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We investigate oscillation and spectral properties (sufficient conditions for discreteness and boundedness below of the spectrum) of difference operators
B(y)n+k = (-1)nwk n (pk n yk).
Sufficient conditions on the function ensuring that the half-linear second order differential equation
possesses a nontrivial solution having at least two zeros in a given interval are obtained. These conditions extend some recently proved conjugacy criteria for linear equations which correspond to the case .
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