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positive solution; nonlinear discrete delayed equation
When mathematical models describing various processes are analysed, the fact of existence of a positive solution is often among the basic features. In this paper, a general delayed discrete equation \[ \Delta u(k+n)=f(k,u(k),u(k+1),\dots ,u(k+ n)) \] is considered. Sufficient conditions concerning $f$ are formulated in order to guarantee the existence of a positive solution for $k\rightarrow \infty $. An upper estimate for it is given as well. The appearance of the positive solution takes its origin in the nature of the equation considered since the results hold only for delayed equations (i.e. for $n>0$) and not for the case of an ordinary equation (with $n=0$).
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