Abstract
This paper is concerned with estimates, unimprovable in a certain sense, for positive solutions to the half-linear differential equation (\({|y^{\prime}|^{\alpha - 1} {\rm sgn} y^{\prime})^{\prime} = p(t) |y|^{\alpha - 1} {\rm sgn} y}\), where p is a continuous nonnegative function on \({[0, \infty)}\) and \({\alpha> 1}\). It is shown that any positive increasing solution y of the equation satisfies \({y(t) \geqq y(0) exp{h \int_{0}^{t} p^{\frac{1}{\alpha}}(s) {\rm d}s}}\), with \({h< (\alpha - 1)^{-\frac{1}{\alpha}}}\), for all t on the complement of a set of finite Lebesgue measure. Under an additional assumption, this estimate holds for all t. Further, a condition is established which guarantees that the equation has exponentially increasing solutions and exponentially decreasing solutions.
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Řehák, P. Exponential estimates for solutions of half-linear differential equations. Acta Math. Hungar. 147, 158–171 (2015). https://doi.org/10.1007/s10474-015-0522-9
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DOI: https://doi.org/10.1007/s10474-015-0522-9