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Extremal solutions to fourth-order functional boundary value problems including multipoint conditions
This paper concerns the fully fourth-order nonlinear functional equation (E) − d d t ( ϕ ∘ u ‴ ) ( t ) = f ( t , u ″ ( t ) , u ‴ ( t ) , u , u ′ , u ″ ) , for a.a. t ∈ I = [ a , b ] , with the functional boundary conditions (BC) B 1 ( u ( b ) , u , u ′ , u ″ ) = 0 = B 2 ( u ′ ( b ) , u , u ′ , u ″...
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| Published in: | Nonlinear analysis: real world applications 2009-08, Vol.10 (4), p.2157-2170 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | This paper concerns the fully fourth-order nonlinear functional equation
(E)
−
d
d
t
(
ϕ
∘
u
‴
)
(
t
)
=
f
(
t
,
u
″
(
t
)
,
u
‴
(
t
)
,
u
,
u
′
,
u
″
)
,
for a.a.
t
∈
I
=
[
a
,
b
]
,
with the functional boundary conditions (BC)
B
1
(
u
(
b
)
,
u
,
u
′
,
u
″
)
=
0
=
B
2
(
u
′
(
b
)
,
u
,
u
′
,
u
″
)
,
B
3
(
u
″
(
a
)
,
u
″
(
b
)
,
u
‴
(
a
)
,
u
‴
(
b
)
,
u
,
u
′
,
u
″
)
=
0
=
L
2
(
u
″
(
a
)
,
u
″
(
b
)
)
,
where
ϕ
:
R
⟶
R
is an increasing homeomorphism,
f
:
I
×
R
2
×
(
C
(
I
)
)
3
⟶
R
,
B
i
,
i
=
1
,
2
,
3
, and
L
2
are suitable functions. The existence of extremal solutions for problem
(E)-(BC) is proved by defining a convenient partial ordering. Some sufficient conditions to obtain lower and upper solutions are given. |
|---|---|
| ISSN: | 1468-1218 1878-5719 |
| DOI: | 10.1016/j.nonrwa.2008.03.026 |