On certain comparison theorems for half-linear dynamic equations on time scales
Summary: We obtain comparison theorems for the second-order half-linear dynamic equation \[ [r(t)\Phi(y^\Delta)]^\Delta+ p(t)\Phi(y^\sigma)= 0, \] where \(\Phi(x)=|x|^{\alpha-1}\text{sgn\,}x\) with \(\alpha> 1\). In particular, it is shown that the nonoscillation of the previous dynamic equation is preserved if we multiply the coefficient \(p\)(t) by a suitable function \(q(t)\) and lower the exponent \(\alpha\) in the nonlinearity \(\Phi\), under certain assumptions. Moreover, we give a generalization of the Hille-Wintner comparison theorem. In addition to the aspect of unification and extension, our theorems provide some new results even in the continuous and the discrete case.
scientific article; zbMATH DE number 2163673