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Oscillatory properties of second order half-linear difference equations

Pavel Řehák (2001)

Czechoslovak Mathematical Journal

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.

Oscillatory properties of third-order semi-noncanonical nonlinear delay difference equations

Govindasamy Ayyappan, George E. Chatzarakis, Thaniarasu Kumar, Ethiraj Thandapani (2023)

Mathematica Bohemica

We study the oscillatory properties of the solutions of the third-order nonlinear semi-noncanonical delay difference equation D 3 y ( n ) + f ( n ) y β ( σ ( n ) ) = 0 , where D 3 y ( n ) = Δ ( b ( n ) Δ ( a ( n ) ( Δ y ( n ) ) α ) ) is studied. The main idea is to transform the semi-noncanonical operator into canonical form. Then we obtain new oscillation theorems for the studied equation. Examples are provided to illustrate the importance of the main results.

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