Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
EXISTENCE
AND
APPROXIMATION
OF
SOLUTIONS
OF
BOUNDARY
VALUE
PROBLEMS
FOR
SECOND
ORDER
ITERATIVE
DIFFERENTIAL
EQUATIONS
∗
Vasile
Berinde
†
Dedicated
to
the
memory
of
Professor
Emeritus
Dr.
Mihail
Megan
(1947-2025)
DOI
10.56082/annalsarscimath.2026.3.185
Abstract
We
study
the
existence
and
approximation
of
solutions
to
the
second-
order
iterative
boundary-value
problem
x

(
t
)
=
f
t,
x
(
t
)
,
x
[2]
(
t
)
,
0
≤
t
≤
1
,
where
x
[2]
(
t
);
=
x
(
x
(
t
)),
with
solutions
satisfying
one
of
the
two
sets
of
boundary
conditions:
(i)
x
(0)
=
0
,
x
(1)
=
1;
(ii)
x
(0)
=
1
,
x
(1)
=
0.
Keywords:
ordinary
differential
equation,
boundary
value
problem,
fixed
point,
contraction
mapping,
nonexpansive
mapping.
MSC:
47H10,
47H05,
54H25,
34K05,
34A12.
1
Introduction
Iterative
differential
equations
represent
a
special
class
of
differential
equa-
tions
with
deviating
arguments
depending
on
both
the
state
variable
x
and
the
time
t
.
∗
Accepted
for
publication
on
July
30,
2026
†
vasile.berinde@mi.utcluj.ro
,
Department
of
Mathematics
and
Computer
Science,
North
University
Center
at
Baia
Mare,
Technical
University
of
Cluj-Napoca,
Victoriei
76,
430122
Baia
Mare,
Romania;
Academy
of
Romanian
Scientists
185
Existence
and
approximation
of
solutions
of
boundary
value
problems
186
Iterative
differential
equations
are
important
in
theory
and
applications,
see
for
example
Yang
[
41
]
and
the
papers
quoted
there,
where
applications
to
insect
population
dynamics
models
in
ecology,
models
of
hematopoiesis
in
hematology,
epidemiological
models
in
epidemiology,
two-body
equations
of
motion
in
classical
electrodynamics,
the
motion
of
charged
particles
with
retarded
interaction
are
indicated,
and
son
on.
One
of
the
first
papers
devoted
to
first
order
iterative
differential
equa-
tions
appears
to
be
due
to
Pethukov
[
28
]
who
studied
the
existence
of
solu-
tions
for
the
equation
x
(
t
)
=
a
x
(
x
(
t
))
,
(1)
with
a
>
0,
by
constructing
ε
-approximate
solutions
and
then
letting
ε
→
0.
Dunkel
[
11
]
studied
the
more
general
equation
x
(
t
)
=
f
(
x
(
h
(
x
(
t
))))
,
(2)
where
f
is
continuous
and
increasing
on
[0
,
+
∞
)
and
f
(0)
=
0,
h
is
contin-
uous,
increasing
and
such
that
0
<
t
1
<
h
(
t
1
),
for
some
t
1
,
and
h
(
x
)
≤
x
.
Dunkel
[
11
]
obtained
existence
and
uniqueness
theorems
for
solutions
of
such
equations,
as
well
as
bounds
on
their
growth
and
criteria
for
their
nonexistence,
using
successive
approximations
and
differential
inequalities.
We
also
note
that
Dunkel
[
11
]
and
Agarwal
[
1
]
used
the
more
suggestive
term
”nested”
to
designate
iterative
differential
equations.
Apparently
unaware
of
the
previous
contributions
by
Pethukov
[
28
],
Dunkel
[
11
]
and
Agarwal
[
1
],
Eder
[
13
]
considered
the
functional
differential
equation
(
1
)
for
the
particular
case
a
=
1
x
(
t
)
=
x
(
x
(
t
))
,
t
∈
A
⊂
R
,
and
has
shown
that
every
solution
either
vanishes
identically
or
is
strictly
monotonic.
Subsequently,
Feˇ
ckan
[
17
]
studied
a
first
order
iterative
differential
equa-
tion
of
the
more
general
form
similar
to
(
2
)
but
not
obtainable
from
(
2
)
x
(
t
)
=
f
(
x
(
x
(
t
)))
,
with
f
∈
C
1
(
R
).
For
some
other
developments
on
this
topic,
see
also
Buic˘
a
[
9
],
Berinde
[
7
],
Eder
[
13
],
Egri
[
14
],
[
15
],
[
16
],
Li
and
Zhang
[
26
],
Li
and
Cheng
[
27
],
Rus
[
30
],
Yang
[
41
]
and
references
therein.
The
present
author
Berinde
[
7
]
studied
the
existence
of
solutions
of
iter-
ative
differential
equations
of
the
general
form
x
(
t
)
=
f
(
t,
x
(
x
(
t
)))
,
V.
Berinde
187
and
also
illustrated
how
one
can
approximate
the
non-unique
solutions
of
such
kind
of
iterative
differential
equations
by
means
of
iterative
fixed
point
techniques
associated
to
nonexpansive
type
mappings.
Second-order
iterative
differential
equations
were
also
studied
by
some
authors.
For
example,
Khemis
et
al.
[
23
]
established
the
existence
of
periodic
solutions
to
a
class
of
second-order
iterative
differential
equations.
Their
method
is
based
on
the
equivalence
of
the
original
problem
with
a
certain
integral
equation
and
then
by
applying
Schauder’s
fixed
point
theorem
and
the
Green’s
functions
method.
