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On Existence and Asymptotic Properties of Kneser Solutions to Singular Second Order ODE.

Jana Vampolová — 2013

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity p ( t ) u ' ( t ) ' + p ( t ) f ( u ( t ) ) = 0 , u ( 0 ) = u 0 , u ' ( 0 ) = 0 on the unbounded domain [ 0 , ) . Function f is locally Lipschitz continuous on and has at least three zeros L 0 < 0 , 0 and L > 0 . The initial value u 0 ( L 0 , L ) { 0 } . Function p is continuous on [ 0 , ) , has a positive continuous derivative on ( 0 , ) and p ( 0 ) = 0 . Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide...

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