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On periodic solutions of systems of linear functional-differential equations

Ivan Kiguradze, Bedřich Půža (1997)

Archivum Mathematicum

This paper deals with the system of functional-differential equations d x ( t ) d t = p ( x ) ( t ) + q ( t ) , where p : C ω ( R n ) L ω ( R n ) is a linear bounded operator, q L ω ( R n ) , ω > 0 and C ω ( R n ) and L ω ( R n ) are spaces of n -dimensional ω -periodic vector functions with continuous and integrable on [ 0 , ω ] components, respectively. Conditions which guarantee the existence of a unique ω -periodic solution and continuous dependence of that solution on the right hand side of the system considered are established.

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.

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