Displaying similar documents to “A role of the coefficient of the differential term in qualitative theory of half-linear equations”

Bounded oscillation of nonlinear neutral differential equations of arbitrary order

Yeter Ş. Yilmaz, Ağacik Zafer (2001)

Czechoslovak Mathematical Journal

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The paper is concerned with oscillation properties of n -th order neutral differential equations of the form [ x ( t ) + c x ( τ ( t ) ) ] ( n ) + q ( t ) f x ( σ ( t ) ) = 0 , t ≥ t 0 > 0 , where c is a real number with | c | ≠ 1 , q ∈ C ( [ t 0 , ∞ ) , ℝ ) , f ∈ C ( ℝ , ℝ ) , τ , σ ∈ C ( [ t 0 , ∞ ) , ℝ + ) with τ ( t ) < t and lim t → ∞ τ ( t ) = lim t → ∞ σ ( t ) = ∞ . Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations...

Oscillatory properties of second order half-linear difference equations

Pavel Řehák (2001)

Czechoslovak Mathematical Journal

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We study oscillatory properties of the second order half-linear difference equation Δ ( r k | Δ y k | α - 2 Δ y k ) - p k | y k + 1 | α - 2 y k + 1 = 0 , α > 1 . ( HL ) It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation Δ ( r k Δ y k ) - p k y k + 1 = 0 . We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.