
and
results
Throughout
this
paper,
N
is
the
set
of
natural
numbers
including
0,
while
N
∗
=
N
\
{
0
}
.
The
symbol
R
+
denotes
the
set
of
all
real
non-negative
numbers.
Let
(
X,
d,
s
)
be
a
b
-metric
space
and
f
:
X
→
X
be
an
operator.
Then,
we
denote
by
Fix
(
f
)
:=
{
x
∈
X
:
x
=
f
(
x
)
}
the
fixed
point
set
of
f,
and
by
f
n
:=
f
◦
·
·
·
◦
f
(n-times)
the
n
-th
iteration
of
f
.
For
x
0
∈
X
,
the
sequence
x
n
:=
f
n
(
x
0
)
,
n
∈
N
,
n
≥
1
is
called
the
sequence
of
Picard
iterates
starting
from
x
0
.
In
the
case
of
a
complete
b
-metric
spaces,
a
(nonlinear)
variant
of
the
Banach-Caccioppoli
Contraction
Principle
was
given
by
Czerwik
in
[
3
].
The
correction
of
a
gap
from
the
original
proof
was
pointed
out
in
[
6
].
We
present
here
a
linear
form
of
the
Czerwik’s
Principle
with
an
additional
conclusion
Theorem
1.
Let
(
X,
d,
s
)
be
a
b
-complete
metric
space
and
f
:
X
→
X
be
an
α
-contraction,
i.e.,
α
∈
(0
,
1)
and
the
following
relation
takes
place
d
(
f
(
x
)
,
f
(
y
))
≤
αd
(
x,
y
)
,
for
every
x,
y
∈
X.
A.
Petru¸
sel,
G.
Petru¸
sel,
J.-C.
Yao
49
Then,
the
following
conclusions
hold:
(1)
Fix
(
f
)
=
{
x
∗
}
;
(2)
for
each
x
0
∈
X
,
the
sequence
x
n
:=
f
n
(
x
0
)
,
n
∈
N
of
Picard
iterates
starting
from
x
0
converges
to
x
∗
;
(3)
if
additionally,
α
<
1
s
,
then
the
following
retraction-displacement
condition
holds
d
(
x
0
,
x
∗
)
≤
s
1
−
sα
d
(
x
0
,
f
(
x
0
))
,
for
each
x
0
∈
X.
Definition
1.
Let
(
X,
d,
s
)
be
a
b
-metric
space
and
f
:
X
→
X
be
an
operator.
Then,
f
is
called
a
Picard
operator
if:
(1)
Fix
(
f
)
=
{
x
∗
}
;
(2)
for
each
x
0
∈
X
,
the
sequence
x
n
:=
f
n
(
x
0
)
,
n
∈
N
of
Picard
iterates
starting
from
x
0
converges
to
x
∗
.
A
Picard
operator
for
which
there
exists
c
>
0
such
that
d
(
x
0
,
x
∗
)
≤
cd
(
x
0
,
f
(
x
0
))
,
for
each
x
0
∈
X,
is
called
a
c
-Picard
operator.
A
Picard
operator
for
which
there
exists
a
map
Ψ
:
R
+
→
R
+
increasing,
continuous
in
0
with
Ψ(0)
=
0
,
such
that
d
(
x
0
,
x
∗
)
≤
Ψ(
d
(
x
0
,
f
(
x
0
)))
,
for
each
x
0
∈
X,
is
called
a
Ψ
-Picard
operator.
Remark
1.
In
terms
of
the
Picard
operator
theory,
the
first
two
conclusions
of
the
above
result
can
be
stated
in
a
concise
form
as
f
is
a
Picard
operator.
All
three
conclusions
of
the
above
theorem
can
be
stated
by
the
sentence
”Then
f
is
a
s
1
−
sα
-Picard
operator.”
The
following
lemma
(see
[
18
])
is
important
for
the
proof
of
our
main
theorems.
Lemma
1.
(Cauchy-Toeplitz
Lemma)
Let
(
a
n
)
n
∈
N
be
a
sequence
in
R
+
,
such
that
the
series
n
≥
0
a
n
is
convergent
and
let
(
b
n
)
n
∈
N
be
a
sequence
with
nonnegative
terms,
such
that
lim
n
→∞
b
n
=
0
.
