Oscillatory and asymptotic behaviour of perturbed quasilinear second-order difference equations (Q2716331)

The authors investigate the oscillatory and asymptotic properties of the forced second-order quasilinear difference equation NEWLINE\[NEWLINE \Delta (a_{n-1}\Phi _{\alpha }(\Delta y))+F(n,y_n)=G(n,y_n,\Delta y_n), \tag{\(*\)} NEWLINE\]NEWLINE where \(\Phi _{\alpha }(s):=|s|^{\alpha -1}s\), \(\alpha >0\), \(a_n\) is a positive sequence and the functions \(F,G\) satisfy some natural restrictions. In the first part of the paper the asymptotic properties of nonoscillatory solutions of \((*)\) are investigated. Here \((*)\) is essentially compared with an (easier) equation NEWLINE\[NEWLINE \Delta (a_{n-1} \Phi _{\alpha }(\Delta y))+q_nf(y)=p_nf(y), \tag{\(**\)} NEWLINE\]NEWLINE where the function \(f\) is in a certain way related to \(F,G\) and satisfies \(yf(y)>0\) for \(y\neq 0\). NEWLINENEWLINENEWLINEIn the second part of the paper, a discrete version of the continuous ``\(H\)-function'' technique [introduced in the continuous case by \textit{Ch. G. Philos}, Arch. Math. 53, 483-492 (1989; Zbl 0661.34030)] is used to derive new oscillation criteria for \((**)\) with \(f(y)=\Phi _{\alpha }(y)\) and \(G\equiv 0\).