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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.
##### MSC:
 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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