Then
lim
n
→∞
(
n
k
=0
a
n
−
k
b
k
)
=
0
.
Fibre
contraction
principle
50
3
Main
results
The
following
general
problem
is
well-known.
Let
X,
Y
be
two
nonempty
sets
and
f
:
X
×
Y
→
X
×
Y,
f
(
x,
y
)
=
(
f
1
(
x,
y
)
,
f
2
(
x,
y
))
be
an
operator.
The
problem
is:
if
f
:
X
×
Y
→
X
×
Y
satisfies
the
following
conditions:
(h1)
f
1
(
·
,
y
)
:
X
→
X
has
at
least
one
fixed
point
for
all
y
∈
Y
;
(h2)
f
2
(
x,
·
)
:
Y
→
Y
has
at
least
one
fixed
point
for
all
x
∈
X,
under
which
conditions
the
map
f
has
at
least
one
fixed
point?
It
is
known
that
if
(
X,
d
)
,
(
Y,
ρ
)
are
two
complete
metric
spaces
there
may
be
mappings
f
:
X
×
Y
→
X
×
Y,
f
(
x,
y
)
=
(
f
1
(
x,
y
)
,
f
2
(
x,
y
))
with
no
fixed
points,
even
if
(a1)
f
1
(
·
,
y
)
:
X
→
X
is
a
contraction,
for
all
y
∈
Y
;
(a2)
f
2
(
x,
·
)
:
Y
→
Y
is
a
contraction,
for
all
x
∈
X
.
For
example,
take
f
(
x,
y
)
=
(
x
+
y
2
+
1
,
x
+
y
2
+
1)
,
x,
y
∈
R
,
see
[
8
].
The
following
results
are
partial
answers
to
the
above
problem
(see
[
8
]).
Theorem
2
(S.B.
Nadler
Jr.
(1968))
.
Let
(
X,
d
)
be
a
complete
metric
space
and
let
(
Y,
ρ
)
be
a
metric
space.
Let
f
:
X
×
Y
→
X
×
Y
be
a
mapping.
Suppose:
(i)
f
is
uniformly
continuous
on
X
×
Y
(in
particular,
if
f
is
Lipschitz);
(ii)
f
is
a
contraction
mapping
in
the
first
variable.
Then
f
has
at
least
one
fixed
point.
Theorem
3
(S.B.
Nadler
Jr.
(1968))
.
Let
(
X,
d
)
be
a
complete
metric
space
and
let
(
Y,
ρ
)
be
a
metric
space
with
the
fixed
point
property.
If
f
:
X
×
Y
→
X
×
Y
is
a
contraction
mapping
in
each
variable
separately,
then
f
has
at
least
one
fixed
point.
A.
Petru¸
sel,
G.
Petru¸
sel,
J.-C.
Yao
51
A
particular
form
of
the
above
mentioned
general
problem
is
the
case
of
triangular
operators.
We
formulate
the
problem
in
the
context
of
b
-
metric
spaces.
Let
(
X
1
,
d,
s
)
and
(
X
2
,
ρ,
t
)
be
two
b
-metric
spaces.
Given
t
1
:
X
1
→
X
1
and
t
2
:
X
1
×
X
2
→
X
2
we
define
the
following
triangular
operator
t
:
X
1
×
X
2
→
X
1
×
X
2
given
by
t
(
x
1
,
x
2
)
:=
(
t
1
(
x
1
)
,
t
2
(
x
1
,
x
2
))
.
Under
which
conditions
on
t
1
and
t
2
the
operator
t
has
at
least
one
fixed
point?
We
present
now
the
first
main
result
of
this
paper,
which
is
an
extension
to
the
case
of
b
-metric
spaces
of
the
classical
result
of
Hirsch
and
Pugh
[
5
].
Theorem
4.
Let
(
X
1
,
d
1
,
r
)
be
a
b
-metric
space
with
coefficient
r
≥
1
and
t
1
:
X
1
→
X
1
be
an
operator
having
an
attractive
fixed
point
x
∗
1
∈
X
1
.
