scholarly journals Nonuniform exponential stability for evolution families on the real line

2015 ◽  
Vol 23 (1) ◽  
pp. 199-212
Author(s):  
Claudia Isabela Morariu ◽  
Petre Preda
Keyword(s):  
Exponential Stability ◽  
Real Line ◽  
Exponential Growth ◽  
Sufficient Conditions ◽  
Evolution Family ◽  
Input Output ◽  
The Real ◽  
Perron Method ◽  
Evolution Families ◽  
Nonuniform Exponential Stability

AbstractThe purpose of the present paper is to investigate the problem of nonuniform exponential stability of evolution families on the real line using the input-output technique known in the literature as the Perron method for the study of exponential stability. In this manuscript we describe an evolution family on the real line and we present sufficient conditions for the nonuniform exponential stability of an evolution family on the real line that does not have exponential growth.

scholarly journals A NOTE ON SOME THEOREMS OF R. DATKO

2017 ◽  
Vol 5 ◽  
pp. 1048-1054
Author(s):  
Cristina Andreea Babaita ◽  
Raluca Moresan ◽  
Petre Preda
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Exponential Stability ◽  
Exponential Growth ◽  
Homogeneous Equation ◽  
Sufficient Conditions ◽  
Uniform Exponential Stability ◽  
Evolution Families ◽  
Similar Characterization ◽  
Nonuniform Exponential Stability

The asymptotic behavior of the evolution families is a widely interesting topic in mathematics over time. In 1930, O. Perron was the first one who established the connection between the asymptotic behavior of the solution of the homogenous differential equation and the associated non-homogeneous equation, in finite dimensional spaces. Further, the result was extended for infinite dimensional spaces. The case of dynamical systems described by evolution processes was studied by C. Chicone and Y. Latushkin. One of the most remarkable results in the theory of stability of dynamical systems has been obtained by R. Datko in 1970 for the particular case of C0-semigroups. Practically, R. Datko defines a characterization for uniform exponential stability of the C0-semigroups. Later, it was proved that a similar characterization is also valid for two-parameter evolution families.In this paper we obtain different versions of a well-known theorem of R. Datko for uniform and nonuniform exponential bounded evolution families. More precisely, we obtain theorems that characterize the nonuniform and uniform exponential stability of evolution families with uniform and nonuniform exponential growth. We show that, if we choose K dependent of t0 in the form of Datko's theorem used by C. Stoica and M. Megan, we obtain a result of nonuniform exponential stability, which is no longer possible in the original form of Datko's theorem.In conclusion, we generalize the results initially obtained by Datko (1972) and Preda and Megan (1985), by presenting some sufficient conditions for the nonuniform exponential stability of evolution families with nonuniform exponential growth.

scholarly journals Exponential dichotomy for evolution families on the real line

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
Adina Luminiţa Sasu
Keyword(s):  
Real Line ◽  
Exponential Dichotomy ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Evolution Family ◽  
The Real ◽  
Evolution Families ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pair(Lp(ℝ,X),Lq(ℝ,X)). We show that the admissibility of the pair(Lp(ℝ,X),Lq(ℝ,X))is equivalent to the uniform exponential dichotomy of an evolution family if and only ifp≥q. As applications we obtain characterizations for uniform exponential dichotomy of semigroups.

Sufficient conditions for long-range count dependence of stationary point processes on the real line

2001 ◽  
Vol 38 (2) ◽  
pp. 570-581 ◽  
Author(s):  
Rafał Kulik ◽  
Ryszard Szekli
Keyword(s):  
Long Range ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Second Moment ◽  
The Real ◽  
Palm Measure ◽  
Stationary Point Processes

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

Sufficient conditions for long-range count dependence of stationary point processes on the real line

2001 ◽  
Vol 38 (02) ◽  
pp. 570-581 ◽  
Author(s):  
Rafał Kulik ◽  
Ryszard Szekli
Keyword(s):  
Long Range ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Second Moment ◽  
The Real ◽  
Palm Measure ◽  
Stationary Point Processes

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

STOCHASTIC EQUIVALENCE OF CONVEX ORDERED DISTRIBUTIONS AND APPLICATIONS

2000 ◽  
Vol 14 (1) ◽  
pp. 33-48 ◽  
Author(s):  
M. C. Bhattacharjee ◽  
R. N. Bhattacharya
Keyword(s):  
Recent Result ◽  
Real Line ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Survival Times ◽  
The Real ◽  
Ordered Alternatives ◽  
Special Cases ◽  
Stochastic Equivalence

We consider sufficient conditions for stochastic equivalence of convex ordered random variables. Our main results apply to all convex ordered distributions on the real line and improve on a recent result of Huang and Lin [8] for equality in distribution of convex ordered survival times. Illustrative applications include testing for equality in distribution with convex ordered alternatives and demonstrating several earlier results on stochastic equivalence as special cases.

Partially one-sided polynomial approxi- mation on the real line

2001 ◽  
Vol 38 (1-4) ◽  
pp. 313-329
Author(s):  
József Szabados ◽  
G. Mastroianni
Keyword(s):  
Error Estimate ◽  
Bounded Variation ◽  
Real Line ◽  
Exponential Growth ◽  
The Real ◽  
Lp Metric

We generalize a theorem of Freud and Szabados [3] on one-sided polynomial approxi- mation in five different directions: we allow functions with exponential growth at infinity, Lp-metric, Freud-type weights instead of Hermite weights, functions with bounded deriva- tives instead of bounded variation, and include the momentum in the error estimate.

scholarly journals Mean Value theorems for multiplicative functions bounded in mean α-power, α >1

1980 ◽  
Vol 29 (2) ◽  
pp. 177-205 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Real Line ◽  
Sufficient Conditions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Multiplicative Functions ◽  
The Real ◽  
Mean Value Theorems ◽  
Necessary And Sufficient

AbstractOn analogy with functions if Lebesuge class Lα on the real line the author considers those multiplicative arthmetic functions which are bounded in mean α>1. Necessary and sufficient conditions are obtained in order that they should have a mean-value, zero or non-zero. An application is made to Ramanujan's τ-function.

ON SMOOTHING PROPER KNOTS IN 3-MANIFOLDS

2003 ◽  
Vol 12 (07) ◽  
pp. 971-985 ◽  
Author(s):  
OLLIE NANYES
Keyword(s):  
Real Line ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Topological Equivalence ◽  
The Real

A topological proper knot is a proper embedding f:ℝ1→M3 of the real line into an open 3-manifold. Two proper knots are equivalent if they can be connected by a topological proper isotopy. In this paper, we answer a question posed by the author in [6] and show that, up to topological equivalence and orientation, all proper knots running between the opposite ends of D2×ℝ1 are equivalent. Then sufficient conditions for a topological proper knot to be equivalent to a piecewise linear proper knot are given.

scholarly journals The ordering on permutations induced by continuous maps of the real line

1987 ◽  
Vol 7 (2) ◽  
pp. 155-160 ◽  
Author(s):  
Chris Bernhardt
Keyword(s):  
Real Line ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Partial Ordering ◽  
The Real ◽  
Continuous Maps ◽  
Necessary And Sufficient ◽  
Natural Way

AbstractContinuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This ordering restricted to cycles is studied.Necessary and sufficient conditions are given for a cycle to have an immediate predecessor. When a cycle has an immediate predecessor it is unique; it is shown how to construct it. Every cycle has immediate successors; it is shown how to construct them.

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