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On property (B) of higher order delay differential equations

Blanka Baculíková, Jozef Džurina (2012)

Archivum Mathematicum

In this paper we offer criteria for property (B) and additional asymptotic behavior of solutions of the n -th order delay differential equations ( r ( t ) [ x ( n - 1 ) ( t ) ] γ ) ' = q ( t ) f ( x ( τ ( t ) ) ) . Obtained results essentially use new comparison theorems, that permit to reduce the problem of the oscillation of the n-th order equation to the the oscillation of a set of certain the first order equations. So that established comparison principles essentially simplify the examination of studied equations. Both cases r - 1 / γ ( t ) t = and r - 1 / γ ( t ) t < are discussed.

On quadratically integrable solutions of the second order linear equation

T. Chantladze, Nodar Kandelaki, Alexander Lomtatidze (2001)

Archivum Mathematicum

Integral criteria are established for dim V i ( p ) = 0 and dim V i ( p ) = 1 , i { 0 , 1 } , where V i ( p ) is the space of solutions u of the equation u ' ' + p ( t ) u = 0 satisfying the condition + u 2 ( s ) s i d s < + .

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