Let
(
X
2
,
d
2
,
s
)
be
a
complete
b
-metric
space
with
coefficient
s
≥
1
and
let
t
2
:
X
1
×
X
2
→
X
2
.
Suppose:
(1)
there
exists
α
∈
(0
,
1
s
)
such
that
t
2
(
x
1
,
·
)
:
X
2
→
X
2
is
an
α
-
contraction,
for
all
x
1
∈
X
1
.
Denote
by
x
∗
2
the
unique
fixed
point
of
t
2
(
x
∗
1
,
·
)
;
(2)
the
operator
t
:
X
1
×
X
2
→
X
1
×
X
2
,
t
(
x
1
,
x
2
)
:=
(
t
1
(
x
1
)
,
t
2
(
x
1
,
x
2
))
is
continuous.
Then
(
x
∗
1
,
x
∗
2
)
is
an
attractive
fixed
point
of
t
.
In
fact,
the
following
more
general
result
can
be
proved.
Theorem
5.
Let
(
X
1
,
d
1
,
r
)
be
a
b
-metric
space
with
coefficient
r
≥
1
and
(
X
2
,
d
2
,
s
)
be
a
complete
b
-metric
space
with
coefficient
s
≥
1
.
Let
t
1
:
X
1
→
X
1
and
t
2
:
X
1
×
X
2
→
X
2
be
two
operators.
Suppose:
(1)
t
1
:
X
1
→
X
1
is
a
Picard
operator
(denote
by
x
∗
1
its
unique
fixed
point);
(2)
there
exists
α
∈
(0
,
1
s
)
such
that
the
operator
t
2
(
x
1
,
·
)
:
X
2
→
X
2
is
an
α
-contraction,
for
all
x
1
∈
X
1
(denote
by
x
∗
2
the
unique
fixed
point
of
t
2
(
x
∗
1
,
·
)
);
(3)
the
operator
t
:
X
1
×
X
2
→
X
1
×
X
2
,
t
(
x
1
,
x
2
)
:=
(
t
1
(
x
1
)
,
t
2
(
x
1
,
x
2
))
is
continuous.
Then
t
is
a
Picard
operator
and
Fix
(
t
)
=
(
x
∗
1
,
x
∗
2
)
.
Proof.
Let
(
x
1
0
,
x
2
0
)
∈
X
1
×
X
2
be
arbitrary.
Define
the
sequences
x
1
n
+1
=
t
1
(
x
1
n
)
and
x
2
n
+1
=
t
2
(
x
1
n
,
x
2
n
)
,
n
∈
N
.
Fibre
contraction
principle
52
By
(1),
we
know
that
Fix
(
t
1
)
=
{
x
∗
1
}
and
the
sequence
(
x
1
n
)
n
∈
N
converges
to
x
∗
1
.
We
will
show
that
the
sequence
(
x
2
n
)
n
∈
N
converges
to
x
∗
2
.
We
successively
have:
d
2
(
x
2
n
+1
,
x
∗
2
)
=
d
2
(
t
2
(
x
1
n
,
x
2
n
)
,
t
2
(
x
∗
1
,
x
∗
2
))
≤
s
d
2
(
t
2
(
x
1
n
,
x
2
n
)
,
t
2
(
x
1
n
,
x
∗
2
))
+
d
2
(
t
2
(
x
1
n
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))
≤
sαd
2
(
x
2
n
,
x
∗
2
)
+
sd
2
(
t
2
(
x
1
n
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))
≤
sα
sαd
2
(
x
2
n
−
1
,
x
∗
2
)
+
sd
2
(
t
2
(
x
1
n
−
1
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))
+
sd
2
(
t
2
(
x
1
n
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))
=
(
sα
)
2
d
2
(
x
2
n
−
1
,
x
∗
2
)+
s
2
αd
2
(
t
2
(
x
1
n
−
1
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))+
sd
2
(
t
2
(
x
1
n
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))
≤
·
·
·
≤
(
sα
)
n
+1
d
2
(
x
2
0
,
x
∗
2
)
+
s
(
sα
)
n
d
2
(
t
2
(
x
1
0
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))
+
·
·
·
+
s
(
sα
)
d
2
(
t
2
(
x
1
n
−
1
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))
+
sd
2
(
t
2
(
x
1
n
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))
≤
(
sα
)
n
+1
d
2
(
x
2
0
,
x
∗
2
)
+
s
[(
sα
)
n
d
2
(
t
2
(
x
1
0
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))
+
·
·
·
+(
sα
)
d
2
(
t
2
(
x
1
n
−
1
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))+
d
2
(
t
2
(
x
1
n
,
x
∗
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))]
→
0
,
n
→
∞
,
where
the
convergence
to
0
follows
by
the
Cauchy-Toeplitz
Lemma
1
.
