Inequalities applicable in the theory of partial finite difference equations (Q2752337)

In the present paper, the author establishes some new two-variable finite difference inequalities, which are suitable for some new applications in the theory of partial finite difference equations. The typical linear inequality under consideration is of the form NEWLINE\[NEWLINE\begin{multlined} u(m, n)\leq a(m, n)+ p(m, n)\varphi[m, n,u(m, n)]+\\ +\sum^{m-1}_{s=0} \sum^{n-1}_{t=0} L(s, t,u(s,t)),\;m,n\in \{0,1,2,\dots\},\end{multlined}\tag{\(*\)}NEWLINE\]NEWLINE here \(\varphi[m, n,u(m,n)]\) is defined by \(\sum^{m-1}_{s=0} b(s,n) u(s,n)\) or \(\sum^{n-1}_{t=0} b(m,t) u(m,t)\), all the functions involved are real-valued nonnegative functions, with \(L(s, t,u(s,t))\) obeys a Lipschitz-like condition due to S. S. Dragomir.NEWLINENEWLINENEWLINESome linear and nonlinear variants of \((*)\) are also given. An application example of a result obtained for inequality \((*)\) to a certain partial finite difference equation is also discussed.NEWLINENEWLINENEWLINEWe note that the results given herein are discrete analogues of some integral inequalities published by the author in the Journal of Mathematical Analysis and Applications in recent years.