Discrete Evolution Family Research Articles

English

The paper investigates the interaction between the notions of expansiveness and admissibility. We consider a polynomially bounded discrete evolution family and define an admissibility notion via the solvability of an associated difference equation. Using the admissibility of weighted Lebesgue spaces of sequences, we give a characterization of discrete evolution families which are polynomially expansive and also those which are expansive in the ordinary sense. Thereafter, we apply the main results in order to infer continuous-time characterizations for the notions of expansiveness through the solvability of an associated integral equation.

Discrete integral conditions for weak trajectories and asymptotic stability for operators in Banach spaces

Let t↦φ(t):R+→R+ be a nondecreasing function which is positive on (0,∞) and let A be a bounded linear operator acting on a complex Banach space X. Let X′ denote the strong dual of X. We prove that if for x∈X and x′∈X′ one has(0.1)sup‖x‖≤1,‖x′‖≤1⁡∑n=0∞φ(|〈Anx,x′〉|)<∞, then there exists a positive number ν=ν(A,φ)∈(0,1) such that the spectral radius of A, denoted by r(A), satisfies the inequality r(A)≤ν. Some theorems on existence for discrete evolution equations in Banach spaces are presented and extensions of these results to discrete evolution families are also made.

Kallman-Rota type inequality for discrete evolution families of bounded linear operators

Kallman-Rota type inequality for discrete evolution families of bounded linear operators

Asymptotic behavior of discrete evolution families in Banach spaces

Let X be a Banach space, be an operator-valued sequence and let be the discrete evolution family associated to In this paper we prove that the family is non-uniformly strongly stable (i.e. for every nonnegative integer m and every if and only if it is -approximative admissible, i.e. for every sequence in and every positive number there exists the sequence in satisfying such that the solution of the discrete Cauchy Problem belongs to Other types of asymptotic behavior of the family are also analyzed.

Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces

In this article, we prove that the ω-periodic discrete evolution family $\Gamma:= \{\rho(n,k): n, k \in\mathbb{Z}_{+}, n\geq k\}$ of bounded linear operators is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions. More precisely, we prove that if for each real number γ and each sequence $(\xi(n))$ taken from some Banach space, the approximate solution of the nonautonomous ω-periodic discrete system $\theta _{n+1} = \Lambda_{n}\theta_{n}$ , $n\in\mathbb{Z}_{+}$ is represented by $\phi _{n+1}=\Lambda_{n}\phi_{n}+e^{i\gamma(n+1)}\xi(n+1)$ , $n\in\mathbb{Z}_{+}$ ; $\phi_{0}=\theta_{0}$ , then the Hyers-Ulam stability of the nonautonomous ω-periodic discrete system $\theta_{n+1} = \Lambda_{n}\theta_{n}$ , $n\in\mathbb{Z}_{+}$ is equivalent to its uniform exponential stability.

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 inequalityhere is the operator norm of a convolution operator which acts on a certain Banach space of X-valued sequences.

Asymptotic Behavior of Linear and Almost Periodic Discrete Evolution Systems on Banach Space $$\mathcal {AAP}_0^r(\mathbb {Z}_+,\mathcal {W})$$ AAP 0 r ( Z + , W )

