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On Lyapunov stability/instability of equilibria of free damped pendulum with periodically oscillating suspension point

Jiří Šremr (2025)

Applications of Mathematics

We discuss Lyapunov stability/instability of both lower and upper equilibria of free damped pendulum with periodically oscillating suspension point. We recall the results of Bogolyubov and Kapitza, provide new effective criteria of stability/instability of the equilibria of pendulum equation, and give the exact and complete proofs. The criteria obtained are formulated in terms of positivity/negativity of Green's functions of the periodic boundary value problems for linearized equations. Furthermore,...

On non-oscillation on semi-axis of solutions of second order deviating differential equations

Sergey Labovskiy, Manuel Alves (2018)

Mathematica Bohemica

We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations u ' ' ( x ) + ∑ i p i ( x ) u ' ( h i ( x ) ) + ∑ i q i ( x ) u ( g i ( x ) ) = 0 without the delay conditions h i ( x ) , g i ( x ) ≤ x , i = 1 , 2 , ... , and u ' ' ( x ) + ∫ 0 ∞ u ' ( s ) d s r 1 ( x , s ) + ∫ 0 ∞ u ( s ) d s r 0 ( x , s ) = 0 .

On oscillation and asymptotic property of a class of third order differential equations

N. Parhi, Seshadev Pardi (1999)

Czechoslovak Mathematical Journal

In this paper, oscillation and asymptotic behaviour of solutions of y ' ' ' + a ( t ) y ' ' + b ( t ) y ' + c ( t ) y = 0 have been studied under suitable assumptions on the coefficient functions a , b , c ∈ C ( [ σ , ∞ ) , R ) , σ ∈ R , such that a ( t ) ≥ 0 , b ( t ) ≤ 0 and c ( t ) < 0 .

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