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Applications of Langenhop inequality to difference equations: Lower bounds and oscillation. (English) Zbl 1043.39005
It is shown that given any nonnegative sequences \(\left\{ u_{n}\right\} _{n=a}^{\infty }\) and \(\left\{ v_{n}\right\} _{n=a}^{\infty }\) and a nondecreaisng function \(g(u)\) on \([0,\infty )\) satisfying \(g(u)>0\) for \(u>0,\) if \[ u_{n}\geq u_{k}-\sum_{i=k}^{n-1}v_{i}g(u_{i}),\;k,n\geq a, \] then \[ u_{n}\geq G^{-1}\left( G(u_{k})-\sum_{i=k}^{n-1}v_{i}\right) , \] for all \(k,n\geq a\) for which \(G(u_{k})-\sum_{i=k}^{n-1}v_{i}\) is in the domain of \(G^{-1},\) where \(G\) is defined by \[ \Delta G(u_{n})=\frac{\Delta u_{n}}{g(u_{n})}. \] By means of the above result, lower bounds on the norms of solutions of difference systems of the form \[ \Delta z_{n}=f(n,z_{n}),\;n\geq a, \] are established, and oscillation criterion for higher order difference equations are obtained.
39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
26D15 Inequalities for sums, series and integrals
39A12 Discrete version of topics in analysis
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