On
the
other
hand,
Kaufmann
[
21
],
studied
the
existence
and
uniqueness
of
solutions
to
the
second-order
iterative
boundary-value
problem
x

(
t
)
=
f
t,
x
(
t
)
,
x
[2]
(
t
)
,
a
≤
t
≤
b,
(3)
with
the
solutions
satisfying
one
of
the
two
sets
of
boundary
conditions:
x
(
a
)
=
a
x
(
b
)
=
b,
(4)
and
x
(
a
)
=
b
x
(
b
)
=
a.
(5)
Starting
from
this
background,
our
aim
in
this
paper
is
to
complement
the
results
in
Kaufmann
[
21
]
by
providing
results
on
the
existence
and
ap-
proximation
of
solutions
of
the
problems
(
3
)+(
4
)
and
(
3
)+(
5
)
under
signifi-
cantly
weaker
conditions
than
the
ones
in
Kaufmann
[
21
].
To
this
end,
we
use
the
powerful
and
more
reliable
technique
of
con-
tractive
operators,
presented
in
the
next
Section.
More
advanced
conver-
gence
theorems
from
the
theory
of
iterative
approximation
of
fixed
points
of
non-expansive
mappings,
illustrated
in
Berinde
[
7
],
see
also
Berinde
[
6
]
and
Chidume
[
10
]
for
more
details,
are
also
used.
2
Basics
of
fixed
point
theory
of
contractive
mappings
In
order
to
prepare
the
main
theoretical
tools
we
should
use
in
the
next
section,
we
present
some
basic
facts
about
the
contraction
mapping
principle.
Although
we
should
need
its
Banach
space
version
for
our
applications,
we
present
it
in
the
more
general
setting
of
a
metric
space.
Existence
and
approximation
of
solutions
of
boundary
value
problems
188
Let
(
X,
d
)
be
a
complete
metric
space
equipped
with
the
distance
func-
tion
d
(
x,
y
).
A
mapping
T
:
X
→
X
for
which
there
exists
a
constant
k
∈
[0
,
1)
such
that
d
(
T
(
x
)
,
T
(
y
))
≤
k
·
d
(
x,
y
),
for
all
x,
y
∈
X
,
is
called
a
k
-
contraction
or
simply
a
strict
contraction
.
An
element
x
∈
X
is
called
a
fixed
point
of
T
,
provided
that
x
=
T
(
x
).
In
many
instances,
in
order
to
solve
a
certain
nonlinear
equation
/
problem,
it
is
advantageous
to
transform
the
nonlinear
equation
/
problem
into
an
equivalent
fixed
point
problem
x
=
T
(
x
)
and
therefore
the
task
of
solving
the
original
nonlinear
equation
/
problem
is
converted
into
that
of
finding
a
fixed
point
of
the
mapping
T
.
One
fundamental
tool
in
doing
so
is
the
famous
Banach
contraction
mapping
principle,
which
in
its
complete
form
can
be
stated
as
follows.
Theorem
1.
Let
(
X,
d
)
be
a
complete
metric
space
and
T
:
X
→
X
a
δ
-contraction.
Then
1)
Fix
(
T
)
=
{
x
}
;
2)
For
any
x
0
∈
X
,
Picard
iteration
{
x
n
}
∞
n
=0
,
x
n
=
T
n
x
0
,
converges
to
x
;
3)
The
following
estimate
holds
d
(
x
n
+
i
−
1
,
x
)
≤
δ
i
1
−
δ
d
(
x
n
,
x
n
−
1
)
,
n
=
0
,
1
,
2
,
.
.
.
;
i
=
1
,
2
,
.
.
.
(6)
Remark
1.
The
estimate
(
6
)
,
which
incorporates
both
a
priori
and
a
poste-
riori
error
estimates,
is
very
important
in
applications,
as
it
offers
stopping
criteria
for
the
Picard
iteration,
see
for
example
Berinde
[
6
]
for
more
details.
We
note
that
Theorem
1
could
be
stated
with
additional
conclusions,
re-
lated
to
data
dependence
on
the
operator
perturbations,
Ulam-Hyers
stability,
De
Blasi
well-posedness
and
so
on,
see
for
example
Rus
[
31
].
An
important
case
of
contractive
mappings
is
obtained
within
the
limit
case
k
=
1
of
the
definition
of
a
k
-contraction.
A
mapping
T
:
X
→
X
is
said
to
be
nonexpansive
if
d
(
Tx,
Ty
)
≤
d
(
x,
y
)
,
∀
x,
y
∈
X.
(7)
Despite
the
fact
that
the
contractive
condition
(
7
)
is
obviously
related
to
that
of
strict
contractions,
the
problem
of
the
existence
and
approximation
of
fixed
points
for
nonexpansive
mappings
is
essentially
different
from
the
case
of
contractions.
This
means
that
a
nonexpansive
mapping
defined
on
a
complete
metric
space
(even
on
a
Banach
space)
need
not
have
a
fixed
point.
Even
if
a
V.
Berinde
189
nonexpansive
mapping
T
has
a
(unique)
fixed
point,
then
Picard
iteration
does
not
converge
in
general
to
that
fixed
point.
In
order
to
get
positive
answers
to
the
problem
of
existence
and
approx-
imation
of
fixed
points
of
nonexpansive
mappings,
we
need
a
more
appro-
priate
setting
that
the
one
of
a
metric
space,
i.e.,
that
of
a
Banach
space
(complete
linear
normed
space).
