On Different Types of Stability for Linear Delay Dynamic Equations

Braverman, Elena

doi:10.4171/zaa/1592

Cited by 2 publications

(6 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…satisfying condition (5). Finally, since 13) is satisfied and therefore, by Theorem 3.5 in Braverman and Karpuz, 23 the trivial equilibrium is globally asymptotically stable.…”

Section: Figurementioning

confidence: 75%

“…If 𝜇p < 0 and (− 𝜇p)k < 1, then the trivial solution of ( 9) is globally asymptotically stable. 24) in Braverman and Karpuz 23 is satisfied. We note that…”

Section: Figurementioning

confidence: 89%

“…Then, α ( t )≤ t for all

t \in 𝕋

and Condition (A1) in Braverman and Karpuz 23 holds. Furthermore,

\int_{t_{0}}^{\infty} A false(η false) normalΔ η = \sum_{n = 0}^{\infty} false(- P false) = \infty

and Condition (24) in Braverman and Karpuz 23 is satisfied. We note that

α_{*} (η) = \inf {α (τ) : τ \in [t, \infty) \cap 𝕋} = ρ^{k} (t),

so that

α_{- 1} (t) = \sup {η : α_{*} (η) \leq t} = σ^{k} (t) .

Thus,

\int_{α_{*} (t)}^{σ (t)} A (η) Δ η = \sum_{n = t}^{σ^{k - 1} false(t false)} (- P) = (- P) k,

satisfying condition (5).…”

Section: Delay Dynamic Equations With One‐periodic Coefficientsmentioning

confidence: 97%

“…Proof Define

α false(t false) = ρ^{k} false(t false)

and

A false(η false) = - p false(η false) = \frac{- C}{μ false(η false)}

, where

C \in ℝ

. Then, α ( t )≤ t for all

t \in 𝕋

and Condition (A1) in Braverman and Karpuz 23 holds. Furthermore,

\int_{t_{0}}^{\infty} A false(η false) normalΔ η = \sum_{n = 0}^{\infty} false(- P false) = \infty

and Condition (24) in Braverman and Karpuz 23 is satisfied.…”

Section: Delay Dynamic Equations With One‐periodic Coefficientsmentioning

confidence: 99%

“…T = {2,3,5,9,12,13,16,18,21,22,23,26,29,32, 36,37, 39,42, 43,46, 47,48, 50,53, 54,56, 58,59, 61,68, 70} and initial values (1.0976, 9.3376, 1.8746, 2.6618). Panel (A) displays the behavior of the solution x, and (B) visualizes the behavior of 𝜇x.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Delay dynamic equations on isolated time scales and the relevance of one‐periodic coefficients

Böhner

Cuchta

Streipert

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

We are motivated by the idea that certain properties of delay differential and difference equations with constant coefficients arise as a consequence of their one-periodic nature. We apply the recently introduced definition of periodicity for arbitrary isolated time scales to linear delay dynamic equations and a class of nonlinear delay dynamic equations. Utilizing a derived identity of higher order delta derivatives and delay terms, we rewrite the considered linear and nonlinear delayed dynamic equations with one-periodic coefficients as a linear autonomous dynamic system with constant matrix. As the simplification of a constant matrix is only obtained for one-periodic coefficients, dynamic equations with one-periodic coefficients are the simplest form compared to the commonly used constant coefficients.

show abstract

“…satisfying condition (5). Finally, since 13) is satisfied and therefore, by Theorem 3.5 in Braverman and Karpuz, 23 the trivial equilibrium is globally asymptotically stable.…”

Section: Figurementioning

confidence: 75%

“…If 𝜇p < 0 and (− 𝜇p)k < 1, then the trivial solution of ( 9) is globally asymptotically stable. 24) in Braverman and Karpuz 23 is satisfied. We note that…”

Section: Figurementioning

confidence: 89%

“…Then, α ( t )≤ t for all

t \in 𝕋

and Condition (A1) in Braverman and Karpuz 23 holds. Furthermore,

\int_{t_{0}}^{\infty} A false(η false) normalΔ η = \sum_{n = 0}^{\infty} false(- P false) = \infty

and Condition (24) in Braverman and Karpuz 23 is satisfied. We note that

α_{*} (η) = \inf {α (τ) : τ \in [t, \infty) \cap 𝕋} = ρ^{k} (t),

so that

α_{- 1} (t) = \sup {η : α_{*} (η) \leq t} = σ^{k} (t) .

Thus,

\int_{α_{*} (t)}^{σ (t)} A (η) Δ η = \sum_{n = t}^{σ^{k - 1} false(t false)} (- P) = (- P) k,

satisfying condition (5).…”

Section: Delay Dynamic Equations With One‐periodic Coefficientsmentioning

confidence: 97%

“…Proof Define

α false(t false) = ρ^{k} false(t false)

and

A false(η false) = - p false(η false) = \frac{- C}{μ false(η false)}

, where

C \in ℝ

. Then, α ( t )≤ t for all

t \in 𝕋

and Condition (A1) in Braverman and Karpuz 23 holds. Furthermore,

\int_{t_{0}}^{\infty} A false(η false) normalΔ η = \sum_{n = 0}^{\infty} false(- P false) = \infty

and Condition (24) in Braverman and Karpuz 23 is satisfied.…”

Section: Delay Dynamic Equations With One‐periodic Coefficientsmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Delay dynamic equations on isolated time scales and the relevance of one‐periodic coefficients

Böhner

Cuchta

Streipert

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

A Test for Global Attractivity of Linear Dynamic Equations with Delay

Alsharif,

Karpuz

2024

Qual. Theory Dyn. Syst.

View full text Add to dashboard Cite

In this paper, we consider the delay dynamic equation Based on Lyapunov’s method, we study global attractivity of the trivial solution of ($$*$$ ∗ ). Our new result generalizes some well-known results in the theory of difference and differential equations to dynamic equations of the form ($$*$$ ∗ ). We also present some examples on nonstandard time scales to illustrate the importance of the new result.

show abstract

On Different Types of Stability for Linear Delay Dynamic Equations

Cited by 2 publications

References 34 publications

Delay dynamic equations on isolated time scales and the relevance of one‐periodic coefficients

Delay dynamic equations on isolated time scales and the relevance of one‐periodic coefficients

A Test for Global Attractivity of Linear Dynamic Equations with Delay

Contact Info

Product

Resources

About