A
similar
result
takes
place
if
we
replace
the
contraction
condition
with
a
´
Ciri´
c-Kannan-Reich-Rus
type
condition
(see
[
2
],
[
12
],
[
18
]).
We
recall
first
a
Kannan
type
fixed
point
theorem
in
complete
b
-metric
spaces.
Theorem
6.
Let
(
X,
d,
s
)
be
a
b
-complete
metric
space
and
f
:
X
→
X
be
a
Kannan
α
-contraction,
i.e.,
α
∈
(0
,
max
{
1
2
,
1
1+
s
}
)
and
the
following
relation
takes
place
d
(
f
(
x
)
,
f
(
y
))
≤
α
(
d
(
x,
f
(
x
))
+
d
(
y,
f
(
y
)))
,
for
every
x,
y
∈
X.
Then,
the
following
conclusions
hold:
(1)
Fix
(
f
)
=
{
x
∗
}
;
(2)
for
each
x
0
∈
X
,
the
sequence
x
n
:=
f
n
(
x
0
)
,
n
∈
N
of
Picard
iterates
starting
from
x
0
converges
to
x
∗
;
(3)
if
we
denote
γ
:=
α
1
−
α
∈
(0
,
1)
,
then
the
following
retraction-
displacement
condition
holds
d
(
x
0
,
x
∗
)
≤
s
1
−
sγ
d
(
x
0
,
f
(
x
0
))
,
for
each
x
0
∈
X.
Remark
2.
In
the
language
of
Picard
operator
theory
the
conclusion
of
the
above
theorem
is
that
f
is
a
s
1
−
sγ
-Picard
operator.
A.
Petru¸
sel,
G.
Petru¸
sel,
J.-C.
Yao
53
Theorem
7.
Let
(
X
1
,
d
1
,
r
)
be
a
b
-metric
space
with
coefficient
r
≥
1
and
(
X
2
,
d
2
,
s
)
be
a
complete
b
-metric
space
with
coefficient
s
≥
1
.
Let
t
1
:
X
1
→
X
1
and
t
2
:
X
1
×
X
2
→
X
2
be
two
operators.
Suppose:
(1)
t
1
:
X
1
→
X
1
is
a
Picard
operator
(denote
by
x
∗
1
its
unique
fixed
point);
(2)
there
exists
α
∈
(0
,
1
2
s
)
and
β
>
0
such
that
d
2
(
t
2
(
x
1
,
x
2
)
t
2
(
y
1
,
y
2
))
≤
βd
1
(
x
1
,
y
1
)+
α
[
d
2
(
t
2
(
x
1
,
x
2
)
,
x
2
)
+
d
2
(
t
2
(
y
1
,
y
2
)
,
y
2
)]
,
for
all
(
x
1
,
x
2
)
,
(
y
1
,
y
2
)
∈
X
1
×
X
2
;
(3)
the
operator
t
:
X
1
×
X
2
→
X
1
×
X
2
,
t
(
x
1
,
x
2
)
:=
(
t
1
(
x
1
)
,
t
2
(
x
1
,
x
2
))
is
continuous.
Then
t
is
a
Picard
operator
and
Fix
(
t
)
=
(
x
∗
1
,
x
∗
2
)
.
Proof.
Let
(
x
1
0
,
x
2
0
)
∈
X
1
×
X
2
be
arbitrary.
Define
the
sequences
x
1
n
+1
=
t
1
(
x
1
n
)
and
x
2
n
+1
=
t
2
(
x
1
n
,
x
2
n
)
,
n
∈
N
.
