Dynamic Picone's identity and its applications
The authors establish Picone type identities and inequalities that are related to the nonlinear dynamic system \[ u^\Delta=Au^\sigma+B\varphi_{1/\alpha}(v), v^\Delta=-C\varphi_\alpha(u^\sigma)-Dv \] on a time scale, where \(\varphi_\alpha(t)=|t|^\alpha\text{sgn}t\), \(\alpha>0\), \(A,B,C,D\) are rd-continuous with \(B>0\), and \(-A\) and \(-D\) are positively regressive. Using these tools they derive Writinger and Hardy type inequalities on time scales which serve to study qualitative properties of the system. Among other, the Reid type roundabout theorem involving disconjugacy and Sturmian theorems are stated. Another aim of the paper is to correct the results from \textit{S. H. Saker} et al. [Appl. Math. Lett. 48, 162--169 (2015; Zbl 1323.26029)].
scientific article; zbMATH DE number 6957994