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On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE

Martin Rohleder — 2012

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The paper investigates the singular initial problem[4pt] ( p ( t ) u ' ( t ) ) ' + q ( t ) f ( u ( t ) ) = 0 , u ( 0 ) = u 0 , u ' ( 0 ) = 0 [4pt] on the half-line [ 0 , ) . Here u 0 [ L 0 , L ] , where L 0 , 0 and L are zeros of f , which is locally Lipschitz continuous on . Function p is continuous on [ 0 , ) , has a positive continuous derivative on ( 0 , ) and p ( 0 ) = 0 . Function q is continuous on [ 0 , ) and positive on ( 0 , ) . For specific values u 0 we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for f , p and q it is shown that the problem has for each specified u 0 a unique...

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