By
(1),
we
know
that
Fix
(
t
1
)
=
{
x
∗
1
}
and
the
sequence
(
x
1
n
)
n
∈
N
converges
to
x
∗
1
as
n
→
∞
.
We
notice
first
that,
using
(2),
we
obtain
that
t
2
(
x
∗
1
,
·
)
:
X
2
→
X
2
satisfies
the
following
relation
d
2
(
t
2
(
x
∗
1
,
x
2
)
,
t
2
(
x
∗
1
,
y
2
))
≤
α
[
d
2
(
t
2
(
x
∗
1
,
x
2
)
,
x
2
)
+
d
2
(
t
2
(
x
∗
1
,
y
2
)
,
y
2
)]
,
for
every
x
2
,
y
2
∈
X
2
.
Thus,
t
2
(
x
∗
1
,
·
)
:
X
2
→
X
2
satisfies
the
assumptions
of
Kannan’s
fixed
point
theorem
(see
Theorem
6
)
and,
as
a
consequence,
there
exists
x
∗
2
∈
X
2
such
that
t
2
(
x
∗
1
,
x
∗
2
)
=
x
∗
2
.
We
will
show
now
that
the
sequence
(
x
2
n
)
n
∈
N
converges
to
x
∗
2
.
We
successively
have:
d
2
(
x
2
n
+1
,
x
∗
2
)
=
d
2
(
t
2
(
x
1
n
,
x
2
n
)
,
t
2
(
x
∗
1
,
x
∗
2
))
≤
βd
1
(
x
1
n
,
x
∗
1
)
+
α
d
2
(
t
2
(
x
1
n
,
x
2
n
)
,
x
2
n
)
+
d
2
(
t
2
(
x
∗
1
,
x
∗
2
)
,
x
∗
2
)
≤
βd
1
(
x
1
n
,
x
∗
1
)
+
sα
d
2
(
x
2
n
+1
,
x
∗
2
)
+
d
2
(
x
∗
2
,
x
2
n
)
.
From
here,
for
each
n
∈
N
,
we
get
d
2
(
x
2
n
+1
,
x
∗
2
)
≤
β
1
−
sα
d
1
(
x
1
n
,
x
∗
1
)
+
sα
1
−
sα
d
2
(
x
2
n
,
x
∗
2
)
.
(1)
Then,
we
successively
have
d
2
(
x
2
n
+1
,
x
∗
2
)
≤
β
1
−
sα
d
1
(
x
1
n
,
x
∗
1
)
+
β
1
−
sα
sα
1
−
sα
d
1
(
x
1
n
−
1
,
x
∗
1
)
+
(
sα
1
−
sα
)
2
d
2
(
x
2
n
−
1
,
x
∗
2
)
Fibre
contraction
principle
54
≤
·
·
·
≤
β
1
−
sα
d
1
(
x
1
n
,
x
∗
1
)
+
β
1
−
sα
[
sα
1
−
sα
d
1
(
x
1
n
−
1
,
x
∗
1
)
+
·
·
·
+
(
sα
1
−
sα
)
n
d
1
(
x
1
0
,
x
∗
1
)]
+
(
sα
1
−
sα
)
n
+1
d
1
(
x
2
0
,
x
∗
2
)
.
Hence,
by
(1)
and
the
Cauchy-Toeplitz
Lemma
1
,
the
right
hand
side
of
the
above
relation
tends
to
0.
The
proof
is
complete.
We
will
conclude
our
study
by
analyzing
some
stability
properties
(see
[
9
],
[
13
],
[
14
])
for
the
above
problem.
Theorem
8.
Let
(
X
1
,
d
1
,
r
)
be
a
complete
b
-metric
space
with
coefficient
r
≥
1
and
(
X
2
,
d
2
,
s
)
be
a
complete
b
-metric
space
with
coefficient
s
≥
1
.
Let
t
1
:
X
1
→
X
1
and
t
2
:
X
1
×
X
2
→
X
2
be
two
operators.
