Constantin Buse

Uniform stability of periodic discrete systems in Banach spaces

Let q>1 be a fixed integer number. We prove that a discrete q-periodic evolution family on a complex Banach space is uniformly asymptotically stable, that is, U(m, n) → 0 in the norm of when (m − n) → ∞, if and only if for each and each x ∈ X one has In particular, we obtain the following result of Datko type. The family is uniformly asymptotically stable if and only if for each one has

Ostrowski's inequality for vector-valued functions and applications

Abstract In this paper, we obtain some Ostrowski type inequalities for vector-valued functions which both generalise and improve the scalar case and show that the midpoint inequality is the best possible inequality in the class. A quadrature formula of the Riemann type for the Bochner integral and the error bounds are considered. Applications to operator inequalities and numerical approximation for the mild solution of inhomogeneous vector-valued differential equations are also pointed out. In the last section, a numerical example is analysed.

A Theorem of Rolewicz's type in solid function spaces

Let R+ be the set of all non-negative real numbers, I ∈ {R,R+} and UI = {U(t, s) : t ≥ s ∈ I} be a strongly measurable and exponentially bounded evolution family of bounded linear operators acting on a Banach space X. Let φ : R+ → R+ be a strictly increasing function and E be a normed function space over I satisfying some properties, see Section 2. We prove that if φ ◦ (χ[s,∞)(·)||U(·, s)x||) defines an element of the space E for every s ∈ I and all x ∈ X and if there exists M > 0 such that sup s∈I |φ ◦ (χ[s,∞)(·)||U(·, s)x||)|E = M 0 for all t > 0 and if there exists K > 0 such that

A refinement of Gruss type inequality for the Bochner integral of vector-valued functions in Hilbert spaces and applications

A refinement of Gruss type inequality for the Bochner integral of vector-valued functions in real or complex Hilbert spaces is given. Related results are obtained. Application for finite Fourier transforms of vector-valued functions and some particular inequalitites are provided.

An inequality concerning the growth bound of a discrete evolution family on a complex Banach space

We prove that the uniform growth bound of a discrete evolution family of bounded linear operators acting on a complex Banach space X satisfies the inequality here is the operator norm of a convolution operator which acts on a certain Banach space of X-valued sequences.

Asymptotic stability and uniform boundedness with respect to parameters for discrete non-autonomous periodic systems

Let X be a complex Banach space and q ≥ 2 be a fixed integer number. Let be a q-periodic discrete evolution family generated by the ℒ(X)-valued, q-periodic sequence (A n ). We prove that the solution of the following discrete problem is bounded (uniformly with respect to the parameter μ ∈ ℝ) for each vector b ∈ X if and only if the Poincare map U(q,0) is stable.

A New Estimation of the Growth Bound of a Periodic Evolution Family on Banach Spaces

Let be a strongly continuous and -periodic evolution family acting on a complex Banach space . We prove that if and then the growth bound of the family is less than or equal to .

Dichotomy and Boundedness of Solutions for Some Discrete Cauchy Problems Constantin Buşe and Akbar Zada

Let us denote by ℤ+ the set of all nonnegative integer numbers. We prove that a square size matrix A of order m having complex entries is dichotomic (i.e., its spectrum does not intersect the set {z∈ℂ:|z| = 1} if and only if there exists a projection P on ℂ m which commutes with A, and for each number μ∈ℝ and each vector b∈ℂ m the solutions of the following two Cauchy problems are bounded:

A surjection problem leading to the Ax-Grothendieck theorem

\left\{ \begin{gathered} x_{n + 1} = Ax_n + e^{i\mu n} Pb, n \in \mathbb{Z}_ + \hfill \\ x_0 = 0 \hfill \\ \end{gathered} \right.