17-03-2014, 10:39 PM
Abstract—This paper introduces a concept of exponential
instability for skew-evolution semiflows in Banach spaces. This
is defined by means of evolution semiflows and evolution
cocycles. The approach is from nonuniform point of view.
Keywords- evolution semiflow; evolution cocycle; skewevolution
semiflow; exponential instability
I. INTRODUCTION
The study of asymptotic properties, such as
exponential stability and exponential instability, considered
basic concepts that appear in the theory of dynamical
systems, plays an important role in the study of stable,
instable and central manifolds. Some of the original results
concerning stability and instability were published in [6], for
a particular case of skew-evolution semiflows defined by
means of semiflows and cocycles.
Concerning previous results, D.R. Latcu and
M.Megan in [2] and M. Megan and C. Buse presents in [3]
the exponential dichotomy for evolutionary processes using
this kind of definition. M. Megan and C.Stoica, in [4] and
[5], give the definition for the following notions: evolution
semiflow, evolution cocycle over the semiflow and skewevolution
semiflow. Caracterization for the nonuniform
exponential stability, instability and dichotomy for skewevolution
semiflows on Banach spaces were obtained by M.
Megan, C. Stoica and L. Buliga in [6]. Caracterizations for
the nonuniform exponential stability, from another point of
view, were obtained by A. Minda, M. Tomescu, C.Anghel
and D. Stoica in [1]. The study of the nonuniform
exponential dichotomy for evolution families was
emphasized by P. Preda and M. Megan in [7].
In this paper we extend the asymptotic properties
of exponential instability for the newly introduced concept
of skew-evolution semiflows defined on Banach spaces,
which can be considered generalizations for evolution
operators and skew-product semiflows. The results
concerning the nonuniform exponential instability are
generalizations of some theorems proved for evolution
operators.
instability for skew-evolution semiflows in Banach spaces. This
is defined by means of evolution semiflows and evolution
cocycles. The approach is from nonuniform point of view.
Keywords- evolution semiflow; evolution cocycle; skewevolution
semiflow; exponential instability
I. INTRODUCTION
The study of asymptotic properties, such as
exponential stability and exponential instability, considered
basic concepts that appear in the theory of dynamical
systems, plays an important role in the study of stable,
instable and central manifolds. Some of the original results
concerning stability and instability were published in [6], for
a particular case of skew-evolution semiflows defined by
means of semiflows and cocycles.
Concerning previous results, D.R. Latcu and
M.Megan in [2] and M. Megan and C. Buse presents in [3]
the exponential dichotomy for evolutionary processes using
this kind of definition. M. Megan and C.Stoica, in [4] and
[5], give the definition for the following notions: evolution
semiflow, evolution cocycle over the semiflow and skewevolution
semiflow. Caracterization for the nonuniform
exponential stability, instability and dichotomy for skewevolution
semiflows on Banach spaces were obtained by M.
Megan, C. Stoica and L. Buliga in [6]. Caracterizations for
the nonuniform exponential stability, from another point of
view, were obtained by A. Minda, M. Tomescu, C.Anghel
and D. Stoica in [1]. The study of the nonuniform
exponential dichotomy for evolution families was
emphasized by P. Preda and M. Megan in [7].
In this paper we extend the asymptotic properties
of exponential instability for the newly introduced concept
of skew-evolution semiflows defined on Banach spaces,
which can be considered generalizations for evolution
operators and skew-product semiflows. The results
concerning the nonuniform exponential instability are
generalizations of some theorems proved for evolution
operators.