Suppose:
(1)
t
1
:
X
1
→
X
1
is
a
k
-contraction
with
k
∈
(0
,
1
r
)
(denote
by
x
∗
1
its
unique
fixed
point);
(2)
there
exists
α
∈
(0
,
1
s
2
)
such
that
the
operator
t
2
(
x
1
,
·
)
:
X
2
→
X
2
is
an
α
-contraction,
for
all
x
1
∈
X
1
(denote
by
x
∗
2
the
unique
fixed
point
of
the
mapping
t
2
(
x
∗
1
,
·
)
);
(3)
there
exists
L
>
0
such
that
the
operator
t
2
(
·
,
x
2
)
:
X
1
→
X
2
is
an
L
-Lipschitz,
for
all
x
2
∈
X
2
.
Then
the
fixed
point
problem
for
t
is
Ulam-Hyers
stable
and
has
the
well-posedness
property
in
the
sense
of
Reich
and
Zaslavski.
Proof.
By
Theorem
4
we
know
that
Fix
(
t
)
=
{
(
x
∗
1
,
x
∗
2
)
}
,
where
x
∗
2
is
the
unique
fixed
point
of
the
mapping
t
2
(
x
∗
1
,
·
)
:
X
2
→
X
2
.
For
the
Ulam-Hyers
property,
take
any
>
0
and
let
(˜
x
1
,
˜
x
2
)
∈
X
1
×
X
2
satisfying
the
relation
d
∞
((˜
x
1
,
˜
x
2
)
,
t
(˜
x
1
,
˜
x
2
))
≤
.
Then,
we
have
max
{
d
1
(˜
x
1
,
t
1
(˜
x
1
))
,
d
2
(˜
x
2
,
t
2
(˜
x
1
,
˜
x
2
))
}
≤
.
We
must
now
estimate
the
following
value
d
∞
((˜
x
1
,
˜
x
2
)
,
(
x
∗
1
,
x
∗
2
))
=
max
{
d
1
(˜
x
1
,
x
∗
1
)
,
d
2
(˜
x
2
,
x
∗
2
)
}
.
First,
we
have
d
1
(˜
x
1
,
x
∗
1
)
≤
r
[
d
1
(˜
x
1
,
t
1
(˜
x
1
))
+
d
1
(
t
1
(˜
x
1
)
,
t
1
(
x
∗
1
))]
≤
r
+
rkd
1
(˜
x
1
,
x
∗
1
)
.
A.
Petru¸
sel,
G.
Petru¸
sel,
J.-C.
Yao
55
Thus,
we
conclude
d
1
(˜
x
1
,
x
∗
1
)
≤
r
1
−
rk
.
(2)
Next,
we
have
d
2
(˜
x
2
,
x
∗
2
)
≤
s
[
d
2
(˜
x
2
,
t
2
(˜
x
1
,
˜
x
2
))
+
d
2
(
t
2
(˜
x
1
,
˜
x
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))]
≤
s
+
s
2
[
d
2
(
t
2
(˜
x
1
,
˜
x
2
)
,
t
2
(
x
∗
1
,
˜
x
2
))
+
d
2
(
t
2
(
x
∗
1
,
˜
x
2
)
,
t
2
(
x
∗
1
,
x
∗
2
))]
≤
s
+
s
2
Ld
1
(˜
x
1
,
x
∗
1
)
+
s
2
αd
2
(˜
x
2
,
x
∗
2
)
.
As
a
consequence,
we
have
d
1
(˜
x
2
,
x
∗
2
)
≤
s
1
−
s
2
α
1
+
s
2
L
1
−
sk
.
(3)
Thus,
d
∞
((˜
x
1
,
˜
x
2
)
,
(
x
∗
1
,
x
∗
2
))
≤
C,
where
C
:=
max
{
r
1
−
rk
,
s
1
−
s
2
α
1
+
s
2
L
1
−
sk
}
.
This
shows
that
the
fixed
point
problem
for
t
is
Ulam-Hyers
stable.
For
the
well-posedness
property,
consider
((
x
1
n
,
x
2
n
))
n
∈
N
in
X
1
×
X
2
with
the
property
that
lim
n
→∞
d
∞
((
x
1
n
,
x
2
n
)
,
t
(
x
1
n
,
x
2
n
))
=
0
.
We
will
show
that
lim
n
→∞
d
∞
((
x
1
n
,
x
2
n
)
,
(
x
∗
1
,
x
∗
2
))
=
0
.
