133 research outputs found
Bohl-Perron type stability theorems for linear difference equations with infinite delay
Relation between two properties of linear difference equations with infinite
delay is investigated: (i) exponential stability, (ii) \l^p-input
\l^q-state stability (sometimes is called Perron's property). The latter
means that solutions of the non-homogeneous equation with zero initial data
belong to \l^q when non-homogeneous terms are in \l^p. It is assumed that
at each moment the prehistory (the sequence of preceding states) belongs to
some weighted \l^r-space with an exponentially fading weight (the phase
space). Our main result states that (i) (ii) whenever and a certain boundedness condition on coefficients is
fulfilled. This condition is sharp and ensures that, to some extent,
exponential and \l^p-input \l^q-state stabilities does not depend on the
choice of a phase space and parameters and , respectively. \l^1-input
\l^\infty-state stability corresponds to uniform stability. We provide some
evidence that similar criteria should not be expected for non-fading memory
spaces.Comment: To be published in Journal of Difference Equations and Application
Stability of Hahnfeldt Angiogenesis Models with Time Lags
Mathematical models of angiogenesis, pioneered by P. Hahnfeldt, are under
study. To enrich the dynamics of three models, we introduced biologically
motivated time-varying delays. All models under study belong to a special class
of nonlinear nonautonomous systems with delays. Explicit conditions for the
existence of positive global solutions and the equilibria solutions were
obtained. Based on a notion of an M-matrix, new results are presented for the
global stability of the system and were used to prove local stability of one
model. For a local stability of a second model, the recent result for a
Lienard-type second-order differential equation with delays was used. It was
shown that models with delays produce a complex and nontrivial dynamics. Some
open problems are presented for further studies
Delay differential equations with Hill's type growth rate and linear harvesting
AbstractFor the equation, N˙(t)=r(t)N(t)1+[N(t)]γ−b(t)N(t)−a(t)N(g(t)),we obtain the following results: boundedness of all positive solutions, extinction, and persistence conditions. The proofs employ recent results in the theory of linear delay equations with positive and negative coefficients
Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space
Consider a linear impulsive equation in a Banach space
with . Suppose each solution of
the corresponding semi-homogeneous equation
(2) is bounded for any bounded sequence .
The conditions are determined ensuring
(a) the solution of the corresponding homogeneous equation has an exponential
estimate;
(b) each solution of (1),(2) is bounded on the half-line for any bounded
and bounded sequence ;
(c) for any tending to
zero;
(d) exponential estimate of implies a similar estimate for .Comment: 19 pages, LaTex-fil
Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization
Let be an underlying space with a non-atomic measure on it (e.g.
and is the Lebesgue measure). We introduce and study a
class of non-commutative generalized stochastic processes, indexed by points of
, with freely independent values. Such a process (field),
, , is given a rigorous meaning through smearing out
with test functions on , with being a
(bounded) linear operator in a full Fock space. We define a set
of all continuous polynomials of , and then define a con-commutative
-space by taking the closure of in the norm
, where is the vacuum in the Fock
space. Through procedure of orthogonalization of polynomials, we construct a
unitary isomorphism between and a (Fock-space-type) Hilbert space
, with
explicitly given measures . We identify the Meixner class as those
processes for which the procedure of orthogonalization leaves the set invariant. (Note that, in the general case, the projection of a
continuous monomial of oder onto the -th chaos need not remain a
continuous polynomial.) Each element of the Meixner class is characterized by
two continuous functions and on , such that, in the
space, has representation
\omega(t)=\di_t^\dag+\lambda(t)\di_t^\dag\di_t+\di_t+\eta(t)\di_t^\dag\di^2_t,
where \di_t^\dag and \di_t are the usual creation and annihilation
operators at point
A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed Equation Δ
A linear (k+1)th-order discrete delayed equation Δx(n)=−p(n)x(n−k) where p(n) a positive sequence is considered for n→∞. This equation is known to have a positive solution if the sequence p(n) satisfies an inequality. Our aim is to show that, in the case of the opposite inequality for p(n), all solutions of the equation considered are oscillating for n→∞
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