Generalized Prüfer angle and oscillation of half-linear differential equations

Abstract In this paper, we introduce a new modification of the half-linear Prufer angle. Applying this modification, we investigate the conditional oscillation of the half-linear second order differential equation ( ∗ ) t α − 1 r ( t ) Φ ( x ′ ) ′ + t α − 1 − p s ( t ) Φ ( x ) = 0 , Φ ( x ) = | x | p − 1 sgn x , where p > 1 , α ≠ p , and r , s are continuous functions such that r ( t ) > 0 for large t . We present conditions on the functions r , s which guarantee that Eq. ( ∗ ) behaves like the Euler type equation [ t α − 1 Φ ( x ′ ) ] ′ + λ t α − 1 − p Φ ( x ) = 0 , which is conditionally oscillatory with the oscillation constant λ 0 = | p − α | p ∕ p p .

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