As
we
have
mentioned
before,
although
the
non-expansive
mappings
are
natural
generalizations
of
α
-contractions,
they
do
not
inherit
the
fixed
point
properties
of
contractive
mappings.
In
the
following
example,
T
is
a
nonex-
pansive
mapping
which
is
fixed
point
free.
Example
1.
(
[
18
],
Example
3.3,
pp.
30)
In
the
space
c
0
(
N
)
the
isometry
T
defined
by
T
(
x
1
,
x
2
,
.
.
.
)
=
(1
,
x
1
,
x
2
,
.
.
.
)
maps
the
unit
ball
into
its
boundary
but
T
has
not
fixed
points.
As
is
shown
by
the
next
example,
even
in
the
cases
where
the
nonexpan-
sive
mapping
T
has
a
unique
fixed
point,
the
Picard
iteration
associated
to
T
(i.e.,
the
Picard
iteration
{
x
n
}
,
for
a
given
x
0
∈
K
),
may
fail
to
converge
to
the
fixed
point.
Example
2.
Let
[0
,
1]
be
the
unit
interval
with
the
usual
norm.
The
function
T
:
[0
,
1]
→
[0
,
1]
given
by
Tx
=
1
−
x
,
for
all
x
∈
[0
,
1]
has
a
unique
fixed
point,
x
=
1
2
but,
except
for
the
trivial
case
x
0
=
1
2
,
the
Picard
iteration
starting
from
x
0
yields
an
oscillatory
sequence.
The
previous
examples
indicate
that,
in
order
to
have
ensured
the
ex-
istence
and
approximation
of
fixed
points
for
nonexpansive
mappings,
we
need
some
richer
geometrical
properties
of
the
ambient
space
E
.
One
such
setting
is
essential
in
the
following
fixed
point
theorem
for
non-
expansive
mappings,
due
to
Browder,
G¨
ohde
and
Kirk,
see
e.g.
[
6
],
is
stated
as
follows.
Theorem
2.
If
K
is
a
nonempty
closed
convex
and
bounded
subset
of
a
uniformly
convex
Banach
space
E
then
any
non-expansive
mapping
T
:
K
→
K
has
a
fixed
point.
Remark
2.
As
one
can
see,
Theorem
2
does
not
provide
information
on
the
approximation
of
fixed
points
of
a
nonexpansive
mapping
T
.
Also,
by
Example
2
,
we
see
that
Picard
iteration
does
not
resolve
the
problem
of
approximating
the
fixed
points
of
such
a
mapping.
Therefore,
we
need
more
reliable
fixed
point
iteration
procedures,
like
Krasnoselskij
iteration,
Mann
or
Ishikawa
iterations,
see
Berinde
[
6
]
and
Chidume
[
10
].
Existence
and
approximation
of
solutions
of
boundary
value
problems
190
Let
K
be
a
convex
subset
of
a
normed
linear
space
E
and
let
T
:
K
→
K
be
a
self-mapping.
Given
x
0
∈
K
given
and
a
real
number
λ
∈
[0
,
1],
the
sequence
{
x
n
}
defined
by
the
formula
x
n
+1
=
(1
−
λ
)
x
n
+
λTx
n
,
n
=
0
,
1
,
2
,
·
·
·
(8)
is
usually
called
Krasnoselskij
iteration
,
or
Krasnoselskij-Mann
iteration
.
Clearly,
(
8
)
reduces
to
Picard
iteration
for
λ
=
1.
For
x
0
∈
K
given,
the
sequence
{
x
n
}
defined
by
the
formula
x
n
+1
=
(1
−
λ
n
)
x
n
+
λ
n
Tx
n
,
n
=
0
,
1
,
2
,
·
·
·
(9)
where
{
λ
n
}
⊂
[0
,
1]
is
a
sequence
of
real
numbers
satisfying
some
appropriate
conditions,
is
called
Mann
iteration
.
It
was
shown
by
Krasnoselskij
[
24
]
in
the
case
λ
=
1
/
2,
and
latter
by
Schaefer
[
32
]
for
λ
∈
(0
,
1)
arbitrary,
that
if
E
is
a
uniformly
convex
Ba-
nach
space
and
K
is
a
convex
and
compact
subset
of
E
containing
fixed
points
of
T
,
then
the
Krasnoselskij
iteration
converges
to
a
fixed
point
of
the
nonexpansive
mapping
T
.
Moreover,
Edelstein
[
12
]
proved
that
strict
convexity
of
E
suffices
for
the
same
conclusion.
The
question
of
whether
or
not
strict
convexity
can
be
removed
has
been
answered
in
the
affirmative
by
Ishikawa
[
20
]
by
the
following
result.
Theorem
3.
Let
K
be
a
subset
of
a
Banach
space
E
and
let
T
:
K
→
K
be
a
non-expansive
mapping.
For
arbitrary
x
0
∈
K
,
consider
the
Mann
iteration
process
{
x
n
}
given
by
(
9
)
under
the
following
assumptions
(a)
x
n
∈
K
for
all
positive
integers
n
;
(b)
0
≤
λ
n
≤
b
<
1
for
all
positive
integers
n
;
(c)
∞
n
=0
λ
n
=
∞
.
If
{
x
n
}
is
bounded,
then
x
n
−
Tx
n
→
0
as
n
→
∞
.
The
following
corollaries
of
Theorem
3
will
be
particularly
important
for
the
application
part
of
our
paper.
Corollary
1.
