Displaying similar documents to “Conjugacy criteria for half-linear differential equations”

On asymptotic properties of solutions of third order linear differential equations with deviating arguments

Ivan Kiguradze (1994)

Archivum Mathematicum

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The asymptotic properties of solutions of the equation u ' ' ' ( t ) = p 1 ( t ) u ( τ 1 ( t ) ) + p 2 ( t ) u ' ( τ 2 ( t ) ) , are investigated where p i : [ a , + ∞ [ → R ( i = 1 , 2 ) are locally summable functions, τ i : [ a , + ∞ [ → R ( i = 1 , 2 ) measurable ones and τ i ( t ) ≥ t ( i = 1 , 2 ) . In particular, it is proved that if p 1 ( t ) ≤ 0 , p 2 2 ( t ) ≤ α ( t ) | p 1 ( t ) | , ∫ a + ∞ [ τ 1 ( t ) - t ] 2 p 1 ( t ) d t < + ∞ and ∫ a + ∞ α ( t ) d t < + ∞ , then each solution with the first derivative vanishing at infinity is of the Kneser type and a set of all such solutions forms a one-dimensional linear space.

Resolvents, integral equations, limit sets

Theodore Allen Burton, D. P. Dwiggins (2010)

Mathematica Bohemica

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In this paper we study a linear integral equation x ( t ) = a ( t ) - ∫ 0 t C ( t , s ) x ( s ) d s , its resolvent equation R ( t , s ) = C ( t , s ) - ∫ s t C ( t , u ) R ( u , s ) d u , the variation of parameters formula x ( t ) = a ( t ) - ∫ 0 t R ( t , s ) a ( s ) d s , and a perturbed equation. The kernel, C ( t , s ) , satisfies classical smoothness and sign conditions assumed in many real-world problems. We study the effects of perturbations of C and also the limit sets of the resolvent. These results lead us to the study of nonlinear perturbations.