Affiliation:
1. Fachbereich Mathematik , Universität Duisburg–Essen , 47048 Duisburg , Germany
Abstract
Abstract
We present sufficient conditions on the existence of solutions, with various specific almost periodicity properties, in the context of nonlinear, generally multivalued, non-autonomous initial value differential equations,
d
u
d
t
(
t
)
∈
A
(
t
)
u
(
t
)
,
t
≥
0
,
u
(
0
)
=
u
0
,
\frac{du}{dt}(t)\in A(t)u(t),\quad t\geq 0,\qquad u(0)=u_{0},
and their whole line analogues,
d
u
d
t
(
t
)
∈
A
(
t
)
u
(
t
)
{\frac{du}{dt}(t)\in A(t)u(t)}
,
t
∈
ℝ
{t\in\mathbb{R}}
,
with a family
{
A
(
t
)
}
t
∈
ℝ
{\{A(t)\}_{t\in\mathbb{R}}}
of ω-dissipative operators
A
(
t
)
⊂
X
×
X
{A(t)\subset X\times X}
in a general Banach space X.
According to the classical DeLeeuw–Glicksberg theory, functions of various generalized almost periodic types uniquely decompose in a “dominating” and a “damping” part.
The second main object of the study – in the above context – is to determine the corresponding “dominating” part
[
A
(
⋅
)
]
a
(
t
)
{[A(\,\cdot\,)]_{a}(t)}
of the operators
A
(
t
)
{A(t)}
, and the corresponding “dominating” differential equation,
d
u
d
t
(
t
)
∈
[
A
(
⋅
)
]
a
(
t
)
u
(
t
)
,
t
∈
ℝ
.
\frac{du}{dt}(t)\in[A(\,\cdot\,)]_{a}(t)u(t),\quad t\in\mathbb{R}.
Reference31 articles.
1. H. Attouch,
Variational Convergence for Functions and Operators,
Pitman, Boston, 1984.
2. B. Aulbach and N. Van Minh,
A sufficient condition for almost periodicity of nonautonomous nonlinear evolution equations,
Nonlinear Anal. 51 (2002), 145–153.
3. P. Cieuta, S. Fatajou and G. N’Guérékata,
Bounded and almost automorphic solutions of some semilinear differential equations in Banach spaces,
Nonlinear Anal. 71 (2009), 74–684.
4. M. G. Crandall and A. Pazy,
Nonlinear evolution equations in Banach spaces,
Israel J. Math. 11 (1972), 57–94.
5. K. de Leeuw and I. Glicksberg,
Almost periodic functions on semigroups,
Acta Math. 105 (1961), 99–140.
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7 articles.
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