Progress in Applied Mathematics
Comparison Theorem for Oscillation of Nonlinear Delay Partial Difference Equations
Abstract
In this paper,we consider certain nonlinear partial difference equations
$${(aA_{m+1,n}+bA_{m,n+1}+cA_{m,n})}^k-{(dA_{m,n})}^k+\sum\limits_{i=1}^{u} p_{i}(m,n)A^k_{m-\sigma_{i},n-\tau_{i}}=0 $$
where $a,b,c,d \in(0,\infty )$, $d>c$, $k=q/p$, p, q are positive odd integers, $u$ is a positive integer, $p_{i}(m,n), (i=0,1,2,\cdots u)$ are positive real sequences. $\sigma_i,\tau_i\in N_{0}=\{1,2,\cdots \}, i=1,2,\cdots,u$. A new comparison theorem for oscillation of the above equation is obtained.
$${(aA_{m+1,n}+bA_{m,n+1}+cA_{m,n})}^k-{(dA_{m,n})}^k+\sum\limits_{i=1}^{u} p_{i}(m,n)A^k_{m-\sigma_{i},n-\tau_{i}}=0 $$
where $a,b,c,d \in(0,\infty )$, $d>c$, $k=q/p$, p, q are positive odd integers, $u$ is a positive integer, $p_{i}(m,n), (i=0,1,2,\cdots u)$ are positive real sequences. $\sigma_i,\tau_i\in N_{0}=\{1,2,\cdots \}, i=1,2,\cdots,u$. A new comparison theorem for oscillation of the above equation is obtained.
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