Asymptotic Stability in Functional Differential Equations with Delay

Ignatyev, Alexander O.

doi:10.1007/978-0-8176-8184-5_17

Cited by 4 publications

(6 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let F 1 t F N t R → R n+m be almost periodic functions. Then for every ε > 0 there exists l = l ε > 0 such that any segment α α + l α ∈ R, contains a number τ such that [9]. If the functional F t ϕ R × C H L → R n+m is Lipschitzian in ϕ and almost periodic in t for every fixed ϕ ∈ C H L , then it is uniformly almost periodic in t.…”

Section: Partial Equiasymptotic Stability Inmentioning

confidence: 95%

“…We also assume that the functional Z t ϕ is Lipschitzian in ϕ and almost periodic in t for every fixed ϕ ∈ C H . Lemma 3.3 [9]. Consider the solution z t 0 ϕ 0 of system (1.1).…”

Section: Partial Equiasymptotic Stability Inmentioning

confidence: 98%

“…A continuous function F t R → R n+m is called almost periodic if for every ε > 0 there exists l = l ε > 0 such that any segment α α + l α ∈ R, contains at least one number τ such that F t + τ − F t < ε for every t ∈ R. A number τ is called an ε-almost period of F. Definition 3.2 [9]. A continuous functional F t ϕ R × C r → R n+m 0 < r < ∞ is called uniformly almost periodic in t if for every ε > 0 there exists l = l ε r > 0 such that any segment α α + l α ∈ R, contains at least one number τ such that F t + τ ϕ − F t ϕ < ε for every t ∈ R, ϕ ∈ C r .…”

Section: Partial Equiasymptotic Stability Inmentioning

confidence: 99%

See 2 more Smart Citations

On the Partial Equiasymptotic Stability in Functional Differential Equations

Ignatyev

2002

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

A system of functional differential equations with delay dz/dt = Z t z t , where Z is the vector-valued functional is considered. It is supposed that this system has a zero solution z = 0. Definitions of its partial stability, partial asymptotical stability, and partial equiasymptotical stability are given. Theorems on the partial equiasymptotical stability are formulated and proved.  2002 Elsevier Science (USA)

show abstract

Section: Partial Equiasymptotic Stability Inmentioning

confidence: 95%

“…We also assume that the functional Z t ϕ is Lipschitzian in ϕ and almost periodic in t for every fixed ϕ ∈ C H . Lemma 3.3 [9]. Consider the solution z t 0 ϕ 0 of system (1.1).…”

Section: Partial Equiasymptotic Stability Inmentioning

confidence: 98%

Section: Partial Equiasymptotic Stability Inmentioning

confidence: 99%

See 1 more Smart Citation

On the Partial Equiasymptotic Stability in Functional Differential Equations

Ignatyev

2002

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…Just as Theorem 3.3 generalizes Theorem 3.2 [9], Theorem 3.4 generalizes Theorem 4.2 [9] for the periodic equation. Also, Theorem 3.2 yields a corollary for almost periodic in t equations, according to the following definition [13]:…”

Section: Complete Instability Of the Zero Solutionmentioning

confidence: 99%

Razumikhin-Type Theorems in the Problem on Instability of Nonautonomous Equations with Finite Delay

Sedova

2004

View full text Add to dashboard Cite

Abstract. The paper provides some theorems on complete instability of zero solution relative to a set for nonautonomous nonlinear equations with delay. The right-hand side of the equation is assumed to satisfy conditions, which provide standard existenceuniqueness-continuous dependence-continuation theory for the equation, as well as precompactness of the collection of translations in time of the right-hand side in a functional space with a metrizable compact open topology. These assumptions allow constructing limiting equations. Using conceptions of Lyapunov-Razumikhin functions and limiting equations, new instability results are obtained, which are applicable, in particular, to autonomous and periodic delayed di¤erential equations and generalize some known results.

show abstract

“…For the system with a large n, several methods have been used, such as the Liapunov functional method, the invariance principle of Liapunov-Rarumikhin type and the monotone method, to investigate the global dynamics of (1.1) (see [2,3,5,6,9]). Moreover, in the application of the Liapunov functional method, the authors often require that the Liapunov functional's derivative along (1.1) is less than or equal to 0 (see [3] and the references cited therein). Before illustrating our study method, we shall recall one recent paper [8] on the global attractivity of the positive steady state of the diffusive Nicholson equation.…”

Section: Introductionmentioning

confidence: 99%