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Discrete nonlinear integral inequalities in two variables and their applications. (English) Zbl 1178.26032

In this paper discrete nonlinear integral inequalities of the following general type \[ u(m,n)\leq a(m,n)+\sum_{i=1}^k\sum_{s=0}^{m-1}\sum_{t=0}^{n-1}f_i(m,n,s,t)w_i(u(s,t)) \] are studied, where \(a\), \(f_1,\dots,f_k\), and \(w_1,\dots,w_k\) are given functions. The main result offers an explicite upper bound for the unknown function \(u\) in terms of the data. Applications to prove boundedness of solutions of difference equations are also presented.

MSC:

26D15 Inequalities for sums, series and integrals
39A13 Difference equations, scaling (\(q\)-differences)
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