Compact almost automorphic solutions to integral equations with infinite delay
Given a ∈ L 1 ( R ) and the generator A of an L 1 -integrable resolvent family of linear bounded operators defined on a Banach space X , we prove the existence of compact almost automorphic solutions of the semilinear integral equation u ( t ) = ∫ − ∞ t a ( t − s ) [ A u ( s ) + f ( s , u ( s ) ) ]...
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| Published in: | Nonlinear analysis Vol. 71; no. 12; pp. 6029 - 6037 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Amsterdam
Elsevier Ltd
15.12.2009
Elsevier |
| Subjects: | |
| ISSN: | 0362-546X, 1873-5215 |
| Online Access: | Get full text |
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| Summary: | Given
a
∈
L
1
(
R
)
and the generator
A
of an
L
1
-integrable resolvent family of linear bounded operators defined on a Banach space
X
, we prove the existence of compact almost automorphic solutions of the semilinear integral equation
u
(
t
)
=
∫
−
∞
t
a
(
t
−
s
)
[
A
u
(
s
)
+
f
(
s
,
u
(
s
)
)
]
d
s
for each
f
:
R
×
X
→
X
compact almost automorphic in
t
, for each
x
∈
X
, and satisfying Lipschitz and Hölder type conditions. In the scalar linear case, we prove that
a
∈
L
1
(
R
)
positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions. |
|---|---|
| Bibliography: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
| ISSN: | 0362-546X 1873-5215 |
| DOI: | 10.1016/j.na.2009.05.042 |