Let
K
be
a
convex
and
compact
subset
of
a
Banach
space
E
and
let
T
:
K
→
K
be
a
non-expansive
mapping.
If
the
Mann
iteration
process
{
x
n
}
given
by
(
9
)
satisfies
assumptions
(a)-(c)
in
Theorem
3
,
then
{
x
n
}
converges
strongly
to
a
fixed
point
of
T
.
Proof.
See
Theorem
6.17
in
Chidume
[
10
].
V.
Berinde
191
Corollary
2.
Let
E
be
a
real
normed
space,
K
a
closed
bounded
convex
subset
of
E
and
let
T
:
K
→
K
be
a
non-expansive
mapping.
If
I
−
T
maps
closed
bounded
subsets
of
E
into
closed
subsets
of
E
and
{
x
n
}
is
the
Mann
iteration
defined
by
(
9
),
with
{
λ
n
}
satisfying
assumptions
(a)-(c)
in
Theorem
3
,
then
{
x
n
}
converges
strongly
to
a
fixed
point
of
T
in
K
.
Proof.
See
Corollary
6.19
in
Chidume
[
10
].
3
Second
order
boundary
value
problems
Our
aim
in
this
section
is
to
obtain
existence
and
approximation
results
for
the
two
point
boundary
value
problem
associated
to
the
second
order
iterative
differential
equation
x

(
t
)
=
f
t,
x
(
t
)
,
x
[2]
(
t
)
,
0
≤
t
≤
1
,
(10)
with
the
solutions
satisfying
one
of
the
two
sets
of
boundary
conditions:
x
(0)
=
0
x
(1)
=
1
,
(11)
and
x
(0)
=
1
x
(1)
=
0
.
(12)
We
shall
need
the
following
two
auxiliary
lemmas.
Lemma
1.
Problem
(
10
)
+
(
11
)
is
equivalent
to
the
integral
equation
x
(
t
)
=
t
−
1
0
G
(
t,
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
ds,
0
≤
t
≤
1
,
(13)
where
G
:
[0
,
1]
×
[0
,
1]
→
R
is
the
Green’s
function
G
(
t,
s
)
=
(1
−
t
)
·
s,
if
0
≤
s
≤
t
≤
1
,
(1
−
s
)
·
t,
if
0
≤
t
≤
s
≤
1
.
(14)
Proof.
Let
x
∈
C
2
[0
,
1]
be
a
solution
of
(
10
)+(
11
).
By
integrating
twice
equation
(
10
),
x

(
t
)
=
f
t,
x
(
t
)
,
x
[2]
(
t
)
,
0
≤
t
≤
1
,
Existence
and
approximation
of
solutions
of
boundary
value
problems
192
we
get
x
(
t
)
=
x
(0)
t
+
t
0
(
t
−
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
ds,
0
≤
t
≤
1
,
(15)
Taking
t
=
1
in
(
15
)
and
solving
the
resulted
identity
for
x
(0)
we
obtain
x
(0)
=
1
−
1
0
(1
−
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
ds
(16)
+
t
0
(
t
−
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
ds.
Now,
by
combining
(
15
)
and
(
16
)
we
get
x
(
t
)
=
t
−
t
·
1
0
(1
−
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
ds
(17)
+
t
0
(
t
−
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
ds.
By
splitting
the
first
integral
in
(
17
)
on
[0
,
t
]
and
[
t,
1]
and
then
by
grouping
the
two
integrals
on
[0
,
t
],
we
obtain
x
(
t
)
=
t
+
t
0
(
t
−
1)
sf
s,
x
(
s
)
,
x
[2]
(
s
)
ds
(18)
+
1
t
(1
−
s
)
tf
s,
x
(
s
)
,
x
[2]
(
s
)
ds,
which
is
exactly
the
integral
equation
(
13
).
This
proves
that
any
solution
of
the
problem
(
10
)+(
11
)
is
a
solution
of
the
integral
equation
(
13
)
and
vice
versa.
Lemma
2.
The
Green’s
function
G
:
[0
,
1]
×
[0
,
1]
→
R
,
defined
by
(
14
)
,
has
the
following
properties:
(i)
0
≤
4
G
(
t,
s
)
≤
1
,
∀
t,
s
∈
[0
,
1]
;
(ii)
8
1
0
G
(
t,
s
)
ds
≤
1
.
Proof.
See
for
example
[
4
].
Now
we
can
state
and
prove
the
first
main
result
of
this
paper.
V.
Berinde
193
Theorem
4.
Assume
that
the
following
conditions
are
satisfied:
(1)
f
∈
C
[0
,
1]
3
,
R
;
(2)
There
exist
the
nonnegative
constants
L
1
,
L
2
such
that
|
f
(
t,
x,
y
)
−
f
(
t,
u,
v
)
|
≤
L
1
|
x
−
u
|
+
L
2
|
y
−
v
|
,
(19)
for
any
(
t,
x,
y
)
,
(
t,
u,
v
)
∈
[0
,
1]
3
.
(3)
L
1
+
L
2
<
8
.
Then,
the
problem
(
10
)
+
(
11
)
has
a
unique
solution
x
in
C
[0
,
1]
and,
for
any
x
0
∈
C
[0
,
1]
satisfying
(
11
)
,
the
sequence
{
x
n
}
defined
by
x
n
+1
(
t
)
=
t
−
1
0
G
(
t,
s
)
f
s,
x
n
(
s
)
,
x
[2]
n
(
s
)
ds,
0
≤
t
≤
1
,
n
≥
0
,
(20)
converges
uniformly
to
x
as
n
→
∞
.
