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Preservation of exponential stability for equations with several delays

Leonid Berezansky;  Elena Braverman — 2011

Mathematica Bohemica

We consider preservation of exponential stability for the scalar nonoscillatory linear equation with several delays x ˙ ( t ) + ∑ k = 1 m a k ( t ) x ( h k ( t ) ) = 0 , a k ( t ) ≥ 0 under the addition of new terms and a delay perturbation. We assume that the original equation has a positive fundamental function; our method is based on Bohl-Perron type theorems. Explicit stability conditions are obtained.

Oscillation properties for a scalar linear difference equation of mixed type

Leonid Berezansky;  Sandra Pinelas — 2016

Mathematica Bohemica

The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type Δ x ( n ) + ∑ k = - p q a k ( n ) x ( n + k ) = 0 , n > n 0 , where Δ x ( n ) = x ( n + 1 ) - x ( n ) is the difference operator and { a k ( n ) } are sequences of real numbers for k = - p , ... , q , and p > 0 , q ≥ 0 . We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

On stability of linear neutral differential equations with variable delays

Leonid Berezansky;  Elena Braverman — 2019

Czechoslovak Mathematical Journal

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays x ˙ ( t ) - a ( t ) x ˙ ( g ( t ) ) + b ( t ) x ( h ( t ) ) = 0 , where | a ( t ) | < 1 , b ( t ) ≥ 0 , h ( t ) ≤ t , g ( t ) ≤ t , and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.

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