×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: EMIS EuDML