Proof.
Let
us
take
X
to
be
the
space
of
continuous
functions
on
[0
,
1]
with
norm
x
=
max
t
∈
[0
,
1]
|
x
(
t
)
|
.
The
space
X
equipped
with
this
norm
is
a
Banach
space,
convergence
in
norm
being
simply
uniform
convergence.
The
set
K
=
{
x
∈
C
[0
,
1]
:
x
([0
,
1])
⊂
[0
,
1]
}
is
a
nonempty
closed
and
convex
subset
of
X
.
Define
the
operator
T
by
(
Tx
)
(
t
)
=
t
−
1
0
G
(
t,
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
ds,
0
≤
t
≤
1
.
Obviously,
(
Tx
)
(
t
)
is
continuous
if
x
(
t
)
is,
which
means
that
T
maps
X
into
X
.
In
particular,
T
maps
K
into
itself.
By
Lemma
1
,
problem
(
10
)+(
11
)
is
equivalent
to
the
fixed
point
problem
x
=
Tx.
For
x,
y
∈
K
,
we
have
|
(
Tx
)
(
t
)
−
(
Ty
)
(
t
)
|
=
1
0
G
(
t,
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
−
f
s,
y
(
s
)
,
y
[2]
(
s
)

ds
≤
1
0
G
(
t,
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
−
f
s,
y
(
s
)
,
y
[2]
(
s
)

ds.
(we
used
property
(i)
in
Lemma
2
)
By
the
Lipschitzian
condition
(2)
we
have
f
s,
x
(
s
)
,
x
[2]
(
s
)
−
f
s,
y
(
s
)
,
y
[2]
(
s
)

Existence
and
approximation
of
solutions
of
boundary
value
problems
194
≤
L
1
·
|
x
(
s
)
−
y
(
s
)
|
+
L
2
·
|
x
[2]
(
s
)
−
y
[2]
(
s
)
|
≤
L
1
·
x
−
y
+
L
2
·
x
−
y
=
(
L
1
+
L
2
)
x
−
y
,
∀
s
∈
[0
,
1]
.
Therefore
|
(
Tx
)
(
t
)
−
(
Ty
)
(
t
)
|
≤
(
L
1
+
L
2
)
x
−
y
·
1
0
G
(
t,
s
)
ds,
∀
t
∈
[0
,
1]
,
(21)
and
using
property
(ii)
in
Lemma
2
and
taking
max
with
respect
to
t
∈
[0
,
1]
in
(
21
)
we
obtain
Tx
−
Ty
≤
L
1
+
L
2
8
·
x
−
y
,
∀
x,
y
∈
K,
which,
by
assumption
(3),
proves
that
T
:
K
→
K
is
a
contraction.
We
apply
the
contraction
mapping
principle
to
get
the
conclusion.
Remark
3.
We
note
that
Theorem
4
significantly
improves
Theorem
3.4
in
Kaufmann
[
21
].
Indeed,
the
corresponding
condition
in
Theorem
3.4,
i.e.,
1
6
(
M
+
N
)(
b
−
a
)
2
<
1
,
would
be
in
our
case
and
with
our
notations
(
a
=
0
,
b
=
1
,
M
:=
L
1
,
N
:=
L
2
):
L
1
+
L
2
<
6
,
which
is
more
restrictive
than
our
condition
L
1
+
L
2
<
8
.
In
case
we
are
working
on
a
interval
[
a,
b
]
rather
than
on
[0
,
1]
,
then
our
condition
will
read
as
follows
1
8
(
L
1
+
L
2
)(
b
−
a
)
2
<
1
,
(22)
From
the
point
of
view
of
applications,
this
means
that
Theorem
4
can
be
applied
in
the
cases
where
Theorem
3.4
in
Kaufmann
[
21
]
does
not
apply,
as
shown
by
Example
3
.
The
corresponding
result
for
the
problem
(
10
)+(
12
)
can
be
stated
and
proven
similarly.
In
the
special
case
L
1
+
L
2
=
8,
the
conclusion
of
Theorem
4
does
not
hold.
In
such
a
case,
we
should
apply
appropriately
Corollaries
1
or
2
.
In
the
following
we
state
an
extension
of
Theorem
4
.
Theorem
5.
Assume
that
conditions
(1)
and
(2)
in
Theorem
4
and
(3
)
L
1
+
L
2
≤
8
are
satisfied.
V.
Berinde
195
For
any
x
0
∈
K
=
{
x
∈
C
[0
,
1]
:
x
([0
,
1])
⊂
[0
,
1]
}
,
consider
the
sequence
{
x
n
}
defined
for
t
∈
[0
,
1]
and
n
≥
0
by
x
n
+1
(
t
)
=
(1
−
λ
n
)
x
n
(
t
)
+
λ
n
t
−
1
0
G
(
t,
s
)
f
s,
x
n
(
s
)
,
x
[2]
n
(
s
)
ds
,
(23)
where
{
λ
n
}
is
a
sequence
in
[0
,
1]
satisfying:
(b)
0
≤
λ
n
≤
b
<
1
for
all
positive
integers
n
;
(c)
∞
n
=0
λ
n
=
∞
.
Then,
the
problem
(
10
)
+
(
11
)
has
solutions
in
K
and
the
sequence
{
x
n
}
converges
uniformly
to
a
solution
in
K
of
(
10
)
+
(
11
)
as
n
→
∞
.
Proof.