Let \(v_{n}\) be the solution of the nonhomogeneous evolution difference equation \(v_{n+1} = \mathcal {A}_nv_{n}+ \alpha _{n+1}\), \(v_0= 0\) for all \(\ n\in \mathbb {Z}_+\), where \(\mathcal {A}_n\) is a sequence of almost periodic (possibly unbounded) linear operators on a Banach space \(\mathcal {W}\). Let \(\mathcal {C}_{00}(\mathbb {Z}_+,\mathcal {W})\) is the space of all \(\mathcal {W}\)-valued bounded sequences which decays at zero and at infinity and \(\mathcal {AP}_0(\mathbb {Z}_+,\mathcal {W})\) is the space of all \(\mathcal {W}\)-valued almost periodic sequences decaying at zero. We consider the space \(\mathcal {AAP}_0^r(\mathbb {Z}_+,\mathcal {W})\) consisting of all sequences \(\alpha (n)\) with relatively compact ranges for which there exists \(\beta (n)\in \mathcal {AP}_0(\mathbb {Z}_+,\mathcal {W})\) and \(\gamma (n) \in \mathcal {C}_{00}(\mathbb {Z}_+,\mathcal {W})\) such that \(\alpha (n) = \beta (n) + \gamma (n)\). We prove that \(v_{n}\) belongs to \(\mathcal {AAP}_0^r(\mathbb {Z}_+,\mathcal {W})\) if and only if for each \(x\in \mathcal {W}\) the discrete evolution family of operators \(E = \{\mathcal {E}(n, m): n,m \in \mathbb {Z}_+, \ n\ge m \}\) is uniformly exponentially stable. Our results are based on evolution semigroup approach.

On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

We prove that a discrete semigroup𝕋={T(n):n∈ℤ+}of bounded linear operators acting on a complex Banach spaceXis uniformly exponentially stable if and only if, for eachx∈AP0(ℤ+,X), the sequencen↦∑k=0n‍T(n-k)x(k):ℤ+→Xbelongs toAP0(ℤ+,X). Similar results for periodic discrete evolution families are also stated.

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.

Connections between the stability of a Poincare map and boundedness of certain associate sequences

Let m ≥ 1 and N ≥ 2 be two natural numbers and let U = {U(p,q)}pq�0 be the N-periodic discrete evolution family of m × m matrices, having complex scalars as entries, generated by L(C m )-valued, N-periodic sequence of m × m matrices (An). We prove that the solution of the following discrete problem yn+1 = Anyn + e iµn b, n ∈ Z+, y0 = 0 is bounded for each µ ∈ R and each m-vector b if the Poincare map U(N,0) is stable. The converse statement is also true if we add a new assumption to the boundedness condition. This new assumption refers to the invertibility for each µ ∈ R of the ma- trix Vµ := P N=1 U(N,�)e iµ� . By an example it is shown that the assumption on invertibility cannot be removed. Finally, a strong variant of Barbashin's type theorem is proved.

Uniform dichotomy and exponential dichotomy of evolution families on the half-line

The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family Φ = { Φ ( m , n ) } ( m , n ) ∈ Δ the subspace X 1 = { x ∈ X : Φ ( ⋅ , 0 ) x ∈ ℓ ∞ ( N , X ) } . Supposing that X 1 is closed and complemented, we prove that the admissibility of the pair ( ℓ ∞ ( N , X ) , ℓ 0 1 ( N , X ) ) implies the uniform dichotomy of Φ. Under the same hypothesis on X 1 , we obtain that the admissibility of the pair ( ℓ ∞ ( N , X ) , ℓ 0 p ( N , X ) ) with p ∈ ( 1 , ∞ ] is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations.

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

Exponential dichotomy and [formula omitted]-admissibility on the half-line

We give necessary and sufficient conditions for uniform exponential dichotomy of discrete evolution families in terms of the admissibility of the pair ( l p ( N , X ) , l 0 q ( N , X ) ) . We prove that the admissibility of the pair ( l p ( N , X ) , l 0 q ( N , X ) ) is a sufficient condition for uniform exponential dichotomy of a discrete evolution family. This condition becomes necessary for discrete evolution families with uniform exponential growth if and only if p ⩾ q . As consequences, we obtain necessary and sufficient conditions for uniform exponential dichotomy of evolution families.

Discrete admissibility and exponential dichotomy for evolution families

Connections between admissibility and uniform exponential dichotomy of discrete evolution families are studied. Discrete and continuous characterizations for uniform exponential dichotomy of evolution families are given. A new version for a theorem due to Van Minh, Räbiger and Schnaubelt, for the discrete case is obtained.