(4)
Indeed,
we
have
d
1
(
x
1
n
,
x
∗
1
)
≤
r
d
1
(
x
1
n
,
t
1
(
x
1
n
))
+
d
1
(
t
1
(
x
1
n
)
,
t
1
(
x
∗
1
))
≤
r
d
1
(
x
1
n
,
t
1
(
x
1
n
))
+
kd
1
(
x
1
n
,
x
∗
1
)
.
As
a
consequence,
we
have
d
1
(
x
1
n
,
x
∗
1
)
≤
r
1
−
rk
d
1
(
x
1
n
,
t
1
(
x
1
n
))
→
0
,
n
→
∞
.
(5)
Similarly,
we
get
d
2
(
x
2
n
,
x
∗
2
)
≤
s
d
2
(
x
2
n
,
t
2
(
x
1
n
,
x
2
n
))
+
d
2
(
t
2
(
x
1
n
,
x
2
n
)
,
t
2
(
x
∗
1
,
x
∗
2
))
≤
sd
2
(
x
2
n
,
t
2
(
x
1
n
,
x
2
n
))+
s
2
d
2
(
t
2
(
x
1
n
,
x
2
n
)
,
t
2
(
x
∗
1
,
x
2
n
))
+
d
2
(
t
2
(
x
∗
1
,
x
2
n
)
,
t
2
(
x
∗
1
,
x
∗
2
))
≤
sd
2
(
x
2
n
,
t
2
(
x
1
n
,
x
2
n
))
+
s
2
Ld
1
(
x
1
n
,
x
∗
1
)
+
s
2
αd
2
(
x
2
n
,
x
∗
2
)
.
As
a
consequence,
we
have
d
2
(
x
2
n
,
x
∗
2
)
≤
s
1
−
s
2
α
d
2
(
x
2
n
,
t
2
(
x
1
n
,
x
2
n
))
+
sLd
1
(
x
1
n
,
x
∗
1
)
→
0
,
n
→
∞
.
(6)
By
(
5
)
and
(
6
)
we
obtain
(
4
).
Fibre
contraction
principle
56
In
fact,
the
following
saturated
form
of
the
fibre
contraction
principle
in
b
-metric
spaces
can
be
established.
Theorem
9.
Let
(
X
1
,
d
1
,
r
)
be
a
b
-metric
space
with
coefficient
r
≥
1
and
(
X
2
,
d
2
,
s
)
be
a
complete
b
-metric
space
with
coefficient
s
≥
1
.
Let
t
1
:
X
1
→
X
1
and
t
2
:
X
1
×
X
2
→
X
2
be
two
operators.
Suppose:
(1)
t
1
:
X
1
→
X
1
is
a
c
1
-Picard
operator
(denote
by
x
∗
1
its
unique
fixed
point);
(2)
there
exists
α
∈
(0
,
1
s
2
)
such
that
the
operator
t
2
(
x
1
,
·
)
:
X
2
→
X
2
is
an
α
-contraction,
for
all
x
1
∈
X
1
(denote
by
x
∗
2
the
unique
fixed
point
of
the
mapping
t
2
(
x
∗
1
,
·
)
);
(3)
there
exists
L
>
0
such
that
the
operator
t
2
(
·
,
x
2
)
:
X
1
→
X
2
is
an
L
-Lipschitz,
for
all
x
2
∈
X
2
.
Then
t
is
a
Ψ
-Picard
operator
with
Ψ(
t
)
:=
max
{
c
1
t,
s
1
−
s
2
α
t
+
sLc
1
t
}
.
4
Conclusions
In
this
paper,
two
fibre
contraction
principles
in
b
-metric
spaces
were
es-
tablished.
The
approach
is
based
on
some
fixed
point
results
for
self
op-
erators
in
complete
b
-metric
spaces
(Banach-Caccioppoli
contractions
and,
respectively
Kannan
contractions)
and
the
Cauchy-Toeplitz
Lemma.
Some
stability
results
(well-posedness
in
the
sense
of
Reich
and
Zaslavski
and
Ulam-Hyers
stability)
are
also
proved.
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