If
in
condition
(3
)
strict
inequality
holds,
then
we
are
in
the
case
of
Theorem
4
.
If
L
1
+
L
2
=
8,
then,
with
the
same
notations
as
in
the
proof
of
Theorem
4
,
X
with
norm
x
=
max
t
∈
[0
,
1]
|
x
(
t
)
|
,
is
a
Banach
space
and
the
set
K
=
{
x
∈
C
[0
,
1]
:
x
([0
,
1])
⊂
[0
,
1]
}
is
a
nonempty
bounded
closed
and
convex
subset
of
X
.
Define
the
operator
T
by
(
Tx
)
(
t
)
=
t
−
1
0
G
(
t,
s
)
f
s,
x
(
s
)
,
x
[2]
(
s
)
ds,
0
≤
t
≤
1
.
Obviously,
(
Tx
)
(
t
)
is
continuous
if
x
(
t
)
is,
which
means
that
T
maps
X
into
X
and
in
particular,
T
maps
K
into
itself
and
is
nonexpansive.
To
reach
the
conclusion,
we
apply
Corollary
2
.
Theorem
6.
Assume
that
conditions
(1)
and
(2)
and
(3’)
in
Theorem
5
are
satisfied.
For
any
x
0
in
the
set
K
=
{
x
∈
C
[0
,
1]
:
x
([0
,
1])
⊂
[0
,
1]
,
|
x
(
t
1
)
−
x
(
t
2
)
|
≤
|
t
1
−
t
2
|
,
∀
t
1
,
t
1
∈
[0
,
1]
}
,
consider
the
sequence
{
x
n
}
defined
by
(
23
)
.
Then,
the
problem
(
10
)
+
(
11
)
has
solutions
in
K
and
the
sequence
{
x
n
}
converges
uniformly
to
a
solution
in
K
of
(
10
)
+
(
11
)
as
n
→
∞
.
Proof.
The
set
K
is
compact
by
Arzel´
a-Ascoli
theorem.
We
prove
that
I
−
T
maps
closed
bounded
subsets
of
X
=
C
[0
,
1]
into
closed
subsets
of
X
.
The
proof
is
essentially
based
on
the
fact
that
typical
integral
operators
act
as
compact
operators
on
standard
Banach
spaces
(such
as
C
[0
,
1]
in
our
case).
Let
A
⊂
X
be
a
closed,
bounded
subset
of
X
.
We
want
to
show
that
the
image
set
(
I
−
T
)(
A
)
is
closed
in
X
.
To
this
end,
take
an
arbitrary
sequence
Existence
and
approximation
of
solutions
of
boundary
value
problems
196
(
y
n
)
∈
(
I
−
T
)(
A
)
that
converges
to
some
element
y
∈
X
.
We
have
to
prove
that
y
∈
(
I
−
T
)(
A
).
Since
(
y
n
)
∈
(
I
−
T
)(
A
),
there
exists
a
corresponding
sequence
(
x
n
)
in
the
domain
subset
A
such
that:
(
I
−
T
)
x
n
=
y
n
=
⇒
x
n
−
Tx
n
=
y
n
.
By
our
assumption,
y
n
→
y
as
n
→
∞
.
Because
A
is
a
bounded
subset
of
X
and
T
is
a
compact
integral
operator,
the
image
set
T
(
A
)
is
relatively
compact,
that
is,
its
closure
is
compact.
By
the
definition
of
relative
compactness,
the
sequence
(
Tx
n
)
must
contain
a
sub-sequence
(
Tx
n
k
)
that
converges
to
some
limit
v
∈
X
,
that
is,
lim
k
→∞
Tx
n
k
=
v.
Now,
we
restrict
ourselves
to
our
chosen
convergent
subsequence
indices
(
n
k
)
and
hence
work
with
the
equation
x
n
k
=
y
n
k
+
Tx
n
k
.
Take
the
limit
of
both
sides
as
k
→
∞
.
As
y
n
→
y
it
follows
that
y
n
k
→
y
.
Using
the
fact
that
Tx
n
k
→
v
and
that
the
sum
of
two
convergent
sequences
is
convergent,
it
follows
that
the
subsequence
(
x
n
k
)
converges
to
a
limit
x
∈
X
:
lim
k
→∞
x
n
k
=
y
+
v
=:
x.
Since
the
subset
A
is
closed
and
the
subsequence
((
x
n
k
)
⊂
A
converges
to
x
,
the
limit
point
x
must
belong
to
A
,
that
is,
(
x
∈
A
).
Therefore,
because
T
is
a
continuous
(bounded)
operator,
we
can
pass
the
limit
inside
the
operator:
lim
k
→∞
Tx
n
k
=
Tx.
This
implies
that
our
earlier
limit
v
is
exactly
equal
to
Tx
.
Substituting
this
back
into
our
limit
equation
yields:
x
=
y
+
Tx
=
⇒
x
−
Tx
=
y
=
⇒
(
I
−
T
)
x
=
y
.
Since
x
∈
A
,
we
have
successfully
shown
that
y
∈
(
I
−
T
)(
A
).
Therefore,
(
I
−
T
)(
A
)
contains
all
its
limit
points
and
hence
it
is
a
closed
subset.
The
conclusion
of
our
Theorem
follows
now
by
applying
Corollary
1
.
Remark
4.
We
note
that
Theorems
5
and
6
ensure
only
existence
and
approximation
of
the
solution
for
the
problem
(
10
)
+
(
11
)
,
while
Theorem
4
ensures
existence,
uniqueness
and
approximation
of
the
solution.
