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Comparison results for multidimensional difference equations. (English) Zbl 0661.39004

Main results of the paper (contained in section 2) are various theorems of comparison type (also in section 5), relating solutions of the vector- valued difference equation of several variables \((*)\quad \Delta^ n_ xu(x)=f(x,u(x))\) with solutions of suitable difference inequalities. Here \(x=(x_ 1,...,x_ n)\in N^ n\), \(N=\{0,1,...\}\); \(u(x)=(u_ 1(x),...,u_ m(x)),\) \(\Delta^ n_ x\) is the n-fold forward difference operator \(\Delta_{x_ 1}...\Delta_{x_ n}\), \(\Delta_{x_ i}u(x)=u(x_ 1,...,x_ i+1,x_{i+1},...,x_ n)-u(x_ 1,...,x_ i,...,x_ n)\), and f is (in most of the cases) nondecreasing with respect to all \(u_ 1,...,u_ m\). The results of section 2 are used to study the dependence of solutions of (*) on initial values and on parameters (in section 3), and to establish some asymptotic properties as e.g. boundedness (in section 4). See for similar considerations the author [J. Math. Phys. Sci. 10, 277-288 (1976; Zbl 0339.39003)], the reviewer [Fasc. Math. 16, 29-42 (1986; Zbl 0616.26012)], R. Redheffer and W. Walter [J. Differ. Equations 44, 111-117 (1982; Zbl 0455.35009)], S. Sugiyama [Proc. Japan Acad. 47, 477-480 (1971; Zbl 0272.39002)].
Reviewer: J.Popenda

MSC:

39A10 Additive difference equations
26D20 Other analytical inequalities
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