Necessary and sufficient conditions for oscillations of hyperbolic and elliptic type partial difference equations (Q2720025)

This paper is concerned with the hyperbolic type and elliptic type delay partial equations NEWLINE\[NEWLINE aA_{m+1,n+1}+bA_{m,n+1}-pA_{m,n}+\sum \limits_{i=1}^{u}q_{i}A_{m-k_{i},n-l_{i}}=0 NEWLINE\]NEWLINE and NEWLINE\[NEWLINE aA_{m+1,n+1}-2aA_{m,n+1}-pA_{m,n}+\sum \limits_{i=1}^{u}q_{i}A_{m-k_{i},n-l_{i}}=0, NEWLINE\]NEWLINE respectively, under some hypotheses for \ \(a,~b,~p,~q_{i},~k_{i},~l_{i}.\) For both equations, each solution \(\left\{ A_{m,n}\right\} \) is oscillatory if and only if the characteristic equation has no positive roots. The characteristic equation is NEWLINE\[NEWLINE a\lambda \mu +b\mu -p+\sum\limits_{i=1}^{u}q_{i}\lambda ^{-k_{i}}\mu ^{-l_{i}}=0 NEWLINE\]NEWLINE in the first case and NEWLINE\[NEWLINE a\lambda \mu -2a\mu -p+\sum\limits_{i=1}^{u}q_{i}\lambda ^{-k_{i}}\mu ^{-l_{i}}=0 NEWLINE\]NEWLINE for the second equation, respectively.