V.
Berinde
197
4
Numerical
examples
Example
3.
Consider
the
nested
(iterative)
differential
equation
x

(
t
)
=
t
+
3
x
(
t
)
+
4
x
[2]
(
t
)
,
0
≤
t
≤
1
,
(24)
with
the
boundary
value
conditions
x
(0)
=
0
x
(1)
=
1
.
(25)
In
this
case
we
have
f
(
t,
y,
z
)
=
t
+3
y
+4
z
which
satisfies
the
Lipschitzian
condition
(
19
)
|
f
(
t,
y,
z
)
−
f
(
t,
u,
v
)
|
≤
L
1
|
y
−
u
|
+
L
2
|
z
−
v
|
,
with
L
1
=
3
and
L
2
=
4
,
for
any
(
t,
y,
z
)
,
(
t,
u,
v
)
∈
[0
,
1]
3
.
Since
L
1
+
L
2
=
7
>
6
,
one
cannot
use
Theorem
3.4
in
Kaufmann
[
21
]
to
decide
about
the
existence
(and
uniqueness)
of
the
solution
of
the
two
point
boundary
value
problem
(
24
)
+
(
25
)
.
But,
based
on
the
fact
that
we
have
L
1
+
L
2
=
7
<
8
,
one
can
apply
Theorem
4
in
our
paper.
Therefore,
according
to
our
result,
the
two
point
boundary
value
problem
(
24
)
+
(
25
)
has
a
unique
solution
x
in
C
[0
,
1]
and,
additionally,
for
any
initial
value
x
0
(
t
)
in
C
[0
,
1]
satisfying
(
25
)
,
the
sequence
of
successive
approximations
x
n
(
t
)
=
t
−
1
0
g
(
t,
s
)
s
+
3
x
n
−
1
(
s
)
+
4
x
n
−
1
x
n
−
1
(
s
)
ds,
n
≥
1
(26)
converges
uniformly
to
x
,
as
n
→
∞
.
In
order
to
compute
simpler
the
iterates
in
(
26
)
,
we
can
write
x
n
(
t
)
=
t
−
(1
−
t
)
t
0
s
s
+
3
x
n
−
1
(
s
)
+
4
x
n
−
1
x
n
−
1
(
s
)
ds
+
t
1
t
(1
−
s
)
s
+
3
x
n
−
1
(
s
)
+
4
x
n
−
1
x
n
−
1
(
s
)
ds
(27)
and
then
proceed
with
the
computation
of
the
integrals.
For
numerical
experiments,
we
consider
the
nested
equation
in
Example
3
.
To
approximate
its
solution,
we
take
as
initial
guess
x
0
(
t
)
=
t,
t
∈
[0
,
1]
.
Existence
and
approximation
of
solutions
of
boundary
value
problems
198
t
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
n
(
t
)
0.2
0.4
0.6
0.8
First
iterations
for
x
0
(
t
)
=
t
x
0
(
t
)
x
1
(
t
)
x
2
(
t
)
x
3
(
t
)
x
4
(
t
)
x
5
(
t
)
Figure
1:
Graph
of
first
iterations
starting
with
x
0
(
t
)
=
t
After
performing
symbolic
computations
with
Matlab
we
obtained
succes-
sively
x
1
(
t
)
=
4
3
t
3
−
1
3
t,
x
2
(
t
)
=
512
4455
t
11
−
32
243
t
9
+
32
567
t
7
+
41
405
t
5
+
2
27
t
3
+
73442
93555
t,
and,
because
the
coefficients
are
decreasing,
we
considered
for
the
next
iter-
ation
a
truncated
expression:
x
3
(
t
)
≈
0
.
476271
t
−
1
.
940010
t
3
−
1
.
574639
t
5
−
0
.
211533
t
7
−
0
.
153887
t
9
+
O
(
t
11
)
,
..
To
illustrate
the
convergence
of
the
Picard
iteration
(
26
)
we
graph
in
Figure
1
the
first
five
iterations
computed.
We
note
that,
in
Figure
1,
due
to
the
approximation
by
truncating
the
polynomial
expressions
of
x
3
(
t
),
x
4
(
t
),
x
5
(
t
),
their
values
at
x
=
1
are
less
than
1.
Example
4.
Consider
the
nested
(iterative)
differential
equation
x

(
t
)
=
t
+
4
x
(
t
)
+
4
x
[2]
(
t
)
,
0
≤
t
≤
1
,
(28)
with
the
boundary
value
conditions
x
(0)
=
0
x
(1)
=
1
.
(29)
V.
Berinde
199
In
this
case
we
have
f
(
t,
y,
z
)
=
t
+4
y
+4
z
which
satisfies
the
Lipschitzian
condition
(
19
)
|
f
(
t,
y,
z
)
−
f
(
t,
u,
v
)
|
≤
L
1
|
y
−
u
|
+
L
2
|
z
−
v
|
,
with
L
1
=
4
and
L
2
=
4
,
for
any
(
t,
y,
z
)
,
(
t,
u,
v
)
∈
[0
,
1]
3
.
Since
L
1
+
L
2
=
8
,
one
cannot
use
neither
Theorem
3.4
in
Kaufmann
[
21
]
nor
our
Theorem
4
to
decide
about
the
existence
of
the
solutions
of
the
two
point
boundary
value
problem
(
28
)
+
(
29
)
.
But,
since
condition
(3
)
is
satisfied,
one
can
apply
Theorem
6
.
We
conclude
that
the
two
point
boundary
value
problem
(
28
)
+
(
29
)
has
solutions
in
the
set
K
=
{
x
∈
C
[0
,
1]
:
x
([0
,
1])
⊂
[0
,
1]
,
|
x
(
t
1
)
−
x
(
t
2
)
|
≤
|
t
1
−
t
2
|
,
∀
t
1
,
t
1
∈
[0
,
1]
}
,
and,
additionally,
for
any
initial
value
x
0
(
t
)
in
K
,
the
Mann
iteration
con-
verges
to
such
a
solution.
t
0.2
0.4
0.6
0.8
x
n
(t)
0.2
0.4
0.6
0.8
SolutionofBVPbyManniteration
Figure
2:
Graph
of
the
solution
x
14
(
t
)
obtained
by
Mann
iteration
process
starting
with
x
0
(
t
)
=
t
and
λ
n
=
1
2
,
n
=
0
,
1
,
2
,
.
.
.
We
performed
numerical
tests
in
Matlab,
using
Mann
iteration
for
λ
n
=
1
2
,
n
=
0
,
1
,
2
,
.
.
.
,
starting
from
x
0
(
t
)
=
t
,
t
∈
[0
,
1]
,
and
with
grid
size
0
.
01
.
The
Mann
iteration
converges
to
the
solution
of
(
28
)
+
(
29
)
in
14
itera-
tions.
The
graph
of
the
obtained
solution
is
plotted
in
Figure
2.
Existence
and
approximation
of
solutions
of
boundary
value
problems
200
5
Conclusions
(1)
We
studed
the
existence
and
approximation
of
solutions
to
the
second-
order
iterative
boundary-value
problem
x

(
t
)
=
f
t,
x
(
t
)
,
x
[2]
(
t
)
,
0
≤
t
≤
1
,
(30)
with
the
boundary
conditions
x
(0)
=
0
,
x
(1)
=
1.
(2)
Our
approach
was
essentially
based
on
the
transformation
of
the
orig-
inal
BVP
into
an
equivalent
fixed
point
problem
x
=
Tx
,
with
T
a
Fredholm
type
operator,
to
which
we
applied
the
contraction
mapping
principle
and
the
principle
of
nonexpansive
mappings,
respectively.
(3)
Our
first
main
result
(Theorem
4
)
significantly
enlarges
the
area
of
application
of
Theorem
3.4
in
Kaufmann
[
21
]
by
imposing
the
weaker
contraction
condition
L
1
+
L
2
<
8
instead
of
L
1
+
L
2
<
6,
where
L
1
and
L
2
are
the
Lipschitz
constants
of
the
function
f
appearing
in
the
righthand
side
of
the
nested
equation
(
30
).
(4)
Our
existence
and
approximation
result
can
be
similarly
formulated
and
proved
for
the
problem
(
30
)
and
the
dual
boundary
conditions
x
(0)
=
1
,
x
(1)
=
0.
(5)
It
appears
that
our
Theorem
4
is
the
first
one
in
literature
to
pro-
vide
not
only
existence
(and
uniqueness)
but
also
approximation
results
for
boundary
value
problems
associated
to
iterative
(nested)
second
order
dif-
ferential
equations.
(6)
In
order
to
illustrate
the
superiority
of
our
findings,
we
considered
and
example
of
a
boundary
value
problem
associated
to
a
nested
second
order
differential
equation
(Example
3
),
for
which
Theorem
3.4
in
Kaufmann
[
21
])
cannot
be
applied
but
our
main
result
(Theorem
4
)
applies.
We
also
considered
the
corresponding
case
of
nonexpansive
mappings
in
Example
4
.
(7)
We
performed
complex
numerical
tests
in
Matlab
and
computed
the
first
5
iterates
of
the
Picard
iteration
(
26
).
Figure
1
plots
these
iterations.
For
the
nonexpansive
case,
in
Figure
2
it
is
plotted
the
solution
obtained
after
n
=
14
iterations
by
the
Mann
iterative
process,
starting
with
x
0
(
t
)
=
t
and
λ
n
=
1
2
,
n
=
0
,
1
,
2
,
.
.
.
.
(8)
As
differential
equations
with
deviating
arguments
have
very
impor-
tant
applications
in
solving
real
world
mathematical
models
from
science
and
technology,
we
suggest
the
readers
to
consider
and
study
such
kind
of
concrete
applications.
(9)
We
also
suggest
readers
to
consider
some
other
approaches
to
study
the
existence
/
non
existence
of
nested
differential
equations,
as
those
studied
V.
Berinde
201
in
Anderson
[
2
],
Bacot
¸iu
[
3
],
Berinde
[
7
],
Brestovansk´
a
et
al.
[
8
],
Guerfi
and
Ardjouni
[
19
],
Karek
and
Ardjouni
[
22
],
Khemis
et
al.
[
23
],
Dunkel
[
11
],
Stanˇ
ek
[
35
]-
[
38
],
Zhang
and
Song
[
42
],
Zhao
and
Feˇ
ckan
[
43
]
etc.
Acknowledgements
.
I
dedicate
this
paper
to
the
memory
of
late
Pro-
fessor
Emeritus
Dr.
Mihail
Megan
(1947-2025),
a
mathematician
of
great
stature,
an
elite
researcher,
a
school
creator
and
an
exceptional
teacher
and
mentor
to
many
generations
of
mathematicians,
who
showed
us
what
the
true
passion
and
enthusiasm
for
doing
mathematics
look
like.
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